Patent Publication Number: US-6985809-B2

Title: Control apparatus, control method, and engine control unit

Description:
BACKGROUND OF THE INVENTION 
   1. Field of the Invention 
   The present invention relates to a control apparatus, a control method, and an engine control unit which calculate a control input to a controlled object based on a ΔΣ modulation algorithm or the like to converge the output of the controlled object to a target value. 
   2. Description of the Prior Art 
   Conventionally, a control apparatus of the type mentioned above is known, for example, from Laid-open Japanese Patent Application No. 2001-154704. This control apparatus comprises detecting means for detecting an output of a controlled object to output the result of detection as a detection signal indicative of a detected analog amount; deviation calculating means for calculating a deviation of the detection signal from a target value of an analog amount inputted from a higher rank apparatus; converting means for converting the calculated deviation to a 1-bit digital signal; and compensating means for compensating the 1-bit digital signal from the converting means to output the compensated signal as a manipulation signal (see  FIG. 6  of the application). 
   In this control apparatus, the deviation calculating means calculates a deviation of a detection signal from a target value (analog amount) which is converted to a 1-bit digital signal by a ΔΣ modulation in the converting means. The converted signal is further compensated by the compensating means before it is inputted to a controlled object as a manipulation signal. In the foregoing configuration, the manipulation amount is generated in the opposite phase to the deviation so as to cancel the deviation of the output of the controlled object from the target value, and inputted to the controlled object. As a result, the output of the controlled object is controlled in feedback to converge to the target value. 
   According to the conventional control apparatus mentioned above, when a dynamic characteristic of a controlled object has a relatively large phase delay, a dead time, or the like, this causes a delay in outputting an output signal, which reflects an input signal from the controlled object, after the controlled object is fed with the input signal, leading to a slippage in control timing between the input and output of the controlled object. As a result, a control system could lose the stability. For example, when an internal combustion engine is controlled for an air/fuel ratio of exhaust gases using a fuel injection amount of the internal combustion engine as an input, a time lag is needed until the air/fuel ratio of the exhaust gases actually change after a fuel has been actually injected, so that the air/fuel ratio control experiences lower stability and controllability, resulting in an instable characteristic of exhaust gases purified by a catalyst. 
   OBJECT AND SUMMARY OF THE INVENTION 
   The present invention has been made to solve the foregoing problem, and it is an object of the invention to provide a control apparatus, a control method, and an engine control unit which are capable of eliminating a slippage in control timing between the input and output of a controlled object, even when the controlled object exhibits a relatively large dynamic characteristic such as a phase delay, a dead time, and the like, and of capable of improving the stability and controllability of control. 
   To achieve the above object, according to a first aspect of the invention, there is provided a control apparatus which is characterized by comprising predicted value calculating means for calculating a predicted value of a value indicative of an output of a controlled object based on a prediction algorithm; and control input calculating means for calculating a control input to the controlled object based on one modulation algorithm selected from a Δ modulation algorithm, a ΔΣ modulation algorithm, and a ΣΔ modulation algorithm for controlling the output of the controlled object in accordance with the calculated predicted value. 
   According to this control apparatus, the control input is calculated in accordance with a predicted value of the value indicative of the output of the controlled object based on one modulation algorithm selected from the Δ modulation algorithm, ΔΣ modulation algorithm, and ΣΔ modulation algorithm. Therefore, a slippage in control timing can be eliminated between the input and output of the controlled object by calculating such a predicted value as a value which reflects a dynamic characteristic of the controlled object, for example, a phase delay, a dead time, or the like. As a result, the control apparatus can ensure the stability of the control and improve the controllability (it should be noted that in this specification, “calculation” in “calculation of a predicted value,” “calculation of a control input” and the like is not limited to a program-based operation but includes hardware-based generation of electric signals indicative of such values). 
   To achieve the above object, according to a second aspect of the invention, there is provided a control method which is characterized by comprising the steps of calculating a predicted value of a value indicative of an output of a controlled object based on a prediction algorithm; and calculating a control input to the controlled object based on one modulation algorithm selected from a Δ modulation algorithm, a ΔΣ modulation algorithm, and a ΣΔ modulation algorithm for controlling the output of the controlled object in accordance with the calculated predicted value. 
   This control method provides the same advantageous effects as described above concerning the control apparatus according to the first aspect of the invention. 
   To achieve the above object, according to a third aspect of the invention, there is provided an engine control unit including a control program for causing a computer to calculate a predicted value of a value indicative of an output of a controlled object based on a prediction algorithm; and calculate a control input to the controlled object based on one modulation algorithm selected from a Δ modulation algorithm, a ΔΣ modulation algorithm, and a ΣΔ modulation algorithm for controlling the output of the controlled object in accordance with the calculated predicted value. 
   This engine control unit provides the same advantageous effects as described above concerning the control apparatus according to the first aspect of the invention. 
   Preferably, in the control apparatus described above, the predicted value calculating means calculates the predicted value in accordance with at least one of the calculated control input and a value which reflects a control input inputted to the controlled object, and the output of the controlled object, based on the prediction algorithm. 
   According to this preferred embodiment of the control apparatus, the predicted value can be calculated while reflecting the state of the control input, so that the predicted value can be correspondingly calculated with an improved accuracy (prediction accuracy). As a result, the control apparatus can ensure the stability of the control and improve the controllability. 
   Preferably, in the control method described above, the step of calculating a predicted value includes calculating the predicted value in accordance with at least one of the calculated control input and a value which reflects a control input inputted to the controlled object, and the output of the controlled object, based on the prediction algorithm. 
   This preferred embodiment of the control method provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the engine control unit described above, the control program causes the computer to calculate a predicted value includes calculating the predicted value in accordance with at least one of the calculated control input and a value which reflects a control input inputted to the controlled object, and the output of the controlled object, based on the prediction algorithm. 
   This preferred embodiment of the engine control unit provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the control apparatus described above, the prediction algorithm is an algorithm based on a controlled object model which has a variable associated with a value indicative of one of the control input and the value which reflects a control input inputted to the controlled object such as an air/fuel ratio deviation and LAF output deviation, and a variable associated with a value indicative of the output of the controlled object. 
   According to this preferred embodiment of the control apparatus, since the predicted value is calculated based on a controlled object model which has a variable associated with a value indicative of one of the control input and the value which reflects a control input inputted to the controlled object, and a variable associated with a value indicative of the output of the controlled object, this controlled object model can be defined as a model which reflects the dynamic characteristic such as a phase delay, a dead time and the like of the controlled object to calculate the predicted value which reflects the dynamic characteristic such as the phase delay, dead time and the like of the controlled object. As a result, the control apparatus can ensure the stability of the control and improve the controllability. 
   Preferably, in the control method described above, the prediction algorithm is an algorithm based on a controlled object model which has a variable associated with a value indicative of one of the control input and the value which reflects a control input inputted to the controlled object, and a variable associated with a value indicative of the output of the controlled object. 
   This preferred embodiment of the control method provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the engine control unit, the prediction algorithm is an algorithm based on a controlled object model which has a variable associated with a value indicative of one of the control input and the value which reflects a control input inputted to the controlled object, and a variable associated with a value indicative of the output of the controlled object. 
   This preferred embodiment of the engine control unit provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the control apparatus described above, the value indicative of the output of the controlled object is an output deviation which is a deviation of the output of the controlled object such as the output of an oxygen concentration sensor from a predetermined target value. 
   Generally, it is known in a controlled object model that the dynamic characteristic of the controlled object model can be fitted more to the actual dynamic characteristic of the controlled object when a deviation of input/output of the controlled object from a predetermined value is defined as a variable representative of the input/output than when an absolute value of the input/output is defined as a variable, because it can more precisely identify or define model parameters. Therefore, according to this preferred embodiment of the control apparatus, the controlled object model employs a variable representative of the output deviation which is a deviation of the output of the controlled object from the predetermined target value, so that the dynamic characteristic of the controlled object model can be fitted more closely to the actual dynamic characteristic of the controlled object, as compared with the case where an absolute value of the output of the controlled object is chosen as a variable, thereby making it possible to calculate the predicted value of the output deviation with a higher accuracy. As a result, the control apparatus can further enhance the ensured stability of the control and the improved controllability. 
   Preferably, in the control method described above, the value indicative of the output of the controlled object is an output deviation which is a deviation of the output of the controlled object from a predetermined target value. 
   This preferred embodiment of the control method provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the engine control unit described above, the value indicative of the output of the controlled object is an output deviation which is a deviation of the output of the controlled object from a predetermined target value. 
   This preferred embodiment of the engine control unit provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the control apparatus described above, the value indicative of one of the control input and the value which reflects a control input inputted to the controlled object is one of a deviation such as an air/fuel ratio deviation of the control input such as a target air/fuel ratio from a predetermined reference value, and a deviation of the value which reflects a control input inputted to the controlled object, such as the output of an LAF sensor, from the predetermined reference value. 
   As described above, in a controlled object model, the dynamic characteristic of the controlled object model can be fitted more to the actual dynamic characteristic of the controlled object when a deviation of input/output of the controlled object from a predetermined value is defined as a variable representative of the input/output than when an absolute value of the input/output is defined as a variable, because it can more precisely identify or define model parameters. Therefore, according to this preferred embodiment of the control apparatus, since the controlled object model employs a variable representative of a deviation of the calculated control input from the predetermined reference value, or a variable representative of a deviation of the value which reflects a control input inputted to the controlled object from the predetermined reference value, the dynamic characteristic of the controlled object model can be fitted more closely to the actual dynamic characteristic of the controlled object than when the controlled object model employs a variable representative of a control input or an absolute value of the value which reflects the control input, thereby further enhancing the ensured stability of the control and the improved controllability. 
   Preferably, in the control method, described above, the value indicative of one of the control input and the value which reflects a control input inputted to the controlled object is one of a deviation of the control input from a predetermined reference value, and a deviation of the value which reflects a control input inputted to the controlled object from the predetermined reference value. 
   This preferred embodiment of the control method provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the engine control unit described above, the value indicative of one of the control input and the value which reflects a control input inputted to the controlled object is one of a deviation of the control input from a predetermined reference value, and a deviation of the value which reflects a control input inputted to the controlled object from the predetermined reference value. 
   This preferred embodiment of the engine control unit provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the control apparatus described above, the control input calculating means calculates an intermediate value in accordance with the predicted value based on the one modulation algorithm, and calculates the control input, such as a target air fuel ratio or an adaptive target air/fuel ratio, based on the calculated intermediate value multiplied by a predetermined gain. 
   Generally, each of the ΔΣ modulation algorithm, ΣΔ modulation algorithm, and Δ modulation algorithm determines a control input on the assumption that a controlled object has a unity gain, so that if the controlled object has an actual gain different from a unity value, the controllability may be degraded due to a failure in calculating an appropriate control input. For example, when the controlled object has an actual gain larger than one, the control input is calculated as a value larger than necessity, resulting in an over-gain condition. On the other hand, according to this preferred embodiment of the control apparatus, the control input is calculated based on the intermediate value, which is calculated based on the one modulation algorithm, multiplied by a predetermined gain, so that a satisfactory controllability can be ensured by setting the predetermined gain to an appropriate value. 
   Preferably, in the control method described above, the step of calculating a control input includes calculating an intermediate value in accordance with the predicted value based on the one modulation algorithm, and calculating the control input based on the calculated intermediate value multiplied by a predetermined gain. 
   This preferred embodiment of the control method provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the engine control unit described above, the control program causes the computer to calculate an intermediate value in accordance with the predicted value based on the one modulation algorithm, and calculate the control input based on the calculated intermediate value multiplied by a predetermined gain. 
   This preferred embodiment of the engine control unit provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, the control apparatus described above further comprises gain parameter detecting means for detecting a gain parameter indicative of a gain characteristic of the controlled object, such as an exhaust gas volume, and gain setting means for setting the predetermined gain in accordance with the detected gain parameter. 
   According to this preferred embodiment of the control apparatus, since the predetermined gain for use in the calculation of the control input is set in accordance with the gain characteristic of the controlled object, the control input can be calculated as a value which has appropriate energy in accordance with the gain characteristic of the controlled object, thereby making it possible to avoid an over-gain condition and the like to ensure a satisfactory controllability. 
   Preferably, the control method described above further comprises the steps of detecting a gain parameter indicative of a gain characteristic of the controlled object; and setting the predetermined gain in accordance with the detected gain parameter. 
   This preferred embodiment of the control method provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the engine control unit described above, the control program further causes the computer to detect a gain parameter indicative of a gain characteristic of the controlled object; and set the predetermined gain in accordance with the detected gain parameter. 
   This preferred embodiment of the engine control unit provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the control apparatus described above, the control input calculating means calculates a second intermediate value, such as ΔΣ modulation control amount, in accordance with the predicted value based on the one modulation algorithm, and adds a predetermined value to the calculated second intermediate value to calculate the control input such as an adaptive target air/fuel ratio. 
   Generally, any of the Δ modulation algorithm, ΔΣ modulation algorithm, and ΣΔ modulation algorithm can only calculate a positive-negative inversion type control input centered at zero. On the contrary, according to this preferred embodiment of the control apparatus, the control input calculating means calculates the control input by adding the predetermined value to the second intermediate value calculated based on the one modulation algorithm, so that the control input calculating means can calculate the control input not only as a value which positively and negatively inverts centered at zero, but also as a value which repeats predetermined increase and decrease about a predetermined value, thereby making it possible to improve the degree of freedom in control. 
   Preferably, in the control method described above, the step of calculating a control input includes calculating a second intermediate value in accordance with the predicted value based on the one modulation algorithm, and adding a predetermined value to the calculated second intermediate value to calculate the control input. 
   This preferred embodiment of the control method provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably in the engine control unit described above, the control program causes the computer to calculate a second intermediate value in accordance with the predicted value based on the one modulation algorithm; and add a predetermined value to the calculated second intermediate value to calculate the control input. 
   This preferred embodiment of the engine control unit provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the control apparatus described above, the predicted value calculating means calculates a prediction time from the time at which the control input is inputted to the controlled object to the time at which the control input is reflected to the output of the controlled object in accordance with a dynamic characteristic of the controlled object, and calculates the predicted value in accordance with the calculated prediction time. 
   According to this preferred embodiment of the control apparatus, the prediction time from the time at which the control input is inputted to the controlled object to the time at which the control input is reflected to the output of the controlled object is calculated in accordance with the dynamic characteristic of the controlled object, and the predicted value is calculated in accordance with the calculated prediction time, so that a slippage in control timing between the input/output of the controlled object, possibly caused by a response delay, a dead time, and the like of the controlled object, can be eliminated without fail by calculating the control input using the predicted value calculated in this manner, thereby making it possible to further improve the controllability. 
   Preferably, in the control method described above, the step of calculating a predicted value includes calculating a predicted value includes calculating a prediction time from the time at which the control input is inputted to the controlled object to the time at which the control input is reflected to the output of the controlled object in accordance with a dynamic characteristic of the controlled object; and calculating the predicted value in accordance with the calculated prediction time. 
   This preferred embodiment of the control method provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the engine control unit described above, the control program causes the computer to calculate a predicted value includes calculating a prediction time from the time at which the control input is inputted to the controlled object to the time at which the control input is reflected to the output of the controlled object in accordance with a dynamic characteristic of the controlled object; and calculate the predicted value in accordance with the calculated prediction time. 
   This preferred embodiment of the engine control unit provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the control apparatus described above, the controlled object comprises a downstream air/fuel ratio sensor, such as an oxygen concentration sensor, disposed at a location downstream of a catalyst in an exhaust pipe of an internal combustion engine for detecting an air/fuel ratio of exhaust gases which have passed through the catalyst, and the output of the controlled object is an output of the downstream air/fuel ratio sensor. The value indicative of the output of the controlled object is an output deviation of an output of the downstream air/fuel ratio sensor from a predetermined target value. The control input to the controlled object is a target air/fuel ratio of an air/fuel mixture supplied to the internal combustion engine. The value reflecting a control input inputted to the controlled object is an output of an upstream air/fuel ratio sensor disposed at a location upstream of the catalyst in the exhaust passage for detecting an air/fuel ratio of exhaust gases which have not passed through the catalyst. The predicted value calculating means calculates the predicted value of the output deviation in accordance with at least one of the target air/fuel ratio of the air/fuel mixture supplied to the internal combustion engine, the output of the upstream air/fuel ratio sensor, and the output of the downstream air/fuel ratio sensor based on the prediction algorithm. The control input calculating means comprises air/fuel ratio calculating means for calculating the target air/fuel ratio of the air/fuel mixture supplied to the internal combustion engine for converging the output of the downstream air/fuel ratio sensor to the predetermined target value in accordance with the calculated predicted value of the output deviation based on the on modulation algorithm. 
   According to this preferred embodiment of the control apparatus, the predicted value of the output deviation, which is a deviation of the output of the downstream air/fuel ratio sensor from the predetermined target value, is calculated in accordance with the target air/fuel ratio of the air/fuel mixture supplied to the internal combustion engine, the output of the upstream air/fuel ratio sensor, and the output of the downstream air/fuel ratio sensor, and the target air/fuel ratio of the air/fuel mixture is calculated based on the one modulation algorithm for converging the output of the downstream air/fuel ratio sensor to the predetermined target value in accordance with the thus calculated predicted value of the output deviation. Since the control input is calculated in the foregoing manner, the air/fuel ratio of exhaust gases can be controlled such that the exhaust gases can be satisfactorily purified by the catalyst by appropriately setting the predetermined target value, resulting in an improved characteristic of the exhaust gases purified by the catalyst (hereinafter called the “post-catalyst exhaust gas characteristic). Also, since the predicted value is calculated in accordance with the output of the upstream air/fuel ratio sensor disposed at a location upstream of the catalyst, the air/fuel ratio of exhaust gases actually supplied to the catalyst can be more appropriately reflected to the predicted value, resulting in a correspondingly improved accuracy in which the predicted value can be calculated. 
   Preferably, in the control method described above, the controlled object comprises a downstream air/fuel ratio sensor disposed at a location downstream of a catalyst in an exhaust passage of an internal combustion engine for detecting an air/fuel ratio of exhaust gases which have passed through the catalyst, and the output of the controlled object is an output of the downstream air/fuel ratio sensor. The value indicative of the output of the controlled object is an output deviation of an output of the downstream air/fuel ratio sensor from a predetermined target value. The control input to the controlled object is a target air/fuel ratio of an air/fuel mixture supplied to the internal combustion engine. The value reflecting a control input inputted to the controlled object is an output of an upstream air/fuel ratio sensor disposed at a location upstream of the catalyst in the exhaust passage for detecting an air/fuel ratio of exhaust gases which have not passed through the catalyst. The step of calculating a predicted value includes calculating the predicted value of the output deviation in accordance with at least one of the target air/fuel ratio of the air/fuel mixture supplied to the internal combustion engine, the output of the upstream air/fuel ratio sensor, and the output of the downstream air/fuel ratio sensor based on the prediction algorithm. The step of calculating a control input includes calculating the target air/fuel ratio of the air/fuel mixture supplied to the internal combustion engine for converging the output of the downstream air/fuel ratio sensor to the predetermined target value in accordance with the calculated predicted value of the output deviation based on the one modulation algorithm. 
   This preferred embodiment of the control method provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the engine control unit described above, the controlled object comprises a downstream air/fuel ratio sensor disposed at a location downstream of a catalyst in an exhaust passage of an internal combustion engine for detecting an air/fuel ratio of exhaust gases which have passed through the catalyst, and the output of the controlled object is an output of the downstream air/fuel ratio sensor. The value indicative of the output of the controlled object is an output deviation of an output of the downstream air/fuel ratio sensor from a predetermined target value. The control input to the controlled object is a target air/fuel ratio of an air/fuel mixture supplied to the internal combustion engine. The value reflecting a control input inputted to the controlled object is an output of an upstream air/fuel ratio sensor disposed at a location upstream of the catalyst in the exhaust passage for detecting an air/fuel ratio of exhaust gases which have not passed through the catalyst. The engine control unit causes the computer to calculate the predicted value of the output deviation in accordance with at least one of the target air/fuel ratio of the air/fuel mixture supplied to the internal combustion engine, the output of the upstream air/fuel ratio sensor, and the output of the downstream air/fuel ratio sensor based on the prediction algorithm; and calculate the target air/fuel ratio of the air/fuel mixture supplied to the internal combustion engine for converging the output of the downstream air/fuel ratio sensor to the predetermined target value in accordance with the calculated predicted value of the output deviation based on the one modulation algorithm. 
   This preferred embodiment of the engine control unit provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, the control apparatus described above further comprises operating condition detecting means for detecting an operating condition, such as an engine rotational speed or an intake pipe inner absolute pressure, of the internal combustion engine, wherein the predicted value calculating means calculates a prediction time from the time at which the air/fuel mixture is supplied to the internal combustion engine in the target air/fuel ratio to the time at which the target air/fuel ratio is reflected to the output of the downstream air/fuel ratio sensor in accordance with the detected operating condition of the internal combustion engine, and calculates the predicted value of the output deviation further in accordance with the calculated prediction time. 
   In this type of control apparatus for controlling the air/fuel ratio, the dynamic characteristic (for example, a response delay and a dead time) of a controlled object including an internal combustion engine and a catalyst varies depending on an operating condition of the internal combustion engine, for example, an exhaust gas volume. On the contrary, according to this preferred embodiment of the control apparatus, the prediction time from the time at which the air/fuel mixture is supplied to the internal combustion engine in the target air/fuel ratio to the time at which the target air/fuel ratio is reflected to the output of the downstream air/fuel ratio sensor is calculated in accordance with the detected operating condition of the internal combustion engine, and the predicted value of the output deviation is calculated further in accordance with the calculated prediction time, so that the control apparatus can eliminate without fail a slippage in control timing between the input and output of the controlled object, caused by the dynamic characteristic of the controlled object, by calculating the control input using the predicted value calculated in this manner, thereby making it possible to further improve the post-catalyst exhaust gas characteristic. 
   Preferably, the control method described above further comprises the step of detecting an operating condition of the internal combustion engine, wherein the step of calculating a predicted value includes calculating a prediction time from the time at which the air/fuel mixture is supplied to the internal combustion engine in the target air/fuel ratio to the time at which the target air/fuel ratio is reflected to the output of the downstream air/fuel ratio sensor in accordance with the detected operating condition of the internal combustion engine; and calculating the predicted value of the output deviation further in accordance with the calculated prediction time. 
   This preferred embodiment of the control method provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the engine control unit described above, the control program further causes the computer to detect an operating condition of the internal combustion engine; calculate a prediction time from the time at which the air/fuel mixture is supplied to the internal combustion engine in the target air/fuel ratio to the time at which the target air/fuel ratio is reflected to the output of the downstream air/fuel ratio sensor in accordance with the detected operating condition of the internal combustion engine; and calculate the predicted value of the output deviation further in accordance with the calculated prediction time. 
   This preferred embodiment of the engine control unit provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, the control apparatus described above further comprises operating condition detecting means for detecting an operating condition of the internal combustion engine, wherein the air/fuel ratio calculating means includes intermediate value calculating means for calculating an intermediate value of the target air/fuel ratio of the air/fuel mixture supplied to the internal combustion engine in accordance with the predicted value of the output deviation based on the one modulation algorithm; gain setting means for setting a gain in accordance with the detected operating condition of the internal combustion engine; and target air/fuel ratio calculating means for calculating the target air/fuel ratio of the air/fuel mixture supplied to the internal combustion engine based on the calculated intermediate value multiplied by the set gain. 
   In this type of control apparatus for controlling the air/fuel ratio, the gain characteristic to the air/fuel ratio of a controlled object including an internal combustion engine and a catalyst varies depending on an operating condition of the internal combustion engine, for example, an exhaust gas volume. In this event, the one modulation algorithm determines the control input on the assumption that the controlled object has a unity gain, as described above, so that if the gain characteristic of the controlled object varies as described above, the target air/fuel ratio of the air/fuel mixture, as a control input, largely deviates from an appropriate value and becomes oscillatory, causing an oscillatory output of the downstream air/fuel ratio sensor at a location downstream of the catalyst. This would result in a degradation in the post-catalyst exhaust gas characteristic. On the contrary, according to this preferred embodiment of the control apparatus, since the target air/fuel ratio of the air/fuel mixture is calculated based on the intermediate value calculated based on the one modulation algorithm, multiplied by the gain, and the gain is set in accordance with an operating condition of the internal combustion engine, the target air/fuel ratio of the air/fuel mixture can be calculated as a value which appropriately reflects a change in the gain characteristic of the controlled object resulting from a change in the operating condition, thereby making it possible to further improve the post-catalyst exhaust gas characteristic. 
   Preferably, the control method described above further comprises the step of detecting an operating condition of the internal combustion engine, wherein the step of calculating the target air/fuel ratio includes calculating an intermediate value of the target air/fuel ratio of the air/fuel mixture supplied to the internal combustion engine in accordance with the predicted value of the output deviation based on the one modulation algorithm; setting a gain in accordance with the detected operating condition of the internal combustion engine; and calculating the target air/fuel ratio of the air/fuel mixture supplied to the internal combustion engine based on the calculated intermediate value multiplied by the set gain. 
   This preferred embodiment of the control method provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the engine control unit described above, the control program further causes the computer to detect an operating condition of the internal combustion engine; calculate an intermediate value of the target air/fuel ratio of the air/fuel mixture supplied to the internal combustion engine in accordance with the predicted value of the output deviation based on the one modulation algorithm; set a gain in accordance with the detected operating condition of the internal combustion engine; and calculate the target air/fuel ratio of the air/fuel mixture supplied to the internal combustion engine based on the calculated intermediate value multiplied by the set gain. 
   This preferred embodiment of the engine control unit provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, the control apparatus described above further comprises multiplying means for multiplying the calculated predicted value of the output deviation by a correction coefficient, and correction coefficient setting means for setting the correction coefficient to a smaller value when the predicted value of the output deviation is equal to or larger than a predetermined value than when the predicted value of the output deviation is smaller than the predetermined value, wherein the air/fuel ratio calculating means calculates the target air/fuel ratio of the air/fuel mixture in accordance with the predicted value of the output deviation multiplied by the correction coefficient based on the one modulation algorithm. 
   According to this preferred embodiment of the control apparatus, the target air/fuel ratio of the air/fuel mixture is calculated in accordance with the predicted value of the output deviation multiplied by the correction coefficient, and the correction coefficient is set to a smaller value when the predicted value of the output deviation is equal to or larger than a predetermined value than when the predicted value of the output deviation is smaller than the predetermined value, so that the output of the downstream air/fuel ratio sensor can be converged at a different rate in accordance with the order of the predicted value of the output deviation with respect to the predetermined value. Therefore, for changing the target air/fuel ratio to be leaner because of the predicted value of the output deviation being equal to or larger than zero, i.e., the output of the downstream air/fuel ratio sensor being larger than a target value when the predetermined value is set, for example, to zero, the correction coefficient is set such that the output of the downstream air/fuel ratio sensor is converged at a lower rate than when the target air/fuel ratio is changed to be richer, thereby providing the effect of suppressing the amount of exhausted NOx by a lean bias. On the other hand, when the target air/fuel ratio is changed to be richer, the correction coefficient is set such that the output of the downstream air/fuel ratio sensor is converted at a higher rate than when the target air/fuel ratio is changed to be leaner, thereby making it possible to sufficiently recover the NOx purifying rate of the catalyst. 
   Preferably, the control method described above further comprises the steps of multiplying the calculated predicted value of the output deviation by a correction coefficient; and setting the correction coefficient to a smaller value when the predicted value of the output deviation is equal to or larger than a predetermined value than when the predicted value of the output deviation is smaller than the predetermined value, wherein the step of calculating the target air/fuel ratio includes calculating the target air/fuel ratio of the air/fuel mixture in accordance with the predicted value of the output deviation multiplied by the correction coefficient based on the one modulation algorithm. 
   This preferred embodiment of the control method provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the engine control unit described above, the control program further causes the computer to multiply the calculated predicted value of the output deviation by a correction coefficient; set the correction coefficient to a smaller value when the predicted value of the output deviation is equal to or larger than a predetermined value than when the predicted value of the output deviation is smaller than the predetermined value; and calculate the target air/fuel ratio of the air/fuel mixture in accordance with the predicted value of the output deviation multiplied by the correction coefficient based on the one modulation algorithm. 
   This preferred embodiment of the engine control unit provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the control apparatus described above, the controlled object comprises an air/fuel ratio sensor disposed at a location downstream of a catalyst in an exhaust pipe of an internal combustion engine for detecting an air/fuel ratio of exhaust gases which have passed through the catalyst, and the output of the controlled object is an output of the downstream air/fuel ratio sensor. The value indicative of the output of the controlled object is an output deviation of an output of the air/fuel ratio sensor from a predetermined target value. The control input to the controlled object is a target air/fuel ratio of an air/fuel mixture supplied to the internal combustion engine. The predicted value calculating means calculates the predicted value of the output deviation in accordance with the target air/fuel ratio of the air/fuel mixture supplied to the internal combustion engine, and the output of the air/fuel ratio sensor based on the prediction algorithm. The control input calculating means includes an air/fuel ratio calculating means for calculating the target air/fuel ratio of the air/fuel mixture supplied to the internal combustion engine for converging the output of the air/fuel ratio sensor to the predetermined target value in accordance with the calculated predicted value of the output deviation based on the one modulation algorithm. 
   According to this preferred embodiment of the control apparatus, the predicted value of the output deviation, which is a deviation of the output of the air/fuel ratio sensor from the predetermined target value, is calculated in accordance with the target air/fuel ratio of the air/fuel mixture supplied to the internal combustion engine, and the output of the air/fuel ratio sensor, and the target air/fuel ratio of the air/fuel mixture for converging the output of the air/fuel ratio sensor to the predetermined target value is calculated in accordance with the predicted value of the output deviation calculated in this manner based on the one modulation algorithm. Since the control input is calculated as described above, it is possible to control the air/fuel ratio of exhaust gases such that the catalyst purifies exhaust gases in a satisfactory manner by appropriately setting the predetermined target value, resulting in an improved post-catalyst exhaust gas characteristic. In addition, the control apparatus can be realized at a relatively low cost because it only requires a single air/fuel ratio sensor. 
   Preferably, in the control method described above, the controlled object comprises an air/fuel ratio sensor disposed at a location downstream of a catalyst in an exhaust pipe of an internal combustion engine for detecting an air/fuel ratio of exhaust gases which have passed through the catalyst, and the output of the controlled object is an output of the downstream air/fuel ratio sensor. The value indicative of the output of the controlled object is an output deviation of an output of the air/fuel ratio sensor from a predetermined target value. The control input to the controlled object is a target air/fuel ratio of an air/fuel mixture supplied to the internal combustion engine. The step of calculating a predicted value includes calculating the predicted value of the output deviation in accordance with the target air/fuel ratio of the air/fuel mixture supplied to the internal combustion engine, and the output of the air/fuel ratio sensor based on the prediction algorithm. The step of calculating a control input includes calculating the target air/fuel ratio of the air/fuel mixture supplied to the internal combustion engine for converging the output of the air/fuel ratio sensor to the predetermined target value in accordance with the calculated predicted value of the output deviation based on the one modulation algorithm. 
   This preferred embodiment of the control method provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the An engine control unit described above, the controlled object comprises an air/fuel ratio sensor disposed at a location downstream of a catalyst in an exhaust pipe of an internal combustion engine for detecting an air/fuel ratio of exhaust gases which have passed through the catalyst, and the output of the controlled object is an output of the downstream air/fuel ratio sensor. The value indicative of the output of the controlled object is an output deviation of an output of the air/fuel ratio sensor from a predetermined target value. The control input to the controlled object is a target air/fuel ratio of an air/fuel mixture supplied to the internal combustion engine. The control program causes the computer to calculate the predicted value of the output deviation in accordance with the target air/fuel ratio of the air/fuel mixture supplied to the internal combustion engine, and the output of the air/fuel ratio sensor based on the prediction algorithm; and calculate the target air/fuel ratio of the air/fuel mixture supplied to the internal combustion engine for converging the output of the air/fuel ratio sensor to the predetermined target value in accordance with the calculated predicted value of the output deviation based on the one modulation algorithm. 
   This preferred embodiment of the engine control unit provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, the control apparatus described above further comprises operating condition detecting means for detecting an operating condition of the internal combustion engine, wherein the predicted value calculating means calculates a prediction time from the time at which the air/fuel mixture is supplied to the internal combustion engine in the target air/fuel ratio to the time at which the target air/fuel ratio is reflected to the output of the air/fuel ratio sensor in accordance with the detected operating condition of the internal combustion engine, and calculates the predicted value of the output deviation further in accordance with the calculated prediction time. 
   According to this preferred embodiment of the control apparatus, the prediction time from the time at which the air/fuel mixture is supplied to the internal combustion engine in the target air/fuel ratio to the time at which the target air/fuel ratio is reflected to the output of the air/fuel ratio sensor is calculated in accordance with the detected operating condition of the internal combustion engine, and the predicted value of the output deviation is calculated further in accordance with the calculated prediction time, so that the control apparatus can eliminate without fail a slippage in control timing between the input and output of the controlled object, caused by the dynamic characteristic of the controlled object, by calculating the control input using the predicted value calculated in this manner, thereby making it possible to further improve the post-catalyst exhaust gas characteristic. 
   Preferably, the control method described above further comprises the step of detecting an operating condition of the internal combustion engine, wherein the step of calculating a predicted value includes calculating a prediction time from the time at which the air/fuel mixture is supplied to the internal combustion engine in the target air/fuel ratio to the time at which the target air/fuel ratio is reflected to the output of the air/fuel ratio sensor in accordance with the detected operating condition of the internal combustion engine; and calculating the predicted value of the output deviation further in accordance with the calculated prediction time. 
   This preferred embodiment of the control method provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the engine control unit described above, the control program further causes the computer to detect an operating condition of the internal combustion engine; calculate a prediction time from the time at which the air/fuel mixture is supplied to the internal combustion engine in the target air/fuel ratio to the time at which the target air/fuel ratio is reflected to the output of the air/fuel ratio sensor in accordance with the detected operating condition of the internal combustion engine; and calculate the predicted value of the output deviation further in accordance with the calculated prediction time. 
   This preferred embodiment of the engine control unit provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, the control apparatus described above further comprises operating condition detecting means for detecting an operating condition of the internal combustion engine, wherein the air/fuel ratio calculating means includes intermediate value calculating means for calculating an intermediate value of the target air/fuel ratio of the air/fuel mixture supplied to the internal combustion engine in accordance with the predicted value of the output deviation based on the one modulation algorithm; gain setting means for setting a gain in accordance with the detected operating condition of the internal combustion engine; and target air/fuel ratio calculating means for calculating the target air/fuel ratio of the air/fuel mixture supplied to the internal combustion engine based on the calculated intermediate value multiplied by the set gain. 
   According to this preferred embodiment of the control apparatus, since the target air/fuel ratio of the air/fuel mixture is calculated based on the intermediate value calculated based on the one modulation algorithm, multiplied by the gain, and the gain is set in accordance with an operating condition, the target air/fuel ratio of the air/fuel mixture can be calculated as a value which appropriately reflects a change in the gain characteristic of the controlled object, thereby making it possible to further improve the post-catalyst exhaust gas characteristic. 
   Preferably, the control method described above further comprises the step of detecting an operating condition of the internal combustion engine, wherein the step of calculating the air/fuel ratio calculating means includes calculating an intermediate value of the target air/fuel ratio of the air/fuel mixture supplied to the internal combustion engine in accordance with the predicted value of the output deviation based on the one modulation algorithm; setting a gain in accordance with the detected operating condition of the internal combustion engine; and calculating the target air/fuel ratio of the air/fuel mixture supplied to the internal combustion engine based on the calculated intermediate value multiplied by the set gain. 
   This preferred embodiment of the control method provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the engine control unit described above, the control program further causes the computer to detect an operating condition of the internal combustion engine; calculate an intermediate value of the target air/fuel ratio of the air/fuel mixture supplied to the internal combustion engine in accordance with the predicted value of the output deviation based on the one modulation algorithm; set a gain in accordance with the detected operating condition of the internal combustion engine; and calculate the target air/fuel ratio of the air/fuel mixture supplied to the internal combustion engine based on the calculated intermediate value multiplied by the set gain. 
   This preferred embodiment of the engine control unit provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, the control apparatus described above further comprises multiplying means for multiplying the calculated predicted value of the output deviation by a correction coefficient, and correction coefficient setting means for setting the correction coefficient to a smaller value when the predicted value of the output deviation is equal to or larger than a predetermined value than when the predicted value of the output deviation is smaller than the predetermined value, wherein the air/fuel ratio calculating means calculates the target air/fuel ratio of the air/fuel mixture in accordance with the predicted value of the output deviation multiplied by the correction coefficient based on the one modulation algorithm. 
   According to this preferred embodiment of the control apparatus, the target air/fuel ratio of the air/fuel mixture is calculated in accordance with the predicted value of the output deviation multiplied by the correction coefficient, and the correction coefficient is set to a smaller value when the predicted value of the output deviation is equal to or larger than a predetermined value than when the predicted value of the output deviation is smaller than the predetermined value, so that the output of the downstream air/fuel ratio sensor can be converged at a different rate in accordance with the order of the predicted value of the output deviation with respect to the predetermined value. Therefore, for changing the air/fuel ratio to be leaner because of the predicted value of the output deviation being equal to or larger than zero, i.e., the output of the downstream air/fuel ratio sensor being larger than a target value when the predetermined value is set, for example to zero, the correction coefficient is set such that the output of the downstream air/fuel ratio sensor is converged at a lower rate than when the air/fuel ratio is changed to be richer, thereby providing the effect of suppressing the amount of exhausted NOx by a lean bias. On the other hand, when the air/fuel ratio is changed to be richer, the correction coefficient is set such that the output of the downstream air/fuel ratio sensor is converted at a higher rate than when the air/fuel ratio is changed to be leaner, thereby making it possible to sufficiently recover the NOx purifying rate of the catalyst. 
   Preferably, the control method described above further comprises the steps of multiplying the calculated predicted value of the output deviation by a correction coefficient; and setting the correction coefficient to a smaller value when the predicted value of the output deviation is equal to or larger than a predetermined value than when the predicted value of the output deviation is smaller than the predetermined value, wherein the step of calculating the target air/fuel ratio includes calculating the target air/fuel ratio of the air/fuel mixture in accordance with the predicted value of the output deviation multiplied by the correction coefficient based on the one modulation algorithm. 
   This preferred embodiment of the control method provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the engine control unit described above, the control program further causes the computer to multiply the calculated predicted value of the output deviation by a correction coefficient; set the correction coefficient to a smaller value when the predicted value of the output deviation is equal to or larger than a predetermined value than when the predicted value of the output deviation is smaller than the predetermined value; and calculate the target air/fuel ratio of the air/fuel mixture in accordance with the predicted value of the output deviation multiplied by the correction coefficient based on the one modulation algorithm. 
   This preferred embodiment of the engine control unit provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   To achieve the above object, according to a fourth aspect of the present invention, there is provided a control apparatus which comprises control input calculating means for calculating a control input, such as a target air/fuel ratio, to a controlled object, based on one modulation algorithm selected from a Δ modulation algorithm, a ΔΣ modulation algorithm, and a ΣΔ modulation algorithm, and a controlled object model which models the controlled object, for controlling an output of the controlled. 
   According to the control apparatus described above, since the control input is calculated based on the one modulation algorithm selected from the Δ modulation algorithm, ΔΣ modulation algorithm, and ΣΔ modulation algorithm, and the controlled object model which models the controlled object, the control input can be calculated as a value which reflects a dynamic characteristic such as a phase delay, a dead time, or the like of the controlled object by defining the controlled object model as appropriately reflecting the dynamic characteristic of the controlled object, consequently making it possible to ensure the stability of the control and improve the controllability. 
   To achieve the above object, according to a fifth aspect of the invention, there is provided a control method which is characterized by comprising the step of calculating a control input to a controlled object based on one modulation algorithm selected from a Δ modulation algorithm, and ΔΣ modulation algorithm, and a ΣΔ modulation algorithm, and a controlled object model which models the controlled object, for controlling an output of the controlled object. 
   This control method provides the same advantageous effects as described above concerning the control apparatus according to the fourth aspect of the invention. 
   To achieve the above object, according to a sixth aspect of the invention, there is provided an engine control unit including a control program for causing a computer to calculate a control input to a controlled object based on one modulation algorithm selected from a Δ modulation algorithm, and ΔΣ modulation algorithm, and a ΣΔ modulation algorithm, and a controlled object model which models the controlled object, for controlling an output of the controlled object. 
   This engine control unit provides the same advantageous effects as described above concerning the control apparatus according to the fourth aspect of the invention. 
   Preferably, in the control apparatus described above, the controlled object model is built as a discrete time system model, and the control apparatus further comprises identifying means for sequentially identifying model parameters of the controlled object model in accordance with one of the calculated control input and a value reflecting a control input inputted to the controlled object, and the output of the controlled object. 
   According to this preferred embodiment of the control apparatus, the model parameters are sequentially identified in accordance with the value which reflects the control input and/or the value reflecting the control input, and the output of the controlled object, i.e., the model parameters are identified in real time, and the control input is calculated based on the controlled object model, the model parameters of which are identified in the foregoing manner. Thus, even if the dynamic characteristic of the controlled object varies due to a changing environment or has been aged, the dynamic characteristic of the controlled object model can be fitted to the actual dynamic characteristic of the controlled object, while avoiding the influence of the variations and aging changes thereof. As a result, the control apparatus can appropriately correct a slippage in control timing between the input and output, caused by the dynamic characteristic of the controlled object, for example, a response delay, a dead time, or the like, thereby making it possible to ensure the stability of the control and improve the controllability. 
   Preferably, in the control method described above, the controlled object model is built as a discrete time system model, and the control method further comprises the step of sequentially identifying model parameters of the controlled object model in accordance with one of the calculated control input and a value reflecting a control input inputted to the controlled object, and the output of the controlled object. 
   This preferred embodiment of the control method provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the engine control unit described above, the controlled object model is built as a discrete time system model, and the control program further causes the computer to sequentially identify model parameters of the controlled object model in accordance with one of the calculated control input and a value reflecting a control input inputted to the controlled object, and the output of the controlled object. 
   This preferred embodiment of the engine control unit provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the control apparatus described above, the identifying means includes identification error calculating means for calculating an identification error of the model parameters; filtering means for filtering the calculated identification error in a predetermined manner; and parameter determining means for determining the model parameters based on the filtered identification error. 
   Generally, an identification algorithm for identifying model parameters based on an identification error, for example, an identification algorithm based on a least-square algorithm, and the like identifies model parameters with the frequency characteristic of the controlled object emphasized in a predetermined frequency band due to a frequency weighting characteristic of the identification algorithm, so that the gain characteristic of the controlled object model may fail to fit to the actual gain characteristic of the controlled object. For example, when a controlled object has a low pass characteristic, model parameters may be identified with a high frequency characteristic of the controlled object which is emphasized due to the frequency weighting characteristic of the identification algorithm, in which case the controlled object model exhibits the gain characteristic which tends to be lower than the actual gain characteristic of the controlled object. Therefore, according to this preferred embodiment of the control apparatus, the model parameters are identified based on the identification error of the filtered model parameters, so that the controlled object model can be matched with the control object in the gain characteristic by appropriately setting the filtering characteristic, for example, in accordance with the frequency characteristic of the controlled object, thereby making it possible to correct a slippage in control timing between the input and output of the controlled object with an improved accuracy. 
   Preferably, in the control method described above, the step of identifying includes calculating an identification error of the model parameters; filtering the calculated identification error in a predetermined manner; and determining the model parameters based on the filtered identification error. 
   This preferred embodiment of the control method provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the engine control unit described above, the control program causes the computer to calculate an identification error of the model parameters; filter the calculated identification error in a predetermined manner; and determine the model parameters based on the filtered identification error. 
   This preferred embodiment of the engine control unit provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the control apparatus described above, the filtering means sets a filtering characteristic for the filtering in accordance with a dynamic characteristic of the controlled object, such as an exhaust gas volume. 
   According to this preferred embodiment of the control apparatus, since the filtering characteristic is set in accordance with the dynamic characteristic of the controlled object, the controlled object model can be matched with the controlled object in the gain characteristic for the reason set forth above, thereby making it possible to correct a slippage in control timing between the input and output of the controlled object with a more improved accuracy. 
   Preferably, in the control method described above, the step of filtering includes setting a filtering characteristic for the filtering in accordance with a dynamic characteristic of the controlled object. 
   This preferred embodiment of the control method provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the engine control unit described above, the control program causes the computer to set a filtering characteristic for the filtering in accordance with a dynamic characteristic of the controlled object. 
   This preferred embodiment of the engine control unit provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the control apparatus described above, the controlled object model comprises an input variable indicative of one of the control input and the value reflecting a control input inputted to the controlled object, and an output variable indicative of the output of the controlled object. The identifying means identifies a model parameter multiplied by the input variable and a model parameter multiplied by the output variable such that the model parameters fall within respective predetermined restriction ranges. 
   Generally, with a sequential identification algorithm, when the input and output of a controlled object enter a steady state, a control system may become instable or oscillatory because a so-called drift phenomenon is more likely to occur, in which absolute values of identified model parameters increase due to a shortage of self excitation condition. On the contrary, according to this preferred embodiment of the control apparatus, since the model parameters of the controlled object model, i.e., the model parameter multiplied by the input variable and the model parameter multiplied by the output variable are sequentially identified such that they fall within respective predetermined restriction ranges, it is possible to avoid the drift phenomenon by appropriately setting the predetermined restriction ranges, to enhance the ensured stability of the control. 
   Preferably, in the control method described above, the controlled object model comprises an input variable indicative of one of the control input and the value reflecting a control input inputted to the controlled object, and an output variable indicative of the output of the controlled object. The step of identifying includes identifying a model parameter multiplied by the input variable and a model parameter multiplied by the output variable such that the model parameters fall within respective predetermined restriction ranges. 
