Patent Publication Number: US-6698186-B2

Title: Air-fuel ratio controller for internal combustion engines

Description:
TECHNICAL FIELD 
     The present invention relates to an apparatus for controlling the air-fuel ratio of an internal combustion engine, and more particularly to an air-fuel ratio control apparatus which is capable of evaluating the deteriorated state of a catalytic converter for purifying exhaust gases. 
     BACKGROUND ART 
     Conventional processes of determining the deteriorated state of a catalytic converter for purifying exhaust gases which is disposed in the exhaust passage of an internal combustion engine are known from Japanese patent publication No. 2,526,640 and Japanese laid-open patent publication No. 7-19033, for example. 
     The disclosed techniques are based on the fact that when the air-fuel ratio of an air-fuel mixture to be combusted by an internal combustion engine is changed from a leaner value to a richer value or from a richer value to a leaner value, the outputs from oxygen concentration sensors that are positioned respectively upstream and downstream of a catalytic converter combined with the internal combustion engine are inverted. More specifically, under certain operating conditions of the internal combustion engine, i.e., when the power output of the internal combustion engine is increased or the fuel supplied to the internal combustion engine is cut off as disclosed in Japanese patent publication No. 2,526,640 or when certain conditions are satisfied, e.g., the load and rotational speed of the internal combustion engine are in predetermined ranges as disclosed in Japanese laid-open patent publication No. 7-19033, the air-fuel ratio is positively changed from a leaner value to a richer value or from a richer value to a leaner value. At this time, the time consumed after the output of the upstream oxygen concentration sensor is inverted until the output of the downstream oxygen concentration sensor is inverted, and the period at which output of the downstream oxygen concentration sensor is inverted are measured, and the deteriorated state of the catalytic converter is evaluated based on the measured values. 
     According to these techniques, when the internal combustion engine is operating under ordinary conditions, i.e., conditions without estimating the deteriorated state of the catalytic converter, the air-fuel ratio is feedback-controlled depending on the inversion of the outputs from the oxygen concentration sensors in order to keep the air-fuel ratio of the internal combustion engine in the vicinity of a stoichiometric air-fuel ratio, for thereby allowing the catalytic converter to keep an appropriate purifying capability. 
     According to the above process of evaluating the deteriorated state of the catalytic converter, however, it is necessary to positively change the air-fuel ratio from a leaner value to a richer value or from a richer value to a leaner value in order to evaluate the deteriorated state of the catalytic converter. Consequently, as long as the air-fuel ratio of the internal combustion engine is feedback-controlled so as to allow the catalytic converter to keep an appropriate purifying capability, it is impossible to evaluate the deteriorated state of the catalytic converter. Therefore, at the time of evaluating the deteriorated state of the catalytic converter, it is difficult for the catalytic converter to keep an appropriate purifying capability. 
     The applicant of the present application has proposed another technique for achieving an appropriate purifying capability of a catalytic converter (see, for example, Japanese laid-open patent publication No. 9-324681, Japanese laid-open patent publication No. 11-153051, U.S. Pat. No. 5,852,930, and U.S. patent application Ser. No. 09/153,300). According to the proposal, an exhaust gas sensor for detecting the concentration of a certain component, e.g., oxygen, of exhaust gases is positioned downstream of the catalytic converter, and the air-fuel ratio of an air-fuel mixture to be combusted by an internal combustion engine is manipulated in order to converge the output of the exhaust gas sensor to a predetermined target value. 
     More specifically, in order to converge the output (the detected value of the oxygen concentration) of the exhaust gas sensor to a predetermined target value (constant value), a target value (target air-fuel ratio) for the air-fuel ratio of the exhaust gases that enter the catalytic converter, or specifically the air-fuel ratio recognized from the oxygen concentration of the exhaust gases, is successively calculated according to a sliding mode control process. The air-fuel ratio of the air-fuel mixture to be combusted by an internal combustion engine is then manipulated depending on the target air-fuel ratio to achieve the appropriate purifying capability of the catalytic converter. 
     Since the above proposed technique is capable of stably keeping the appropriate purifying capability of the catalytic converter by controlling the air-fuel ratio as described above, it is desirable to be able to evaluate the deteriorated state of the catalytic converter while performing the above air-fuel control process. 
     It is therefore an object of the present invention to provide an apparatus for controlling the air-fuel ratio of an internal combustion engine, which is capable of appropriately evaluating the deteriorated state of a catalytic converter while keeping a desired purifying capability of the catalytic converter that is disposed in the exhaust passage of the internal combustion engine. 
     DISCLOSURE OF THE INVENTION 
     To achieve the above object, there is provided in accordance with the present invention an apparatus for controlling the air-fuel ratio of an internal combustion engine, comprising an exhaust gas sensor disposed downstream of a catalytic converter which is disposed in an exhaust passage of an internal combustion engine, for detecting the concentration of a particular component of a exhaust gas emitted from the internal combustion engine and having passed through the catalytic converter, air-fuel ratio manipulated variable generating means for sequentially generating a manipulated variable to determine the air-fuel ratio of the exhaust gas entering the catalytic converter to converge an output of the exhaust gas sensor to a predetermined target value, air-fuel ratio manipulating means for manipulating the air-fuel ratio of an air-fuel mixture to be combusted in the internal combustion engine depending on the manipulated variable, and deteriorated state evaluating means for sequentially determining the value of a deterioration evaluating linear function from time-series data of the output of the exhaust gas sensor, the deterioration evaluating linear function having variable components represented by the time-series data of the output of the exhaust gas sensor, and evaluating a deteriorated state of the catalytic converter based on the determined value of the deterioration evaluating linear function, while the air-fuel ratio of the air-fuel mixture is being manipulated by the air-fuel ratio manipulating means. 
     The inventors of the present invention have found that while a manipulated variable for the air-fuel ratio (e.g., a target value for the air-fuel ratio) of the exhaust gas entering the catalytic converter to converge the output of the exhaust gas sensor downstream of the catalytic converter to a given target value is being sequentially generated and the air-fuel ratio of the air-fuel mixture is being manipulated depending on the manipulated variable, when an appropriate linear function whose variable components are represented by time-series data of the output of the exhaust gas sensor, i.e., a linear function expressed as a linear coupling of the time-series data of the output of the exhaust gas sensor, is determined from the time-series data of the output of the exhaust gas sensor, the value of the linear function has a tendency to exhibit a characteristic correlation between itself and the degree to which the deterioration of the catalytic converter progresses. 
     For example, when the catalytic converter is substantially brand-new, the value of the linear function tends to be accumulated in the vicinity of a certain value. As the deterioration of the catalytic converter progresses, the value of the linear function tends to go away from the certain value. Stated otherwise, as the deterioration of the catalytic converter progresses, the degree to which the value of the linear function varies becomes larger. 
     The deteriorated state evaluating means employs the above linear function as the deterioration evaluating linear function, and sequentially determines the value of the deterioration evaluating linear function from the time-series data of the output of the exhaust gas sensor. The deteriorated state evaluating means determines the deteriorated state of the catalytic converter based on the value of the deterioration evaluating linear function. 
     The value of the deterioration evaluating linear function as a basis for evaluating the deteriorated state of the catalytic converter is determined from time-series data of the output of the exhaust gas sensor while the air-fuel ratio manipulating means is manipulating the air-fuel ratio of the air-fuel mixture depending on the manipulated variable that is generated by the air-fuel ratio manipulated variable generating means in order to converge the output of the exhaust gas sensor to the target value. Therefore, the value of the deterioration evaluating linear function is obtained while the air-fuel ratio is being manipulated to maintain a desired purifying capability of the catalytic converter. 
     Thus, the deteriorated state of the catalytic converter can be evaluated while maintaining the desired purifying capability of the catalytic converter, by evaluating the deteriorated state of the catalytic converter based on the value of the deterioration evaluating linear function. 
     In the present invention, the above tendency of the value of the deterioration evaluating linear function depending on the degree to which the deterioration of the catalytic converter progresses is likely to be clear when the manipulated variable is generated by the air-fuel ratio manipulated variable generating means according to a sliding mode control process as one feedback control process. If the air-fuel ratio manipulated variable generating means comprises means for generating the manipulated variable according to the sliding mode control process, then the deterioration evaluating linear function which is highly correlated to the deteriorated state of the catalytic converter is closely related to a switching function used in the sliding mode control process. Therefore, a linear function determined depending on the switching function is preferably used as the deterioration evaluating linear function. 
     More specifically, the sliding mode control process used by the air-fuel ratio manipulated variable generating means employs a linear function whose variable components are represented by time-series data of the difference between the output of the exhaust gas sensor and the target value, for example, as the switching function. If this switching function is used in the sliding mode control process, then the deterioration evaluating linear function preferably comprises a linear function which has coefficients of variable components that are identical to coefficients of the variable components of the switching function. The linear function may be the switching function itself for the sliding mode control process. 
     With the linear function determined depending on the switching function for the sliding mode control process being used as the deterioration evaluating linear function, the correlation between the value of the deterioration evaluating linear function and the deteriorated state of the catalytic converter is manifested, allowing the deteriorated state of the catalytic converter to be evaluated appropriately based on the value of the deterioration evaluating linear function. 
     Even if the manipulated variable is generated by any of various other feedback control processes including a PID (proportional plug integral plus derivative) control process, when a linear function similar to the switching function for the sliding mode control process is determined as the deterioration evaluating linear function, it is possible to provide a correlation of the above tendency between the value of the deterioration evaluating linear function and the deteriorated state of the catalytic converter. 
     According to the present invention, the deteriorated state evaluating means preferably comprises means for determining data representing the degree to which time-series data of the value of the deterioration evaluating linear function vary, as a deterioration evaluating parameter from the time-series data of the value of the deterioration evaluating linear function, and evaluating the deteriorated state of the catalytic converter based on the value of the determined deterioration evaluating parameter. 
     Specifically, as the deterioration of the catalytic converter progresses, the value of the deterioration evaluating linear function is liable to change away from a certain value, resulting in a large variation of the value of the deterioration evaluating linear function. According to the present invention, the data representing the degree to which the time-series data of the value of the deterioration evaluating linear function vary is used as the deterioration evaluating parameter, and the deterioration evaluating parameter is determined from the time-series data of the value of the deterioration evaluating linear function. When the deterioration evaluating parameter is thus determined, the correlation between its value and the deteriorated state of the catalytic converter is made clear, making it possible to evaluate the deteriorated state of the catalytic converter more appropriately based on the value of the deterioration evaluating parameter. 