   This preferred embodiment of the control method provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the engine control unit described above, the controlled object model comprises an input variable indicative of one of the control input and the value reflecting a control input inputted to the controlled object, and an output variable indicative of the output of the controlled object. The control program causes the computer to identify a model parameter multiplied by the input variable and a model parameter multiplied by the output variable such that the model parameters fall within respective predetermined restriction ranges. 
   This preferred embodiment of the engine control unit provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the control apparatus described above, the output variable comprises a plurality of time-series data of output variables which are multiplied by a plurality of model parameters, respectively, and the identifying means identifies the plurality of model parameters such that a combination of the model parameters falls within the predetermined restriction range. 
   With this type of identification algorithm, when a plurality of model parameters are identified independently of one another such that they fall within a predetermined restriction range in which a control system is stable, the control system may become instable or oscillatory depending on a combination of the model parameters. On the contrary, according to this preferred embodiment of the control apparatus, since the plurality of model parameters are identified such that a combination of the model parameters falls within the predetermined restriction range, the control system can be more securely held in a stable state by appropriately setting the predetermined restriction range, as compared with an identification algorithm which identifies a plurality of model parameters independently of one another. 
   Preferably, in the control method described above, the output variable comprises a plurality of time-series data of output variables which are multiplied by a plurality of model parameters, respectively, and the step of identifying includes identifying the plurality of model parameters such that a combination of the model parameters falls within the predetermined restriction range. 
   This preferred embodiment of the control method provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the engine control unit described above, the output variable comprises a plurality of time-series data of output variables which are multiplied by a plurality of model parameters, respectively, and the control program causes the computer to identify the plurality of model parameters such that a combination of the model parameters falls within the predetermined restriction range. 
   This preferred embodiment of the engine control unit provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the control apparatus described above, the identifying means sets the predetermined restriction range in accordance with a dynamic characteristic of the controlled object. 
   According to this preferred embodiment of the control apparatus, since the restriction range for restricting the model parameters is set in accordance with the dynamic characteristic of the controlled object, the control input can be calculated as a value which can ensure the stability of the controlled object by calculating the control input based on the controlled object model which uses the model parameters that are set in the foregoing manner, thereby making it possible to enhance the ensured stability of the control. 
   Preferably, in the control method described above, the step of identifying further includes setting the predetermined restriction range in accordance with a dynamic characteristic of the controlled object. 
   This preferred embodiment of the control method provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the engine control unit described above, the control program causes the engine to set the predetermined restriction range in accordance with a dynamic characteristic of the controlled object. 
   This preferred embodiment of the engine control unit provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the control apparatus described above, the output variable is a deviation of the output of the controlled object from a predetermined target value, and the input variable is one of a deviation of the control input from a predetermined reference value, and a deviation of the value reflecting a control input inputted to the controlled object from the predetermined reference value. 
   As described above, the dynamic characteristic of a controlled object model can be fitted more closely to the actual dynamic characteristic of a controlled object when a deviation of the input/output of the controlled object from a predetermined value is defined as a variable indicative of the input/output than when the input/output itself is defined as a variable. Therefore, according to this preferred embodiment of the control apparatus, since the controlled object model has a variable associated with a deviation of a control input and/or a value reflecting the control input inputted to the controlled object from a predetermined reference value, and a variable associated with a deviation of the output of the controlled object from a predetermined target value, the dynamic characteristic of the controlled object model can be fitted more closely to the actual dynamic characteristic of the controlled object, as compared with a controlled object model which has a variable associated with an absolute value of a control input and/or a value reflecting a control input, and a variable associated with an absolute value of the output of the controlled object. It is therefore possible to enhance the ensured stability of the control by calculating the control input based on the controlled object model as described above. 
   Preferably, in the control method described above, the output variable is a deviation of the output of the controlled object from a predetermined target value, and the input variable is one of a deviation of the control input from a predetermined reference value, and a deviation of the value reflecting a control input inputted to the controlled object from the predetermined reference value. 
   This preferred embodiment of the control method provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the engine control unit described above, the output variable is a deviation of the output of the controlled object from a predetermined target value, and the input variable is one of a deviation of the control input from a predetermined reference value, and a deviation of the value reflecting a control input inputted to the controlled object from the predetermined reference value. 
   This preferred embodiment of the engine control unit provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the control apparatus described above, the identifying means further includes identifying the model parameters based on a weighted identification algorithm which uses weighting parameters for determining behaviors of the model parameters, and setting the weighting parameters in accordance with a dynamic characteristic of the controlled object. 
   In this type of control apparatus, the output of a controlled object is more likely to be oscillatory under a condition in which the dynamic characteristic of the controlled object varies, particularly, under a condition in which a response delay and a dead time become larger, causing associated variations in identified model parameters. On the contrary, according to this preferred embodiment of the present invention, since the weighting parameters for determining the behaviors of the model parameters are set in accordance with the dynamic characteristic of the controlled object, the weighting parameters can be appropriately set to stabilize the behaviors of the model parameters even under a condition in which a response delay and a dead time of the controlled object become larger, thereby making it possible to further enhance the ensured stability of the control. 
   Preferably, in the control method described above, the step of identifying further includes identifying the model parameters based on a weighted identification algorithm which uses weighting parameters for determining behaviors of the model parameters, and setting the weighting parameters in accordance with a dynamic characteristic of the controlled object. 
   This preferred embodiment of the control method provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the engine control unit described above, the control program causes the computer to identify the model parameters based on a weighted identification algorithm which uses weighting parameters for determining behaviors of the model parameters; and set the weighting parameters in accordance with a dynamic characteristic of the controlled object. 
   This preferred embodiment of the engine control unit provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the control apparatus described above, the identifying means further includes dead time setting means for setting a dead time between one of the control input inputted to the controlled object and the value reflecting the control input inputted to the controlled object and the output of the controlled object in accordance with a dynamic characteristic of the controlled object, wherein the dead time is used in the identification algorithm. 
   This type of identification algorithm can increase an identification accuracy for a model parameter multiplied by the control input of the controlled object model when a dead time between a control input or a value reflecting the control input inputted to the controlled object and the output of the control object is set to be highly correlated to an actual input to the controlled object. Therefore, according to this preferred embodiment of the control apparatus, since the dead time between the control input to the controlled object or the value reflecting the control input inputted to the controlled object, and the output of the controlled object, used in the identification algorithm, is set in accordance with the dynamic characteristic of the controlled object, the model parameter multiplied by the control input of the controlled object model can be identified with a higher accuracy to more accurately calculate the control input. 
   Preferably, in the control method described above, the step of identifying further includes setting a dead time between one of the control input inputted to the controlled object and the value reflecting the control input inputted to the controlled object and the output of the controlled object in accordance with a dynamic characteristic of the controlled object, wherein the dead time is used in the identification algorithm. 
   This preferred embodiment of the control method provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the engine control unit described above, the control program causes the computer to set a dead time between one of the control input inputted to the controlled object and the value reflecting the control input inputted to the controlled object and the output of the controlled object in accordance with a dynamic characteristic of the controlled object, wherein the dead time is used in the identification algorithm. 
   This preferred embodiment of the engine control unit provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the control apparatus described above, the control input calculating means calculates a predicted value of a value indicative of the output of the controlled object based on a prediction algorithm which applies the controlled object model, and calculates the control input in accordance with the calculated predicted value based on the one modulation algorithm. 
   According to this preferred embodiment of the control apparatus, the predicted value of the value indicative of the output of the controlled object is calculated based on the predication algorithm which applies the controlled object model, and the control input is calculated in accordance with the calculated predicted value based on the one modulation algorithm. In this event, since the dynamic characteristic of the controlled object model can be fitted to the actual dynamic characteristic of the controlled object by using the model parameters identified by the identifying means as described above, the predicted value can be calculated as a value reflecting the actual dynamic characteristic of the controlled object by calculating the predicted value based on the prediction algorithm which applies the controlled object model as described above. As a result, the control apparatus can more appropriately correct a slippage in control timing between the control input and the output of the controlled object to further improve the stability of the control and the controllability. 
   Preferably, in the control method described above, the step of calculating a control input includes calculating a predicted value of a value indicative of the output of the controlled object based on a prediction algorithm which applies the controlled object model; and calculating the control input in accordance with the calculated predicted value based on the one modulation algorithm. 
   This preferred embodiment of the control method provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the engine control unit described above, the control program causes the computer to calculate a predicted value of a value indicative of the output of the controlled object based on a prediction algorithm which applies the controlled object model; and calculate the control input in accordance with the calculated predicted value based on the one modulation algorithm. 
   This preferred embodiment of the engine control unit provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the control apparatus described above, the control input calculating means calculates a prediction time from the time at which the control input is inputted to the controlled object to the time at which the control input is reflected to the output of the controlled object in accordance with a dynamic characteristic of the controlled object, and calculates the predicted value in accordance with the calculated prediction time based on the prediction algorithm. 
   According to this preferred embodiment of the control apparatus, the prediction time from the time at which the control input is inputted to the controlled object to the time at which the control input is reflected to the output of the controlled object is calculated in accordance with the dynamic characteristic of the controlled object, and the predicted value is calculated in accordance with the calculated prediction time, so that a slippage in control timing between the input/output of the controlled object, possibly caused by a response delay, a dead time, and the like of the controlled object, can be eliminated without fail by calculating the control input calculated in this manner, thereby making it possible to further improve the controllability. 
   Preferably, in the control method described above, the step of calculating a control input includes calculating a prediction time from the time at which the control input is inputted to the controlled object to the time at which the control input is reflected to the output of the controlled object in accordance with a dynamic characteristic of the controlled object; and calculating the predicted value in accordance with the calculated prediction time based on the prediction algorithm. 
   This preferred embodiment of the control method provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the engine control unit described above, the control program causes the computer to calculate a prediction time from the time at which the control input is inputted to the controlled object to the time at which the control input is reflected to the output of the controlled object in accordance with a dynamic characteristic of the controlled object; and calculate the predicted value in accordance with the calculated prediction time based on the prediction algorithm. 
   This preferred embodiment of the engine control unit provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the control apparatus described above, the control input calculating means calculates an intermediate value based on the controlled object model and the one modulation algorithm, and calculates the control input based on the calculated intermediate value multiplied by a predetermined gain. 
   According to this preferred embodiment of the control apparatus, the control input is calculated based on the intermediate value calculated based on the controlled object model and one modulation algorithm multiplied by a predetermined gain, so that a satisfactory controllability can be ensured by setting the predetermined gain to an appropriate value. 
   Preferably, in the control method described above, the step of calculating a control input includes calculating an intermediate value based on the controlled object model and the one modulation algorithm; and calculating the control input based on the calculated intermediate value multiplied by a predetermined gain. 
   This preferred embodiment of the control method provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the engine control unit described above, the control program causes the computer to calculate an intermediate value based on the controlled object model and the one modulation algorithm; and calculate the control input based on the calculated intermediate value multiplied by a predetermined gain. 
   This preferred embodiment of the engine control unit provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, the control apparatus described above further comprises gain parameter detecting means for detecting a gain parameter indicative of a gain characteristic of the controlled object; and gain setting means for setting the predetermined gain in accordance with the detected gain parameter. 
   According to this preferred embodiment of the control apparatus, since the predetermined gain for use in the calculation of the control input is set in accordance with the gain characteristic of the controlled object, the control input can be calculated as a value which has appropriate energy in accordance with the gain characteristic of the controlled object, thereby making it possible to avoid an over-gain condition and the like to ensure a satisfactory controllability. 
   Preferably, the control method described above further comprises the steps of detecting a gain parameter indicative of a gain characteristic of the controlled object; and setting the predetermined gain in accordance with the detected gain parameter. 
   This preferred embodiment of the control method provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the engine control unit described above, the control program further causes the computer to detect a gain parameter indicative of a gain characteristic of the controlled object; and set the predetermined gain in accordance with the detected gain parameter. 
   This preferred embodiment of the engine control unit provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the control apparatus described above, the control input calculating means calculates a second intermediate value in accordance with the predicted value based on the one modulation algorithm, and calculates the control input by adding a predetermined value to the calculated second intermediate value. 
   According to this preferred embodiment of the control apparatus, the control input calculating means calculates the control input by adding the predetermined value to the second intermediate value calculated based on one modulation algorithm, so that the control input calculating means can calculate the control input not only as a value which positively and negatively inverts centered at zero, but also as a value which repeats predetermined increase and decrease about a predetermined value, thereby making it possible to improve the degree of freedom in control. 
   Preferably, in the control method described above, the step of calculating a control input includes calculating a second intermediate value in accordance with the predicted value based on the one modulation algorithm; and calculating the control input by adding a predetermined value to the calculated second intermediate value. 
   This preferred embodiment of the control method provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the engine control unit described above, the control program causes the computer to calculate a second intermediate value in accordance with the predicted value based on the one modulation algorithm; and calculate the control input by adding a predetermined value to the calculated second intermediate value. 
   This preferred embodiment of the engine control unit provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the control apparatus described above, the controlled object comprises a downstream air/fuel ratio sensor disposed at a location downstream of a catalyst in an exhaust pipe of an internal combustion engine for detecting an air/fuel ratio of exhaust gases which have passed through the catalyst, and the output of the controlled object is an output of the downstream air/fuel ratio sensor. The control input to the controlled object is a target air/fuel ratio of an air/fuel mixture supplied to the internal combustion engine. The value reflecting a control input inputted to the controlled object is an output of an upstream air/fuel ratio sensor disposed at a location upstream of the catalyst in the exhaust passage of the internal combustion engine for detecting an air/fuel ratio of exhaust gases which have not passed through the catalyst. The controlled object model is a model which has a variable associated with a value indicative of the output of the downstream air/fuel ratio sensor, and a variable associated with one of a value indicative of the target air/fuel ratio and the output of the upstream air/fuel ratio sensor. The identifying means sequentially identifies a model parameter multiplied by the value indicative of the output of the downstream air/fuel ratio sensor, and a model parameter multiplied by one of the value indicative of the target air/fuel ratio and a value indicative of the output of the upstream air/fuel ratio sensor in accordance with one of the output of the upstream air/fuel ratio sensor and the target air/fuel ratio, and the output of the downstream air/fuel ratio sensor. The control input calculating means includes air/fuel ratio calculating means for calculating the target air/fuel ratio of the air/fuel mixture supplied to the internal combustion engine for converging the output of the downstream air/fuel ratio sensor to a predetermined target value based on the one modulation algorithm and the controlled object model. 
   According to this preferred embodiment of the control apparatus, the model parameters are sequentially identified in accordance with the output of the upstream air/fuel ratio sensor and the output of the downstream air/fuel ratio sensor, i.e., the model parameters are identified in real time, and the target air/fuel ratio of the air/fuel mixture supplied to the internal combustion engine is calculated based on the controlled object model,the model parameters of which are identified in the foregoing manner, and one modulation algorithm. Thus, even if the characteristics of the catalyst and both air/fuel ratio sensors vary due to a changing environment or have been aged, the output of the downstream air/fuel ratio sensor can be converged to the predetermined target value, while avoiding the influence of the variations and aging changes of the characteristics. Also, since the model parameters are identified in accordance with the upstream air/fuel ratio sensor disposed at a location upstream of the catalyst, the model parameters can be identified while more precisely reflecting exhaust gases actually supplied to the catalyst, thereby making it possible to identify the model parameters with an improved accuracy. Consequently, the control apparatus can appropriately correct a slippage in control timing of the air/fuel ratio control, caused by a response delay, a dead time, and the like of exhaust gases with respect to the air/fuel mixture supplied to the internal combustion engine, thereby making it possible to improve the post-catalyst exhaust gas characteristic. 
   Preferably, in the control method described above, the controlled object comprises a downstream air/fuel ratio sensor disposed at a location downstream of a catalyst in an exhaust passage of an internal combustion engine for detecting an air/fuel ratio of exhaust gases which have passed through the catalyst, and the output of the controlled object is an output of the downstream air/fuel ratio sensor. The control input to the controlled object is a target air/fuel ratio of an air/fuel mixture supplied to the internal combustion engine. The value reflecting a control input inputted to the controlled object is an output of an upstream air/fuel ratio sensor disposed at a location upstream of the catalyst in the exhaust passage of the internal combustion engine for detecting an air/fuel ratio of exhaust gases which have not passed through the catalyst. The controlled object model is a model which has a variable associated with a value indicative of the output of the downstream air/fuel ratio sensor, and a variable associated with one of a value indicative of the target air/fuel ratio and the output of the upstream air/fuel ratio sensor. The step of identifying includes sequentially identifying a model parameter multiplied by the value indicative of the output of the downstream air/fuel ratio sensor, and a model parameter multiplied by one of the value indicative of the target air/fuel ratio and a value indicative of the output of the upstream air/fuel ratio sensor in accordance with one of the output of the upstream air/fuel ratio sensor and the target air/fuel ratio, and the output of the downstream air/fuel ratio sensor. The step of calculating a control input includes calculating the target air/fuel ratio of the air/fuel mixture supplied to the internal combustion engine for converging the output of the downstream air/fuel ratio sensor to a predetermined target value based on the one modulation algorithm and the controlled object model. 
   This preferred embodiment of the control method provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the engine control unit described above, the controlled object comprises a downstream air/fuel ratio sensor disposed at a location downstream of a catalyst in an exhaust passage of an internal combustion engine for detecting an air/fuel ratio of exhaust gases which have passed through the catalyst, and the output of the controlled object is an output of the downstream air/fuel ratio sensor. The control input to the controlled object is a target air/fuel ratio of an air/fuel mixture supplied to the internal combustion engine. The value reflecting a control input inputted to the controlled object is an output of an upstream air/fuel ratio sensor disposed at a location upstream of the catalyst in the exhaust passage of the internal combustion engine for detecting an air/fuel ratio of exhaust gases which have not passed through the catalyst. The controlled object model is a model which has a variable associated with a value indicative of the output of the downstream air/fuel ratio sensor, and a variable associated with one of a value indicative of the target air/fuel ratio and the output of the upstream air/fuel ratio sensor. The control program causes the computer to sequentially identify a model parameter multiplied by the value indicative of the output of the downstream air/fuel ratio sensor, and a model parameter multiplied by one of the value indicative of the target air/fuel ratio and a value indicative of the output of the upstream air/fuel ratio sensor in accordance with one of the output of the upstream air/fuel ratio sensor and the target air/fuel ratio, and the output of the downstream air/fuel ratio sensor; and calculate the target air/fuel ratio of the air/fuel mixture supplied to the internal combustion engine for converging the output of the downstream air/fuel ratio sensor to a predetermined target value based on the one modulation algorithm and the controlled object model. 
   This preferred embodiment of the engine control unit provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the control apparatus described above, the value indicative of the output of the downstream air/fuel ratio sensor is an output deviation which is a deviation of the output of the downstream air/fuel ratio sensor from the predetermined target value. The value indicative of the output of the upstream air/fuel ratio sensor is an upstream output deviation which is a deviation of the output of the upstream air/fuel ratio sensor from a predetermined reference value. The value indicative of the target air/fuel ratio is an air/fuel ratio deviation which is a deviation of the target air/fuel ratio from the predetermined reference value. The controlled object model is a model which has a variable associated with the output deviation, and a variable associated with one of the air/fuel ratio deviation and the upstream output deviation. The identifying means identifies a model parameter multiplied by the output deviation, and a model parameter multiplied by one of the air/fuel ratio deviation and the upstream output deviation such that the parameters fall within respective predetermined restriction ranges. 
   According to this preferred embodiment of the control apparatus, since the controlled object model has a variable associated with the output deviation, and a variable associated with one of the air/fuel ratio deviation and upstream output deviation, the dynamic characteristic of the controlled object model can be fitted to the actual dynamic characteristic of the controlled object because the model parameters can be more precisely identified or defined for the controlled object model, for the reason set forth above, as compared with a controlled object model which has a variable associated with an absolute value of the output of the downstream air/fuel ratio sensor, and a variable associated with one of an absolute value of the target air/fuel ratio and an absolute value of the output of the upstream air/fuel ratio sensor. Also, as described above, with a sequential identification algorithm, when the input and output of a controlled object enter a steady state, a control system may become instable or oscillatory because a so-called drift phenomenon is more likely to occur, in which absolute values of identified model parameters increase due to a shortage of self excitation condition. On the contrary, according to this preferred embodiment of the control apparatus, since the model parameter multiplied by the output deviation and the model parameter multiplied by one of the air/fuel ratio deviation and upstream output deviation are identified such that they fall within respective predetermined restriction ranges, it is possible to avoid the drift phenomenon by appropriately setting the predetermined restriction ranges, to securely ensure the stability of the air/fuel ratio control and improve the post-catalyst exhaust gas characteristic. 
   Preferably, in the control method described above, the value indicative of the output of the downstream air/fuel ratio sensor is an output deviation which is a deviation of the output of the downstream air/fuel ratio sensor from the predetermined target value. The value indicative of the output of the upstream air/fuel ratio sensor is an upstream output deviation which is a deviation of the output of the upstream air/fuel ratio sensor from a predetermined reference value. The value indicative of the target air/fuel ratio is an air/fuel ratio deviation which is a deviation of the target air/fuel ratio from the predetermined reference value. The controlled object model is a model which has a variable associated with the output deviation, and a variable associated with one of the air/fuel ratio deviation and the upstream output deviation. The step of identifying includes identifying a model parameter multiplied by the output deviation, and a model parameter multiplied by one of the air/fuel ratio deviation and the upstream output deviation such that the parameters fall within respective predetermined restriction ranges. 
   This preferred embodiment of the control method provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the engine control unit described above, the value indicative of the output of the downstream air/fuel ratio sensor is an output deviation which is a deviation of the output of the downstream air/fuel ratio sensor from the predetermined target value. The value indicative of the output of the upstream air/fuel ratio sensor is an upstream output deviation which is a deviation of the output of the upstream air/fuel ratio sensor from a predetermined reference value. The value indicative of the target air/fuel ratio is an air/fuel ratio deviation which is a deviation of the target air/fuel ratio from the predetermined reference value. The controlled object model is a model which has a variable associated with the output deviation, and a variable associated with one of the air/fuel ratio deviation and the upstream output deviation. The control program causes the computer to identify a model parameter multiplied by the output deviation, and a model parameter multiplied by one of the air/fuel ratio deviation and the upstream output deviation such that the parameters fall within respective predetermined restriction ranges. 
   This preferred embodiment of the engine control unit provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the control apparatus described above, the output deviation comprises a plurality of time-series data of the output deviation. The control apparatus further comprises operating condition detecting means for detecting an operating condition of the internal combustion engine. The identifying means further includes restriction range setting means for identifying a plurality of model parameters respectively multiplied by the plurality of time-series data of the output deviation such that a combination of the model parameters falls within the predetermined restriction range, and setting the predetermined restriction range in accordance with the detected operating condition of the internal combustion engine. 
   As described above, with this type of identification algorithm, when a plurality of model parameters are identified independently of one another, the control system may become instable or oscillatory depending on a combination of the model parameters. In addition, generally, as an operating condition of an internal combustion engine changes, its stable limit also changes. For example, in a low load operating condition, a reduction in exhaust gas volume causes an increase in a response delay, a dead time, and the like of exhaust gases with respect to a supplied air/fuel mixture, so that the downstream air/fuel ratio sensor is likely to generate an oscillatory output. As a result, identified parameters are also likely to fluctuate associated with the oscillatory output of the downstream air/fuel ratio sensor, so that the post-catalyst exhaust gas characteristic becomes instable. On the contrary, according to this preferred embodiment of the control apparatus, since the plurality of model parameters are identified such that a combination of the model parameters falls within the predetermined restriction range, and the predetermined restriction range is set in accordance with a detected operating condition of the internal combustion engine, the control apparatus can avoid the instable post-catalyst exhaust gas characteristic as described above to further improve the post-catalyst exhaust gas characteristic and further improve the stability of the air/fuel ratio control. 
   Preferably, in the control method described above, the output deviation comprises a plurality of time-series data of the output deviation. The control method further comprises the step of detecting an operating condition of the internal combustion engine, wherein step of identifying further includes identifying a plurality of model parameters respectively multiplied by the plurality of time-series data of the output deviation such that a combination of the model parameters falls within the predetermined restriction range, and setting the predetermined restriction range in accordance with the detected operating condition of the internal combustion engine. 
   This preferred embodiment of the control method provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the engine control unit described above, the output deviation comprises a plurality of time-series data of the output deviation. The control program further causes the computer to detect an operating condition of the internal combustion engine; identify a plurality of model parameters respectively multiplied by the plurality of time-series data of the output deviation such that a combination of the model parameters falls within the predetermined restriction range; and set the predetermined restriction range in accordance with the detected operating condition of the internal combustion engine. 
   This preferred embodiment of the engine control unit provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, the control apparatus described above further comprises an operating condition detecting means for detecting an operating condition of the internal combustion engine, wherein the identifying means further includes weighting parameter setting means for identifying the model parameters based on a weighted identification algorithm which uses weighting parameters for determining behaviors of the model parameters, and setting the weighting parameters in accordance with the detected operating condition of the internal combustion engine. 
   As described above, when an internal combustion engine is in a low load operating condition, a reduction in the exhaust gas volume causes susceptibility to an oscillatory output of the downstream air/fuel ratio sensor, and to an instable control system. On the contrary, according to this preferred embodiment of the control apparatus, since the model parameters are identified based on the weighted identification algorithm using weighting parameters for determining the behaviors of the model parameters, and the weighting parameters are set in accordance with the detected operating condition of the internal combustion engine, the post-catalyst exhaust gas characteristic can be improved during a low load operation of the internal combustion engine by appropriately setting the weighting parameters to values which stabilize the behaviors of the model parameters during the low load operating condition. 
   Preferably, the control method described above further comprises the step of detecting an operating condition of the internal combustion engine, wherein the step of identifying further includes identifying the model parameters based on a weighted identification algorithm which uses weighting parameters for determining behaviors of the model parameters, and setting the weighting parameters in accordance with the detected operating condition of the internal combustion engine. 
   This preferred embodiment of the control method provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the engine control unit described above, the control program further causes the computer to detect an operating condition of the internal combustion engine; identify the model parameters based on a weighted identification algorithm which uses weighting parameters for determining behaviors of the model parameters; and set the weighting parameters in accordance with the detected operating condition of the internal combustion engine. 
   This preferred embodiment of the engine control unit provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, the control apparatus described above further comprises an operating condition detecting means for detecting an operating condition of the internal combustion engine, wherein the identifying means further includes dead time setting means for identifying the model parameters based on an identification algorithm which uses a dead time between the output of the upstream air/fuel ratio sensor and the output of the downstream air/fuel ratio sensor, and setting the dead time in accordance with the detected operating condition of the internal combustion engine. 
   This type of control apparatus can increase an identification accuracy for a model parameter multiplied by the input of the controlled object model when a dead time between the input and output of the controlled object model is set to be highly correlated to an actual input/output of the controlled object, as compared with when the dead time is set to be lowly correlated to the actual input/output of the controlled object. In addition, the dynamic characteristic such as a dead time, a response delay, and the like in an exhaust system of the internal combustion engine, including the catalyst, varies in accordance with an operating condition, i.e., an exhaust gas volume of the internal combustion engine. Therefore, according to this preferred embodiment of the control apparatus, since the dead time between the output of the upstream air/fuel ratio sensor and the output of the downstream air/fuel ratio sensor, used for identifying the model parameters, is set in accordance with a detected operating condition of the internal combustion engine, the control apparatus can calculate the control input based on the controlled object model with an improved accuracy to more accurately correct a slippage in control timing of the air/fuel ratio control. 
   Preferably, the control method described above, further comprises the step of detecting an operating condition of the internal combustion engine, wherein the step of identifying further includes identifying the model parameters based on an identification algorithm which uses a dead time between the output of the upstream air/fuel ratio sensor and the output of the downstream air/fuel ratio sensor, and setting the dead time in accordance with the detected operating condition of the internal combustion engine. 
   This preferred embodiment of the control method provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the engine control unit described above, the control program further causes the computer to detect an operating condition of the internal combustion engine; identify the model parameters based on an identification algorithm which uses a dead time between the output of the upstream air/fuel ratio sensor and the output of the downstream air/fuel ratio sensor; and set the dead time in accordance with the detected operating condition of the internal combustion engine. 
   This preferred embodiment of the engine control unit provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, the control apparatus described above further comprises an operating condition detecting means for detecting an operating condition of the internal combustion engine, wherein the air/fuel ratio calculating means includes prediction time calculating means for calculating a prediction time from the time at which the air/fuel mixture is supplied to the internal combustion engine in the target air/fuel ratio to the time at which the target air/fuel ratio is reflected to the output of the downstream air/fuel ratio sensor in accordance with the detected operating condition of the internal combustion engine; predicted value calculating means for calculating a predicted value of the value indicative of the target air/fuel ratio in accordance with the calculated prediction time based on a prediction algorithm which applies the controlled target model; and target air/fuel ratio calculating means for calculating the target air/fuel ratio in accordance with the calculated predicted value based on the one modulation algorithm. 
   According to this preferred embodiment of the control apparatus, the prediction time from the time at which the air/fuel mixture is supplied to the internal combustion engine in the target air/fuel ratio to the time at which the target air/fuel ratio is reflected to the output of the downstream air/fuel ratio sensor is calculated in accordance with the detected operating condition of the internal combustion engine, the predicted value of the value indicative of the target air/fuel ratio is calculated in accordance with the calculated prediction time, and the target air/fuel ratio is calculated in accordance with the calculated predicted value, so that the target air/fuel ratio can be calculated while reflecting a response delay and a dead time between the input and output of the controlled object, i.e., a response delay and a dead time of the output of the downstream air/fuel ratio sensor with respect to the air/fuel mixture supplied to the internal combustion engine, thereby making it possible to more securely eliminate a slippage in control timing of the air/fuel ratio control. 
   Preferably, the control method described above further comprises the step of detecting an operating condition of the internal combustion engine, wherein the step of calculating the target air/fuel ratio includes calculating a prediction time from the time at which the air/fuel mixture is supplied to the internal combustion engine in the target air/fuel ratio to the time at which the target air/fuel ratio is reflected to the output of the downstream air/fuel ratio sensor in accordance with the detected operating condition of the internal combustion engine; calculating a predicted value of the value indicative of the target air/fuel ratio in accordance with the calculated prediction time based on a prediction algorithm which applies the controlled object model; and calculating the target air/fuel ratio in accordance with the calculated predicted value based on the one modulation algorithm. 
   This preferred embodiment of the control method provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the engine control unit described above, the control program further causes the computer to detect an operating condition of the internal combustion engine; calculate a prediction time from the time at which the air/fuel mixture is supplied to the internal combustion engine in the target air/fuel ratio to the time at which the target air/fuel ratio is reflected to the output of the downstream air/fuel ratio sensor in accordance with the detected operating condition of the internal combustion engine; calculate a predicted value of the value indicative of the target air/fuel ratio in accordance with the calculated prediction time based on a prediction algorithm which applies the controlled object model; and calculate the target air/fuel ratio in accordance with the calculated predicted value based on the one modulation algorithm. 
   This preferred embodiment of the engine control unit provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, the control apparatus described above further comprises multiplying means for multiplying the predicted value by a correction coefficient; and correction coefficient setting means for setting the correction coefficient to be a smaller value when the predicted value is equal to or larger than a predetermined value than when the predicted value is smaller than the predetermined value, wherein the air/fuel ratio calculating means calculates the target air/fuel ratio of the air/fuel mixture in accordance with the predicted value multiplied by the correction coefficient based on the one modulation algorithm. 
   According to this preferred embodiment of the control apparatus, the target air/fuel ratio of the air/fuel mixture is calculated in accordance with the predicted value of the output deviation multiplied by the correction coefficient, and the correction coefficient is set to a smaller value when the predicted value of the output deviation is equal to or larger than a predetermined value than when the predicted value of the output deviation is smaller than the predetermined value, so that the output of the downstream air/fuel ratio sensor can be converged at a different rate in accordance with the order of the predicted value of the output deviation with respect to the predetermined value. Therefore, for changing the air/fuel ratio to be leaner because of the predicted value of the output deviation being equal to or larger than zero, i.e., the output of the downstream air/fuel ratio sensor being larger than a target value when the predetermined value is set, for example to zero, the correction coefficient is set such that the output of the downstream air/fuel ratio sensor is converged at a lower rate than when the air/fuel ratio is changed to be richer, thereby providing the effect of suppressing the amount of exhausted NOx by a lean bias. On the other hand, when the air/fuel ratio is changed to be richer, the correction coefficient is set such that the output of the downstream air/fuel ratio sensor is converted at a higher rate than when the air/fuel ratio is changed to be leaner, thereby making it possible to sufficiently recover the NOx purifying rate of the catalyst. 
   Preferably, the control method described above further comprises the steps of multiplying the predicted value by a correction coefficient; and setting the correction coefficient to be a smaller value when the predicted value is equal to or larger than a predetermined value than when the predicted value is smaller than the predetermined value, wherein the step of calculating the target air/fuel ratio includes calculating the target air/fuel ratio of the air/fuel mixture in accordance with the predicted value multiplied by the correction coefficient based on the one modulation algorithm. 
   This preferred embodiment of the control method provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the engine control unit described above, the control program further causes the computer to multiply the predicted value by a correction coefficient; set the correction coefficient to be a smaller value when the predicted value is equal to or larger than a predetermined value than when the predicted value is smaller than the predetermined value; and calculate the target air/fuel ratio of the air/fuel mixture in accordance with the predicted value multiplied by the correction coefficient based on the one modulation algorithm. 
   This preferred embodiment of the engine control unit provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, the control apparatus described above further comprises operating condition detecting means for detecting an operating condition of the internal combustion engine, wherein the air/fuel ratio calculating means further includes intermediate value calculating means for calculating an intermediate value of the target air/fuel ratio of the air/fuel mixture supplied to the internal combustion engine in accordance with the predicted value of the output deviation based on the controlled object model and the one modulation algorithm; gain setting means for setting a gain in accordance with the detected operating condition of the internal combustion engine; and target air/fuel ratio calculating means for calculating the target air/fuel ratio based on the calculated intermediate value multiplied by the set gain. 
   Generally, in this type of internal combustion engine, the gain characteristic between the input and output of the controlled object, i.e., between the target air/fuel ratio and the output of the downstream air/fuel ratio sensor varies in response to a change in an operating condition, i.e., an exhaust gas volume of the internal combustion engine. Therefore, according to this preferred embodiment of the control apparatus, since the target air/fuel ratio is calculated based on the intermediate value multiplied by a predetermined gain set in accordance with the operating condition of the internal combustion engine, the target air/fuel ratio can be calculated while reflecting a change in the dynamic characteristic such as a dead time, a response delay, or the like associated with a change in the operating condition, i.e., the exhaust gas volume of the internal combustion engine. It is therefore possible to ensure the stability of the air/fuel ratio control, suppress unnecessary fluctuations in the air/fuel ratio to maintain satisfactorily purified exhaust gases by the catalyst, and avoid surging due to the fluctuations in the air/fuel ratio, for example, in a high load operation. 
   Preferably, the control method described above further comprises the step of detecting an operating condition of the internal combustion engine, wherein the step of calculating the target air/fuel ratio further includes calculating an intermediate value of the target air/fuel ratio of the air/fuel mixture supplied to the internal combustion engine based on the controlled object model and the one modulation algorithm; setting a gain in accordance with the detected operating condition of the internal combustion engine; and calculating the target air/fuel ratio based on the calculated intermediate value multiplied by the set gain. 
   This preferred embodiment of the control method provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the engine control unit described above, the control program further causes the computer to detect an operating condition of the internal combustion engine; calculate an intermediate value of the target air/fuel ratio of the air/fuel mixture supplied to the internal combustion engine based on the controlled object model and the one modulation algorithm; set a gain in accordance with the detected operating condition of the internal combustion engine; and calculate the target air/fuel ratio based on the calculated intermediate value multiplied by the set gain. 
   This preferred embodiment of the engine control unit provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the control apparatus described above, the controlled object comprises an air/fuel ratio sensor disposed at a location downstream of a catalyst in an exhaust pipe of an internal combustion engine for detecting an air/fuel ratio of exhaust gases which have passed through the catalyst, and the output of the controlled object is an output of the downstream air/fuel ratio sensor. The control input to the controlled object is a target air/fuel ratio of an air/fuel mixture supplied to the internal combustion engine. The controlled object model is a model which has a variable associated with a value indicative of the output of the air/fuel ratio sensor, and a variable associated with a value indicative of the target air/fuel ratio. The identifying means sequentially identifies a model parameter multiplied by the value indicative of the output of the air/fuel ratio sensor, and a model parameter multiplied by the value indicative of the target air/fuel ratio in accordance with the output of the air/fuel ratio sensor and the target air/fuel ratio of the air/fuel mixture. The control input calculating means includes air/fuel ratio calculating means for calculating the target air/fuel ratio of the air/fuel mixture supplied to the internal combustion engine for converging the output of the air/fuel ratio sensor to a predetermined target value based on the one modulation algorithm and the controlled object model. 
   According to this preferred embodiment of the control apparatus, the model parameters of the controlled object model are sequentially identified in accordance with the target air/fuel ratio and the output of the air/fuel ratio sensor, i.e., identified in real time, and the target/air fuel ratio of the air/fuel mixture supplied to the internal combustion engine is calculated based on the controlled object model, the model parameters of which are identified in this manner, and the one modulation algorithm. Thus, even if the characteristics of the catalyst and air/fuel ratio sensor vary due to a changing environment or have been aged, the output of the air/fuel ratio sensor can be converged to the predetermined target value, while avoiding the influence of the variations and aging changes of the characteristics. Consequently, the control apparatus can appropriately correct a slippage in control timing of the air/fuel ratio control caused by a response delay, a dead time, and the like of exhaust gases with respect to the air/fuel mixture supplied to the internal combustion engine to improve the post-catalyst exhaust gas characteristic. In addition, the control apparatus can be realized at a relatively low cost because it only requires a single air/fuel ratio sensor. 
   Preferably, in the control method described above, the controlled object comprises an air/fuel ratio sensor disposed at a location downstream of a catalyst in an exhaust pipe of an internal combustion engine for detecting an air/fuel ratio of exhaust gases which have passed through the catalyst, and the output of the controlled object is an output of the downstream air/fuel ratio sensor. The control input to the controlled object is a target air/fuel ratio of an air/fuel mixture supplied to the internal combustion engine. The controlled object model is a model which has a variable associated with a value indicative of the output of the air/fuel ratio sensor, and a variable associated with a value indicative of the target air/fuel ratio. The step of identifying includes sequentially identifying a model parameter multiplied by the value indicative of the output of the air/fuel ratio sensor, and a model parameter multiplied by the value indicative of the target air/fuel ratio in accordance with the output of the air/fuel ratio sensor and the target air/fuel ratio of the air/fuel mixture. The step of calculating a control input includes calculating the target air/fuel ratio of the air/fuel mixture supplied to the internal combustion engine for converging the output of the air/fuel ratio sensor to a predetermined target value based on the one modulation algorithm and the controlled object model. 
   This preferred embodiment of the control method provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the engine control unit described above, the controlled object comprises an air/fuel ratio sensor disposed at a location downstream of a catalyst in an exhaust pipe of an internal combustion engine for detecting an air/fuel ratio of exhaust gases which have passed through the catalyst, and the output of the controlled object is an output of the downstream air/fuel ratio sensor. The control input to the controlled object is a target air/fuel ratio of an air/fuel mixture supplied to the internal combustion engine. The controlled object model is a model which has a variable associated with a value indicative of the output of the air/fuel ratio sensor, and a variable associated with a value indicative of the target air/fuel ratio. The control program causes the computer to sequentially identify a model parameter multiplied by the value indicative of the output of the air/fuel ratio sensor, and a model parameter multiplied by the value indicative of the target air/fuel ratio in accordance with the output of the air/fuel ratio sensor and the target air/fuel ratio of the air/fuel mixture; and calculate the target air/fuel ratio of the air/fuel mixture supplied to the internal combustion engine for converging the output of the air/fuel ratio sensor to a predetermined target value based on the one modulation algorithm and the controlled object model. 
   This preferred embodiment of the engine control unit provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the control apparatus described above, the value indicative of the output of the air/fuel ratio sensor is an output deviation which is a deviation of the output of the air/fuel ratio sensor from the predetermined target value. The value indicative of the target air/fuel ratio is an air/fuel ratio deviation which is a deviation of the target air/fuel ratio from a predetermined reference value. The controlled object model is a model which has variables associated with the output deviation and the air/fuel ratio deviation. The identifying means identifies a model parameter multiplied by the output deviation, and a model parameter multiplied by the air/fuel ratio deviation such that the model parameters fall within respective predetermined restriction ranges. 
   According to this preferred embodiment of the control apparatus, since the controlled object model has a variable associated with the output deviation, and a variable associated with the air/fuel ratio deviation, the dynamic characteristic of the controlled object model can be fitted to the actual dynamic characteristic of the controlled object because the model parameters can be more precisely identified or defined for the controlled object model, for the reason set forth above, as compared with a controlled object model which has a variable associated with an absolute value of the output of the air/fuel ratio sensor, and a variable associated with an absolute value of the target air/fuel ratio. Also, as described above, with a sequential identification algorithm, when the input and output of a controlled object enter a steady state, a control system may become instable or oscillatory because a so-called drift phenomenon is more likely to occur, in which absolute values of identified model parameters increase due to a shortage of self excitation condition. On the contrary, according to this preferred embodiment of the control apparatus, since the model parameter multiplied by the output deviation and the model parameter multiplied by the air/fuel ratio deviation are identified such that they fall within respective predetermined restriction ranges, it is possible to avoid the drift phenomenon by appropriately setting the predetermined restriction ranges, to securely ensure the stability of the air/fuel ratio control and improve the post-catalyst exhaust gas characteristic. 
   Preferably, in the control method described above, the value indicative of the output of the air/fuel ratio sensor is an output deviation which is a deviation of the output of the air/fuel ratio sensor from the predetermined target value. The value indicative of the target air/fuel ratio is an air/fuel ratio deviation which is a deviation of the target air/fuel ratio from a predetermined reference value. The controlled object model is a model which has variables associated with the output deviation and the air/fuel ratio deviation. The step of identifying includes identifying a model parameter multiplied by the output deviation, and a model parameter multiplied by the air/fuel ratio deviation such that the model parameters fall within respective predetermined restriction ranges. 
   This preferred embodiment of the control method provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the engine control unit described above, the value indicative of the output of the air/fuel ratio sensor is an output deviation which is a deviation of the output of the air/fuel ratio sensor from the predetermined target value. The value indicative of the target air/fuel ratio is an air/fuel ratio deviation which is a deviation of the target air/fuel ratio from a predetermined reference value. The controlled object model is a model which has variables associated with the output deviation and the air/fuel ratio deviation. The control program causes the computer to identify a model parameter multiplied by the output deviation, and a model parameter multiplied by the air/fuel ratio deviation such that the model parameters fall within respective predetermined restriction ranges. 
   This preferred embodiment of the engine control unit provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the control apparatus described above, the output deviation comprises a plurality of time-series data of the output deviation. The control apparatus further comprises operating condition detecting means for detecting an operating condition of the internal combustion engine. The identifying means further includes restriction range setting means for identifying a plurality of model parameters respectively multiplied by the plurality of time-series data of the output deviation such that a combination of the model parameters falls within the predetermined restriction range, and setting the predetermined restriction range in accordance with the detected operating condition of the internal combustion engine. 
   According to this preferred embodiment of the control apparatus, since the plurality of model parameters are identified such that a combination of the model parameters falls within the predetermined restriction range, and the predetermined restriction range is set in accordance with a detected operating condition of the internal combustion engine, the control system can avoid the instable post-catalyst exhaust gas characteristic as described above, further improve the post-catalyst exhaust gas characteristic, and further improve the stability of the air/fuel ratio control. 
   Preferably, in the control method described above, the output deviation comprises a plurality of time-series data of the output deviation. The control method further comprises the step of detecting an operating condition of the internal combustion engine. The step of identifying further includes identifying a plurality of model parameters respectively multiplied by the plurality of time-series data of the output deviation such that a combination of the model parameters falls within the predetermined restriction range, and setting the predetermined restriction range in accordance with the detected operating condition of the internal combustion engine. This preferred embodiment of the control method provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the engine control unit described above, the output deviation comprises a plurality of time-series data of the output deviation. The control program further causes the computer to detect an operating condition of the internal combustion engine; identify a plurality of model parameters respectively multiplied by the plurality of time-series data of the output deviation such that a combination of the model parameters falls within the predetermined restriction range; and set the predetermined restriction range in accordance with the detected operating condition of the internal combustion engine. 
   This preferred embodiment of the engine control unit provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, the control apparatus described above further comprises operating condition detecting means for detecting an operating condition of the internal combustion engine, wherein the identifying means further includes weighting parameter setting means for identifying the model parameters based on a weighted identification algorithm which uses weighting parameters for determining behaviors of the model parameters, and setting the weighting parameters in accordance with the detected operating condition of the internal combustion engine. 
   According to this preferred embodiment of the control apparatus, since the model parameters are identified based on the weighted identification algorithm using weighting parameters for determining the behaviors thereof, and the weighting parameters are set in accordance with the detected operating condition of the internal combustion engine, the post-catalyst exhaust gas characteristic can be improved during a low load operation of the internal combustion engine by appropriately setting the weighting parameters to values which stabilize the behaviors of the model parameters during the low load operating condition. 
   Preferably, the control method described above further comprises the step of detecting an operating condition of the internal combustion engine, wherein the step of identifying further includes identifying the model parameters based on a weighted identification algorithm which uses weighting parameters for determining behaviors of the model parameters, and setting the weighting parameters in accordance with the detected operating condition of the internal combustion engine. 
   This preferred embodiment of the control method provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the engine control unit described above, the control program further causes the computer to detect an operating condition of the internal combustion engine; identify the model parameters based on a weighted identification algorithm which uses weighting parameters for determining behaviors of the model parameters; and set the weighting parameters in accordance with the detected operating condition of the internal combustion engine. 
   This preferred embodiment of the engine control unit provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, the control apparatus described above further comprises operating condition detecting means for detecting an operating condition of the internal combustion engine, wherein the air/fuel ratio calculating means includes prediction time calculating means for calculating a prediction time from the time at which the air/fuel mixture is supplied to the internal combustion engine in the target air/fuel ratio to the time at which the target air/fuel ratio is reflected to the output of the air/fuel ratio sensor in accordance with the detected operating condition of the internal combustion engine; predicted value calculating means for calculating a predicted value of the value indicative of the target air/fuel ratio in accordance with the calculated prediction time based on a prediction algorithm which applies the controlled target model; and target air/fuel ratio calculating means for calculating the target air/fuel ratio in accordance with the calculated predicted value based on the one modulation algorithm. 
   According to this preferred embodiment of the control apparatus, the prediction time from the time at which the air/fuel mixture is supplied to the internal combustion engine in the target air/fuel ratio to the time at which the target air/fuel ratio is reflected to the output of the downstream air/fuel ratio sensor is calculated in accordance with the detected operating condition of the internal combustion engine, the predicted value of the value indicative of the target air/fuel ratio is calculated in accordance with the calculated prediction time, and the target air/fuel ratio is calculated in accordance with the calculated predicted value, so that the target air/fuel ratio can be calculated while reflecting a response delay and a dead time between the input and output of the controlled object, i.e., a response delay and a dead time of the output of the downstream air/fuel ratio sensor with respect to the air/fuel mixture supplied to the internal combustion engine, thereby making it possible to more securely eliminate a slippage in control timing of the air/fuel ratio control. 
   Preferably, the control method described above further comprises the step of detecting an operating condition of the internal combustion engine, wherein the step of calculating the air/fuel ratio includes calculating a prediction time from the time at which the air/fuel mixture is supplied to the internal combustion engine in the target air/fuel ratio to the time at which the target air/fuel ratio is reflected to the output of the air/fuel ratio sensor in accordance with the detected operating condition of the internal combustion engine; calculating a predicted value of the value indicative of the target air/fuel ratio in accordance with the calculated prediction time based on a prediction algorithm which applies the controlled target model; and calculating the target air/fuel ratio in accordance with the calculated predicted value based on the one modulation algorithm. 
   This preferred embodiment of the control method provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the engine control unit described above, the control program further causes the computer to detect an operating condition of the internal combustion engine; calculate a prediction time from the time at which the air/fuel mixture is supplied to the internal combustion engine in the target air/fuel ratio to the time at which the target air/fuel ratio is reflected to the output of the air/fuel ratio sensor in accordance with the detected operating condition of the internal combustion engine; calculate a predicted value of the value indicative of the target air/fuel ratio in accordance with the calculated prediction time based on a prediction algorithm which applies the controlled target model; and calculate the target air/fuel ratio in accordance with the calculated predicted value based on the one modulation algorithm. 
   This preferred embodiment of the engine control unit provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, the control apparatus described above further comprises multiplying means for multiplying the predicted value by a correction coefficient; and correction coefficient setting means for setting the correction coefficient to be a smaller value when the predicted value is equal to or larger than a predetermined value than when the predicted value is smaller than the predetermined value, wherein the target air/fuel ratio calculating means calculates the target air/fuel ratio of the air/fuel mixture in accordance with the predicted value multiplied by the correction coefficient based on the one modulation algorithm. 
   According to this preferred embodiment of the control apparatus, the target air/fuel ratio of the air/fuel mixture is calculated in accordance with the predicted value of the output deviation multiplied by the correction coefficient, and the correction coefficient is set to a smaller value when the predicted value of the output deviation is equal to or larger than a predetermined value than when the predicted value of the output deviation is smaller than the predetermined value, so that the output of the downstream air/fuel ratio sensor can be converged at a different rate in accordance with the order of the predicted value of the output deviation with respect to the predetermined value. Therefore, for changing the air/fuel ratio to be leaner because of the predicted value of the output deviation being equal to or larger than zero, i.e., the output of the downstream air/fuel ratio sensor being larger than a target value when the predetermined value is set, for example to zero, the correction coefficient is set such that the output of the downstream air/fuel ratio sensor is converged at a lower rate than when the air/fuel ratio is changed to be richer, thereby providing the effect of suppressing the amount of exhausted NOx by a lean bias. On the other hand, when the air/fuel ratio is changed to be richer, the correction coefficient is set such that the output of the downstream air/fuel ratio sensor is converted at a higher rate than when the air/fuel ratio is changed to be leaner, thereby making it possible to sufficiently recover the NOx purifying rate of the catalyst. 
   Preferably, the control method described above further comprises the steps of multiplying the predicted value by a correction coefficient; and setting the correction coefficient to be a smaller value when the predicted value is equal to or larger than a predetermined value than when the predicted value is smaller than the predetermined value, wherein the step of calculating the target air/fuel ratio includes calculating the target air/fuel ratio of the air/fuel mixture in accordance with the predicted value multiplied by the correction coefficient based on the one modulation algorithm. 
   This preferred embodiment of the control method provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the engine control unit described above, the control program further causes the computer to multiply the predicted value by a correction coefficient; set the correction coefficient to be a smaller value when the predicted value is equal to or larger than a predetermined value than when the predicted value is smaller than the predetermined value; calculate the target air/fuel ratio of the air/fuel mixture in accordance with the predicted value multiplied by the correction coefficient based on the one modulation algorithm. 
   This preferred embodiment of the engine control unit provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, the control apparatus described above further comprises operating condition detecting means for detecting an operating condition of the internal combustion engine, wherein the air/fuel ratio calculating means further includes intermediate value calculating means for calculating an intermediate value of the target air/fuel ratio of the air/fuel mixture supplied to the internal combustion engine based on the controlled object model and the one modulation algorithm; gain setting means for setting a gain in accordance with the detected operating condition of the internal combustion engine; and target air/fuel ratio calculating means for calculating the target air/fuel ratio based on the calculated intermediate value multiplied by the set gain. 
   According to this preferred embodiment of the control apparatus, since the target air/fuel ratio is calculated based on the intermediate value multiplied by a predetermined gain set in accordance with the operating condition of the internal combustion engine, the target air/fuel ratio can be calculated while reflecting a change in the dynamic characteristic such as a dead time, a response delay, or the like associated with a change in the operating condition, i.e., the exhaust gas volume of the internal combustion engine. It is therefore possible to ensure the stability of the air/fuel ratio control, suppress unnecessary fluctuations in the air/fuel ratio to maintain satisfactorily purified exhaust gases by the catalyst, and avoid surging due to fluctuations in the air/fuel ratio, for example, in a high load operation. 
   Preferably, the control method described above further comprises the step of detecting an operating condition of the internal combustion engine, wherein the step of calculating the target air/fuel ratio further includes calculating an intermediate value of the target air/fuel ratio of the air/fuel mixture supplied to the internal combustion engine based on the controlled object model and the one modulation algorithm; setting a gain in accordance with the detected operating condition of the internal combustion engine; and calculating the target air/fuel ratio based on the calculated intermediate value multiplied by the set gain. 
   This preferred embodiment of the control method provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the engine control unit described above, the control program further causes the computer to detect an operating condition of the internal combustion engine; calculate an intermediate value of the target air/fuel ratio of the air/fuel mixture supplied to the internal combustion engine based on the controlled object model and the one modulation algorithm; set a gain in accordance with the detected operating condition of the internal combustion engine; and calculate the target air/fuel ratio based on the calculated intermediate value multiplied by the set gain. 
   This preferred embodiment of the engine control unit provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, the control apparatus described above further comprises parameter detecting means for detecting a dynamic characteristic parameter indicative of a change in a dynamic characteristic of the controlled object; and model parameter setting means for setting model parameters of the controlled object model in accordance with the detected dynamic characteristic parameter. 
   According to this preferred embodiment of the control apparatus, the parameter detecting means detects the dynamic characteristic parameter indicative of a change in a dynamic characteristic of the controlled object, and the model parameter setting means sets the model parameters of the controlled object model in accordance with the detected dynamic characteristic parameter, so that the control apparatus can rapidly fit the dynamic characteristic of the controlled object model to the actual dynamic characteristic of the controlled object. As a result, the control apparatus can rapidly and appropriately correct a slippage in control timing between the input and output of the controlled object, caused by the dynamic characteristic of the controlled object, for example, a response delay, a dead time, or the like, to improve the stability of the control and the controllability. 
   Preferably, the control method described above further comprises the steps of detecting a dynamic characteristic parameter indicative of a change in a dynamic characteristic of the controlled object; and setting model parameters of the controlled object model in accordance with the detected dynamic characteristic parameter. 
   This preferred embodiment of the control method provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the engine control unit described above, the control program further causes the computer to detect a dynamic characteristic parameter indicative of a change in a dynamic characteristic of the controlled object; and set model parameters of the controlled object model in accordance with the detected dynamic characteristic parameter. 
   This preferred embodiment of the engine control unit provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the control apparatus described above, the control input calculating means calculates a predicted value of a value indicative of the output of the controlled object based on a prediction algorithm which applies the controlled object model, and calculates the control input in accordance with the calculated predicted value based on the one modulation algorithm. 
   According to this preferred embodiment of the control apparatus, the predicted value of the value indicative of the output of the controlled object is calculated based on the predication algorithm which applies the controlled object model, and the control input is calculated in accordance with the calculated predicted value based on the one modulation algorithm. In this event, since the dynamic characteristic of the controlled object model can be fitted to the actual dynamic characteristic of the controlled object by using the model parameters identified by the identifying means as described above, the predicted value can be calculated as a value reflecting the actual dynamic characteristic of the controlled model by calculating the predicted value based on the prediction algorithm which applies the controlled object model as described above. As a result, the control apparatus can more appropriately correct a slippage in control timing between the control input and the output of the controlled object to further improve the stability of the control and the controllability. 
   Preferably, in the control method described above, the step of calculating a control input includes calculating a predicted value of a value indicative of the output of the controlled object based on a prediction algorithm which applies the controlled object model; and calculating the control input in accordance with the calculated predicted value based on the one modulation algorithm. 
   This preferred embodiment of the control method provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the engine control unit described above, the control program causes the computer to calculate a predicted value of a value indicative of the output of the controlled object based on a prediction algorithm which applies the controlled object model; and calculate the control input in accordance with the calculated predicted value based on the one modulation algorithm. 
   This preferred embodiment of the engine control unit provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the control apparatus described above, the control input calculating means calculates a prediction time from the time at which the control input is inputted to the controlled object to the time at which the control input is reflected to the output of the controlled object in accordance with the dynamic characteristic parameter of the controlled object, and calculates the predicted value in accordance with the calculated prediction time based on the prediction algorithm. 
   According to this preferred embodiment of the control apparatus, the prediction time from the time at which the control input is inputted to the controlled object to the time at which the control input is reflected to the output of the controlled object is calculated in accordance with the dynamic characteristic of the controlled object, and the predicted value is calculated in accordance with the calculated prediction time, so that a slippage in control timing between the input/output of the controlled object, possibly caused by a response delay, a dead time, and the like of the controlled object, can be eliminated without fail by calculating the control input calculated in this manner, thereby making it possible to further improve the controllability. 
   Preferably, in the control method described above, the step of calculating a control input includes calculating a prediction time from the time at which the control input is inputted to the controlled object to the time at which the control input is reflected to the output of the controlled object in accordance with the dynamic characteristic parameter of the controlled object; and calculating the predicted value in accordance with the calculated prediction time based on the prediction algorithm. 
   This preferred embodiment of the control method provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the engine control unit described above, the control program causes the computer to calculate a prediction time from the time at which the control input is inputted to the controlled object to the time at which the control input is reflected to the output of the controlled object in accordance with the dynamic characteristic parameter of the controlled object; and calculate the predicted value in accordance with the calculated prediction time based on the prediction algorithm. 
   This preferred embodiment of the engine control unit provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the control apparatus described above, the control input calculating means calculates an intermediate value based on the controlled object model and the one modulation algorithm, and calculates the control input based on the calculated intermediate value multiplied by a predetermined gain. 
   According to this preferred embodiment of the control apparatus, the control input is calculated based on the intermediate value calculated based on one modulation algorithm multiplied by a predetermined gain, so that a satisfactory controllability can be ensured by setting the predetermined gain to an appropriate value. 
   Preferably, in the control method described above, the step of calculating a control input includes calculating an intermediate value based on the controlled object model and the one modulation algorithm; and calculating the control input based on the calculated intermediate value multiplied by a predetermined gain. 
   This preferred embodiment of the control method provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the engine control unit described above, the control program causes the computer to calculate an intermediate value based on the controlled object model and the one modulation algorithm; and calculate the control input based on the calculated intermediate value multiplied by a predetermined gain. 
   This preferred embodiment of the engine control unit provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, the control apparatus described above further comprises gain parameter detecting means for detecting a gain parameter indicative of a gain characteristic of the controlled object; and gain setting means for setting the predetermined gain in accordance with the detected gain parameter. 
   According to this preferred embodiment of the control apparatus, since the predetermined gain for use in the calculation of the control input is set in accordance with the gain characteristic of the controlled object, the control input can be calculated as a value which has appropriate energy in accordance with the gain characteristic of the controlled object, thereby making it possible to avoid an over-gain condition and the like to ensure a satisfactory controllability. 
   Preferably, the control method described above further comprises the steps of detecting a gain parameter indicative of a gain characteristic of the controlled object; and setting the predetermined gain in accordance with the detected gain parameter. 
   This preferred embodiment of the control method provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the engine control unit described above, the control program further causes the computer to detect a gain parameter indicative of a gain characteristic of the controlled object; and set the predetermined gain in accordance with the detected gain parameter. 
   This preferred embodiment of the engine control unit provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the control apparatus described above, the control input calculating means calculates a second intermediate value in accordance with the predicted value based on the one modulation algorithm, and calculates the control input by adding a predetermined value to the calculated second intermediate value. 
   According to this preferred embodiment of the control apparatus, the control input calculating means calculates the control input by adding the predetermined value to the second intermediate value calculated based on one modulation algorithm, so that the control input calculating means can calculate the control input not only as a value which positively and negatively inverts centered at zero, but also as a value which repeats predetermined increase and decrease about a predetermined value, thereby making it possible to improve the degree of freedom in control. 
   Preferably, in the control method described above, the step of calculating a control input includes calculating a second intermediate value in accordance with the predicted value based on the one modulation algorithm; and calculating the control input by adding a predetermined value to the calculated second intermediate value. 
   This preferred embodiment of the control method provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the engine control unit described above, the control program causes the computer to calculate a second intermediate value in accordance with the predicted value based on the one modulation algorithm; and calculate the control input by adding a predetermined value to the calculated second intermediate value. 
   This preferred embodiment of the engine control unit provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the control apparatus described above, the controlled object model has a variable associated with at least one of a deviation of the control input from a predetermined reference value, and the value reflecting a control input inputted to the controlled object from the predetermined reference value, and a variable associated with a deviation of the output of the controlled object from a predetermined target value. 
   According to this preferred embodiment of the control apparatus, since the controlled object model has a variable associated with at least one of the deviation of the control input from the predetermined reference value, and the value reflecting a control input inputted to the controlled object from the predetermined reference value, and a variable associated with the deviation of the output of the controlled object from the predetermined target value, the dynamic characteristic of the controlled object model can be fitted more closely to the actual dynamic characteristic of the controlled object, as compared with a controlled object model which has a variable associated with an absolute value of a value reflecting a control input and/or a control output, and a variable associated with an absolute value of the output of the controlled object. It is therefore possible to more securely ensure the stability of the control by calculating the control input based on the controlled object model as described above. 
   Preferably, in the control method described above, the controlled object model has a variable associated with at least one of a deviation of the control input from a predetermined reference value, and the value reflecting a control input inputted to the controlled object from the predetermined reference value, and a variable associated with a deviation of the output of the controlled object from a predetermined target value. 
   This preferred embodiment of the control method provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the engine control unit described above, the controlled object model has a variable associated with at least one of a deviation of the control input from a predetermined reference value, and the value reflecting a control input inputted to the controlled object from the predetermined reference value, and a variable associated with a deviation of the output of the controlled object from a predetermined target value. 
   This preferred embodiment of the engine control unit provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the control apparatus described above, the controlled object comprises a downstream air/fuel ratio sensor disposed at a location downstream of a catalyst in an exhaust pipe of an internal combustion engine for detecting an air/fuel ratio of exhaust gases which have passed through the catalyst, and the output of the controlled object is an output of the downstream air/fuel ratio sensor. The control input to the controlled object is the target air/fuel ratio of the air/fuel mixture supplied to the internal combustion engine. The controlled object model is a model representative of a relationship between the output of the downstream air/fuel ratio sensor and the target air/fuel ratio. The parameter detecting means comprises operating condition detecting means for detecting an operating condition of the internal combustion engine. The model parameter setting means sets model parameters of the controlled object model in accordance with the detected operating condition of the internal combustion engine. The control apparatus further comprises an upstream air/fuel ratio sensor disposed at a location upstream of the catalyst in the exhaust passage of the internal combustion engine. The control input calculating mean includes predicted value calculating means for calculating a predicted value of a value indicative of the output of the downstream air/fuel ratio sensor in accordance with the output of the downstream air/fuel ratio sensor, the output of the upstream air/fuel ratio sensor, and the target air/fuel ratio of the air/fuel mixture based on a prediction algorithm which applies the controlled object model; and air/fuel ratio calculating means for calculating the target air/fuel ratio of the air/fuel mixture supplied to the internal combustion engine for converging the output of the downstream air/fuel ratio sensor to a predetermined target value in accordance with the calculated predicted value based on the one modulation algorithm. 
   According to this preferred embodiment of the control apparatus, since the model parameters are set in accordance with the detected operating condition of the internal combustion engine, the model parameters can be rapidly calculated, even when the operating condition of the internal combustion engine suddenly changes, while precisely reflecting exhaust gases supplied to the catalyst. In addition, since the target air/fuel ratio is calculated for the air/fuel mixture supplied to the internal combustion engine based on the controlled object model, the model parameters of which are calculated in this manner, and the one modulation algorithm, the output of the downstream air/fuel ratio sensor can be rapidly converged to the predetermined target value. Further, since the predicted value is calculated in accordance with the output of the upstream air/fuel ratio sensor disposed at a location upstream of the catalyst, the air/fuel ratio of exhaust gases actually supplied to the catalyst can be more appropriately reflected to the predicted value, with a corresponding improvement on the accuracy of calculating the predicted value. Consequently, the control apparatus can rapidly and appropriately correct a slippage in control timing of the air/fuel ratio control, caused by a response delay, a dead time, and the like of exhaust gases with respect to the air/fuel mixture supplied to the internal combustion engine, to improve the stability of the air/fuel ratio control, and the post-catalyst exhaust gas characteristic. 
   Preferably, in the control method described above, the controlled object comprises a downstream air/fuel ratio sensor disposed at a location downstream of a catalyst in an exhaust pipe of an internal combustion engine for detecting an air/fuel ratio of exhaust gases which have passed through the catalyst, and the output of the controlled object is an output of the downstream air/fuel ratio sensor. The control input to the controlled object is the target air/fuel ratio of the air/fuel mixture supplied to the internal combustion engine. The controlled object model is a model representative of a relationship between the output of the downstream air/fuel ratio sensor and the target air/fuel ratio. The step of detecting a parameter includes detecting an operating condition of the internal combustion engine. The step of setting model parameters includes setting model parameters of the controlled object model in accordance with the detected operating condition of the internal combustion engine. The step of calculating a control input includes calculating a predicted value of a value indicative of the output of the downstream air/fuel ratio sensor in accordance with the output of the downstream air/fuel ratio sensor, an output of an upstream air/fuel ratio sensor disposed at a location upstream of the catalyst in the exhaust passage of the internal combustion engine, and the target air/fuel ratio of the air/fuel mixture based on a prediction algorithm which applies the controlled object model; and calculating the target air/fuel ratio of the air/fuel mixture supplied to the internal combustion engine for converging the output of the downstream air/fuel ratio sensor to a predetermined target value in accordance with the calculated predicted value based on the one modulation algorithm. 
   This preferred embodiment of the control method provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the engine control unit described above, the controlled object comprises a downstream air/fuel ratio sensor disposed at a location downstream of a catalyst in an exhaust pipe of an internal combustion engine for detecting an air/fuel ratio of exhaust gases which have passed through the catalyst, and the output of the controlled object is an output of the downstream air/fuel ratio sensor. The control input to the controlled object is the target air/fuel ratio of the air/fuel mixture supplied to the internal combustion engine. The controlled object model is a model representative of a relationship between the output of the downstream air/fuel ratio sensor and the target air/fuel ratio. The control program causes the computer to detect an operating condition of the internal combustion engine; set model parameters of the controlled object model in accordance with the detected operating condition of the internal combustion engine; calculate a predicted value of a value indicative of the output of the downstream air/fuel ratio sensor in accordance with the output of the downstream air/fuel ratio sensor, an output of an upstream air/fuel ratio sensor disposed at a location upstream of the catalyst in the exhaust passage of the internal combustion engine, and the target air/fuel ratio of the air/fuel mixture based on a prediction algorithm which applies the controlled object model; and calculate the target air/fuel ratio of the air/fuel mixture supplied to the internal combustion engine for converging the output of the downstream air/fuel ratio sensor to a predetermined target value in accordance with the calculated predicted value based on the one modulation algorithm. 
   This preferred embodiment of the engine control unit provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the control apparatus described above, the predicted value calculating means calculates a prediction time from the time at which the air fuel mixture is supplied to the internal combustion engine in the target air/fuel ratio to the time at which the target air/fuel ratio is reflected to the output of the downstream air/fuel ratio sensor, in accordance with an operating condition of the internal combustion engine, and calculates the predicted value further in accordance with the calculated prediction time. 
   According to this preferred embodiment of the control apparatus, the prediction time from the time at which the air/fuel mixture is supplied to the internal combustion engine in the target air/fuel ratio to the time at which the target air/fuel ratio is reflected to the output of the downstream air/fuel ratio sensor is calculated in accordance with the detected operating condition of the internal combustion engine, and the predicted value of the output deviation is calculated further in accordance with the calculated prediction time, so that the control apparatus can eliminate without fail a slippage in control timing between the input and output of the controlled object, caused by the dynamic characteristic of the controlled object, by calculating the control input using the predicted value calculated in this manner, thereby making it possible to further improve the post-catalyst exhaust gas characteristic. In addition, since the model parameters can be rapidly calculated, the control apparatus can rapidly ensure a satisfactory post-catalyst exhaust gas characteristic. 
   Preferably, in the control method described above, the step of calculating a predicted value includes calculating a prediction time from the time at which the air fuel mixture is supplied to the internal combustion engine in the target air/fuel ratio to the time at which the target air/fuel ratio is reflected to the output of the downstream air/fuel ratio sensor, in accordance with an operating condition of the internal combustion engine; and calculating the predicted value further in accordance with the calculated prediction time. 
   This preferred embodiment of the control method provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the engine control unit described above, the control program causes the computer to calculate a prediction time from the time at which the air fuel mixture is supplied to the internal combustion engine in the target air/fuel ratio to the time at which the target air/fuel ratio is reflected to the output of the downstream air/fuel ratio sensor, in accordance with an operating condition of the internal combustion engine; and calculate the predicted value further in accordance with the calculated prediction time. 
   This preferred embodiment of the engine control unit provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the control apparatus described above, the air/fuel ratio calculating means includes intermediate value calculating means for calculating an intermediate value of the target air/fuel ratio of the air/fuel mixture supplied to the internal combustion engine in accordance with the calculated predicted value based on the controlled object model and the one modulation algorithm; gain setting means for setting a gain in accordance with an operating condition of the internal combustion engine; and target air/fuel ratio calculating means for calculating the target air/fuel ratio of the air/fuel mixture supplied to the internal combustion engine for converging the output of the downstream air/fuel ratio sensor to a predetermined target value based on the calculated intermediate value multiplied by the set gain. 
   According to this preferred embodiment of the control apparatus, since the target air/fuel ratio of the air/fuel mixture is calculated based on the intermediate value calculated based on the one modulation algorithm, multiplied by the gain, and the gain is set in accordance with an operating condition, the target air/fuel ratio of the air/fuel mixture can be calculated as a value which appropriately reflects a change in the gain characteristic of the controlled object, thereby making it possible to further improve the post-catalyst exhaust gas characteristic. In addition, since the model parameters can be rapidly calculated, the control apparatus can rapidly ensure a satisfactory post-catalyst exhaust gas characteristic. 
   Preferably, in the control method described above, the step of calculating the target air/fuel ratio includes calculating an intermediate value of the target air/fuel ratio of the air/fuel mixture supplied to the internal combustion engine in accordance with the calculated predicted value based on the one modulation algorithm; setting a gain in accordance with an operating condition of the internal combustion engine; and calculating the target air/fuel ratio of the air/fuel mixture supplied to the internal combustion engine for converging the output of the downstream air/fuel ratio sensor to a predetermined target value based on the calculated intermediate value multiplied by the set gain. 
   This preferred embodiment of the control method provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the engine control unit described above, the control program causes the computer to calculate an intermediate value of the target air/fuel ratio of the air/fuel mixture supplied to the internal combustion engine in accordance with the calculated predicted value based on the one modulation algorithm; set a gain in accordance with an operating condition of the internal combustion engine; and calculate the target air/fuel ratio of the air/fuel mixture supplied to the internal combustion engine for converging the output of the downstream air/fuel ratio sensor to a predetermined target value based on the calculated intermediate value multiplied by the set gain. 
   This preferred embodiment of the engine control unit provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, the control apparatus described above further comprises multiplying means for multiplying the predicted value by a correction coefficient; and correction coefficient setting means for setting the correction coefficient to be a smaller value when the predicted value is equal to or larger than a predetermined value than when the predicted value is smaller than the predetermined value, wherein the air/fuel ratio calculating means calculates the target air/fuel ratio of the air/fuel mixture in accordance with the predicted value multiplied by the correction coefficient based on the one modulation algorithm. 
   According to this preferred embodiment of the control apparatus, the target air/fuel ratio of the air/fuel mixture is calculated in accordance with the predicted value of the output deviation multiplied by the correction coefficient, and the correction coefficient is set to a smaller value when the predicted value of the output deviation is equal to or larger than a predetermined value than when the predicted value of the output deviation is smaller than the predetermined value, so that the output of the downstream air/fuel ratio sensor can be converged at a different rate in accordance with the order of the predicted value of the output deviation with respect to the predetermined value. Therefore, for changing the air/fuel ratio to be leaner because of the predicted value of the output deviation being equal to or larger than zero, i.e., the output of the downstream air/fuel ratio sensor being larger than a target value when the predetermined value is set, for example to zero, the correction coefficient is set such that the output of the downstream air/fuel ratio sensor is converged at a lower rate than when the air/fuel ratio is changed to be richer, thereby providing the effect of suppressing the amount of exhausted NOx by a lean bias. On the other hand, when the air/fuel ratio is changed to be richer, the correction coefficient is set such that the output of the downstream air/fuel ratio sensor is converted at a higher rate than when the air/fuel ratio is changed to be leaner, thereby making it possible to sufficiently recover the NOX purifying rate of the catalyst. In addition, since the model parameters can be rapidly calculated, the control apparatus can rapidly ensure a satisfactory post-catalyst exhaust gas characteristic. 
   Preferably, the control method described above further comprises the steps of multiplying the predicted value by a correction coefficient; and setting the correction coefficient to be a smaller value when the predicted value is equal to or larger than a predetermined value than when the predicted value is smaller than the predetermined value, wherein the step of calculating the target air/fuel ratio calculating means includes calculating the target air/fuel ratio of the air/fuel mixture in accordance with the predicted value multiplied by the correction coefficient based on the one modulation algorithm. 
   This preferred embodiment of the control method provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the engine control unit described above, the control program further causes the computer to multiply the predicted value by a correction coefficient; set the correction coefficient to be a smaller value when the predicted value is equal to or larger than a predetermined value than when the predicted value is smaller than the predetermined value; and calculate the target air/fuel ratio of the air/fuel mixture in accordance with the predicted value multiplied by the correction coefficient based on the one modulation algorithm. 
   This preferred embodiment of the engine control unit provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the control apparatus described above, the controlled object comprises an air/fuel ratio sensor disposed at a location downstream of a catalyst in an exhaust pipe of an internal combustion engine for detecting an air/fuel ratio of exhaust gases which have passed through the catalyst, and the output of the controlled object is an output of the downstream air/fuel ratio sensor. The control input to the controlled object is the target air/fuel ratio of the air/fuel mixture supplied to the internal combustion engine. The controlled object model is a model representative of a relationship between the output of the downstream air/fuel ratio sensor and the target air/fuel ratio. The parameter detecting means comprises operating condition detecting means for detecting an operating condition of the internal combustion engine. The model parameter setting means sets model parameters of the controlled object model in accordance with the detected operating condition of the internal combustion engine. The control input calculating means includes air/fuel ratio calculating means for calculating the target air/fuel ratio of the air/fuel mixture supplied to the internal combustion engine for converging the output of the downstream air/fuel ratio sensor to a predetermined target value based on the one modulation algorithm and the controlled object model. 
   According to this preferred embodiment of the control apparatus, since the model parameters are set in accordance with the detected operating condition of the internal combustion engine, the model parameters can be rapidly calculated, even when the operating condition of the internal combustion engine suddenly changes, while precisely reflecting exhaust gases supplied to the catalyst. In addition, since the target air/fuel ratio is calculated for the air/fuel mixture supplied to the internal combustion engine based on the controlled object model, the model parameters of which are calculated in this manner, and the one modulation algorithm, the output of the air/fuel ratio sensor can be rapidly converged to the predetermined target value. Consequently, the control apparatus can rapidly and appropriately correct a slippage in control timing of the air/fuel ratio control, caused by a response delay, a dead time, and the like of exhaust gases with respect to the air/fuel mixture supplied to the internal combustion engine, to improve the stability of the air/fuel ratio control, and the post-catalyst exhaust gas characteristic. Further, the control apparatus can be realized at a relatively low cost because it only requires a single air/fuel ratio sensor. 
   Preferably, in the control method described above, the controlled object comprises an air/fuel ratio sensor disposed at a location downstream of a catalyst in an exhaust pipe of an internal combustion engine for detecting an air/fuel ratio of exhaust gases which have passed through the catalyst, and the output of the controlled object is an output of the downstream air/fuel ratio sensor. The control input to the controlled object is the target air/fuel ratio of the air/fuel mixture supplied to the internal combustion engine. The controlled object model is a model representative of a relationship between the output of the air/fuel ratio sensor and the target air/fuel ratio. The step of detecting a parameter includes detecting an operating condition of the internal combustion engine. The step of setting model parameters includes setting model parameters of the controlled object model in accordance with the detected operating condition of the internal combustion engine. The step of calculating a control includes calculating the target air/fuel ratio of the air/fuel mixture supplied to the internal combustion engine for converging the output of the air/fuel ratio sensor to a predetermined target value based on the one modulation algorithm and the controlled object model. 
   This preferred embodiment of the control method provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the engine control unit described above, the controlled object comprises an air/fuel ratio sensor disposed at a location downstream of a catalyst in an exhaust pipe of an internal combustion engine for detecting an air/fuel ratio of exhaust gases which have passed through the catalyst, and the output of the controlled object is an output of the downstream air/fuel ratio sensor. The control input to the controlled object is the target air/fuel ratio of the air/fuel mixture supplied to the internal combustion engine. The controlled object model is a model representative of a relationship between the output of the air/fuel ratio sensor and the target air/fuel ratio. The control program causes the computer to detect a parameter includes detecting an operating condition of the internal combustion engine; set model parameters of the controlled object model in accordance with the detected operating condition of the internal combustion engine; and calculate the target air/fuel ratio of the air/fuel mixture supplied to the internal combustion engine for converging the output of the air/fuel ratio sensor to a predetermined target value based on the one modulation algorithm and the controlled object model. 
   This preferred embodiment of the engine control unit provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the control apparatus described above, the air/fuel ratio calculating means includes predicted value calculating means for calculating a predicted value of a value indicative of the output of the air/fuel ratio sensor in accordance with the output of the air/fuel ratio sensor and the target air/fuel ratio based on a prediction algorithm which applies the controlled object model; and target air/fuel ratio calculating means for calculating the target air/fuel ratio of the air/fuel mixture supplied to the internal combustion engine in accordance with the calculated predicted value based on the one modulation algorithm. 
   According to this preferred embodiment of the control apparatus, the prediction time from the time at which the air/fuel mixture is supplied to the internal combustion engine in the target air/fuel ratio to the time at which the target air/fuel ratio is reflected to the output of the downstream air/fuel ratio sensor is calculated in accordance with the detected operating condition of the internal combustion engine, and the predicted value of the output deviation is calculated further in accordance with the calculated prediction time, so that the control apparatus can eliminate without fail a slippage in control timing between the input and output of the controlled object, caused by the dynamic characteristic of the controlled object, by calculating the control input using the predicted value calculated in this manner, thereby making it possible to further improve the post-catalyst exhaust gas characteristic. In addition, since the model parameters can be rapidly calculated, the control apparatus can rapidly ensure a satisfactory post-catalyst exhaust gas characteristic. 
   Preferably, in the control method described above, the step of calculating the target air/fuel ratio includes calculating a predicted value of a value indicative of the output of the air/fuel ratio sensor in accordance with the output of the air/fuel ratio sensor and the target air/fuel ratio based on a prediction algorithm which applies the controlled object model; and calculating the target air/fuel ratio of the air/fuel mixture supplied to the internal combustion engine in accordance with the calculated predicted value based on the one modulation algorithm. 
   This preferred embodiment of the control method provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the engine control unit described above, the control program causes the computer to calculate a predicted value of a value indicative of the output of the air/fuel ratio sensor in accordance with the output of the air/fuel ratio sensor and the target air/fuel ratio based on a prediction algorithm which applies the controlled object model; and calculate the target air/fuel ratio of the air/fuel mixture supplied to the internal combustion engine in accordance with the calculated predicted value based on the one modulation algorithm. 
   This preferred embodiment of the engine control unit provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the control apparatus described above, the predicted value calculating means calculates a prediction time from the time at which the air/fuel mixture is supplied to the internal combustion engine in the target air/fuel ratio to the time at which the target air/fuel ratio is reflected to the output of the air/fuel ratio sensor in accordance with an operating condition of the internal combustion engine, and calculates a predicted value of a value indicative of the output of the air/fuel ratio sensor further in accordance with the calculated prediction time. 
   According to this preferred embodiment of the control apparatus, since the target air/fuel ratio of the air/fuel mixture is calculated based on the intermediate value calculated based on the one modulation algorithm, multiplied by the gain, and the gain is set in accordance with an operating condition, the target air/fuel ratio of the air/fuel mixture can be calculated as a value which appropriately reflects a change in the gain characteristic of the controlled object, thereby making it possible to further improve the post-catalyst exhaust gas characteristic. In addition, since the model parameters can be rapidly calculated, the control apparatus can rapidly ensure a satisfactory post-catalyst exhaust gas characteristic. 
   Preferably, in the control method described above, the step of calculating a predicted value includes calculating a prediction time from the time at which the air/fuel mixture is supplied to the internal combustion engine in the target air/fuel ratio to the time at which the target air/fuel ratio is reflected to the output of the air/fuel ratio sensor in accordance with an operating condition of the internal combustion engine; and calculating a predicted value of a value indicative of the output of the air/fuel ratio sensor further in accordance with the calculated prediction time. 
   This preferred embodiment of the control method provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the engine control unit described above, the control program causes the computer to calculate a prediction time from the time at which the air/fuel mixture is supplied to the internal combustion engine in the target air/fuel ratio to the time at which the target air/fuel ratio is reflected to the output of the air/fuel ratio sensor in accordance with an operating condition of the internal combustion engine; and calculate a predicted value of a value indicative of the output of the air/fuel ratio sensor further in accordance with the calculated prediction time. 
   This preferred embodiment of the engine control unit provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the control apparatus described above, the target air/fuel ratio calculating means includes intermediate value calculating means for calculating an intermediate value of the target air/fuel ratio of the air/fuel mixture supplied to the internal combustion engine in accordance with the predicted value based on the controlled object model and the one modulation algorithm; gain setting means for setting a gain in accordance with the operating condition of the internal combustion engine; and target air/fuel ratio determining means for determining a target air/fuel ratio of the air/fuel mixture supplied to the internal combustion engine based on the calculated intermediate value multiplied by the set gain. 
   According to this preferred embodiment of the control apparatus, the prediction time from the time at which the air/fuel mixture is supplied to the internal combustion engine in the target air/fuel ratio to the time at which the target air/fuel ratio is reflected to the output of the downstream air/fuel ratio sensor is calculated in accordance with the detected operating condition of the internal combustion engine, and the predicted value of the output deviation is calculated further in accordance with the calculated prediction time, so that the control apparatus can eliminate without fail a slippage in control timing between the input and output of the controlled object, caused by the dynamic characteristic of the controlled object, by calculating the control input using the predicted value calculated in this manner, thereby making it possible to further improve the post-catalyst exhaust gas characteristic. In addition, since the model parameters can be rapidly calculated, the control apparatus can rapidly ensure a satisfactory post-catalyst exhaust gas characteristic. 
   Preferably, in the control method described above, the step of calculating the target air/fuel ratio includes calculating an intermediate value of the target air/fuel ratio of the air/fuel mixture supplied to the internal combustion engine in accordance with the predicted value based the one modulation algorithm; setting a gain in accordance with the operating condition of the internal combustion engine; and determining a target air/fuel ratio of the air/fuel mixture supplied to the internal combustion engine based on the calculated intermediate value multiplied by the set gain. 
   This preferred embodiment of the control method provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the engine control unit described above, the control program causes the computer to calculate an intermediate value of the target air/fuel ratio of the air/fuel mixture supplied to the internal combustion engine in accordance with the predicted value based the one modulation algorithm; set a gain in accordance with the operating condition of the internal combustion engine; and determine a target air/fuel ratio of the air/fuel mixture supplied to the internal combustion engine based on the calculated intermediate value multiplied by the set gain. 
   This preferred embodiment of the engine control unit provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, the control apparatus described above further comprises multiplying means for multiplying the predicted value by a correction coefficient; and correction coefficient setting means for setting the correction coefficient to be a smaller value when the predicted value is equal to or larger than a predetermined value than when the predicted value is smaller than the predetermined value, wherein the target air/fuel ratio calculating means calculates the target air/fuel ratio of the air/fuel mixture in accordance with the predicted value multiplied by the correction coefficient based on the one modulation algorithm. 
   According to this preferred embodiment of the control apparatus, the target air/fuel ratio of the air/fuel mixture is calculated in accordance with the predicted value of the output deviation multiplied by the correction coefficient, and the correction coefficient is set to a smaller value when the predicted value of the output deviation is equal to or larger than a predetermined value than when the predicted value of the output deviation is smaller than the predetermined value, so that the output of the downstream air/fuel ratio sensor can be converged at a different rate in accordance with the order of the predicted value of the output deviation with respect to the predetermined value. Therefore, for changing the air/fuel ratio to be leaner because of the predicted value of the output deviation being equal to or larger than zero, i.e., the output of the downstream air/fuel ratio sensor being larger than a target value when the predetermined value is set, for example to zero, the correction coefficient is set such that the output of the downstream air/fuel ratio sensor is converged at a lower rate than when the air/fuel ratio is changed to be richer, thereby providing the effect of suppressing the amount of exhausted NOx by a lean bias. On the other hand, when the air/fuel ratio is changed to be richer, the correction coefficient is set such that the output of the downstream air/fuel ratio sensor is converted at a higher rate than when the air/fuel ratio is changed to be leaner, thereby making it possible to sufficiently recover the NOX purifying rate of the catalyst. In addition, since the model parameters can be rapidly calculated, the control apparatus can rapidly ensure a satisfactory post-catalyst exhaust gas characteristic. 
   Preferably, the control method described above further comprises the steps of multiplying the predicted value by a correction coefficient; and setting the correction coefficient to be a smaller value when the predicted value is equal to or larger than a predetermined value than when the predicted value is smaller than the predetermined value, wherein the step of calculating the target air/fuel ratio includes calculating the target air/fuel ratio of the air/fuel mixture in accordance with the predicted value multiplied by the correction coefficient based on the one modulation algorithm. 
   This preferred embodiment of the control method provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 
   Preferably, in the engine control unit described above, the control program further causes the computer to multiply the predicted value by a correction coefficient; set the correction coefficient to be a smaller value when the predicted value is equal to or larger than a predetermined value than when the predicted value is smaller than the predetermined value; and calculate the target air/fuel ratio of the air/fuel mixture in accordance with the predicted value multiplied by the correction coefficient based on the one modulation algorithm. 
   This preferred embodiment of the engine control unit provides the same advantageous effects provided by the corresponding preferred embodiment of the control apparatus. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
       FIG. 1  is a block diagram generally illustrating a control apparatus according to a first embodiment of the present invention, and an internal combustion engine to which the control apparatus is applied; 
       FIG. 2  is a graph showing an exemplary result of measurements made for HC and NOx purification percentages of a first catalyzer and an output Vout of an O2 sensor  15 , with respect to an output KACT of an LAF sensor, when a deteriorated and a normal first catalyzer are used; 
       FIG. 3  is a block diagram illustrating the configuration of an ADSM controller and a PRISM controller in the control apparatus according to the first embodiment; 
       FIG. 4  shows exemplary equations which express a prediction algorithm associated with a state predictor; 
       FIG. 5  shows exemplary equations which express an identification algorithm associated with an on-board identifier; 
       FIG. 6  shows other exemplary equations which express an identification algorithm associated with the on-board identifier; 
       FIG. 7  is a block diagram illustrating the configuration of a controller which executes and ΔΣ modulation, and a control system which comprises the controller; 
       FIG. 8  is a timing chart showing an exemplary result of control conducted by the control system in  FIG. 7 ; 
       FIG. 9  is a timing chart for explaining the principles of an adaptive prediction type ΔΣ modulation control conducted by the ADSM controller in the first embodiment; 
       FIG. 10  is a block diagram illustrating the configuration of a DSM controller in the ADSM controller; 
       FIG. 11  shows equations which express a sliding mode control algorithm; 
       FIG. 12  shows equations which express a sliding mode control algorithm for the PRISM controller; 
       FIG. 13  is a flow chart illustrating a routine for executing fuel injection control processing for an internal combustion engine; 
       FIGS. 14 and 15  are flow charts illustrating in combination a routine for executing adaptive air/fuel ratio control processing; 
       FIG. 16  is a flow chart illustrating a routine for executing launch determination processing at step  21  in  FIG. 14 ; 
       FIG. 17  is a flow chart illustrating a routine for executing PRISM/ADSM processing execution determination processing at step  23  in  FIG. 14 ; 
       FIG. 18  is a flow chart illustrating a routine for executing the processing for determining whether or not the identifier should execute its operation at step  24  in  FIG. 14 ; 
       FIG. 19  is a flow chart illustrating a routine for executing the processing for calculating a variety of parameters at step  25  in  FIG. 14 ; 
       FIG. 20  shows an exemplary table for use in calculating dead times CAT_DELAY, KACT_D; 
       FIG. 21  shows an exemplary table for use in calculating a weighting parameter λ 1 ; 
       FIG. 22  shows an exemplary table for use in calculating limit values X_IDA2L, X_IDB1L, X_IDB1H for limiting ranges of model parameters a 1 , a 2 , b 1 ; 
       FIG. 23  shows an exemplary table for use in calculating a filter order n; 
       FIG. 24  is a flow chart illustrating a routine for executing the operation of the identifier at step  31  in  FIG. 14 ; 
       FIG. 25  is a flow chart illustrating a routine for executing θ(k) stabilization processing at step  94  in  FIG. 24 ; 
       FIG. 26  is a flow chart illustrating a routine for executing the processing for limiting identified values a 1 ′ and a 2 ′ at step  101  in  FIG. 25 ; 
       FIG. 27  is a diagram showing a restriction range in which a combination of the identified values a 1 ′ and a 2 ′ is restricted by the processing of  FIG. 26 ; 
       FIG. 28  is a flow chart illustrating a routine for executing the processing for limiting an identified value b 1 ′ at step  102  in  FIG. 25 ; 
       FIG. 29  is a flow chart illustrating the operation performed by the state predictor at step  33  in  FIG. 15 ; 
       FIG. 30  is a flow chart illustrating a routine for executing the processing for calculating a control amount Us 1  at step  34  in  FIG. 15 ; 
       FIG. 31  is a flow chart illustrating a routine for executing the processing for calculating an integrated value of a prediction switching function σPRE; 
       FIGS. 32 and 33  are flow charts illustrating in combination a routine for executing the processing for calculating a sliding mode control amount DKCMDSLD at step  36  in  FIG. 15 ; 
       FIG. 34  is a flow chart illustrating a routine for executing the processing for calculating a ΔΣ modulation control amount DKCMDDSM at step  37  in  FIG. 15 ; 
       FIG. 35  shows an exemplary table for use in calculating a gain KDSM; 
       FIG. 36  is a flow chart illustrating a routine for executing the processing for calculating an adaptive target air/fuel ratio KCMDSLD at step  38  in  FIG. 15 ; 
       FIG. 37  is a flow chart illustrating a routine for executing the processing for calculating an adaptive correction term FLAFADP at step  39  in  FIG. 15 ; 
       FIG. 38  is a block diagram generally illustrating the configuration of a control apparatus according to a second embodiment; 
       FIG. 39  is a block diagram generally illustrating the configuration of a control apparatus according to a third embodiment; 
       FIG. 40  is a block diagram generally illustrating the configuration of a control apparatus according to a fourth embodiment; 
       FIG. 41  shows an exemplary table for use in calculating model parameters in a parameter scheduler in the control apparatus according to the fourth embodiment; 
       FIG. 42  is a block diagram generally illustrating the configuration of an SDM controller in a control apparatus according to a fifth embodiment; 
       FIG. 43  is a block diagram generally illustrating the configuration of an DM controller in a control apparatus according to a sixth embodiment; 
       FIG. 44  is a block diagram generally illustrating a control apparatus according to a seventh embodiment, and an internal combustion engine to which the control apparatus is applied; 
       FIG. 45  is a block diagram generally illustrating the configuration of a control apparatus according to a seventh embodiment; and 
       FIG. 46  is a block diagram generally illustrating the configuration of a control apparatus according to an eighth embodiment. 
   