     The deterioration evaluating parameter may comprise the square or absolute value of the difference between the value of the deterioration evaluating linear function and a predetermined value. Preferably, however, the deterioration evaluating parameter is determined by effecting a low-pass filtering process on the square or absolute value of the difference between each of the time-series data of the value of the deterioration evaluating linear function and a predetermined value as a central value of the value of the deterioration evaluating linear function. 
     When the deterioration evaluating parameter is determined by effecting the low-pass filtering process on the square or absolute value of the difference, the value of the deterioration evaluating parameter is adequate as a value representing the degree to which the value of the deterioration evaluating linear function varies. As the deterioration of the catalytic converter increases, the value of the deterioration evaluating parameter increases monotonously, clearly indicating a correlation between itself and the deteriorated state of the catalytic converter. Therefore, it is possible to evaluate the deteriorated state of the catalytic converter with high reliability based on the value of the deterioration evaluating parameter. 
     The low-pass filtering process preferably comprises a filtering process according to a sequential statistic algorithm. 
     By determining the deterioration evaluating parameter in the filtering process according to the sequential statistic algorithm, the deterioration evaluating parameter can be stored in a reduced memory capacity without the need for a memory for storing many data of the difference and its square or absolute value. 
     The sequential statistic algorithm may preferably comprise a method of least squares, a method of weighted least squares, a degressive gain method, a fixed gain method, etc. 
     The deterioration evaluating parameter increases monotonously as the deterioration of the catalytic converter progresses. Therefore, the deteriorated state evaluating means may comprise means for comparing the deterioration evaluating parameter with a predetermined threshold to determine whether the catalytic converter is deteriorated to an extent corresponding to the threshold or not. 
     Depending on a change in the rate of flow of the exhaust gas entering the catalytic converter, it may not be possible to obtain the value of the deterioration evaluating linear function or the value of the deterioration evaluating parameter which is suitable for evaluating the deteriorated state of the catalytic converter. Therefore, the deteriorated state evaluating means may comprise means for determining whether the deteriorated state of the catalytic converter is to be evaluated or not depending on a change in the rate of flow of the exhaust gas entering the catalytic converter. 
     Specifically, when the rate of flow of the exhaust gas is maintained stably at a constant level, since disturbance are small, the output of the exhaust gas sensor is likely to be kept stably at the target value steadily according to the feedback control process such as the sliding mode control process. In such a situation, even when the deterioration of the catalytic converter has progressed, the value of the deterioration evaluating linear function is apt to be accumulated near a certain value, making it difficult to distinguish the catalytic converter from a brand-new catalytic converter. 
     To alleviate such a drawback, the deteriorated state evaluating means may comprise means for failing to evaluate the deteriorated state of the catalytic converter if the rate of flow of the exhaust gas entering the catalytic converter is maintained at a substantially constant level, and evaluating the deteriorated state of the catalytic converter if the rate of flow of the exhaust gas entering the catalytic converter is not maintained at the substantially constant level. 
     Since the deteriorated state of the catalytic converter is evaluated only when the rate of flow of the exhaust gas changes properly, i.e., when the rate of flow of the exhaust gas undergoes a certain variation, the evaluation of the deteriorated state of the catalytic converter is made highly reliable. 
     The deteriorated state evaluating means may comprise means for failing to determine the deterioration evaluating parameter if the rate of flow of the exhaust gas entering the catalytic converter is maintained at a substantially constant level, and determining the deterioration evaluating parameter if the rate of flow of the exhaust gas entering the catalytic converter is not maintained at the substantially constant level. With this arrangement, the deteriorated state of the catalytic converter can be evaluated using only the deterioration evaluating parameter that is calculated while the rate of flow of the exhaust gas is appropriate, and the evaluation of the deteriorated state of the catalytic converter is made highly reliable. 
     The air-fuel ratio manipulated variable generating means may comprise estimating means for sequentially determining data representing an estimated value of the output of the exhaust gas sensor after a dead time of a system ranging from a position upstream of the catalytic converter to the exhaust gas sensor, and means for generating the manipulated variable using the data determined by the estimating means. 
     Alternatively, the air-fuel ratio manipulated variable generating means may comprise estimating means for sequentially determining data representing an estimated value of the output of the exhaust gas sensor after a total data time which is the sum of a dead time of a system ranging from a position upstream of the catalytic converter to the exhaust gas sensor and a dead time of a system comprising the air-fuel ratio manipulating system and the internal combustion engine, and means for generating the manipulated variable using the data determined by the estimating means. 
     The system ranging from the position upstream of the catalytic converter to the exhaust gas sensor, i.e., a system for generating the output of the exhaust gas sensor from the air-fuel ratio of the exhaust gas determined by the manipulated variable (hereinafter referred to as “object exhaust system”), generally has a relatively long dead time due to the catalytic converter included in the object exhaust system. When the rotational speed of the internal combustion engine is comparatively low, the system comprising the air-fuel ratio manipulating system and the internal combustion engine, i.e., a system for generating the air-fuel ratio of the exhaust gas entering the catalytic converter from the manipulated variable (hereinafter referred to as “air-fuel ratio manipulating system”), also has a relatively long dead time. These dead times should preferably be compensated for as they may possibly adversely affect the process of converting the output of the exhaust gas sensor to the target value. 
     The estimating means determines data representing the estimated value of the output of the exhaust gas sensor after the dead time of the object exhaust system, or data representing the estimated value of the output of the exhaust gas sensor after the total data time which is the sum of the above dead time and the dead time of the air-fuel ratio manipulating system, and the manipulated variable is generated using the data determined by the estimating means. 
     Because the effect of the dead time can thus be compensated for, the process of converting the output of the exhaust gas sensor to the target value can well be performed. As a result, the deteriorated state of the catalytic converter can be evaluated while well maintaining the desired purifying capability or performance of the catalytic converter. 
     With the estimating means thus employed, the air-fuel ratio manipulated variable generating means comprises means for generating the manipulated variable to converge the estimated value of the output of the exhaust gas sensor represented by the data determined by the estimating means to the target value, according to the sliding mode control process. The process of converting the output of the exhaust gas sensor to the target value is thus performed highly stably, and hence the desired purifying capability of the catalytic converter can be maintained stably. 
     The data representing the estimated value of the output of the exhaust gas sensor can be generated using the output of the exhaust gas sensor and the manipulated variable generated in the past by the air-fuel ratio manipulated variable generating means or the detected value of the air-fuel ratio of the exhaust gas upstream of the catalytic converter which depends on the manipulated variable. 
     The sliding mode control process comprises an adaptive sliding mode control process. 
     The adaptive sliding mode control process is a combination of an ordinary sliding mode control process and a control law referred to as an adaptive law (adaptive algorithm). When the manipulated variable is generated according to the adaptive sliding mode control process, the reliability of the manipulated variable is increased, allowing the output of the exhaust gas sensor to be converged to the target value with a quick response. The effect of a simple disturbance other than the deteriorated state of the catalytic converter on the value of the deterioration evaluating linear function determined depending on the switching function for the sliding mode control process is suppressed. Consequently, the reliability of the evaluation of the deteriorated state of the catalytic converter based on the deterioration evaluating parameter which represents the degree to which the value of the deterioration evaluating linear function varies is increased. 
     The manipulated variable generated by the air-fuel ratio manipulated variable generating means comprises a target air-fuel ratio for the exhaust gas entering the catalytic converter, and the apparatus further comprises an air-fuel ratio sensor disposed upstream of the catalytic converter for detecting the air-fuel ratio of the exhaust gas entering the catalytic converter, the air-fuel ratio manipulating means comprising means for manipulating the air-fuel ratio of the air-fuel mixture according to a feedback control process to converge the output of the air-fuel ratio sensor to the target air-fuel ratio. 
     With the manipulated variable comprising the target air-fuel ratio for the exhaust gas entering the catalytic converter, the output of the air-fuel ratio sensor which detects the air-fuel ratio of the exhaust gas entering the catalytic converter, i.e., the detected value of the air-fuel ratio, is feedback-controlled at the target air-fuel ratio. Therefore, the output of the exhaust gas sensor can well be converged to the target value, and hence the desired purifying capability of the catalytic converter can well be maintained. 
     The manipulated variable may comprise a corrective quantity for the amount of fuel supplied to the internal combustion engine, for example, other than the target air-fuel ratio. The air-fuel ratio of the air-fuel mixture may be manipulated depending on the manipulated variable according to a feed-forward control process based on the manipulated variable. For maintaining the optimum purifying capability of the catalytic converter and evaluating the deteriorated state of the catalytic converter, the exhaust gas sensor should preferably be an oxygen concentration sensor, i.e., an O 2  sensor, and a target value therefor should preferably be a constant value. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 is a block diagram of an overall system arrangement of an apparatus for controlling the air-fuel ratio of an internal combustion engine according to a first embodiment of the present invention; 
     FIG. 2 is a diagram showing output characteristics of an O 2  sensor and an air-fuel ratio sensor used in the apparatus shown in FIG. 1; 
     FIG. 3 is a block diagram showing a basic arrangement of an exhaust-side main processor of the apparatus shown in FIG. 1; 
     FIG. 4 is a diagram illustrative of a sliding mode control process employed by the apparatus shown in FIG. 1; 
     FIG. 5 is a diagram illustrative of a process of evaluating the deteriorated state of a catalytic converter employed by the apparatus shown in FIG. 1; 
     FIG. 6 is a diagram illustrative of the process of evaluating the deteriorated state of the catalytic converter employed by the apparatus shown in FIG. 1; 
     FIG. 7 is a diagram illustrative of the process of evaluating the deteriorated state of the catalytic converter employed by the apparatus shown in FIG. 1; 
     FIG. 8 is a diagram illustrative of the process of evaluating the deteriorated state of the catalytic converter employed by the apparatus shown in FIG. 1; 
     FIG. 9 is a block diagram of an adaptive controller employed in the apparatus shown in FIG. 1; 
     FIG. 10 is a flowchart of a processing sequence of an engine-side control unit of the apparatus shown in FIG. 1; 
     FIG. 11 is a flowchart of a subroutine of the processing sequence shown in FIG. 10; 
     FIG. 12 is a flowchart of a processing sequence of an exhaust-side control unit of the apparatus shown in FIG. 1; 
     FIG. 13 is a flowchart of a subroutine of the processing sequence shown in FIG. 12; 
     FIG. 14 is a flowchart of a subroutine of the processing sequence shown in FIG. 12; 
     FIG. 15 is a flowchart of a subroutine of the processing sequence shown in FIG. 12; 
     FIG. 16 is a flowchart of a subroutine of the processing sequence shown in FIG. 12; 
     FIG. 17 is a flowchart of a subroutine of the processing sequence shown in FIG. 16; 
     FIG. 18 is a flowchart of a subroutine of the processing sequence shown in FIG. 16; 
     FIG. 19 is a flowchart of a subroutine of the processing sequence shown in FIG. 16; 
     FIG. 20 is a block diagram of an engine-side control unit of an apparatus for controlling the air-fuel ratio of an internal combustion engine according to a second embodiment of the present invention; 
     FIG. 21 is a flowchart of a processing sequence of the exhaust-side control unit of the apparatus shown in FIG. 20; and 
     FIG. 22 is a diagram showing a data table employed in the processing sequence shown in FIG.  21 . 
    
    
     BEST MODE FOR CARRYING OUT THE INVENTION 
     A first embodiment of the present invention will be described below with reference to FIGS. 1 through 19. 
     FIG. 1 shows in block form the apparatus according to the embodiment. As shown in FIG. 1, an internal combustion engine  1  such as a four-cylinder internal combustion engine is mounted as a propulsion source on an automobile or a hybrid vehicle, for example. When a mixture of fuel and air is combusted in each cylinder of the internal combustion engine  1 , an exhaust gas is generated and emitted from each cylinder into a common discharge pipe  2  positioned near the internal combustion engine  1 , from which the exhaust gas is discharged into the atmosphere. Two three-way catalytic converters  3 ,  4  are mounted in the common exhaust pipe  2  at successively downstream locations thereon. 
     The upstream catalytic converter  3  is evaluated for its deteriorated state according to the present invention. The downstream catalytic converter  4  may be dispensed with. 
     The apparatus serves to control the air-fuel ratio of the exhaust gas entering the catalytic converter  3 , or specifically the air-fuel ratio recognized from the oxygen concentration in the exhaust gas entering the catalytic converter  3  (hereinafter referred to as the “air-fuel ratio of the internal combustion engine”) in order to achieve an optimum purifying capability of the catalytic converter  3 . While controlling the air-fuel ratio, the apparatus also evaluates the deteriorated state of the catalytic converter  3 . 
     In order to perform the above processing, the apparatus has an air-fuel ratio sensor  5  mounted on the exhaust pipe  2  upstream of the catalytic converter  3 , or more precisely at a position where exhaust gases from the cylinders of the internal combustion engine  1  are put together, an O 2  sensor (oxygen concentration sensor)  6  mounted as an exhaust gas sensor on the exhaust pipe  2  downstream of the catalytic converter  3  and upstream of the catalytic converter  4 , and a control unit  7  for carrying out a control process (described later on) and evaluating the deteriorated state of the catalytic converter  3  based on detected outputs from the sensors  5 ,  6 . 
     The control unit  7  is supplied with detected outputs from the sensors  5 ,  6  and also detected outputs from various other sensors for detecting operating conditions of the internal combustion engine  1 , including a engine speed sensor, an intake pressure sensor, a coolant temperature sensor, etc. 
     The O 2  sensor  6  comprises an ordinary O 2  sensor for generating an output VO 2 /OUT having a level depending on the oxygen concentration in the exhaust gas that has passed through the catalytic converter  3 , i.e., an output VO 2 /OUT representing a detected value of the oxygen concentration of the exhaust gas. The oxygen concentration in the exhaust gas is commensurate with the air-fuel ratio of an air-fuel mixture which, when combusted, produces the exhaust gas. The output VO 2 /OUT from the O 2  sensor  6  will change with high sensitivity in proportion to the oxygen concentration in the exhaust gas, with the air-fuel ratio corresponding to the oxygen concentration in the exhaust gas being in a range Δ close to a stoichiometric air-fuel ratio, as indicated by the solid-line curve a in FIG.  2 . At oxygen concentrations corresponding to air-fuel ratios outside of the range Δ, the output VO 2 /OUT from the O 2  sensor  6  is saturated and is of a substantially constant level. 
     The air-fuel ratio sensor  5  generates an output KACT representing a detected value of the air-fuel ratio of the internal combustion engine  1  which is recognized from the concentration of oxygen in the exhaust gas that enters the catalytic converter  3 . The air-fuel ratio sensor  5  comprises a wide-range air-fuel ration sensor disclosed in detail in Japanese laid-open patent publication No. 4-369471, which corresponds to U.S. Pat. No. 5,391,282. As indicated by the solid-line curve b in FIG. 2, the air-fuel ratio sensor  5  generates an output whose level is proportional to the concentration of oxygen in the exhaust gas in a wider range than the O 2  sensor  6 . Stated otherwise, the air-fuel ratio sensor  5  (hereinafter referred to as “LAF sensor  5 ”) generates an output whose level corresponding to the concentration of oxygen in the exhaust gas in a wide range of air-fuel ratios. 
     The control unit  7  comprises a control unit  7   a  (hereinafter referred to as an “exhaust-side control unit  7   a ”) for performing a process of calculating a target air-fuel ratio KCMD as a manipulated variable for determining the air-fuel ratio of the internal combustion engine  1  and carrying out a process of evaluating the deteriorated state of the catalytic converter  3 , and a control unit  7   b  (hereinafter referred to as an “engine-side control unit  7   b ”) for carryout out a process of controlling the air-fuel ratio of an air-fuel ratio to be combusted in the internal combustion engine  1  based on the target air-fuel ratio KCMD by adjusting the fuel injection quantity (fuel supply quantity) of the internal combustion engine  1  depending on the target air-fuel ratio KCMD. 
     The control units  7   a ,  7   b  comprise a microcomputer, and perform their respective control processes in given control cycles. In the present embodiment, each of the control cycles in which the exhaust-side control unit  7   a  performs its processes of generating the target air-fuel ratio KCMD and evaluating the deteriorated state of the catalytic converter  3  has a predetermined period (e.g., ranging from 30 to 100 ms) in view of the dead time (described later on) present in the catalytic converter  3 , calculating loads, etc. 
     The process of adjusting the fuel injection quantity, which is carried out by the engine-side control unit  7   b , needs to be in synchronism with the rotational speed of the internal combustion engine  1 , or more specifically combustion cycles of the internal combustion engine  1 . Therefore, each of the control cycles in which the engine-side control unit  7   b  performs its process has a period in synchronism with the crankshaft angle period (so-called TDC) of the internal combustion engine  1 . 
     The constant period of the control cycles of the exhaust-side control unit  7   a  is longer than the crankshaft angle period (so-called TDC) of the internal combustion engine  1 . 
     The engine-side control unit  7   b  will further be described below with reference to FIG.  1 . The engine-side control unit  7   b  has, as its main functions, a basic fuel injection quantity calculator  8  for determining a basic fuel injection quantity Tim to be injected into the internal combustion engine  1 , a first correction coefficient calculator  9  for determining a first correction coefficient KTOTAL to correct the basic fuel injection quantity Tim, and a second correction coefficient calculator  10  for determining a second correction coefficient KCMDM to correct the basic fuel injection quantity Tim. 
     The basic fuel injection quantity calculator  8  determines a reference fuel injection quantity (fuel supply quantity) from the rotational speed NE and intake pressure PB of the internal combustion engine  1  using a predetermined map, and corrects the determined reference fuel injection quantity depending on the effective opening area of a throttle valve (not shown) of the internal combustion engine  1 , thereby calculating a basic fuel injection quantity Tim. 
     The first correction coefficient KTOTAL determined by the first correction coefficient calculator  9  serves to correct the basic fuel injection quantity Tim in view of an exhaust gas recirculation ratio of the internal combustion engine  1 , i.e., the proportion of an exhaust gas contained in an air-fuel mixture introduced into the internal combustion engine  1 , an amount of purged fuel supplied to the internal combustion engine  1  when a canister (not shown) is purged, a coolant temperature, an intake temperature, etc. 
     The second correction coefficient KCMDM determined by the second correction coefficient calculator  10  serves to correct the basic fuel injection quantity Tim in view of the charging efficiency of an air-fuel mixture due to the cooling effect of fuel flowing into the internal combustion engine  1  depending on a target air-fuel ratio KCMD determined by the exhaust-side control unit  7   a , as described later on. 
     The engine-side control unit  7   b  corrects the basic fuel injection quantity Tim with the first correction coefficient KTOTAL and the second correction coefficient KCMDM by multiplying the basic fuel injection quantity Tim by the first correction coefficient KTOTAL and the second correction coefficient KCMDM, thus producing a demand fuel injection quantity Tcyl for the internal combustion engine  1 . 
     Specific details of processes for calculating the basic fuel injection quantity Tim, the first correction coefficient KTOTAL, and the second correction coefficient KCMDM are disclosed in detail in Japanese laid-open patent publication No. 5-79374 and U.S. Pat. No. 5,253,630, and will not be described below. 
     The engine-side control unit  7   b  also has, in addition to the above functions, a feedback controller  14  for feedback-controlling the air-fuel ratio of the air-fuel mixture to be combusted in the internal combustion engine  1  by adjusting a fuel injection quantity of the internal combustion engine  1  so as to converge the output KACT of the LAF sensor  5  (the detected air-fuel ratio of the internal combustion engine  1 ) toward the target air-fuel ratio KCMD which is calculated by the exhaust-side control unit  7   a.    
     The feedback controller  14  comprises a general feedback controller  15  for feedback-controlling a total fuel injection quantity for all the cylinders of the internal combustion engine  1  and a local feedback controller  16  for feedback-controlling a fuel injection quantity for each of the cylinders of the internal combustion engine  1 . 
     The general feedback controller  15  sequentially determines a feedback correction coefficient KFB to correct the demand fuel injection quantity Tcyl (by multiplying the demand fuel injection quantity Tcyl) so as to converge the output KACT from the LAF sensor  5  toward the target air-fuel ratio KCMD. 