   DETAILED DESCRIPTION OF THE EMBODIMENTS 
   In the following, a control apparatus according to a first embodiment of the present invention will be described with reference to the accompanying drawings. The control apparatus according to the first embodiment is configured to control, by way of example, an air/fuel ratio of an internal combustion engine.  FIG. 1  generally illustrates the configuration of the control apparatus  1  and an internal combustion engine (hereinafter called the “engine”)  3  which applies the control apparatus  1 . As illustrated, the control apparatus  1  comprises an electronic control unit (ECU)  2  which controls the air/fuel ratio of an air/fuel mixture supplied to the engine  3  in accordance with an operating condition thereof. 
   The engine is an in-line four-cylinder gasoline engine equipped in a vehicle, not shown, and has four, a first to a fourth cylinder # 1 -# 4 . A throttle valve opening sensor  10 , for example, comprised of a potentiometer or the like, is provided near a throttle valve  5  in an intake pipe  4  of the engine  3 . The throttle valve opening sensor  10  detects an opening θTH of the throttle valve  5  (hereinafter called the “throttle valve opening”), and sends a detection signal indicative of the throttle valve opening θTH to the ECU  2 . 
   An absolute intake pipe inner pressure sensor  11  is further provided at a location of the intake pipe  4  downstream of the throttle valve  5 . The absolute intake pipe inner pressure sensor  11 , which implements gain parameter detecting means, operating condition detecting means, and dynamic characteristic parameter detecting means, is comprised, for example, of a semiconductor pressure sensor or the like for detecting an absolute intake pipe inner pressure PBA within the intake pipe  4  to output a detection signal indicative of the absolute intake pipe inner pressure PBA to the ECU  2 . 
   The intake pipe  4  is connected to the four cylinders # 1 -# 4 , respectively, through four branches  4   b  of an intake manifold  4   a . An injector  6  is attached to each of the branches  4   b  at a location upstream of an intake port, not shown. Each injector  6  is controlled by a driving signal from the ECU  2  in terms of a final fuel injection amount TOUT, which indicates a valve opening time, and an injection timing when the engine  3  is in operation. 
   A water temperature sensor  12  comprised, for example, of a thermistor or the like is attached to the body of the engine  3 . The water temperature sensor  12  detects an engine water temperature TW, which is the temperature of cooling water that circulates within a cylinder block of the engine  3 , and outputs a detection signal indicative of the engine water temperature TW to the ECU  2 . 
   A crank angle sensor  13  is mounted on a crank shaft (not shown) of the engine  3 . The crank angle sensor  13 , which implements gain parameter detecting means, operating condition detecting means, and dynamic characteristic parameter detecting means, outputs a CRK signal and a TDC signal, both of which are(pulse signals, to the ECU  2  as the crank shaft is rotated. 
   The CRK signal generates one pulse every predetermined crank angle (for example, 30°). The ECU  2  calculates a rotational speed NE of the engine  3  (hereinafter called the “engine rotational speed”) in response to the CRK signal. The TDC signal in turn indicates that a piston (not shown) of each cylinder is present at a predetermined crank angle position which is slightly in front of a TDC (top dead center) position in an intake stroke, and generates one pulse every predetermined crank angle. 
   At locations downstream of an exhaust manifold  7   a  in an exhaust pipe (exhaust passage), a first and a second catalyzer  8   a ,  8   b  (catalysts) are provided in this order from the upstream side, spaced apart from each other. Each catalyzer  8   a ,  8   b  is a combination of an NOx catalyst and a three-way catalyst. The NOx catalyst is made up of an iridium catalyst (a sintered product of iridium supported on silicon carbide whisker powder, and silica) coated on the surface of a base material in honeycomb structure, and a perovskite double oxide (a sintered product of LaCoO 3  powder and silica) coated on the iridium catalyst. The catalyzers  8   a ,  8   b  purify NOx in exhaust gases during a lean burn operation through oxidation/reduction actions of the NOx catalyst, and purify CO, HC and NOx in exhaust gases during an operation other than the lean burn operation through oxidation/reduction actions of the three-way catalyst. It should be noted that the catalyzers  8  are not limited to a combination of NOx catalyst and three-way catalyst, but may be made of any material as long as it can purify CO, HC and NOx in exhaust gases. For example, the catalyzers  8   a ,  8   b  may be made of a non-metal catalyst such as a perovskite catalyst and the like, and/or a metal-based catalyst such as a three-way catalyst and the like. 
   An oxygen concentration sensor (hereinafter called the “O2 sensor)  15  is mounted between the first and second catalyzers  8   a ,  8   b . The O2 sensor  15  (which implements a downstream air/fuel ratio sensor) is made of zirconium, a platinum electrode, and the like, and sends an output Vout to the ECU  2  based on the oxygen concentration in exhaust gases downstream of the first catalyzer  8   a . The output Vout of the O2 sensor  15  (output of a controlled object) goes to a voltage value at high level (for example, 0.8 V) when an air/fuel mixture richer than the stoichiometric air/fuel ratio is burnt, and goes to a voltage value at low level (for example, 0.2 V) when the air/fuel mixture is lean. Also, the output Vout goes to a predetermined target value Vop (for example, 0.6 V) when the air/fuel mixture is near the stoichiometric air/fuel ratio (see FIG.  2 ). 
   An LAF sensor  14  (which implements an upstream air/fuel ratio sensor) is mounted near a junction of the exhaust manifold  7   a  upstream of the first catalyzer  8   a . The LAF sensor  14  is comprised of a sensor similar to the O2 sensor  15 , and a detecting circuit such as a linearizer in combination for linearly detecting an oxygen concentration in exhaust gases over a wide range of the air/fuel ratio extending from a rich region to a lean region to send an output KACT proportional to the detected oxygen concentration to the ECU  2 . The output KACT is represented as an equivalent ratio proportional to an inverse of the air/fuel ratio. 
   Next, referring to  FIG. 2 , description will be made on the relationship between an exhaust gas purifying percentage provided by the first catalyzer  8   a  and the output Vout (voltage value) of the O2 sensor  15 .  FIG. 2  shows exemplary results of measuring the HC and NOx purifying percentage provided by the first catalyzer  8   a  and the output Vout of the O2 sensor  15  when the output KACT of the LAF sensor  14 , i.e., the air/fuel ratio of an air/fuel mixture supplied to the engine  3  varies near the stoichiometric air/fuel ratio, for two cases where the first catalyzer  8   a  is deteriorated due to a long-term use and therefore has degraded capabilities of purifying, and where the first catalyzer  8   a  is not deteriorated and therefore has high capabilities of purifying. In  FIG. 2 , data indicated by broken lines show the results of measurements when the first catalyzer  8   a  is not deteriorated, and data indicated by solid lines show the results of measurements when the first catalyzer  8   a  is deteriorated.  FIG. 2  also shows that the air/fuel ratio of the air/fuel mixture is richer as the output KACT of the LAF sensor  14  is larger. 
   As shown in  FIG. 2 , when the first catalyzer  8   a  is deteriorated, its capabilities of purifying exhaust gases are degraded, as compared with the one not deteriorated, so that the output Vout of the O2 sensor  15  crosses the target value Vop when the output KACT of the LAF sensor  14  is at a value KACT1 deeper in a lean region. On the other hand, the first catalyzer  8   a  has the characteristic of most efficiently purifying HC and NOx when the output Vout of the O2 sensor  15  is at the target value Vop, irrespective of whether the first catalyzer  8   a  is deteriorated or not. It is therefore appreciated that exhaust gases can be most efficiently purified by the first catalyzer  8   a  by controlling the air/fuel ratio of the air/fuel mixture to bring the output Vout of the O2 sensor  15  to the target value Vop. For this reason, in the air/fuel control later described, a target air/fuel ratio KCMD is controlled such that the output Vout of the O2 sensor  15  converges to the target value Vop. 
   The ECU  2  is further connected to an accelerator opening sensor  16 , an atmospheric pressure sensor  17 , an intake air temperature sensor  18 , a vehicle speed sensor  19 , and the like. The accelerator opening sensor  16  detects an amount AP by which the driver treads on an accelerating pedal, not shown, of the vehicle (hereinafter called the “accelerator opening”), and outputs a detection signal indicative of the accelerator opening AP to the ECU  2 . Likewise, the atmospheric pressure sensor  17 , intake air temperature sensor  18  and vehicle speed sensor  19  detect the atmospheric pressure PA, an intake air temperature TA, and a vehicle speed VP, respectively, and output detection signals indicative of the respective detected values to the ECU  2 . 
   Next, description will be made on the ECU  2  which implements predicted value calculating means, control input calculating means, gain parameter detecting means, gain setting means, air/fuel ratio calculating means, operating state detecting means, an intermediate value calculating means, target air/fuel ratio calculating means, multiplying means, correction coefficient setting means, identifying means, identification error calculating means, filtering means, parameter determining means, dead time setting means, restriction range setting means, weighting parameter setting means, dynamic characteristic parameter detecting means, and model parameter setting means. 
   The ECU  2 , based on a microcomputer which comprises an I/O interface, a CPU, a RAM, a ROM, and the like, determines an operating condition of the engine  3  in accordance with the outputs of the variety of sensors  10 - 19  mentioned above, and calculates the target air/fuel ratio KCMD (control input) by executing adaptive air/fuel ratio control processing or map search processing, later described, in accordance with a control program previously stored in the ROM and data stored in the RAM. Further, as will be described later, the ECU  2  calculates the final fuel injection amount TOUT of the injector  6  for each cylinder based on the calculated target air/fuel ratio KCMD, and drives the injector  6  using a driving signal based on the calculated final fuel injection amount TOUT to control the air/fuel ratio of the air/fuel mixture. 
   As illustrated in  FIG. 3 , the control apparatus  1  comprises an ADSM controller  20  for calculating the target air/fuel ratio KCMD, and a PRISM controller  21 . Specifically, both controllers  20 ,  21  are implemented by the ECU  2 . 
   In the following, the ADSM controller  20  (which implements control input calculating means) will be described. The ADSM controller  20  calculates the target air/fuel ratio KCMD for converging the output Vout of the O2 sensor  15  to the target value Vop in accordance with a control algorithm of adaptive prediction ΔΣ modulation control (hereinafter abbreviated as “ADSM”), later described. The ADSM controller  20  comprises a state predictor  22 , an on-board identifier  23 , and a DSM controller  24 . A specific program for executing the ADSM processing will be described later. 
   Description will first be made on the state predictor  22  (which implements predicted value calculating means). The state predictor  22  predicts (calculates) a predicted value PREVO2 of an output deviation VO2 in accordance with a prediction algorithm, later described. Assume, in this embodiment, that a control input to a controlled object is the target air/fuel ratio KCMD of an air/fuel mixture; the output of the controlled object is the output Vout of the O2 sensor  15 ; and the controlled object is a system from an intake system of the engine  3  including the injectors  6  to the O2 sensor  15  downstream of the first catalyzer  8   a  in an exhaust system including the first catalyzer  8   a . Then, this controlled object is modelled, as expressed by the following equation (1), as an ARX model (auto-regressive model with exogenous input) which is a discrete time system model.
 