     The general feedback controller  15  comprises a PID controller  17  for generating a feedback manipulated variable KLAF as the feedback correction coefficient KFB depending on the difference between the output KACT from the LAF sensor  5  and the target air-fuel ratio KCMD according to a known PID control process, and an adaptive controller  18  (indicated by “STR” in FIG. 1) for adaptively determining a feedback manipulated variable KSTR for determining the feedback correction coefficient KFB in view of changes in operating conditions of the internal combustion engine  1  or characteristic changes thereof from the output KACT from the LAF sensor  5  and the target air-fuel ratio KCMD. 
     In the present embodiment, the feedback manipulated variable KLAF generated by the PID controller  17  is of “1” and can be used directly as the feedback correction coefficient KFB when the output KACT (the detected air-fuel ratio) from the LAF sensor  5  is equal to the target air-fuel ratio KCMD. The feedback manipulated variable KSTR generated by the adaptive controller  18  becomes the target air-fuel ratio KCMD when the output KACT from the LAF sensor  5  is equal to the target air-fuel ratio KCMD. A feedback manipulated variable kstr (=KSTR/KCMD) which is produced by dividing the feedback manipulated variable KSTR by the target air-fuel ratio KCMD with a divider  19  can be used as the feedback correction coefficient KFB. 
     The feedback manipulated variable KLAF generated by the PID controller  17  and the feedback manipulated variable kstr which is produced by dividing the feedback manipulated variable KSTR from the adaptive controller  18  by the target air-fuel ratio KCMD are selected one at a time by a switcher  20 . A selected one of the feedback manipulated variable KLAF and the feedback manipulated variable KSTR is used as the feedback correction coefficient KFB. The demand fuel injection quantity Tcyl is corrected by being multiplied by the feedback correction coefficient KFB. Details of the general feedback controller  15  (particularly, the adaptive controller  18 ) will be described later on. 
     The local feedback controller  16  comprises an observer  21  for estimating real air-fuel ratios #nA/F (n=1, 2, 3, 4) of the respective cylinders from the output KACT from the LAF sensor  5 , and a plurality of PID controllers  22  (as many as the number of the cylinders) for determining respective feedback correction coefficients #nKLAF for fuel injection quantities for the cylinders from the respective real air-fuel ratios #nA/F estimated by the observer  21  according to a PID control process so as to eliminate variations of the air-fuel ratios of the cylinders. 
     Briefly stated, the observer  21  estimates a real air-fuel ratio #nA/F of each of the cylinders as follows: A system from the internal combustion engine  1  to the LAF sensor  5  (where the exhaust gases from the cylinders are combined) is considered to be a system for generating an air-fuel ratio detected by the LAF sensor  5  from a real air-fuel ratio #nA/F of each of the cylinders, and is modeled in view of a detection response delay (e.g., a time lag of first order) of the LAF sensor  5  and a chronological contribution of the air-fuel ratio of each of the cylinders to the air-fuel ratio detected by the LAF sensor  5 . Based on the modeled system, a real air-fuel ratio #nA/F of each of the cylinders is estimated from the output KACT from the LAF sensor  5 . 
     Details of the observer  21  are disclosed in Japanese laid-open patent publication No. 7-83094 and U.S. Pat. No. 5,531,208, and will not be described below. 
     Each of the PID controllers  22  of the local feedback controller  16  divides the output signal KACT from the LAF sensor  5  by an average value of the feedback correction coefficients #nKLAF determined by the respective PID controllers  22  in a preceding control cycle to produce a quotient value, and uses the quotient value as a target air-fuel ratio for the corresponding cylinder. Each of the PID controllers  22  then determines a feedback correction coefficient #nKLAF in a present control cycle so as to eliminate any difference between the target air-fuel ratio and the corresponding real air-fuel ratio #nA/F determined by the observer  21 . 
     The local feedback controller  16  multiplies a value, which has been produced by multiplying the demand fuel injection quantity Tcyl by the selected feedback correction coefficient KFB produced by the general feedback controller  15 , by the feedback correction coefficient #nKLAF for each of the cylinders, thereby determining an output fuel injection quantity #nTout (n=1, 2, 3, 4) for each of the cylinders. 
     The output fuel injection quantity #nTout thus determined for each of the cylinders is corrected for accumulated fuel particles on intake pipe walls of the internal combustion engine  1  by a fuel accumulation corrector  23  in the engine-side control unit  7   b . The corrected output fuel injection quantity #nTout is applied to each of fuel injectors (not shown) of the internal combustion engine  1 , which injects fuel into each of the cylinders with the corrected output fuel injection quantity #nTout. 
     The correction of the output fuel injection quantity in view of accumulated fuel particles on intake pipe walls is disclosed in detail in Japanese laid-open patent publication No. 8-21273 and U.S. Pat. No. 5,568,799, and will not be described in detail below. 
     A sensor output selector  24  shown in FIG. 1 serves to select the output KACT from the LAF sensor  5 , which is suitable for the estimation of a real air-fuel ratio #nA/F of each cylinder with the observer  21 , depending on the operating conditions of the internal combustion engine  1 . Details of the sensor output selector  24  are disclosed in detail in Japanese laid-open patent publication No. 7-259588 and U.S. Pat. No. 5,540,209, and will not be described in detail below. 
     The exhaust-side control unit  7   a  has a subtractor  11  for determining a difference kact (=KACT−FLAF/BASE) between the output KACT from the LAF sensor  5  and a predetermined reference value FLAF/BASE and a subtractor  12  for determining a difference VO 2  (=VO 2 /OUT−VO 2 /TARGET) between the output VO 2 /OUT from the O 2  sensor  6  and a target value VO 2 /TARGET therefor. 
     The catalytic converter  3  achieves an optimum purifying capability irrespective of its deteriorated state at the air-fuel ratio of the internal combustion engine  1  which causes the output VO 2 /OUT from the O 2  sensor  6  to settle on a certain constant value VO 2 /TARGET (see FIG.  2 ). In the present embodiment, therefore, the constant value VO 2 /TARGET is used as the target value VO 2 /TARGET for the output VO 2 /OUT from the O 2  sensor  6 . The reference value FLAF/BASE with respect to the output KACT from the LAF sensor  5  is set to a “stoichiometric air-fuel ratio”. 
     The differences kact, VO 2  determined respectively by the subtractors  11 ,  12  are referred to as a differential output kact of the LAF sensor  5  and a differential output VO 2  of the O 2  sensor  6 , respectively. 
     The exhaust-side control unit  7   a  also has an exhaust-side main processor  13  which is supplied with the data of the differential outputs kact, VO 2  as the data of the output from the LAF sensor  5  and the output of the O 2  sensor  6 . 
     As shown in FIG. 3, the exhaust-side main processor  13  comprises, as its functions, a target air-fuel ratio calculating means  13   a  as an air-fuel ratio manipulated variable determining means for sequentially determining a target air-fuel ratio KCMD for the internal combustion engine  1  based on the data of the differential outputs kact, VO 2 , and a deteriorated state evaluating means  13   b  for evaluating the deteriorated state of the catalytic converter  3  based on the data of the differential output VO 2  of the O 2  sensor  6 . 
     The target air-fuel ratio calculating means  13   a  serves to control an object exhaust system (denoted by E in FIG. 1) including the catalytic converter  3 , which ranges from the LAF sensor  5  to the O 2  sensor  6  along the exhaust pipe  2 . The target air-fuel ratio calculating means  13   a  sequentially determines the target air-fuel ratio KCMD for the internal combustion engine  1  so as to converge the output VO 2 /OUT of the O 2  sensor  6  to the target value VO 2 /TARGET therefor according to a sliding mode control process, or specifically an adaptive sliding mode control process, in view of a dead time present in the object exhaust system E to be controlled, a dead time present in the internal combustion engine  1  and the engine-side control unit  7   b , and behavioral changes of the object exhaust system E. 
     The deteriorated state evaluating means  13   b  serves to evaluate the deteriorated state of the catalytic converter  3  based on the value of a deterioration evaluating linear function, described later on, which is determined from time-series data of the differential output VO 2  of the O 2  sensor  6 , and control the operation of a deterioration indicator  29  connected to the air-fuel ratio control apparatus depending on the evaluation of the deteriorated state of the catalytic converter  3 . The deterioration indicator  29  may comprise a lamp, a buzzer, or a display unit for displaying characters, a graphic image, etc. to indicate the deteriorated state of the catalytic converter  3 . 
     The target air-fuel ratio calculating means  13   a  and the deteriorated state evaluating means  13   b  will further be described below. 
     In order to carry out the process of the target air-fuel ratio calculating means  13   a , according to present embodiment, the object exhaust system E is regarded as a system for generating the output VO 2 /OUT of the O 2  sensor  6  (the detected value of the oxygen concentration of the exhaust gas having passed through the catalytic converter  3 ) from the output KACT of the LAF sensor  5  (the detected value of the air-fuel ratio of the internal combustion engine  1 ) via a dead time element and a response delay element, and the behavior of the system is modeled as a discrete time system. 
     In addition, the system comprising the internal combustion engine  1  and the engine-side control unit  7   b  is regarded as a system (hereinafter referred to as “air-fuel ratio manipulating system”) for generating the output KACT of the LAF sensor  5  from the target air-fuel ratio KCMD via a dead time element, and the behavior of this system is modeled as a discrete time system. 
     The model (hereinafter referred to as “exhaust system model”) in which the behavior of the object exhaust system E is expressed as a discrete time system is expressed, using the differential output kact (=KACT−FLAF/BASE) from the LAF sensor  5  and the differential output VO 2  (=VO 2 /OUT−VO 2 /TARGET) from the O 2  sensor  6 , instead of the output KACT of the LAF sensor  5  and the output VO 2 /OUT of the O 2  sensor  6 , according to the following equation (1): 
     
       
         VO 2 ( k +1)= a   1 ·VO 2 ( k )+ a   2 ·VO 2 ( k −1)+ b   1 ·kact( k−d   1 )  (1) 
       
     
     According to the equation (1), the object exhaust system E is regarded as a system for generating the differential output VO 2  from the O 2  sensor  6  from the differential output kact from the LAF sensor  5  via a dead time element and a response delay element, and expresses the behavior of the object exhaust system E with the model of a discrete time system (more specifically, an autoregressive model having a dead time in the differential output kact as an input to the exhaust system. 
     In the equation (1), “k” represents the ordinal number of a discrete-time control cycle of the exhaust-side control unit  7   a , and “d 1 ” the dead time (more specifically, the dead time required until the air-fuel ratio detected at each point of time by the LAF sensor  5  is reflected in the output VO 2 /OUT of the O 2  sensor  6 ) of the object exhaust system E as represented by the number of control cycles. The dead time of the object exhaust system E is generally equal to the time of 3-10 control cycles (d 1 =3-10) if the period (constant in the present embodiment) of control cycles of the exhaust-side control unit  7   a  ranges from 30 to 100 ms. In the present embodiment, a preset constant value (d 1 =7, for example) which is equal to or slightly longer than the actual dead time of the object exhaust System E is used as the dead time d 1  in the discrete-system model of the object exhaust system E as represented by the equation (1). 
     The first and second terms of the right side of the equation (1) correspond to a response delay element of the object exhaust system E, the first term being a primary autoregressive term and the second term being a secondary autoregressive term. In the first and second terms, “a 1 ”, “a 2 ” represent respective gain coefficients of the primary autoregressive term and the secondary autoregressive term. Stated otherwise, these gain coefficients a 1 , a 2  are relative to the differential output VO 2  of the O 2  sensor  6  as an output of the control system E. 
     The third term of the right side of the equation (1) represents the differential output kact of the LAF sensor  5  as an input to the object exhaust system E, including the dead time d 1  of the object exhaust system E. In the third term, “b 1 ” represents a gain coefficient relative to the input to the object exhaust system E, i.e., the differential output kact of the LAF sensor  5 . These gain coefficients “a 1 ”, “a 2 ”, “b 1 ” are parameters which define the behavior of the exhaust system model, and are sequentially identified by an identifier which will be described later on. 
     The model (hereinafter referred to as “air-fuel ratio manipulating system model”) of the discrete time system of the air-fuel ratio manipulating system which comprises the internal combustion engine  1  and the engine-side control unit  7   b  is expressed, using the differential output kact (=KACT−FLAF/BASE) from the LAF sensor  5  instead of the output KACT from the LAF sensor  5  and also using a difference kcmd (=KCMD−FLAF/BASE, which corresponds to a target value for the differential output kact of the LAF sensor  5 , and will be referred to as “target differential air-fuel ratio kcmd”) between the target air-fuel ratio KCMD and the reference value FLAF/BASE instead of the target air-fuel ratio KCMD, according to the following equation (2): 
     