 V 02( k )= a   1 · V 02( k− 1)+ a   2 · VO 2( K− 2)+ b   1   DKCMD ( k−dt )  (1)
 
where VO2 represents an output deviation which is a deviation (Vout-Vop) between the output Vout of the O2 sensor  15  and the aforementioned target value Vop; DKCMD represents an air/fuel ratio deviation which is a deviation (KCMD-FLAFBASE) between a target air/fuel ratio KCMD (=φop) and a reference value FLAFBASE; and a character k represents the order of each data in a sampling cycle. The reference value FLAFBASE is set to a predetermined fixed value. Model parameters a 1 , a 2 , b 1  are sequentially identified by the on-board identifier  23  in a manner described below.
 
   dt in the equation (1) represents a prediction time period from the time at which an air/fuel mixture set at the target air/fuel ratio KCMD is supplied to the intake system by the injectors  6  to the time at which the target air/fuel ratio KCMD is reflected to the output Vout of the O2 sensor  15 , and is defined by the following equation (2):
 
 dt=d+d′+dd   (2)
 
where d represents a dead time in the exhaust system from the LAF sensor  14  to the O2 sensor  15 ; d′, a dead time in an air/fuel ratio manipulation system from the injectors  6  to the LAF sensor  14 ; and dd represents a phase delay time between the exhaust system and air/fuel ratio manipulation system, respectively (it should be noted that in a control program for the adaptive air/fuel ratio control processing, later described, the phase delay time dd is set to zero (dd=0) for calculating the target air/fuel ratio KCMD while switching between the ADSM processing and PRISM processing).
 