       
         kact( k )=kcmd( k−d   2 )  (2) 
       
     
     The equation (2) expresses the air-fuel ratio manipulating system as the model of a discrete time system, regarding the air-fuel ratio manipulating system as a system for generating the differential output kact from the LAF sensor  5  from the target differential air-fuel ratio kcmd via a dead time element, i.e., a system in which the differential output kact in each control cycle is equal to the target differential air-fuel ratio kcmd prior to the dead time. 
     In the equation (2), “d 2 ” represents the dead time of the air-fuel ratio manipulating system (more specifically, the time required until the target air-fuel ratio KCMD at each point of time is reflected in the output signal KACT of the LAF sensor  5 ) in terms of the number of control cycles of the exhaust-side control unit  7   a . The dead time of the air-fuel ratio manipulating system varies with the rotational speed NE of the internal combustion engine  1 , and is longer as the rotational speed NE of the internal combustion engine  1  is lower. In the present embodiment, in view of the above characteristics of the dead time of the air-fuel ratio manipulating system, a preset constant value (for example, d 2 =3) which is equal to or slightly longer than the actual dead time of the air-fuel ratio manipulating system at an idling rotational speed of the internal combustion engine  1 , which is a rotational speed in a low speed range of the internal combustion engine  1  (the actual dead time is a maximum dead time which can be taken by the air-fuel ratio manipulating system at an arbitrary rotational speed of the internal combustion engine  1 ), used as the value of the dead time d 2  in the air-fuel ratio manipulating system model expressed by the equation (2). 
     The air-fuel ratio manipulating system actually includes a dead time element and a response delay element of the internal combustion engine  1 . Since a response delay of the output KACT of the LAF sensor  5  with respect to the target air-fuel ratio KCMD is basically compensated for by the feedback controller  14  (particularly the adaptive controller  18 ) of the engine-side control unit  7   b , there will arise no problem if a response delay element of the internal combustion engine  1  is not taken into account in the air-fuel ratio manipulating system as viewed from the exhaust-side control unit  7   a.    
     The target air-fuel ratio calculating means  13   a  of the exhaust-side main processor  13  carries out the process for calculating the target air-fuel ratio KCMD based on the exhaust system model and the air-fuel ratio manipulating system model expressed respectively by the equations (1), (2) in constant control cycles of the exhaust-side control unit  7   a . In order to carry out the above process, the target air-fuel ratio calculating means  13   a  has its functions as shown in FIG.  3 . 
     As shown in FIG. 3, the target air-fuel ratio calculating means  13   a  comprises an identifier  25  for sequentially determining in each control cycle identified values a 1  hat, a 2  hat, b 1  hat of the gain coefficients a 1 , a 2 , b 1  (hereinafter referred to as “identified gain coefficients a 1  hat, a 2  hat, b 1  hat”) that are parameters to be established for the exhaust system model (the equation (1)), an estimator  26  for sequentially determining in each control cycle an estimated value VO 2  bar of the differential output VO 2  from the O 2  sensor  6  (hereinafter referred to as “estimated differential output VO 2  bar”) after the total dead time d (=d 1 +d 2 ) which is the sum of the dead time d 1  of the object exhaust system E and the dead time d 2  of the air-fuel ratio manipulating system, and a sliding mode controller  27  for sequentially determining in each control cycle a target air-fuel ratio KCMD according to an adaptive slide mode control process. 
     The algorithm of a processing operation to be carried out by the identifier  25 , the estimator  26 , and the sliding mode controller  27  is constructed as follows: 
     The identifier  25  serves to identify the gain coefficients a 1 , a 2 , b 1  sequentially on a real-time basis for the purpose of minimizing a modeling error of the actual object exhaust system E of the exhaust system model expressed by the equation (1). The identifier  25  carries out its identifying process as follows: 
     In each control cycle, the identifier  25  determines an identified value VO 2 (k) hat of the differential output VO 2  (the output of the exhaust system model) from the O 2  sensor  6  (hereinafter referred to as “identified differential output VO 2 (k) hat”) on the exhaust system model, using the identified gain coefficients a 1  hat, a 2  hat, b 1  hat of the presently established exhaust system model, i.e., identified gain coefficients a 1 (k−1) hat, a 2 (k−1) hat, b 1 (k−1) hat determined in a preceding control cycle, and past data kact(k−d 1 −1), VO 2 (k−1), VO 2 (k−2) of the differential output kact from the LAF sensor  5  and the differential output VO 2  from the O 2  sensor  6 , according to the following equation (3): 
     
       
         VÔ 2 ( k )= â   1 ( k −1)·VO 2 ( k −1)+ â   2 ( k −1)·VO 2 ( k −2)+ {circumflex over (b)}   1 ( k −1)·kact( k−d   1 −1)  (3) 
       
     
     The equation (3) corresponds to the equation (1) which is shifted into the past by one control cycle with the gain coefficients a 1 , a 2 , b 1  being replaced with the respective identified gain coefficients a 1 (k−1) hat, a 2 (k−1) hat, b 1 (k−1) hat. The constant value (d 1 =7) established as described above is used as the value of the dead time d 1  of the object exhaust system E in the third term of the equation (3). 
     If vectors Θ, ξ defined by the following equations (4), (5) are introduced (the letter T in the equations (4), (5) represents a transposition), then the equation (3) is expressed by the equation (6): 
     
       
         Θ T ( k )=[ â   1 ( k ) â   2 ( k ) {circumflex over (b)}   1 ( k )]  (4) 
       
     
     
       
         ξ T ( k )=[VO 2 ( k −1)VO 2 ( k −2)kact( k−d   1 −1)]  (5) 
       
     
     
       
         VÔ 2 ( k )=Θ T ( k −1)·ξ( k )  (6) 
       
     
     The identifier  25  also determines a difference id/e(k) between the identified differential output VO 2 (k) hat from the O 2  sensor  6  which is determined by the equation (3) or (6) and the present differential output VO 2 (k) from the O 2  sensor  6 , as representing a modeling error of the exhaust system model with respect to the actual object exhaust system E (hereinafter the difference id/e will be referred to as “identified error id/e”), according to the following equation (7): 
     
       
           id/e ( k )=VO 2 ( k )−VÔ 2 ( k )  (7) 
       
     
     The identifier  25  further determines new identified gain coefficients a 1 (k) hat, a 2 (k) hat, b 1 (k) hat, stated otherwise, a new vector Θ(k) having these identified gain coefficients as elements (hereinafter the new vector Θ(k) will be referred to as “identified gain coefficient vector Θ”), in order to minimize the identified error id/e, according to the equation (8) given below. That is, the identifier  25  varies the identified gain coefficients a 1  hat (k−1), a 2  hat (k−1), b 1  hat (k−1) determined in the preceding control cycle by a quantity proportional to the identified error id/e(k) for thereby determining the new identified gain coefficients a 1 (k) hat, a 2 (k) hat, b 1 (k) hat. 
     
       
         Θ( k )=Θ( k −1)+ K θ( k )· id/e ( k )  (8) 
       