   The controlled object model is comprised of time series data of the output deviation VO2 and the air/fuel ratio deviation DKCMD as described above for the reason set forth below. It is generally known in a controlled object model that the dynamic characteristic of the controlled object model can be fitted more closely to the actual dynamic characteristic of the controlled object when a deviation of input/output between the controlled object and a predetermined value is defined as a variable representative of the input/output than when an absolute value of the input/output is defined as a variable, because it can more precisely identify or define model parameters. Therefore, as is done in the control apparatus  1  of this embodiment, when the controlled object model is comprised of the time series data of the output deviation VO2 and the air/fuel ratio deviation DKCMD, the dynamic characteristic of the controlled object model can be fitted more closely to the actual dynamic characteristic of the controlled object, as compared with the case where absolute values of the output Vout of the O2 sensor  15  and target air/fuel ratio KCMD are chosen as variables, thereby making it possible to calculate the predicted value PREVO2 with a higher accuracy. 
   The predicted value PREVO2 in turn shows a predicted output deviation VO2(k+dt) after the lapse of the prediction time period dt from the time at which the air/fuel mixture set at the target air/fuel ratio KCMD has been supplied to the intake system. When an equation for calculating the predicted value PREVO2 is derived based on the aforementioned equation (1), the following equation (3) is defined: 
                     PREVO2   ⁢           ⁢     (   k   )       ≈       ⁢     VO2   ⁢           ⁢     (     k   +   dt     )                   =       ⁢         a1   ·   VO2     ⁢           ⁢     (     k   +   dt   -   1     )       +                     ⁢         a2   ·   VO2     ⁢           ⁢     (     k   +   dt   -   2     )       +       b1   ·   DKCMD     ⁢           ⁢     (   k   )                       (   3   )             
 
   In this equation (3), it is necessary to calculate VO2(k+dt−1), VO2(k+dt−2) corresponding to future values of the output deviation VO2 (k), so that actual programming of the equation (3) is difficult. Therefore, matrixes A, B are defined using the model parameters a 1 , a 2 , b 1 , as equations (4), (5) shown in  FIG. 4 , and a recurrence formula of the equation (3) is repeatedly used to transform the equation (3) to derive equation (6) shown in FIG.  4 . When the equation (6) is used as a prediction algorithm, i.e., an equation for calculating the predicted value PREVO2, the predicted value PREVO2 is calculated from the output deviation VO2 and air/fuel ratio deviation DKCMD. 
   Next, when an LAF output deviation DKACT is defined as a deviation (KACT-FLAFBASE) between the output KACT (=φin) of the LAF sensor  14  and the reference value FLAFBASE, a relationship expressed by DKACT(k)=DKCMD(k−d′) is established. Equation (7) shown in  FIG. 4  is derived by applying this relationship to the equation (6) in FIG.  4 . 
   The target air/fuel ratio KCMD can be calculated while appropriately compensating for a response delay and a dead time between the input/output of the controlled object by calculating the target air/fuel ratio KCMD using the predicted value PREVO2 calculated by the foregoing equation (6) or (7), as will be described later. Particularly, when the equation (7) is used as the prediction algorithm, the predicted value PREVO2 is calculated from the LAF output deviation DKACT and target air/fuel ratio KCMD, so that the predicted value PREVO2 can be calculated as a value which reflects the air/fuel ratio of exhaust gases actually supplied to the first catalyzer  8   a , thereby improving the calculation accuracy, i.e., the prediction accuracy more than when the equation (6) is used. Also, if d′ can be regarded to be smaller than 1 (d′≦1) when the equation (7) is used, the predicted value PREVO2 can be calculated only from the output deviation VO2 and LAF output deviation DKACT without using the air/fuel ratio deviation DKCMD. In this embodiment, since the engine  3  is provided with the LAF sensor  14 , the equation (7) is employed as the prediction algorithm. 
   The controlled object model expressed by the equation (1) can be defined as a model which employs the output deviation VO2 and LAF output deviation DKACT as variables by applying a relationship expressed by DKACT(k)=DKCMD(k−d′) to the equation (1). 
   Next, description will be made on the on-board identifier  23  (which implements identifying means, identification error calculating means, filtering means, dead time setting means, restriction range setting means, weighting parameter setting means, and parameter determining means) . The on-board identifier  23  identifies (calculates) the model parameters a 1 , a 2 , b 1  in the aforementioned equation (1) in accordance with a sequential identification algorithm described below. Specifically, a vector θ(k) for model parameters is calculated by equations (8), (9) shown in FIG.  5 . In the equation (8) in  FIG. 5 , KP(k) is a vector for a gain coefficient, and ide_f(k) is an identification error filter value. In the equation (9), θ(k)T represents a transposed matrix of θ(k), and a 1 ′(k), a 2 ′(k) and b 1 ′(k) represent model parameters before they are limited in range in limit processing, later described. In the following description, the term “vector” is omitted if possible. 
   An identification error filter value ide_f(k) in the equation (8) is derived by applying moving average filtering processing expressed by equation (10) in  FIG. 5  to an identification error ide(k) calculated by equations (11)-(13) shown in  FIG. 5. n  in the equation (10) in  FIG. 5  represents the order of filtering (an integer equal to or larger than one) in the moving average filtering processing, and VO2HAT(k) in the equation (12) represents an identified value of the output deviation VO2. 
   The identification error filter value ide_f(k) is used for the reason set forth below. Specifically, the controlled object in this embodiment has the target air/fuel ratio KCMD as a control input, and the output Vout of the O2 sensor  15  as an output. The controlled object also has a low pass frequency characteristic. In such a controlled object having the low pass characteristic, model parameters are identified while the high frequency characteristic of the controlled object is emphasized due to a frequency weighting characteristic of the identification algorithm of the on-board identifier  23 , more specifically, a weighted least-square algorithm, later described, so that the controlled object model tends to have a lower gain characteristic than the actual gain characteristic of the controlled object. As a result, when the ADSM processing or PRISM processing is executed by the control apparatus  1 , the control system can diverge and therefore become instable due to an excessive gain possibly resulting from the processing. 
   Therefore, in this embodiment, the control apparatus  1  appropriately corrects the weighted least-square algorithm for the frequency weighting characteristic, and uses the identification error filter value ide_f(k) applied with the moving average filtering processing for the identification error ide(k), as well as sets the filter order n of the moving average filtering processing in accordance with an exhaust gas volume AB_SV in order to match the gain characteristic of the controlled object model with the actual gain characteristic of the controlled object, as will be later described. 
   Further, the vector KP(k) for the gain coefficient in the equation (8) in  FIG. 5  is calculated by equation (14) in FIG.  5 . P(k) in the equation 14 is a third-order square matrix as defined by equation (15) in FIG.  5 . 
   In the identification algorithm described above, one is selected from the following four identification algorithms by setting weighting parameters λ 1 , λ 2  in the equation (15):
         λ 1 =1, λ 2 =0: Fixed Gain Algorithm;   λ 1 =1, λ 2 =1: Least-Square Algorithm;   λ 1 =1, λ 2 =λ: Gradually Reduced Gain Algorithm; and   λ 1 =λ, λ 2 =1: Weighted Least-Square Algorithm.
 
where λ is a predetermined value set in a range of 0&lt;λ&lt;1.
       

   This embodiment employs the weighted least-square algorithm from among the four identification algorithms. This is because the weighted least-square algorithm can appropriately set an identification accuracy, and a rate at which a model parameter converges to an optimal value, by setting the weighting parameter λ 1  in accordance with an operating condition of the engine  3 , more specifically, the exhaust gas volume AB_SV. For example, when the engine  3  is lightly loaded in operation, a high identification accuracy can be ensured by setting the weighting parameter λ 1  to a value close to one in accordance with this operating condition, i.e., by setting the algorithm close to the least-square algorithm. On the other hand, when the engine  3  is heavily loaded in operation, the model parameter can be rapidly converged to an optimal value by setting the weighting parameter λ 1  to a value smaller than that during the low load operation. By setting the weighting parameter λ 1  in accordance with the exhaust gas volume AB_SV in the foregoing manner, it is possible to appropriately set the identification accuracy, and the rate at which the model parameter converges to an optimal value, thereby improving the post-catalyst exhaust gas characteristic. 
   When the aforementioned relationship, DKACT(k)=DKCMD(k−d′) is applied in the identification algorithm expressed by the equations (8)-(15), an identification algorithm is derived as expressed by equations (16)-(23) shown in FIG.  6 . In this embodiment, since the engine  3  is provided with the LAF sensor  14 , these equations (16)-(23) are employed. When these equations (16)-(23) are employed, the model parameter can be identified as a value which more reflects the air/fuel ratio of exhaust gases actually fed to the first catalyzer  8   a  to a higher degree, for the reason set forth above, and accordingly, the model parameter can be identified with a higher accuracy than when using the identification algorithm expressed by the equations (8)-(15). 
   Also, the on-board identifier  23  applies the limit processing, later described, to the model parameters a 1 ′(k), a 2 ′(k), b 1 ′(k) calculated by the foregoing identification algorithm to calculate the model parameters a 1 (k), a 2 (k), b 1 (k). Further, the aforementioned state predictor  22  calculates the predicted value PREVO2 based on the model parameters a 1 (k), a 2 (k), b 1 (k) after they have been limited in range in the limit processing. 
   Next, the DSM controller  24  will be described. The DSM controller  24  generates (calculates) the control input φop(k) (=target air/fuel ratio KCMD) in accordance with a control algorithm applied with the ΔΣ modulation algorithm, based on the predicted value PREVO2 calculated by the state predictor  22 , and inputs the calculated control input φop(k) to the controlled object to control the output Vout of the O2 sensor  15 , as the output of the controlled object, such that it converges to the target value Vop. 
   First, a general ΔΣ modulation algorithm will be described with reference to FIG.  7 .  FIG. 7  illustrates the configuration of a control system which controls a controlled object  27  by a controller  26  to which the ΔΣ modulation algorithm is applied. As illustrated, in the controller  26 , a subtractor  26   a  generates a deviation signal δ(k) as a deviation between a reference signal r(k) and a DSM signal u(k−1) delayed by a delay element  26   b . Next, an integrator  26   c  generates an integrated deviation value σ d (k) as a signal indicative of the sum of the deviation signal δ(k) and an integrated deviation value σ d (k−1) delayed by a delay element  26   d . Next, a quantizer  26   e  (sign function) generates a DSM signal u(k) as a sign of the integrated deviation value σ d (k) . Consequently, the DSM signal u(k) thus generated is inputted to the controlled object  27  which responsively delivers an output signal y(k). 
   The foregoing ΔΣ modulation algorithm is expressed by the following equations (24)-(26):
 
δ( k )= r ( k )− u ( k− 1)  (24)
 
σ d ( k )=σ d ( k− 1)+δ( k )  (25)
 
 u ( k )= sgn (σ d ( k ))  (26)
 
where the value of the sign function sgn(σ d (k)) takes 1 (sgn(σ d (k) )=1) when σ d (k)≧0, and −1 (sgn(σ d (k))=−1) when σ d (k)&lt;0 (sgn(σ d (k)) may be set to zero (sgn(σ d (k))=0) when σ d (k)=0).
 
     FIG. 8  shows the result of control simulation performed for the foregoing control system. As shown, when the sinusoidal reference signal r(k) is inputted to the control system, the DSM signal u(k) is generated as a square-wave signal and is fed to the controlled object  27  which responsively outputs the output signal y(k) which has a different amplitude from and the same frequency as the reference signal r(k), and is generally in a similar waveform though noise is included. As described, the ΔΣ modulation algorithm is characterized in that the DSM signal u(k) can be generated when the controlled object  27  is fed with the DSM signal u(k) generated from the reference signal r(k) such that the controlled object  27  generates the output y(k) which has a different amplitude from and the same frequency as the reference signal r(k) and is generally similar in waveform to the reference signal r(k). In other words, the ΔΣ modulation algorithm is characterized in that the DSM signal u(k) can be generated (calculated) such that the reference signal r(k) is reproduced in the actual output y(k) of the controlled object  27 . 
   The DSM controller  24  takes advantage of such characteristic of the ΔΣ modulation algorithm to calculate the control input φop(k) for converging the output Vout of the O2 sensor  15  to the target value Vop. Describing the principles of the calculation, when the output deviation VO2 fluctuates with respect to the value of zero, for example, as indicated by a one-dot chain line in  FIG. 9  (i.e., the output Vout of the O2 sensor  15  fluctuates with respect to the target value Vop), the control input φop(k) may be generated to produce an output deviation VO2* having an opposite phase waveform to cancel the output deviation VO2 as indicated by a broken line in  FIG. 9 , in order to converge the output deviation VO2 to zero (i.e., to converge the output Vout to the target value Vop). 
   However, as described above, the controlled object in this embodiment experiences a time delay equal to the prediction time period dt from the time at which the target air/fuel ratio KCMD is inputted to the controlled object as the control input φop(k) to the time at which it is reflected to the output Vout of the O2 sensor  15 . Therefore, an output deviation VO2# derived when the control input φop(k) is calculated based on the current output deviation VO2 delays from the output deviation V02*, as indicated by a solid line in  FIG. 9 , thereby causing a slippage in control timing. To compensate the control timing for the slippage, the DSM controller  24  in the ADSM controller  20  according to this embodiment employs the predicted value PREVO2 of the output deviation VO2 to generate the control input φop(k) as a signal which generates an output deviation (an output deviation similar to the output deviation VO2* in opposite phase waveform) that cancels the current output deviation VO2 without causing a slippage in control timing. 
   Specifically, as illustrated in  FIG. 10 , an inverting amplifier  24   a  in the DSM controller  24  generates the reference signal r(k) by multiplying the value of −1, a gain G d  for the reference signal, and the predicted value PREVO2(k). Next, a subtractor  24   b  generates the deviation signal δ(k) as a deviation between the reference signal r(k) and a DSM signal u″(k−1) delayed by a delay element  24   c.    
   Next, an integrator  24   d  generates the integrated deviation value σ d (k) as the sum of the deviation signal δ(k) and an integrated deviation value σ d (k−1) delayed by a delay element  24   e . Then, a quantizer  24   f  (sign function) generates a DSM signal u″(k) as a sign of the integrated deviation value σ d (k). An amplifier  24   g  next generates an amplified DSM signal u(k) by amplifying the DSM signal u″(k) by a predetermined gain F d . Finally, an adder  24   h  adds the amplified DSM signal u(k) to a predetermined reference value FLAFBASE to generate the control input φop(k). 
   The control algorithm of the DSM controller  24  described above is expressed by the following equations (27)-(32):
 
 r ( k )=−1· Gd·PREVO 2( k )  (27)
 
δ( k )= r ( k )− u ″( k− 1)  (28)
 
σ d ( k )=σ d ( k− 1)+δ( k )  (29)
 
 u ″( k )= sgn (σ d ( k ))  (30)
 
 u ( k )= F   d   ·u ″( k )  (31)
 
 op ( k )= FLAFBASE+u ( k )  (32)
 
where G d , F d  represents gains. The value of the sign function sgn(σ d (k)) takes 1 (sgn(σ d (k))=1) when σ d (k)≧0, and −1 (sgn(σ d(k))= 1) when σ d (k)&lt;0 (sgn(σ d (k)) may be set to zero (sgn(σ d (k))=0) when σ d (k)=0).
 
   The DSM controller  24  calculates the control input φop(k) as a value which generates the output deviation VO2* that cancels the output deviation VO2 without causing a slippage in control timing, as described above. In other words, the DSM controller  24  calculates the control input φop(k) as a value which can converge the output Vout of the O2 sensor  15  to the target value Vop. Also, since the DSM controller  24  calculates the control input φop(k) by adding the amplified DSM signal u(k) to the predetermined reference value FLAFBASE, the resulting control input φop(k) not only inverts in the positive and negative directions about the value of zero, but also repeatedly increases and decreases about the reference value FLAFBASE. This can increase the degree of freedom for the control, as compared with a general ΣΔ modulation algorithm. 
   Next, the aforementioned PRISM controller  21  will be described with reference again to FIG.  3 . The PRISM controller  21  relies on a control algorithm for on-board identification sliding mode control processing (hereinafter called the “PRISM processing”), later described, to calculate the target air/fuel ratio KCMD for converging the output Vout of the O2 sensor  15  to the target value Vop. The PRISM controller  21  comprises the state predictor  22 , on-board identifier  23 , and sliding mode controller (hereinafter called the “SLD controller”)  25 . A specific program for executing the PRISM processing will be described later. 
   Since the state predictor  22  and on-board identifier  23  have been described in the PRISM controller  21 , the following description will be centered on the SLD controller  25 . The SLD controller  25  performs the sliding mode control based on the sliding mode control algorithm. In the following, a general sliding mode control algorithm will be described. Since the sliding mode control algorithm uses the aforementioned discrete time system model expressed by the equation (1) as a controlled object model, a switching function σ is set as a linear function of a time series data of the output deviation VO2 as expressed by the following equation (33): 
    σ( k )= S   1   ·VO 2( k )+ S   2   ·VO 2( k− 1)  (33) 
   where S 1 , S 2  are predetermined coefficients which are set to satisfy a relationship represented by −1&lt;(S 2 /S 1 )&lt;1. 
   Generally, in the sliding mode control algorithm, when the switching function σ is made up of two state variables (time series data of the output deviation VO2 in this embodiment), a phase space defined by the two state variables forms a two-dimensional phase space in which the two state variables are represented by the vertical axis and horizontal axis, respectively, so that a combination of values of the two state variables satisfying σ=0 rests on a line called a “switching line.” Therefore, both the two state variables can be converged (slid) to a position of equilibrium at which the state variables take the value of zero by appropriately determining a control input to a controlled object such that a combination of the two state variables converges to (rests on) the switching line. Further, the sliding mode control algorithm can specify the dynamic characteristic, more specifically, convergence behavior and convergence rate of the state variables by setting the switching function σ. For example, when the switching function σ is made up of two state variables as in this embodiment, the state variables converge slower as the slope of the switching line is brought closer to one, and faster as it is brought closer to zero. 
   In this embodiment, as shown in the aforementioned equation (33), the switching function σ is made up of two time series data of the output deviation VO2 i.e., a current value VO2(k) and the preceding value VO2(k−1) of the output deviation VO2, so that the control input to the controlled object, i.e., the target air/fuel ratio KCMD may be set such that a combination of these current value VO2(k) and preceding vale VO2(k−1) of the output deviation VO2(k) is converged onto the switching line. Specifically, assuming that the sum of a control amount Usl(k) and the reference value FLAFBASE is equal to the target air/fuel ratio KCMD, the control amount Usl(k) for converging the combination of the current value VO2(k) and preceding value VO2(k−1) onto the switching line is set as a total sum of an equivalent control input Ueq(k), an reaching law input Urch(k), and an adaptive law input Uadp(k), as shown in equation (34) shown in  FIG. 11 , in accordance with an adaptive sliding mode control algorithm. 
   The equivalent control input Ueq(k) is provided for restricting the combination of the current value VO2 (k) and preceding value VO2(k−1) of the output deviation VO2 on the switching line, and specifically is defined as equation (35) shown in FIG.  11 . The reaching law input Urch(k) is provided for converging the combination of the current value VO2 (k) and preceding value VO2 (k−1) of the output deviation VO2 onto the switching line if it deviates from the switching line due to disturbance, a modelling error or the like, and specifically is defined as equation (36) shown in FIG.  11 . In the equation (36), F represents a gain. 
   The adaptive law input Uadp(k) is provided for securely converging the combination of the current value VO2 (k) and preceding value VO2(k−1) of the output deviation VO2 onto a switching hyperplane while preventing the influence of a steady-state deviation of the controlled object, a modelling error, and disturbance, and specifically defined as equation (37) shown in FIG.  11 . In the equation (37), G represents a gain, and ΔT a control period, respectively. 
   As described above, the SLD controller  25  in the PRISM controller  21  according to this embodiment uses the predicted value PREVO2 instead of the output deviation VO2, so that the algorithm expressed by the equations (33)-(37) is rewritten to equations (38)-(42) shown in  FIG. 12  for use in the control by applying a relationship expressed by PREVO2(k)≈VO2(k+dt). σPRE in the equation (38) represents the value of the switching function when the predicted value PREVO2 is used (hereinafter called the “prediction switching function”). In other words, the SLD controller  25  calculates the target air/fuel ratio KCMD by adding the control amount Usl(k) calculated in accordance with the foregoing algorithm to the reference value FLAFBASE. 
   In the following, the processing for calculating a fuel injection amount, executed by the ECU  2 , will be described with reference to FIG.  13 . In the following description, the symbol (k) indicative of the current value is omitted as appropriate.  FIG. 13  illustrates a main routine of this control processing which is executed in synchronism with an inputted TDC signal as an interrupt. In this processing, the ECU  2  uses the target air/fuel ratio KCMD calculated in accordance with adaptive air/fuel ratio control processing or map search processing, later described, to calculate the fuel injection amount TOUT for each cylinder. 
   First at step  1  (abbreviated as “S 1 ” in the figure. The same applies to subsequent figures), the ECU  2  reads outputs of the variety of aforementioned sensors  10 - 19 , and stores the read data in the RAM. 
   Next, the routine proceeds to step  2 , where the ECU  2  calculates a basic fuel injection amount Tim. In this processing, the ECU  2  calculates the basic fuel injection amount Tim by searching a map, not shown, in accordance with the engine rotational speed NE and absolute intake pipe inner pressure PBA. 
   Next, the routine proceeds to step  3 , where the ECU  2  calculates a total correction coefficient KTOTAL. For calculating the total correction coefficient KTOTAL, the ECU  2  calculates a variety of correction coefficients by searching a variety of tables and maps in accordance with a variety of operating parameters (for example, the intake air temperature TA, atmospheric pressure PA, engine water temperature TW, accelerator opening AP, and the like), and multiplies these correction coefficients by one another. 
   Next, the routine proceeds to step  4 , where he ECU  2  sets an adaptive control flag F_PRISMON. Though details of this processing are not shown in the figure, specifically, when the following conditions (a)-(f) are fully satisfied, the ECU  2  sets the adaptive control flag F_PRISMON to “1,” determining that the condition is met for using the target air/fuel ratio KCMD calculated in the adaptive air/fuel ratio control processing. On the other hand, if any of the conditions (a)-(f) is not satisfied, the ECU  2  sets the adaptive control flag F_PRISMON to “0.”
         (a) The LAF sensor  14  and O2 sensor  15  are both activated;   (b) the engine  3  is not in a lean burn operation;   (c) the throttle valve  5  is not fully opened;   (d) the ignition timing is not controlled to be retarded;   (e) the engine  3  is not in a fuel cut operation; and   (f) the engine rotational speed NE and absolute intake pipe inner pressure PBA are both within their respective predetermined ranges.       

   Next, the routine proceeds to step  5 , where it is determined whether or not the adaptive control flag F_PRISMON set at step  4  is “1.” If the result of determination at step  5  is YES, the routine proceeds to step  6 , where the ECU  2  sets the target air/fuel ratio KCMD to an adaptive target air/fuel ratio KCMDSLD which is calculated by adaptive air/fuel ratio control processing, later described. 
   On the other hand, if the result of determination at step  5  is NO, the routine proceeds to step  7 , where the ECU  2  sets the target air/fuel ratio KCMD to a map value KCMDMAP. The map value KCMDMAP is calculated by searching a map, not shown, in accordance with the engine rotational speed NE and intake pipe inner absolute pressure PBA. 
   At step  8  subsequent to the foregoing step  6  or  7 , the ECU  2  calculates an observer feedback correction coefficient #nKLAF for each cylinder. The observer feedback correction coefficient #nKLAF is provided for correcting variations in the actual air/fuel ratio for each cylinder. Specifically, the ECU  2  calculates the observer feedback correction coefficient #nKLAF based on a PID control in accordance with an actual air/fuel ratio estimated by an observer for each cylinder from the output KACT of the LAF sensor  14 . The symbol #n in the observer feedback correction coefficient #nKLAF represents the cylinder number # 1 -# 4 . The same applies as well to a required fuel injection amount #nTCYL and a final fuel injection amount #nTOUT, later described. 
   Next, the routine proceeds to step  9 , where the ECU  2  calculates a feedback correction coefficient KFB. Specifically, the ECU  2  calculates the feedback coefficient KFB in the following manner. The ECU  2  calculates a feedback coefficient KLAF based on a PID control in accordance with a deviation of the output KACT of the LAF sensor  14  from the target air/fuel ratio KCMD. Also, the ECU  2  calculates a feedback correction coefficient KSTR by calculating the feedback correction coefficient KSTR by a self tuning regulator type adaptive controller, not shown, and dividing the feedback correction coefficient KSTR by the target air/fuel ratio KCMD. Then, the ECU  2  sets one of these two feedback coefficient KLAF and feedback correction coefficient KSTR as the feedback correction coefficient KFB in accordance with an operating condition of the engine  3 . 
   Next, the routine proceeds to step  10 , where the ECU  2  calculates a corrected target air/fuel ratio KCMDM. This corrected target air/fuel ratio KCMDM is provided for compensating a change in filling efficiency due to a change in the air/fuel ratio A/F. The ECU  2  calculates the corrected target air/fuel ratio KCMDM by searching a table, not shown, in accordance with the target air/fuel ratio KCMD calculated at step  6  or  7 . 
   Next, the routine proceeds to step  11 , where the ECU  2  calculates the required fuel injection amount #nTCYL for each cylinder in accordance with the following equation (43) using the basic fuel injection amount Tim, total correction coefficient KTOTAL, observer feedback correction coefficient #nKLAF, feedback correction coefficient KFB, and corrected target air/fuel ratio KCMDM, which have been calculated as described above.
 
# nTCYL=Tim·KTOTAL·KCMDM·KFB·#nKLAF   (43)
 
   Next, the routine proceeds to step  12 , where the ECU  2  corrects the required fuel injection amount #nTCYL for sticking to calculate the final fuel injection amount #nTOUT. Specifically, the ECU  2  calculates this final fuel injection amount #nTOUT by calculating the proportion of fuel injected from the injector  6  which is stuck to the inner wall of the combustion chamber in the current combustion cycle in accordance with an operating condition of the engine  3 , and correcting the required fuel injection amount #nTCYL based on the proportion thus calculated. 
   Next, the routine proceeds to step  13 , where the ECU  2  outputs a driving signal based on the final fuel injection amount #nTOUT calculated in the foregoing manner to the injector  6  of a corresponding cylinder, followed by termination of this processing. 
   Next, the adaptive air/fuel ratio control processing including the ADSM processing and PRISM processing will be described with reference to  FIGS. 14 and 15  which illustrate routines for executing the ADSM and PRISM processing, respectively. This processing is executed at a predetermined period (for example, every 10 msec). Also, in this processing, the ECU  2  calculates the target air/fuel ratio KCMD in accordance with an operating condition of the engine  3  by the ADSM processing, PRISM processing, or processing for setting a sliding mode control amount DKCMDSLD to a predetermined value SLDHOLD. 
   First, in this processing, the ECU  2  executes post-F/C determination processing at step  20 . Though not shown in detail in the figure, during a fuel cut operation, the ECU  2  sets a F/C post-determination flag F_AFC to “1” for indicating that the engine  3  is in a fuel cut operation. When a predetermined time X_TM_TM_AFC has elapsed after the end of the fuel cut operation, the ECU  2  sets the post-F/C determination flag F_AFC to “0” for indicating this situation. 
   Next, the routine proceeds to step  21 , where the ECU  2  executes start determination processing based on the vehicle speed VP for determining whether or not the vehicle equipped with the engine  3  has started. As illustrated in  FIG. 16  showing a routine for executing the start determination processing, it is first determined at step  49  whether or not an idle operation flag F_IDLE is “1.” The idle operation flag F_IDLE is set to “1” during an idle operation and otherwise to “0.” 
   If the result of determination at step  49  is YES, indicating the idle operation, the routine proceeds to step  50 , where it is determined whether or not the vehicle speed VP is lower than a predetermined vehicle speed VSTART (for example, 1 km/h) . If the result of determination at step  50  is YES, indicating that the vehicle is stopped, the routine proceeds to step  51 , where the ECU  2  sets a time value TMVOTVST of a fist launch determination timer of down-count type to a first predetermined time TVOTVST (for example, 3 msec). 
   Next, the routine proceeds to step  52 , where the ECU  2  sets a timer value TMVST of a second launch determination timer of down-count type to a second predetermined time TVST (for example, 500 msec) longer than the first predetermined time TVOTVST. Then, at steps  53 ,  54 , the ECU  2  sets a first and a second launch flag F_VOTVST, F_VST to “0,” followed by termination of the processing. 
   On the other hand, if the determination result at step  49  or  50  is NO, i.e., when the vehicle is not in an idle operation or when the vehicle has been launched, the routine proceeds to step  55 , where it is determined whether or not the timer value TMVOTVST of the first launch determination timer is larger than zero. If the result of determination at step  55  is YES, indicating that the first predetermined time TVOVST has not elapsed after the end of the idle operation or after the vehicle was launched, the routine proceeds to step  56 , where the ECU  2  sets the first launch flag F_VOTVST to “1” for indicating that the vehicle is now in a first launch mode. 
   On the other hand, if the result of determination at step  55  is NO, indicating that the first predetermined time TVOTVST has elapsed after the end of the idle operation or after the vehicle was launched, the routine proceeds to step  57 , where the ECU  2  sets the first launch flag F_VOTVST to “0” for indicating that the first launch mode has been terminated. 
   At step  58  subsequent to step  56  or  57 , it is determined whether or not the timer value TMVST of the second launch determination timer is larger than zero. If the result of determination at step  58  is YES, i.e., when the second predetermined time TVST has not elapsed after the end of the idle operation or after the vehicle was launched, the routine proceeds to step  59 , where the ECU  2  sets the second launch flag F_VST to “1,” indicating that the vehicle is now in a second launch mode, followed by termination of this processing. 
   On the other hand, if the result of determination at step  58  is NO, i.e., when the second predetermined time TVST has elapsed after the end of the idle operation or after the vehicle was launched, the ECU  2  executes the aforementioned step  54 , regarding that the second launch mode has been terminated, followed by termination of this processing. 
   Turning back to  FIG. 14 , at step  22  subsequent to step  21 , the ECU  2  executes processing for setting state variables. Though not shown, in this processing, the ECU  2  shifts all of the target air/fuel ratio KCMD, the output KACT of the LAF sensor  14 , and time series data of the output deviation VO2 stored in the RAM, to the past by one sampling cycle. Then, the ECU  2  calculates current values of KCMD, KACT and VO2 based on the latest values of KCMD, KACT and time series data of VO2 the reference value FLAFBASE, and an adaptive correction term FLFADP, later described. 
   Next, the routine proceeds to step  23 , where it is determined whether or not the PRISM/ADSM processing should be executed. This processing determines whether or not the condition for executing the PRISM processing or ADSM processing is satisfied. Specifically, the processing is executed along a flow chart illustrated in FIG.  17 . 
   More specifically, at steps  60 - 63  in  FIG. 17 , when the following conditions (g)-(j) are fully satisfied, the ECU  2  sets a PRISM/ADSM execution flag F_PRISMCAL to “1”, at step  64 , for indicating that the vehicle is in an operating condition in which the PRISM processing or ADSM processing should be executed, followed by termination of this processing. On the other hand, if any of the conditions (g)-(j) is not satisfied, the ECU  2  sets the PRISM/ADSM execution flag F_PRISMCAL to “0” at step  65 , for indicating that the vehicle is not in an operating condition in which the PRISM processing or ADSM processing should be executed, followed by termination of this processing.
         (g) The O2 sensor  15  is activated;   (h) the LAF sensor  14  is activated;   (i) the engine  3  is not in a lean burn operation; and   (j) the ignition timing is not controlled to be retarded.       

   Turning back to  FIG. 14 , at step  24  subsequent to step  23 , the ECU  2  executes processing for determining whether or not the identifier  23  should executes the operation. ECU  2  determines whether or not conditions are met for the on-board identifier  23  to identify parameters through this processing which is executed specifically along a flow chart illustrated in FIG.  18 . 
   When the results of determinations at step  70  and  71  in  FIG. 18  are both NO, in other words, when the throttle valve opening θTH is not fully opened and the engine  3  is not in a fuel cut operation, the routine proceeds to step  72 , where the ECU  2  sets an identification execution flag F_IDCAL to “1,” determining that the engine  3  is in an operating condition in which the identification of parameters should be executed, followed by termination of the processing. On the other hand, if the result of determination at step  70  or  71  is YES, the routine proceeds to step  73 , where the ECU  2  sets the identification execution flag F_IDCAL to “0,” determining that the engine  3  is not in an operating condition in which the identification of parameters should be executed, followed by termination of the processing. 
   Turning back to  FIG. 14 , at step  25  subsequent to step  24 , the ECU  2  calculates a variety of parameters (exhaust gas volume AB_SV and the like). Specific details of this calculation will be described later. 
   Next, the routine proceeds to step  26 , where it is determined whether or not the PRISM/ADSM execution flag F_PRISMCAL set at step  23  is “1.” If the result of determination at step  26  is YES, i.e., when conditions are met for executing the PRISM processing or ADSM processing, the routine proceeds to step  27 , where it is determined whether or not the identification execution flag F_IDCAL set at step  24  is “1.” 
   If the result of determination at step  27  is YES, i.e., when the engine  3  is in an operating condition in which the on-board identifier  23  should execute the identification of parameters, the routine proceeds to step  28 , where it is determined whether or not a parameter initialization flag F_IDRSET is “1.” If the result of determination at step  28  is NO, i.e., when the initialization is not required for the model parameters a 1 , a 2 , b 1  stored in the RAM, the routine proceeds to step  31 , later described. 
   On the other hand, if the result of determination at step  28  is YES, i.e., when the initialization is required for the model parameters a 1 , a 2 , b 1 , the routine proceeds to step  29 , where the ECU  2  sets the model parameters a 1 , a 2 , b 1  to their respective initial values. Then, the routine proceeds to step  30 , where the ECU  2  sets the parameter initialization flag F_IDRSET to “0” for indicating that the model parameters a 1 , a 2 , b 1  have been set to the initial values. 
   At step  31  subsequent to step  30  or  28 , the on-board identifier  23  executes the operation to identify the model parameters a 1 , a 2 , b 1 , followed by the routine proceeding to step  32  in  FIG. 15 , later described. Specific details on the operation of the on-board identifier  23  will be described later. 
   On the other hand, if the result of determination at step  27  is NO, i.e., when the engine  3  is not in an operating condition in which the identification of the parameters should not be executed, the routine skips the foregoing steps  28 - 31 , and proceeds to step  32  in FIG.  15 . At step  32  subsequent to step  27  or  31 , the ECU  2  selects identified values or predetermined values for the model parameters a 1 , a 2 , b 1 . Though details on this operation are not shown, specifically, the model parameters a 1 , a 2 , b 1  are set to the identified values identified at step  31  when the identification execution flag F_IDCAL set at step  24  is “1.” On the other hand, when the identification execution flag F_IDCAL is “0,” the model parameters a 1 , a 2 , b 1  are set to the predetermined values. 
   Next, the routine proceeds to step  33 , where the state predictor  22  executes the operation to calculate the predicted value PREVO2 as later described. Subsequently, the routine proceeds to step  34 , where the ECU  2  calculates the control amount Usl, as later described. 
   Next, the routine proceeds to step  35 , where the ECU  2  executes processing for determining whether or not the SLD controller  25  is stable. Though details on this processing are not shown, specifically, the ECU  2  determines based on the value of the prediction switching function cPRE to determine whether or not the sliding mode control conducted by the SLD controller  25  is stable. 
   Next, at steps  36  and  37 , the SLD controller  25  and DSM controller  24  calculate the sliding mode control amount DKCMDSLD and ΔΣ modulation control amount DKCMDDSM, respectively, as described later. 
   Next, the routine proceeds to step  38 , where the ECU  2  calculates the adaptive target air/fuel ratio KCMDSLD using the sliding mode control amount DKCMDSLD calculated by the SLD controller  25  or the ΔΣ modulation control amount DKCMDDSM calculated by the DSM controller  24 . Subsequently, the routine proceeds to step  39 , where the ECU  2  calculates an adaptive correction term FLAFADP, as later described, followed by termination of the processing. 
   Turning back again to  FIG. 14 , if the result of determination at step  26  is NO, i.e., when conditions are not met for executing either the PRISM processing or the ADSM processing, the routine proceeds to step  40 , where the ECU  2  sets the parameter initialization flag F_IDRSET to “1.” Next, the routine proceeds to step  41  in  FIG. 15 , where the ECU  2  sets the sliding mode control amount DKCMDSLD to a predetermined value SLDHOLD. Then, after executing the aforementioned steps  38 ,  39 , the processing is terminated. 
   Next, the processing for calculating a variety of parameters at step  25  will be described with reference to  FIG. 19  which illustrates a routine for executing this processing. First, in this processing, the ECU  2  calculates the exhaust gas volume AB_SV (estimated value of a space velocity) in accordance with the following equation (44) at step  80 :
 
 AB   —   SV =( NE/ 1500)· PBA·X   —   SVPRA   (44)
 
where X_SVPRA is a predetermined coefficient which is determined based on the displacement of the engine  3 .
 
   Next, the routine proceeds to step  81 , where the ECU  2  calculates a dead time KACT_D (=d′) in the aforementioned air/fuel ratio manipulation system, a dead time CAT_DELAY (=d) in the exhaust system, and a prediction time dt. Specifically, by searching a table shown in  FIG. 20  in accordance with the exhaust gas volume AB_SV calculated at step  80 , the ECU  2  calculates the dead times KACT_D, CAT_DELAY, respectively, and sets the sum of these dead times (KACT_D+CAT_DELAY) as the prediction time dt. In other words, in this control program, the phase delay time dd is set to zero. 
   In the table shown in  FIG. 20 , the dead times KACT_D, CAT_DELAY are set to smaller values as the exhaust gas volume AB_SV is larger. This is because the dead times KACT_D, CAT_DELAY are shorter as the exhaust gas volume AB_SV is larger since exhaust gases flow faster. As described above, since the dead times KACT_D, CAT_DELAY and prediction time dt are calculated in accordance with the exhaust gas volume AB_SV, it is possible to eliminate a slippage in control timing between the input and output of the controlled object by calculating the adaptive target air/fuel ratio KCMDSLD, later described, based on the predicted value PREVO2 of the output deviation VO2 which has been calculated using them. Also, since the model parameters a 1 , a 2 , b 1  are fixed using the dead time CAT_DELAY, the dynamic characteristic of the controlled object model can be fitted to the actual dynamic characteristic of the controlled object, thereby making it possible to more fully eliminate the slippage in control timing between the input and output of the controlled object. 
   Next, the routine proceeds to step  82 , where the ECU  2  calculates weighting parameters λ 1 , λ 2  of the identification algorithm. Specifically, the ECU  2  sets the weighting parameter λ 2  to one, and simultaneously calculates the weighting parameter λ 1  by searching a table shown in  FIG. 21  in accordance with the exhaust gas volume AB_SV. 
   In the table shown in  FIG. 21 , the weighting parameter λ 1  is set to a smaller value as the exhaust gas volume AB_SV is larger. In other words, the weighting parameter λ 1  is set to a larger value closer to one as the exhaust gas volume AB_SV is smaller. This setting is made for the following reason. Since the model parameters must be more rapidly identified as the exhaust gas volume AB_SV is larger, or in other words, as the engine  3  is more heavily loaded in operation, the model parameters are converged to optimal values faster by setting the weighting parameter λ 1  to a smaller value. In addition, as the exhaust gas volume AB_SV is smaller, i.e., as the engine  3  is more lightly loaded in operation, the air/fuel ratio is more susceptible to fluctuations, causing the post-catalyst exhaust gas characteristic to become instable, so that a high accuracy must be ensured for the identification of the model parameters. Thus, the weighting parameter λ 1  is brought closer to one (to the least square algorithm) to improve the identification accuracy for the model parameters. 
   Next, the routine proceeds to step  83 , where the ECU  2  calculates a lower limit value X_IDA2L for limiting allowable ranges of the model parameters a 1 , a 2 , and a lower limit value X_IDB1L and an upper limit value X_IDB1H for limiting an allowable range of the model parameter b 1  by searching a table shown in  FIG. 22  in accordance with the exhaust gas volume AB_SV. 
   In the table shown in  FIG. 22 , the lower limit value X_IDA2L is set to a larger value as the exhaust gas volume AB_SV is larger. This is because an increase and/or a decrease in the dead times resulting from a change in the exhaust gas volume AB_SV causes a change in a combination of the model parameters a 1 , a 2  which provide a stable state in the control system. Likewise, the lower limit value X_IDB1L and upper limit value X_IDB1H are set to larger values as the exhaust gas volume AB_SV is larger. This is because a pre-catalyst air/fuel ratio (air/fuel ratio of exhaust gases upstream of the first catalyzer  8   a ) affects more the output Vout of the O2 sensor  15 , i.e., the gain of the controlled object becomes larger as the exhaust gas volume AB_SV is larger. 
   Next, the routine proceeds to step  84 , where the ECU  2  calculates the filter order n of the moving average filtering processing, followed by termination of the processing. Specifically, the ECU  2  calculates the filter order n by searching a table shown in  FIG. 23  in accordance with the exhaust gas volume AB_SV. 
   In the table shown in  FIG. 23 , the filter order n is set to a smaller value as the exhaust gas volume AB_SV is larger. This setting is made for the reason set forth below. As described above, a change in the exhaust gas volume AB_SV causes fluctuations in the frequency characteristic, in particular, the gain characteristic of the controlled object, so that the weighted least square algorithm must be appropriately corrected for the frequency weighting characteristic in accordance with the exhaust gas volume AB_SV for fitting the gain characteristic of the controlled object model to the actual gain characteristic of the controlled object. Therefore, by setting the filter order n of the moving average filtering processing in accordance with the exhaust gas volume AB_SV as in the table shown in  FIG. 23 , constant identification weighting can be ensured in the identification algorithm irrespective of a change in the exhaust gas volume AB_SV, and the controlled object model can be matched with the controlled object in the gain characteristic, thereby making it possible to improve the identification accuracy. 
   Next, the operation performed by the on-board identifier  23  at step  31  will be described with reference to  FIG. 24  which illustrates a routine for executing the processing. As illustrated in  FIG. 24 , in this operation, the on-board identifier  23  first calculates the gain coefficient KP(k) in accordance with the aforementioned equation (22) at step  90 . Next, the routine proceeds to step  91 , where the on-board identifier  23  calculates the identified value VO2HAT (k) for the output deviation VO2 in accordance with the aforementioned equation (20). 
   Next, the routine proceeds to step  92 , where the on-board identifier  23  calculates the identification error filter value ide_f(k) in accordance with the aforementioned equations (18), (19). Next, the routine proceeds to step  93 , where the on-board identifier  23  calculates the vector θ(k) for model parameters in accordance with the aforementioned equation (16), followed by the routine proceeding to step  94 , where the on-board identifier  23  executes processing for stabilizing the vector θ(k) for the model parameters. The stabilization processing will be described later. 
   Next, the routine proceeds to step  95 , where the on-board identifier  23  calculates the next value P(k+1) for the square matrix P(k) in accordance with the aforementioned equation (23). This next value P(k+1) is used as the value for the square matrix P(k) in the calculation in the next loop. 
   In the following, the processing for stabilizing the vector θ(k) for the model parameters at step  94  will be described with reference to FIG.  25 . As illustrated in  FIG. 25 , the ECU  2  first sets three flags F_A1STAB, F_A2STAB, F_B1STAB to “0” at step  100 . 
   Next, the routine proceeds to step  101 , where the ECU  2  limits the identified values a 1 ′, a 2 ′, as described later. Next, at step  102 , the ECU  2  limits the identified value b 1 ′, as later described, followed by termination of the processing for stabilizing the vector θ(k) for the model parameters. 
   In the following, the processing involved in limiting the identified values a 1 ′, a 2 ′ at step  101  will be described with reference to  FIG. 26  which illustrates a routine for executing the processing. As illustrated, it is first determined at step  110  whether or not the identified value a 2 ′ for the model parameter calculated at step  93  is equal to or larger than the lower limit value X_IDA2L calculated at step  83  in FIG.  19 . If the result of determination at step  110  is NO, the routine proceeds to step  111 , where the ECU  2  sets the model parameter a 2  to the lower limit value X_IDA2L for stabilizing the control system, and simultaneously sets the flag F_A2STAB to “1” for indicating that the stabilization has been executed for the model parameter a 2 . On the other hand, if the result of determination at step  110  is YES, indicating that a 2 ′≧X_IDA2L, the routine proceeds to step  112 , where the ECU  2  sets the model parameter a 2  to the identified value a 2 ′. 
   At step  113  subsequent to the foregoing step  111  or  112 , it is determined whether or not the identified value a 1 ′ for the model parameter calculated at step  93  is equal to or larger than a predetermined lower limit value X_IDA1L (for example, a constant value equal to or larger than −2 and smaller than 0). If the result of determination at step  113  is NO, the routine proceeds to step  114 , where the ECU  2  sets the model parameter al to the lower limit value X_IDA1L for stabilizing the control system, and simultaneously sets the flag F_A1STAB to “1” for indicating that the stabilization has been executed for the model parameter al. 
   On the other hand, if the result of determination at step  113  is YES, the routine proceeds to step  115 , where it is determined whether or not the identified value all is equal to or lower than a predetermined upper limit value X_IDA1H (for example, 2). If the result of determination at step  115  is YES, indicating that X_IDA1L≦a 1 ′≦X_IDA1H, the routine proceeds to step  116 , where the ECU  2  sets the model parameter a 1  to the identified value a 1 ′. On the other hand, if the result of determination at step  115  is NO, indicating that X_IDA1H&lt;a 1 ′, the routine proceeds to step  117 , where the ECU  2  sets the model parameter a 1  to the upper limit value X_IDA1H, and simultaneously sets the flag F_A1STAB to “1” for indicating that the stabilization has been executed for the model parameter a 1 . 
   At step  118  subsequent to the foregoing steps  114 ,  116  or  117 , it is determined whether or not the sum of the absolute value of the model parameter a 1  calculated in the manner described above and the model parameter a 2  (|a 1 |+a 2 ) is equal to or smaller than a predetermined determination value X_A2STAB (for example, 0.9). If the result of determination at step  118  is YES, the processing for limiting the identified values a 1 ′, a 2 ′ is terminated without further processing, on the assumption that a combination of the model parameters a 1 , a 2  is within a range (a restriction range indicated by hatchings in  FIG. 27 ) in which the stability can be ensured for the control system. 
   On the other hand, if the result of determination at step  118  is NO, the routine proceeds to step  119 , where it is determined whether or not the model parameter a 1  is equal to or smaller than a value calculated by subtracting the lower limit value X_IDA2L from the determination value X_A2STAB (X_A2STAB-X_IDA2L) . If the result of determination at step  119  is YES, the routine proceeds to step  120 , where the ECU  2  sets the model parameter a 2  to a value calculated by subtracting the absolute value of the model parameter a 1  from the determination value X_A2STAB (X_A2STAB-|a 1 |), and simultaneously sets the flag F_A2STAB to “1” for indicating that the stabilization has been executed for the model parameter a 2 , followed by termination of the processing for limiting the identified values a 1 ′, a 2 ′. 
   No the other hand, if the result of determination at step  119  is NO, indicating that a 1 &gt;(X_A2STAB-X_IDA2L), the routine proceeds to step  121 , where the ECU  2  sets the model parameter a 1  to the value calculated by subtracting the lower limit value X_IDA2L from the determination value X_A2STAB (X_A2STAB-X_IDA2L) for stabilizing the control system, and sets the model parameter a 2  to the lower limit value X_IDA2L. Simultaneously with these settings, the ECU  2  sets both flags F_A1STAB, F_A2STAB to “1” for indicating that the stabilization has been executed for the model parameters a 1 , a 2 , followed by termination of the processing for limiting the identified values a 1 ′, a 2 ′. 
   As described above, in the sequential identification algorithm, when the input and output of a controlled object enter a steady state, a control system may become instable or oscillatory because a so-called drift phenomenon is more likely to occur, in which absolute values of identified model parameters increase due to a shortage of self excitation condition. Also, its stability limit varies depending on the operating condition of the engine  3 . For example, during a low load operating condition, the exhaust gas volume AB_SV becomes smaller to cause an increase in a response delay, a dead time and the like of exhaust gases with respect to a supplied air/fuel mixture, resulting in a high susceptibility to an oscillatory output Vout of the O2 sensor  15 . 
   In contrast, the foregoing a 1 ′ and a 2 ′ limit processing sets a combination of model parameters a 1 , a 2  within the restriction range indicated by hatchings in  FIG. 27 , and sets the lower limit value X_IDA2L for determining this restriction range in accordance with the exhaust gas volume AB_SV, so that this restriction range can be set as an appropriate stability limit range which reflects a change in the stability limit associated with a change in the operating condition of the engine  3 , i.e., a change in the dynamic characteristic of the controlled object. With the use of the model parameters a 1 , a 2  which are restricted to fall within such a restriction range, it is possible to avoid the occurrence of the drift phenomenon to ensure the stability of the control system. In addition, by setting the combination of model parameters a 1 , a 2  as values within the restriction range in which the stability can be ensured for the control system, it is possible to avoid an instable state of the control system which would otherwise be seen when the model parameters a 1 , a 2  are restricted independently of each other. With the foregoing strategy, it is possible to improve the stability of the control system and the post-catalyst exhaust gas characteristic. 
   Next, the b 1 ′ limit processing at step  102  will be described with reference to  FIG. 28  which illustrates a routine for executing this processing. As illustrated, it is determined at step  130  whether or not the identified value b 1 ′ for the model parameter calculated at step  93  is equal to or larger than the lower limit value X_IDB1L calculated at step  83  in FIG.  19 . 
   If the result of determination at step  130  is YES, indicating that b 1 ′&gt;X_IDB1L, the routine proceeds to step  131 , where it is determined whether or not the identified value b 1 ′ for the model parameter is equal to or smaller than the upper limit value X_IDB1H calculated at step  83  in FIG.  19 . If the result of determination at step  131  is YES, indicating that X_IDB1L≦b 1 ′≦X_IDB1H, the routine proceeds to step  132 , where the ECU  2  sets the model parameter b 1  to the identified value b 1 ′, followed by termination of the b 1 ′ limit processing. 
   On the other hand, if the result of determination at step  131  is NO, indicating that b 1 ′&gt;X_IDB1H, the routine proceeds to step  133 , where the ECU  2  sets the model parameter b 1  to the upper limit value X_IDB1H, and simultaneously sets a flag F_B1LMT to “1” for indicating this setting, followed by termination of the b 1 ′ limiting processing. 
   On the other hand, if the result of determination at step  130  is NO, indicating that b 1 ′&lt;X_IDB1L, the routine proceeds to step  134 , where the ECU  2  sets the model parameter b 1  to the lower limit value X_IDB1L, and simultaneously sets the F_B1LMT to “1” for indicating this setting, followed by termination of the b 1 ′ limit processing. 
   By executing the foregoing b 1 ′ limit processing, the model parameter b 1  can be restricted within the restriction range from X_IDB1L to X_IDB1H, thereby avoiding the drift phenomenon caused by the sequential identification algorithm. Further, as described above, these upper and lower limit values X_IDB1H, X_IDB1L are set in accordance with the exhaust gas volume AB_SV, so that the restriction range can be set as an appropriate stability limit range which reflects a change in the stability limit associated with a change in the operating condition of the engine  3 , i.e., a change in the dynamic characteristic of the controlled object. With the use of the model parameter b 1  restricted in such a restriction range, the stability can be ensured for the control system. The foregoing strategy can provide an improvement in the stability of the control system and a resulting improvement in the post-catalyst exhaust gas characteristic. 
   Next, the aforementioned operation performed by the state predictor  22  at step  33  will be described with reference to  FIG. 29  which illustrates a routine for executing this processing. First, the state predictor  22  calculates matrix elements α 1 , α 2 , βi, βj in the aforementioned equation (7) at step  140 . Then, the routine proceeds to step  141 , where the state predictor  22  applies the matrix elements α 1 , α 2 , βi, βj calculated at step  140  to the equation (7) to calculate the predicted value PREVO2 of the output deviation VO2 followed by termination of the processing. 
   Next, the aforementioned processing for calculating the control amount Usl at step  34  in  FIG. 15  will be described with reference to  FIG. 30  which illustrates a routine for executing this processing. First, at step  150 , the ECU  2  calculates the prediction switching function σPRE in accordance with the aforementioned equation (38) in FIG.  12 . 
   Then, the routine proceeds to step  151 , where the ECU  2  calculates an integrated value SUMSIGMA of the prediction switching function σPRE. As illustrated in  FIG. 31 , in the calculation of the integrated value SUMSIGMA, it is first determined at step  160  whether or not at least one of the following three conditions (l)-(n) is satisfied:
         (l) the adaptive control flag F_PRISMON is “1”;   (m) an integrated value holding flag F_SS_HOLD, later described, is “0”; and   (n) an ADSM execution end flag F_KOPR, later described, is “0.”       

   If the result of determination at step  160  is YES, i.e., when the condition is satisfied for calculating the integrated value SUMSIGMA, the routine proceeds to step  161 , where the ECU  2  sets a current value SUMSIGMA (k) of the integrated value SUMSIGMA to a value which is calculated by adding the product of a control period ΔT and the prediction switching function σPRE to the preceding value SUMSIGMA(k−1) [SUMSIGMA(k−1)+ΔT·σPRE]. 
   Next, the routine proceeds to step  162 , where it is determined whether or not the current value SUMSIGMA(k) calculated at step  161  is larger than a predetermined lower limit value SUMSL. If the result of determination at step  162  is YES, the routine proceeds to step  163 , where it is determined whether or not the current value SUMSIGMA(k) is smaller than a predetermined upper limit value SUMSH. If the result of determination at step  163  is YES, indicating that SUMSL&lt;SUMSIGMA(k)&lt;SUMSH, the processing for calculating the prediction switching function σPRE is terminated without further processing. 
   On the other hand, if the result of determination at step  163  is NO, indicating that SUMSIGMA(k)≧SUMSH, the routine proceeds to step  164 , where the ECU  2  sets the current value SUMSIGMA(k) to the upper limit value SUMSH, followed by termination of the processing for calculating the prediction switching function σPRE. On the other hand, if the result of determination at step  162  is NO, indicating SUMSIGMA(k)≦SUMSL, the routine proceeds to step  165 , where the ECU  2  sets the current value SUMSIGMA(k) to the lower limit value SUMSL, followed by termination of the processing for calculating the prediction switching function σPRE. 
   On the other hand, if the result of determination at step  160  is NO, i.e., when any of the three conditions (l)-(n) is not satisfied to result in a failed establishment of the condition for calculating the integrated value SUMSIGMA, the routine proceeds to step  166 , where the ECU  2  sets the current value SUMSIGMA(k) to the preceding value SUMSIGMA(k−1). In other words, the integrated value SUMSIGMA is held unchanged. Subsequently, the processing for calculating the prediction switching function σPRE is terminated. 
   Turning back to  FIG. 30 , at steps  152 - 154  subsequent to step  151 , the ECU  2  calculates the equivalent control input Ueq, reaching law input Urch, and adaptive law input Uadp in accordance with the aforementioned equations (40)-(42), respectively, in FIG.  12 . 
   Next, the routine proceeds to step  155 , where the ECU  2  sets the sum of these equivalent control input Ueq, reaching law input Urch, and adaptive law input Uadp as the control amount Usl, followed by termination of processing for calculating the control amount Usl. 
   Next, the aforementioned processing for calculating the sliding mode control amount DKCMDSLD at step  36  in  FIG. 15  will be described in detail with reference to  FIGS. 32 ,  33  which illustrate routines for executing this processing. First, at step  170 , the ECU  2  executes processing for calculating a limit value for the control amount Usl. In this processing, though detailed description is omitted, the ECU  2  calculates upper and lower limit values Usl_ahf, Usl_alf for non-idle operation, as well as upper and lower limit values Usl_ahfi, Usl_alfi for idle operation, respectively, based on the result of determination for determining the stability of the controller at step  35 , and adaptive upper and lower limit values Usl_ah, Usl_al, later described, for the control amount Usl. 
   Next, the routine proceeds to step  171 , where it is determined whether or not an idle operation flag F_IDLE is “0.” If the result of determination at step  171  is YES, indicating that the engine  3  is not in an idle operation, the routine proceeds to step  172 , where it is determined whether or not the control amount Usl calculated in the aforementioned processing of  FIG. 30  is equal to or smaller than the lower limit value Usl_alf for non-idle operation. 
   If the result of determination at step  172  is NO, indicating that Usl&gt;Usl_alf, the routine proceeds to step  173 , where it is determined whether or not the control amount Usl is equal to or larger than the upper limit value Usl_ahf for non-idle operation. If the result of determination at step  173  is NO, indicating that Usl_alf&lt;Usl&lt;Usl_ahf, the routine proceeds to step  174 , where the ECU  2  sets the sliding mode control amount DKCMDSLD to the control amount Usl, and simultaneously sets the integrated value holding flag F_SS_HOLD to “0.” 
   Next, the routine proceeds to step  175 , where the ECU  2  sets the current value Usl_al(k) of the adaptive lower limit value to a value [Usl_al(k−1)+X_AL_DEC] which is calculated by adding a predetermined decrement value X_AL_DEC to the preceding value Usl_al(k−1), and simultaneously sets the current value Usl_ah(k) of the adaptive upper limit value to a value which is calculated by subtracting the predetermined decrement value X_AL_DEC from the preceding value Usl_ah(k−1) [Usl_al(k−1)−X_AL_DEC], followed by termination of the processing for calculating the sliding mode control amount DKCMDSLD. 
   On the other hand, if the result of determination at step  173  is YES, indicating that Usl≧Usl_ahf, the routine proceeds to step  176 , where the ECU  2  sets the sliding mode control amount DKCMDSLD to the adaptive upper limit value Usl_ahf for non-idle operation, and simultaneously sets the integrated value holding flag F_SS_HOLD to “1.” 
   Next, the routine proceeds to step  177 , where it is determined whether or not a post-start timer presents a timer value TMACR smaller than a predetermined time X_TMAWAST, or whether or not an post-F/C determination flag F_AFC is “1.” This post-start timer is an up-count type timer for measuring a time elapsed after the start of the engine  3 . 
   If the result of determination at step  177  is YES, i.e., when a predetermined time X_TMAWAST has not elapsed after the start of the engine  3 , or when a predetermined time X_TM_AFC has not elapsed after a fuel cut operation is terminated, the processing for calculating the sliding mode control amount DKCMDSLD is terminated without further processing. 
   On the other hand, if the result of determination at step  177  is NO, i.e., when the predetermined time X_TMAWAST has elapsed after the start of the engine  3 , and when the predetermined time X_TM_AFC has elapsed after a fuel cut operation, the routine proceeds to step  178 , where the ECU  2  sets the current value Usl_al(k) of the adaptive lower limit value to a value which is calculated by adding the decrement value X_AL_DEC to the preceding value Usl_al(k−1) [Usl_al(k−1)+X_AL_DEC], and simultaneously sets the current value Usl_ah(k) of the adaptive upper limit value to a value which is calculated by adding a predetermined increment value X_AL_INC to the preceding value Usl_ah(k−1) [Usl_ah(k−1)+X_AL_INC], followed by termination of the processing for calculating the sliding mode control amount DKCMDSLD. 
   On the other hand, if the result of determination at step  172  is YES, indicating that Usl≦Usl_alf, the routine proceeds to step  179 , where the ECU  2  sets the sliding mode control amount DKCMDSLD to the adaptive lower limit value Usl_alf for non-idle operation, and simultaneously sets the integrated value holding flag F_SS_HOLD to “1.” 
   Next, the routine proceeds to step  180 , where it is determined whether or not a second launch flag F_VST is “1.” If the result of determination at step  180  is YES, i.e., when a second predetermined time TVST has not elapsed after the launch of the vehicle so that the vehicle is still in a second launch mode, the processing for calculating the sliding mode control amount DKCMDSLD is terminated without further processing. 
   On the other hand, if the result of determination at step  180  is NO, i.e., when the second predetermined time TVST has elapsed after the launch of the vehicle so that the second launch mode has been terminated, the routine proceeds to step  181 , where the ECU  2  sets the current value Usl_al(k) of the adaptive lower limit value to a value which is calculated by subtracting the increment value X_AL_INC from the preceding value Usl_al(k−1) [Usl_al(k−1)−X_AL_INC], and simultaneously sets the current value Usl-ah(k) of the adaptive upper limit value to a value which is calculated by subtracting the decrement value X_AL_DEC from the preceding value Usl_ah(k−1) [Usl_ah(k−1)−X_AL_DEC], followed by termination of the processing for calculating the sliding mode control amount DKCMDSLD. 
   On the other hand, if the result of determination at step  171  is NO, indicating that the engine  3  is in an idle operation, the routine proceeds to step  182  in  FIG. 33 , where it is determined whether or not the control amount Usl is equal to or smaller than the lower limit value Usl_alfi for idle operation. If the result of determination at step  182  is NO, indicating that Usl&gt;Usl_alfi, the routine proceeds to step  183 , where it is determined whether or not the control amount Us1 is equal to or larger than the upper limit value Usl_ahfi for idle operation. 
   If the result of determination at step  183  is NO, indicating that Usl_alfi&lt;Usl&lt;Usl_ahfi, the routine proceeds to step  184 , where the ECU  2  sets the sliding mode control amount DKCMDSLD to the control amount Usl, and simultaneously sets the integrated value holding flag F_SS_HOLD to “0,” followed by termination of the processing for calculating the sliding mode control amount DKCMDSLD. On the other hand, if the result of determination at step  183  is YES, indicating that Usl≧Usl_ahfi, the routine proceeds to step  185 , where the ECU  2  sets the sliding mode control amount DKCMDSLD to the upper limit value Usl-ahfi for idle operation, and simultaneously sets the integrated value holding flag F_SS_HOLD to “1,” followed by termination of the processing for calculating the sliding mode control amount DKCMDSLD. 
   On the other hand, if the result of determination at step  182  is YES, indicating that Usl≦Usl_alfi, the routine proceeds to step  186 , where the ECU  2  sets the sliding mode control amount DKSMDSLD to the lower limit value Usl_alfi for idle operation, and simultaneously sets the integrated value holding flag F_SS_HOLD to “1,” followed by termination of the processing for calculating the sliding mode control amount DKCMDSLD. 
   Next, the processing for calculating the ΔΣ modulation control amount DKCMDDSM at step  37  in  FIG. 15  will be described with reference to  FIG. 34  which illustrates a routine for executing this processing. As illustrated, at step  190 , the ECU  2  first sets a current value DSMSGNS(k) [=u″(k)] of a DSM signal value calculated in the preceding loop, which is stored in the RAM, as the preceding value DSMSGNS(k−1) [=u″(k−1)]. 
   Next, the routine proceeds to step  191 , where the ECU  2  sets a current value DSMSIGMA(k) [=σ d (k)] of a deviation integrated value calculated in the preceding loop and stored in the RAM as the preceding value DSMSIGMA(k−1) [=σ d (k−1)]. 
   Next, the routine proceeds to step  192 , where it is determined whether or not the predicted value PREVO2 (k) of the output deviation is equal to or larger than zero. If the result of determination at step  192  is YES, the routine proceeds to step  193 , where a gain KRDSM (=Gd) for reference signal value is set to a leaning coefficient KRDSML, on the assumption that the engine  3  is in an operating condition in which the air/fuel ratio of the air-fuel mixture should be changed to be leaner. Then, the routine proceeds to step  195 , later described. 
   On the other hand, if the result of determination at step  192  is NO, the routine proceeds to step  194 , where the gain KRDSM for reference signal value is set to an enriching coefficient KRDSMR, larger than the leaning coefficient KRDSML, on the assumption that the engine  3  is in an operating condition in which the air/fuel ratio of the air-fuel mixture should be changed to be richer. Then, the routine proceeds to step  195 . 
   The leaning coefficient KRDSML and the enriching coefficient KRDSMR are set to values different from each other, as described above, for the reason set forth below. For changing the air/fuel ratio of the air/fuel mixture to be leaner, the leaning coefficient KRDSML is set to a value smaller than the enriching coefficient KRDSMR for effectively suppressing the amount of exhausted NOx by lean biasing to ensure an NOx purification percentage of the first catalyzer  8   a . Thus, the air/fuel ratio is controlled such that the output Vout of the O2 sensor  15  converges to the target value Vop slower than when the air/fuel ratio is changed to be richer. On the other hand, for changing the air/fuel ratio of the air/fuel mixture to be richer, the enriching coefficient KRDSMR is set to a value larger than the leaning coefficient KRDSML for sufficiently recovering the NOx purification percentage of the first and second catalyzers  8   a ,  8   b . Thus, the air/fuel ratio is controlled such that the output Vout of the O2 sensor  15  converges to the target value Vop faster than when the air/fuel ratio is changed to be leaner. In the foregoing manner, a satisfactory post-catalyst exhaust gas characteristic can be ensured whenever the air/fuel ratio of the air/fuel mixture is changed to be either leaner or richer. 
   At step  195  subsequent to step  193  or  194 , the ECU  2  sets a value calculated by subtracting the preceding value DSMSGNS(k−1) of the DSM signal value calculated at the aforementioned step  190  from the product of a value of −1, the gain KRDSM for reference signal value, and the current value PREVO2(k) of the predicted value [−1·KRDSM·PREVO2(k)−DSMSGNS(k−1)] as a deviation signal value DSMDELTA [=δ(k)]. This setting corresponds to the aforementioned equations (27), (28). 
   Next, the routine proceeds to step  196 , where the ECU  2  sets the current value DSMSIGMA(k) of the deviation integrated value to the sum of the preceding value DSMSIGMA(k−1) calculated at step  191  and the deviation signal value DSMDELTA calculated at step  195  [DSMSIGMA(k−1)+DSMDELTA]. This setting corresponds to the aforementioned equation (29). 
   Next, in a sequence of steps  197 - 199 , the ECU  2  sets the current value DSMSGNS (k) of the DSM signal value to 1 when the current value DSMSIGMA(k) of the deviation integrated value calculated at step  196  is equal to or larger than 0, and sets the current value DSMSGNS(k) of the DSM signal value to −1 when the current value DSMSIGMA(k) of the deviation integrated value is smaller than 0. The setting in this sequence of steps  197 - 199  corresponds to the aforementioned equation (30). 
   Next, the ECU  2  calculates a gain KDSM (=F d ) for the DSM signal value at step  200  by searching a table shown in  FIG. 35  in accordance with the exhaust gas volume AB_SV. As shown in  FIG. 35 , the gain KDSM is set to a larger value as the exhaust gas volume AB_SV is smaller. This is because the responsibility of the output Vout of the O2 sensor  15  is degraded as the exhaust gas volume AB_SV is smaller, i.e., as the engine  3  is operating with a smaller load, so that the gain KDSM is set larger to compensate for the degraded responsibility of the output Vout. By thus setting the gain KDSM, the ΔΣ modulation control amount DKCMDDSM can be appropriately calculated in accordance with an operating condition of the engine  3 , while avoiding, for example, an over-gain state, thereby making it possible to improve the post-catalyst exhaust gas characteristic. 
   The table for use in the calculation of the gain KDSM is not limited to the table of  FIG. 35  which sets the gain KDSM in accordance with the exhaust gas volume AB_SV, but any table may be used instead as long as it previously sets the gain KDSM in accordance with a parameter indicative of an operating load of the engine  3  (for example, a basic fuel injection time Tim). Also, when a deterioration determining unit is provided for the catalyzers  8   a ,  8   b , the gain KDSM may be corrected to a smaller value as the catalyzers  8   a ,  8   b  are deteriorated to a higher degree, as determined by the deterioration determining unit. 
   Next, the routine proceeds to step  201 , where the ECU  2  sets the ΔΣ modulation control amount DKCMDDSM to the product of the gain KDSM for DSM signal value and the current value DSMSGNS(k) of the DSM signal value [KDSM·DSMSGNS(k)], followed by termination of the processing for calculating the ΔΣ modulation control amount DKCMDDSM. The setting at step  201  corresponds to the aforementioned equation (31). 
   Next, the aforementioned processing for calculating the adaptive target air/fuel ratio KCMDSLD at step  38  in  FIG. 15  will be described with reference to  FIG. 36  which illustrates a routine for executing this processing. As illustrated, it is first determined at step  210  whether or not the idle operation flag F_IDLE is “1” and whether or not an idle time ADSM execution flag F_SWOPRI is “1.” The idle time ADSM execution flag F_SWOPRI is set to “1” when the engine  3  is idling in an operating condition in which the ADSM processing should be executed, and otherwise to “0.” 
   If the result of determination at step  210  is YES, i.e., when the engine  3  is idling in an operating condition in which the adaptive target air/fuel ratio KCMDSLD should be calculated by the ADSM processing, the routine proceeds to step  211 , where the ECU  2  sets the adaptive target air/fuel ratio KCMDSLD to the sum of the reference value FLAFBASE and the ΔΣ modulation control amount DKCMDDSM [FLAFBASE+DKCMDDSM]. This setting corresponds to the aforementioned equation (32). 
   Next, the routine proceeds to step  212 , where the ECU  2  sets an ADSM execution end flag F_KOPR to “1” for indicating that the ADSM processing has been executed, followed by termination of the processing for calculating the adaptive target air/fuel ratio KCMDSLD. 
   On the other hand, if the result of determination at step  210  is NO, the routine proceeds to step  213 , where it is determined whether or not a catalyst/O2 sensor flag F_FCATDSM is “1.” This catalyst/O2 sensor flag F_FCATDSM is set to “1” when at least one of the four following conditions (o)-(r) is satisfied, and otherwise to “0”:
         (o) the first catalyzer  8   a  has a catalyst capacity equal to or higher than a predetermined value;   (p) the first catalyzer  8   a  has a noble metal content equal to or larger than a predetermined value;   (q) the LAF sensor  14  is not provided in the exhaust pipe  7  of the engine  3 ; and   (r) the O2 sensor  15  is provided downstream of the second catalyzer  8   b.          

   If the result of determination at step  213  is YES, the routine proceeds to step  214 , where it is determined whether or not a first launch flag F_VOTVST and a post-launch ADSM execution flag F_SWOPRVST are both “1.” The post-launch ADSM execution flag F_SWOPRVST is set to “1” when the engine  3  is in an operating condition in which the ADSM processing should be executed after the vehicle has been launched, and otherwise to “0.” 
   If the result of the determination at step  214  is YES, i.e., when a first predetermined time TVOTVST has elapsed after the vehicle was launched and when the engine  3  is in an operating condition in which the ADSM processing should be executed, the ECU  2  executes steps  211 ,  212 , in the manner described above, followed by termination of the processing for calculating the adaptive target air/fuel ratio KCMDSLD. 
   On the other hand, if the result of determination at step  214  is NO, the routine proceeds to step  215 , where it is determined whether or not the following conditions are both satisfied: the exhaust gas volume AB_SV is equal to or smaller than a predetermined value OPRSVH, and a small-exhaust-period ADSM execution flag F_SWOPRSV is “1.” The small-exhaust-period ADSM execution flag F_SWOPRSV is set to “1” when the engine  3  has a small exhaust gas volume AB_SV and when the engine  3  is in an operating condition in which the ADSM processing should be executed, and otherwise to “0.” 
   If the result of determination at step  215  is YES, i.e., when the exhaust gas volume AB_SV is small and when the engine  3  is in an operating condition in which the ADSM processing should be executed, the ECU  2  executes steps  211 ,  212  in the manner described above, followed by termination of the processing for calculating the adaptive target air/fuel ratio KCMDSLD. 
   On the other hand, if the result of determination at step  215  is NO, the routine proceeds to step  216 , on the assumption that the engine  3  is in an operating condition in which the PRISM processing should be executed, where the ECU  2  sets the adaptive target air/fuel ratio KCMDSLD to the sum of the reference value FLAFBASE, the adaptive correction term FLAFADP, and the sliding mode control amount DKCMDSLD [FLAFBASE+FLAFADP+DKCMDSLD]. Next, the routine proceeds to step  217 , where the ECU  2  sets the ADSM execution end flag F_KOPR to “0” for indicating that the PRISM processing has been executed, followed by termination of the processing for calculating the adaptive target air/fuel ratio KCMDSLD. 
   On the other hand, if the result of determination at step  213  is NO, i.e., when any of the four conditions (o)-(r) is not satisfied, the ECU  2  skips steps  214 ,  215 , and executes the aforementioned steps  216 ,  217 , followed by termination of the processing for calculating the adaptive target air/fuel ratio KCMDSLD. In the foregoing manner, in the processing for calculating the adaptive target air/fuel ratio KCMDSLD, the ECU  2  calculates the adaptive target air/fuel ratio KCMDSLD for the ADSM processing or PRISM processing, switched in accordance with an operating condition of the engine  3 . 
   Next, the processing for calculating the adaptive correction term FLAFADP at step  39  in  FIG. 15  will be described with reference to  FIG. 37  which illustrates a routine for executing this processing. As illustrated in  FIG. 37 , it is first determined at step  220  whether or not the output deviation VO2 is within a predetermined range (ADL&lt;VO2&lt;ADH). If the result of determination at step  220  is YES, i.e., when the output deviation VO2 is small so that the output Vout of the O2 sensor  15  is near the target value Vop, the routine proceeds to step  221 , where it is determined whether or not the adaptive law input Uadp is smaller than a predetermined lower limit value NRL. 
   If the result of determination at step  221  is NO, indicating that Uadp≧NRL, the routine proceeds to step  222 , where it is determined whether or not the adaptive law input Uadp is larger than a predetermined upper limit value NRH. If the result of determination at step  222  is NO, indicating that NRL≦Uadp≦NRH, the routine proceeds to step  223 , where the ECU  2  sets the current value FLAFADP(k) of the adaptive correction term to the preceding value FLAFADP(k−1). In other words, the current value of the adaptive correction term FLAFADP is held. Then, the processing for calculating the adaptive correction term FLAFADP is terminated. 
   On the other hand, if the result of determination at step  222  is YES, indicating that Uadp&gt;NRH, the routine proceeds to step  224 , where the ECU  2  sets the current value FLAFADP(k) of the adaptive correction term to the sum of the preceding value FLAFADP(k−1) and a predetermined update value X_FLAFDLT [FLAFADP(k−1)+X_FLAFDLT], followed by termination of the processing for calculating the adaptive correction term FLAFADP. 
   On the other hand, if the result of determination at step  221  is YES, indicating that Uadp&lt;NRL, the routine proceeds to step  225 , where the ECU  2  sets the current value FLAFADP(k) of the adaptive correction term to a value calculated by subtracting the predetermined update value X_FLAFDLT from the preceding value FLAFADP(k−1) [FLAFADP(k−1)−X_FLAFDLT], followed by termination of the processing for calculating the adaptive correction term FLAFADP. 
   As described above, the control apparatus  1  according to the first embodiment can appropriately eliminate a slippage in control timing between the input and output of a controlled object which has the target air/fuel ratio KCMD as a control input and the output Vout of the O2 sensor  15  as the output, and exhibits the dynamic characteristic with relatively large phase delay, dead time and the like, thereby making it possible to improve the stability and controllability of the control and accordingly improve the post-catalyst exhaust gas characteristic. 
   In the following, control apparatuses according to a second through an eighth embodiment of the present invention will be described with reference to  FIGS. 38-46 . In the following description on the respective embodiments, components identical or equivalent to those in the first embodiment are designated the same reference numerals, and description thereon will be omitted as appropriate. 
   First, a control apparatus according to a second embodiment will be described with reference to FIG.  38 . The control apparatus  201  in the second embodiment differs from the control apparatus  1  in the first embodiment only in the on-board identifier  23 . Specifically, the on-board identifier  23  in the first embodiment calculates the model parameters a 1 , a 2 , b 1  based on KACT, Vout, and φop (KCMD), whereas the on-board identifier  23  in the second embodiment calculates the model parameters a 1 , a 2 , b 1  based on Vout and φop. 
   More specifically, the on-board identifier  23  calculates identified values a 1 ′, a 2 ′, b 1 ′ for the model parameters in accordance with the identification algorithm expressed by the equations (8)-(15) in  FIG. 5  in place of the identification algorithm expressed by the equations (16)-(23) in  FIG. 6  used in the first embodiment, and limits the identified values a 1 ′, a 2 ′, b 1 ′, as illustrated in  FIGS. 26 ,  28 , to calculate the model parameters a 1 , a 2 , b 1 . Though no specific program is shown for the processing performed by the on-board identifier  23 , such a program may be organized substantially similar to that used in the first embodiment. The control apparatus  201  according to the second embodiment can provide similar advantages to the control apparatus  1  according to the first embodiment. 
   Next, a control apparatus according to a third embodiment will be described with reference to FIG.  39 . As illustrated, the control apparatus  301  in the third embodiment differs from the control apparatus  1  in the first embodiment only in the state predictor  22 . Specifically, the state predictor  22  in the first embodiment calculates the predicted value PREVO2 based on a 1 , a 2 , b 1 , KACT, Vout, and φop(KCMD), whereas the state predictor  22  in the third embodiment calculates the predicted value PREVO2 based on a 1 , a 2 , b 1 , Vout, and φop. 
   More specifically, the state predictor  22  in the third embodiment calculates the predicted value PREVO2 of the output deviation VO2 in accordance with the prediction algorithm expressed by the equation (6) in  FIG. 4 , in place of the prediction algorithm expressed by the equation (7) in  FIG. 4  used in the first embodiment. Though no specific program is shown for the processing performed by the state predictor  22 , such a program may be organized substantially similar to that used in the first embodiment. The control apparatus  301  according to the third embodiment can provide similar advantages to the control apparatus  1  according to the first embodiment. 
   Next, a control apparatus according to a fourth embodiment will be described with reference to FIG.  40 . As illustrated, the control apparatus  401  according to the fourth embodiment differs from the control apparatus  1  according to the first embodiment only in that a schedule type DSM controller  20 A, a schedule type state prediction sliding mode controller  21 A, and a parameter scheduler  28  (model parameter setting means) are used to calculate the model parameters a 1 , a 2 , b 1  in place of the ADSM controller  20 , PRISM controller  21 , and on-board identifier  23 . 
   The parameter scheduler  28  first calculates the exhaust gas volume AB_SV in accordance with the aforementioned equation (44) based on the engine rotational speed NE and intake pipe inner absolute pressure PBA. Next, the parameter scheduler  28  calculates the model parameters a 1 , a 2 , b 1  in accordance with the exhaust gas volume AB_SV using a table shown in FIG.  41 . 
   In the table sown in  FIG. 41 , the model parameter a 1  is set to a smaller value as the exhaust gas volume AB_SV is larger. Contrary to the model parameter a 1 , the model parameters a 2 , b 1  are set to larger values as the exhaust gas volume AB_SV is larger. This is because the output of the controlled object, i.e., the output Vout of the O2 sensor  15  becomes more stable as the exhaust gas volume AB_SV is increased, whereas the output Vout of the O2 sensor becomes oscillatory as the exhaust gas volume AB_SV is decreased. 
   The schedule type DSM controller  20 A calculates the target air/fuel ratio KCMD in a DSM controller  24  similar to that in the first embodiment, using the model parameters a 1 , a 2 , b 1  calculated as described above. Likewise, the schedule type state prediction sliding mode controller  21 A calculates the target air/fuel ratio KCMD in an SLD controller  25  similar to that in the first embodiment, using the model parameters a 1 , a 2 , b 1  calculated as described above. 
   The control apparatus  401  according to the fourth embodiment can provide similar advantages to the control apparatus  1  according to the first embodiment. In addition, the model parameters a 1 , a 2 , b 1  can be more rapidly calculated using the parameter scheduler  28  than using the on-board identifier  23 . It is therefore possible to improve the responsibility of the control and more rapidly ensure a favorable post-catalyst exhaust gas characteristic. 
   Next, a control apparatus according to a fifth embodiment will be described with reference to FIG.  42 . The control apparatus  501  according to the fifth embodiment differs from the control apparatus  1  according to the first embodiment only in that an SDM controller  29  is used in place of the DSM controller  24  of the control apparatus  1  in the first embodiment. The SDM controller  29  calculates the control input φop(k) in accordance with a control algorithm which applies the ΣΔ modulation algorithm based on the predicted value PREVO2(k). 
   Specifically, in the SDM controller  29  illustrated in  FIG. 42 , an inverting amplifier  29   a  generates a reference signal r(k) as the product of the value of −1, gain G d  for reference signal, and predicted value PREVO2(k). Next, an integrator  29   b  generates a reference signal integrated value σ d r(k) as the sum of a reference signal integrated value σ d r(k−1) delayed by a delay element  29   c  and the reference signal r(k). On the other hand, an integrator  29   d  generates an SDM signal integrated value σ d u(k) as the sum of an SDM signal integrated value σ d u(k−1) delayed by a delay element  29   e , and an SDM signal u″(k−1) delayed by a delay element  29   j . Then, a subtractor  29   f  generates a deviation signal δ″(k) of the SDM signal integrated value σ d u(k) from the reference signal integrated value σ d r(k). 
   Next, a quantizer  29   g  (sign function) generates an SDM signal u″(k) as the sign of the deviation signal δ″(k). Then, an amplifier  29   h  generates an amplified SDM signal u(k) by amplifying the SDM signal u″(k) by a predetermined gain F d . Then, an adder  29   i  generates the control input φop(k) as the sum of the amplified SDM signal u(k) and a predetermined reference value FLAFBASE. 
   The foregoing control algorithm of the SDM controller  29  is expressed by the following equations (45)-(51):
 
 r ( k )=−1 ·Gd·PREVO 2( k )  (45)
 
σ d   r ( k )=σ d   r ( k− 1)+ r ( k )  (46)
 
σ d   u ( k )=σ d   u ( k− 1)+ u ″( k− 1)  (47)
 
δ″( k )=σ d   r ( k )−σ d   u ( k )  (48)
 
  u ″( k )= sgn (δ″( k ))  (49)
 
 u ( k )= F   d   ·u ″( k )  (50)
 
Φ op ( k )= FLAFBASE+u ( k )  (51)
 
where G d  and F d  represent gains. The sign function sgn(δ″(k)) takes the value of 1 (sgn(δ″(k) )=1) when δ″(k)≧0, and −1 (sgn(δ″(k))=−1) when δ″(k)&lt;0 (alternatively, sgn(δ″(k)) may be set to 0 (sgn(δ″(k)=0) when δ″(k)=0.
 
   The ΣΔ modulation algorithm in the control algorithm of the SDM controller  29  is characterized in that the SDM signal u(k) can be generated (calculated) such that the reference signal r(k) is reproduced at the output of the controlled object when the SDM signal u(k) is inputted to the control object, as is the case with the aforementioned ΔΣ modulation algorithm. In other words, the SDM controller  29  has the characteristic of generating the control input φop(k) similar to the aforementioned DSM controller  24 . Therefore, the control apparatus  501  according to the fifth embodiment, which utilizes the SDM controller  29 , can provide similar advantages to the control apparatus  1  according to the first embodiment. Though no specific program is shown for the SDM controller  29 , such a program may be organized substantially similar to the DSM controller  24 . 
   Next, a control apparatus according to a sixth embodiment will be described with reference to FIG.  43 . The control apparatus  601  according to the sixth embodiment differs from the control apparatus  1  according to the first embodiment only in that a DM controller  30  is used in place of the DSM controller  24 . The DM controller  30  calculates the control input φop(k) in accordance with a control algorithm which applies a Δ modulation algorithm based on the predicted value PREVO2(k). 
   Specifically, as illustrated in  FIG. 43 , in the DM controller  30 , an inverting amplifier  30   a  generates the reference signal r(k) as the product of the value of −1, gain Gd for reference signal, and predicted value PREVO2 (k). An integrator  30   b  generates a DM signal integrated value δ d u(k) as the sum of a DM signal integrated value δ d u(k−1) delayed by a delay element  30  and a DM signal u″(k−1) delayed by a delay element  30   h . Then, a subtractor  30   d  generates a deviation signal δ″(k) of the DM signal integrated value δ d u(k) from the reference signal r(k). 
   Next, a quantizer  30   e  (sign function) generates a DM signal u″(k) as a sign of the deviation signal δ″(k). Then, an amplifier  30   f  generates an amplified DM signal u(k) by amplifying the DM signal u″(k) by a predetermined gain F d . Next, an adder  30   g  generates the control input φop(k) as the sum of the amplified DM signal u(k) and the predetermined reference value FLAFBASE. 
   The foregoing control algorithm of the DM controller  30  is expressed by the following equations (52)-(57):
 
 r ( k )=−1 ·Gd·PREVO 2( k )  (52)
 
σ d   u ( k )=σ d   u (i k−1)+ u ″( k− 1)  (53)
 
δ″( k )= r ( k )−σ d   u ( k )  (54)
 
 u ″( k )= sgn (δ″( k ))  (55)
 
 u ( k )= F   d   ·u ″( k )  (56)
 
Φ op ( k )= FLAFBASE+u ( k )  (57)
 
where G d  and F d  represents gains. The sign function sgn(δ″(k)) takes the value of 1 (sgn(δ″(k) )=1) when δ″(k)≧0, and −1 (sgn(δ″(k) )=−1) when δ″(k)&lt;0 (alternatively, sgn(δ″(k) may be set to 0 (sgn(δ″(k)=0) when δ″(k)=0.
 
   The control algorithm of the DM controller  30 , i.e., the Δ modulation algorithm is characterized in that the DM signal u(k) can be generated (calculated) such that the reference signal r(k) is reproduced at the output of the controlled object when the DM signal u(k) is inputted to the controlled object, as is the case with the aforementioned ΔΣ modulation algorithm and ΣΔ modulation algorithm. In other words, the DM controller  30  has the characteristic of generating the control input φop(k) similar to the aforementioned DSM controller  24  and SDM controller  29 . Therefore, the control apparatus  601  according to the sixth embodiment, which utilizes the DM controller  30 , can provide similar advantages to the control apparatus  1  according to the first embodiment. Though no specific program is shown for the DM controller  30 , such a program may be organized substantially similar to the DSM controller  24 . 
   Next, a control apparatus according to a seventh embodiment will be described with reference to  FIGS. 44 and 45 . As illustrated in  FIG. 44 , the control apparatus  701  according to the seventh embodiment differs from the control apparatus  1  according to the first embodiment only in that the engine  3  is not provided with the LAF sensor  14 , and the O2 sensor  15  is disposed downstream of the second catalyzer  8   b.    
   Since the LAF sensor  14  is not provided, the control apparatus  701  relies on the on-board identifier  23  to calculate the model parameters a 1 , a 2 , b 1  based on the output Vout of the O2 sensor  15 , and the control input φop(k) (target air/fuel ratio KCMD), as illustrated in FIG.  45 . In other words, the on-board identifier  23  calculates the identified values a 1 ′, a 2 ′, b 1 ′ for the model parameters in accordance with the identification algorithm expressed by the equation (8)-(15) in  FIG. 5 , and limits these identified values in the manner described above to calculate the model parameters a 1 , a 2 , b 1 . 
   Further, the state predictor  22  calculates the predicted value PREVO2 of the output deviation VO2 based the model parameters a 1 , a 2 , b 1 , output Vout of the O2 sensor  15 , and control input φop. In other words, the state predictor  22  calculates the predicted value PREVO2 of the output deviation VO2 in accordance with the prediction algorithm expressed by the equation (6) in FIG.  4 . Though no specific programs are shown for the processing performed by the state predictor  22  and on-board identifier  23 , such programs may be organized substantially similar to those in the first embodiment. Other programs may also be organized in a similar manner to those in the first embodiment. 
   The control apparatus  701  according to the seventh embodiment as described above can provide similar advantages to the control apparatus  1  according to the first embodiment. Particularly, when the air/fuel ratio is controlled only by the O2 sensor  15 , as in the seventh embodiment, by setting the gain KRDSM for reference signal value to different values at steps  192 - 194  in  FIG. 34  for controlling exhaust gases to be leaner and richer to converge the target air/fuel ratio KCMD to the target value Vop at different rates, the control apparatus  701  can provide a satisfactory post-catalyst exhaust gas characteristic without fail for changing the air/fuel ratio of the air/fuel mixture to be richer and leaner. In addition, since the suitable post-catalyst exhaust gas characteristic can be ensured without using the LAF sensor  14 , the manufacturing cost can be saved correspondingly. 
   Next, a control apparatus according to an eighth embodiment will be described with reference to FIG.  46 . As illustrated, the control apparatus  801  according to the eighth embodiment differs from the control apparatus  701  according to the seventh embodiment in that the ADSM controller  20 , PRISM controller  21 , and on-board identifier  23  in the seventh embodiment are replaced with the schedule type DSM controller  20 A, schedule type state prediction sliding mode controller  21 A, and parameter scheduler  28  in the fourth embodiment. These controllers  20 A,  21 A and parameter scheduler  28  are configured in a manner similar to those in the fourth embodiment. The control apparatus  801  according to the eighth embodiment can provide similar advantages to the control apparatus  701  according to the seventh embodiment. In addition, the model parameters a 1 , a 2 , b 1  can be calculated faster when the parameter scheduler  28  is used than when the on-board identifier  23  is used. This can improve the responsibility of the control and more rapidly ensure a satisfactory post-catalyst exhaust gas characteristic. 
   The foregoing embodiments have illustrated exemplary configurations of the control apparatus according to the present invention for controlling the air/fuel ratio of the internal combustion engine  3 . It should be understood, however, that the present invention is not limited to the foregoing embodiments, but can be widely applied to control apparatuses for controlling other arbitrary controlled objects. In addition, the ADSM controller  20  and PRISM controller  21  may be implemented in hardware in place of the programs as illustrated in the embodiments. 
   As described above, the control apparatus according to the present invention can eliminate a slippage in control timing between the input and output of a controlled object, even when the controlled object exhibits the dynamic characteristic with relatively large phase delay, dead time, and the like, thereby improving the stability and controllability of the control.