     
     where Kθ represents a cubic vector determined by the following equation (9), i.e., a gain coefficient vector for determining a change depending on the identified error id/e of the identified gain coefficients a 1  hat, a 2  hat, b 1  hat):                K                   θ        (   k   )         =         P        (     k   -   1     )       ·     ξ        (   k   )           1   +         ξ   T          (   k   )       ·     P        (     k   -   1     )       ·     ξ        (   k   )                     (   9   )                         
     where P represents a cubic square matrix determined by a recursive formula expressed by the following equation (10):                      P        (   k   )       =                  1       λ   1          (   k   )         ·                                [     I   -             λ   2          (   k   )       ·     P        (     k   -   1     )              ·     ξ        (   k   )              ·       ξ   T          (   k   )                 λ   1          (   k   )       +         l   2          (   k   )       ·       ξ   T          (   k   )       ·     P        (     k   -   1     )       ·     ξ        (   k   )               ]     ·     P        (     k   -   1     )                       (   10   )                         
     where I represents a unit matrix. 
     In the equation (10), λ 1 , λ 2  are established to satisfy the conditions 0&lt;λ 1 ≦1 and 0≦λ 2 &lt;2, and an initial value P(0) of P represents a diagonal matrix whose diagonal components are positive numbers. 
     Depending on how λ 1 , λ 2  in the equation (10) are established, any one of various specific algorithms including a fixed gain method, a degressive gain method, a method of weighted least squares, a method of least squares, a fixed tracing method, etc. may be employed. According to the present embodiment, a method of least squares (λ 1 =λ 2 =1), for example, is employed. 
     Basically, the identifier  25  sequentially determines in each control cycle the identified gain coefficients a 1  hat, a 2  hat, b 1  hat of the exhaust system model in order to minimize the identified error id/e according to the above algorithm (calculating operation). Through this operation, it is possible to sequentially obtain the identified gain coefficients a 1  hat, a 2  hat, b 1  hat which match the actual object exhaust system E. 
     The algorithm described above is the basic algorithm that is carried out by the identifier  25 . 
     The estimator  26  sequentially determines in each control cycle the estimated differential output VO 2  bar which is an estimated value of the differential output VO 2  from the O 2  sensor  6  after the total dead time d (=d 1 +d 2 ) in order to compensate for the effect of the dead time d 1  of the object exhaust system E and the effect of the dead time d 2  of the air-fuel ratio manipulating system for the calculation of the target air-fuel ratio KCMD with the sliding mode controller  27  as described in detail later on. The algorithm for the estimator  26  to determine the estimated differential output VO 2  bar is constructed as follows: 
     If the equation (2) expressing the model of the air-fuel ratio manipulating system is applied to the equation (1) expressing the model of the object exhaust system E, then the equation (1) can be rewritten as the following equation (11):                      VO2        (     k   +   1     )       =                  a1   ·     VO2        (   k   )         +     a2   ·     VO2        (     k   -   1     )         +                              b1   ·     kcmd        (     k   -   d1   -   d2     )                     =                  a1   ·     VO2        (   k   )         +     a2   ·     VO2        (     k   -   1     )         +                              b1   ·     kcmd        (     k   -   d     )                       (   11   )                         
     The equation (11) expresses the behavior of a system which is a combination of the object exhaust system E and the air-fuel manipulating system as the model of a discrete time system, regarding such a system as a system for generating the differential output VO 2  from the O 2  sensor  6  from the target differential air-fuel ratio kcmd via dead time elements of the object exhaust system E and the air-fuel manipulating system and a response delay element of the object exhaust system E. 
     By using the equation (11), the estimated differential output VO 2 (k+d) bar which is an estimated value of the differential output VO 2 (k+d) of the O 2  sensor  6  after the total dead time d in each control cycle can be expressed using time-series data VO 2 (k), VO 2 (k−1) of present and past values of the differential output VO 2  of the O 2  sensor  6  and time-series data kcmd(k−j) (j=1, 2, . . . , d) of the past values of the target differential air-fuel ratio kcmd (=KCMD−FLAF/BASE) which corresponds to the target air-fuel ratio KCMD determined by the sliding mode controller  27  (its specific process of determining the target air-fuel ratio KCMD will be described later on), according to the following equation (12):                      VO2        (     k   +   d     )       =                  α                   1   ·     VO2        (   k   )           +     α                   2   ·   VO2          (     k   -   1     )       +                                ∑     j   =   1     d          β                   j   ·     kcmd        (     k   -   j     )                           (   12   )                         
     where 
     α 1 =the first-row, first-column element of A d , 
     α 2 =the first-row, second-column element of A d , 
     βj=the first-row elements of A j−1 ·B        A   =     [         a1       a2           1       0         ]             B   =     [         b1           0         ]                     
     In the equation (12), “α1”, “α2” represent the first-row, first-column element and the first-row, second-column element, respectively, of the dth power A d  (d: total dead time) of the matrix A defined as described above with respect to the equation (12), and “βj” (j=1, 2, . . . , d) represents the first-row elements of the product A j−1 ·B of the (j−1)th power A j−1  (j=1, 2, . . . , d) of the matrix A and the vector B defined as described above with respect to the equation (12). 
     Of the time-series data of the past values of the target combined differential air-fuel ratio kcmd according to the equation (12), the time-series data kcmd(k−d 2 ), kcmd(k−d 2 −1), . . . , kcmd(k−d) from the present prior to the dead time d 2  of the air-fuel manipulating system can be replaced respectively with data kact(k), kact(k−1), . . . , kact(k−d+d 2 ) obtained prior to the present time of the differential output kact of the LAF sensor  5 . When the time-series data are thus replaced, the following equation (13) is obtained:                        VO2   _          (     k   +   d     )       =                  α                   1   ·     VO2        (   k   )           +     α                   2   ·     VO2        (     k   -   1     )           +                                  ∑     j   =   1       d2   -   1            β                   j   ·     kcmd        (     k   -   j     )             +       ∑     i   =   0       d   -   d2            β                 i       +     d2   ·     kact        (     k   -   i     )                       =                  α                   1   ·     VO2        (   k   )           +     α                   2   ·     VO2        (     k   -   1     )           +                                  ∑     j   =   1       d2   -   1            β                   j   ·     kcmd        (     k   -   j     )             +       ∑     i   =   0     d1          β                 i       +     d2   ·     kact        (     k   -   i     )                         (   13   )                         
     The equation (13) is a basic formula for the estimator  26  to determine the estimated differential output VO 2 (k+d) bar. Stated otherwise, the estimator  26  determines, in each control cycle, the estimated differential output VO 2 (k+d) bar of the O 2  sensor  6  according to the equation (13), using the time-series data VO 2 (k), VO 2 (k−1) of the differential output VO 2  of the O 2  sensor  6 , the time-series data kcmd(k−j) (j=1, . . . , d 2 −1) of the past values of the target differential air-fuel ratio kcmd which represents the target air-fuel ratio KCMD determined in the past by the sliding mode controller  27 , and the time-series data kact(k−i) (i=0, . . . , d 1 ) of the present and past values of the differential output kact of the LAF sensor  5 . 
     In the present embodiment, the values of the coefficients α1, α2, βj (j=1, 2, . . . , d) required to calculate the estimated differential output VO 2 (k+d) bar according to the equation (13) are basically calculated using the identified gain coefficients a 1  hat, a 2  hat, b 1  hat which are the identified values of the gain coefficients a 1 , a 2 , b 1  (which are elements of the vectors A, B defined with respect to the equation (12)). The values of the dead times d 1 , d 2  required in the equation (13) comprise the preset values as described above. 
     The estimated differential output VO 2 (k+d) bar may be determined according to the equation (12) without using the data of the differential output kact of the LAF sensor  5 . For increasing the reliability of the estimated differential output VO 2 (k+d) bar, however, it is preferable to determine the estimated differential output VO 2 (k+d) bar according to the equation (13) using the data of the differential output kact of the LAF sensor  5  which reflects the actual behavior of the internal combustion engine  1 . If the dead time d 2  of the air-fuel ratio manipulating system can be set to “1”, all the time-series data kcmd(k−j) (j=1, 2, . . . , d) of the past values of the target differential air-fuel ratio kcmd in the equation (12) may be replaced with the time-series data kact(k), kact(k−1), . . . , kact(k−d+d 2 ), respectively, prior to the present time, of the differential output kact of the LAF sensor  5 . In this case, the estimated differential output VO 2 (k+d) bar can be determined according to the following equation (14) which does not include the data of the target differential air-fuel ratio kcmd:                        VO2   _          (     k   +   d     )       =                  α                   1   ·     VO2        (   k   )           +     α                   2   ·     VO2        (     k   -   1     )           +                                  ∑     k   =   0       d   -   1            β                 j       +     1   ·     kact        (     k   -   j     )                         (   14   )                         
     The sliding mode controller  27  will be described in detail below. 
     The sliding mode controller  27  determines an input quantity to be given to the object exhaust system E (which is specifically a target value for the difference between the output KACT of the LAF sensor  5  (the detected value of the air-fuel ratio) and the reference value FLAF/BASE, which target value is equal to the target differential air-fuel ratio kcmd) (the input quantity will be referred to as “SLD manipulating input Us 1 ”) in order to cause the output VO 2 /OUT of the O 2  sensor  6  to settle on the target value VO 2 /TARGET, i.e., to converge the differential output VO 2  of the O 2  sensor  6  to “0” according to an adaptive sliding mode control process which incorporates an adaptive control law for minimizing the effect of a disturbance, in a normal sliding mode control process, and determines the target air-fuel ratio KCMD from the determined SLD manipulating input Usl. An algorithm for carrying out the adaptive sliding mode control process is constructed as follows: 
     A switching function required for the algorithm of the adaptive sliding mode control process carried out by the sliding mode controller  27  and a hyperplane defined by the switching function (also referred to as a slip plane) will first be described below. 
     According to a basic concept of the sliding mode control process, the differential output VO 2 (k) of the O 2  sensor  6  obtained in each control cycle and the differential output VO 2 (k−1) obtained in a preceding control cycle are used as a state quantity to be controlled (controlled quantity), and a switching function σ for the sliding mode control process is defined as a linear function whose variable components are represented by the differential outputs VO 2 (k), VO 2 (k−1), according to the following equation (15):                      σ        (   k   )       =                  s1   ·     VO2        (   k   )         +     s2   ·     VO2        (     k   -   1     )                       =                S   ·   X                   (   15   )                         
     where 
     S=[s 1  s 2 ],        X   =     [           VO2        (   k   )                 VO2        (     k   -   1     )             ]                     
     A vector X defined above with respect to the equation (15) as a vector whose elements are represented by the differential outputs VO 2 (k), VO 2 (k−1) will hereinafter be referred to as a state quantity X. 
     The coefficients s 1 , s 2  of the switching function σ is set in order to meet the condition of the following equation (16):                  -   1     &lt;     s2   s1     &lt;   1                        (   16   )                         
     (when s 1 =1, −1&lt;s 2 &lt;1) 
     In the present embodiment, for the sake of brevity, the coefficient s 1  is set to s 1 =1 (s 2 /s 1 =s 2 ), and the coefficient s 2  is established to satisfy the condition: −1&lt;s 2 &lt;1. 
     With the switching function σ thus defined, the hyperplane for the sliding mode control process is defined by the equation σ=0. Since the state quantity X is of the second degree, the hyperplane σ=0 is represented by a straight line as shown in FIG.  4 . At this time, the hyperplane is called a switching line or a switching plane depending on the degree of a topological space. 
     In the present embodiment, the time-series data of the estimated differential output VO 2  bar determined by the estimator  26  is used as the variable components of the switching function for the sliding mode control process, as described later on. 
     The adaptive sliding mode control process serves to converge the state quantity X onto the hyperplane σ=0 according to a reaching control law which is a control law for converging the state quantity X (=VO 2 (k), VO 2 (k−1)) onto the hyperplane σ=0, and an adaptive control law (adaptive algorithm) which is a control law for compensating for the effect of a disturbance in converging the state quantity X onto the hyperplane σ=0 (mode  1  in FIG.  4 ). While holding the state quantity X onto the hyperplane σ=0 according to an equivalent control input, the state quantity X is converged to a balanced point on the hyperplane σ=0 where VO 2 (k)=VO 2 (k−1)=0, i.e., a point where time-series data VO 2 /OUT(k), VO 2 /OUT(k−1) of the output VO 2 /OUT of the O 2  sensor  6  are equal to the target value VO 2 /TARGET (mode  2  in FIG.  4 ). 
     The SLD manipulating input Usl (=the target differential air-fuel ratio kcmd) to be generated by the sliding mode controller  27  for converging the state quantity X toward the balanced point on the hyperplane σ=0 is expressed as the sum of an equivalent control input Ueq to be applied to the object exhaust system E according to the control law for converging the state quantity X onto the hyperplane σ=0, an input Urch (hereinafter referred to as “reaching control law input Urch”) to be applied to the object exhaust system E according to the reaching control law, and an input Uadp (hereinafter referred to as “adaptive control law Uadp”) to be applied to the object exhaust system E according to the adaptive control law (see the following equation (17)). 
     
       
         Usl=Ueq+Urch+Uadp  (17) 
       
     
     The equivalent control input Ueq, the reaching control law input Urch, and the adaptive control law input Uadp are determined on the basis of the model of the discrete time system expressed by the equation (11), i.e., the model in which the differential output kact(k−d 1 ) of the LAF sensor  5  in the equation (11) is replaced with the target differential air-fuel ratio kcmd(k−d) using the total dead time  3 , as follows: 
     The equivalent control input Ueq which is an input component to be applied to the object exhaust system E for converging the state quantity X onto the hyperplane σ=0 is the target differential air-fuel ratio kcmd which satisfies the condition: σ(k+1)=σ(k)=0. Using the equations (11), (15), the equivalent control input Ueq which satisfies the above condition is given by the following equation (18):                      Ueq        (   k   )       =                  -       (     S   ·   B     )       -   1         ·     {     S   ·     (     A   -   1     )       }     ·     X        (     k   +   d     )                     =                    -   1     s1b1     ·     {         [       s1   ·     (     a1   -   1     )       +   s2     ]     ·     VO2        (     k   +   d     )         +                                    (       s1   ·   a2     -   s2     )     ·     VO2        (     k   +   d   -   1     )         }                 (   18   )                         
     The equation (18) is a basic formula for determining the equivalent control law input Ueq(k) in each control cycle. 
     According to the present embodiment, the reaching control law input Urch is basically determined according to the following equation (19):                      Urch        (   k   )       =                  -       (     S   ·   B     )       -   1         ·   F   ·     σ        (     k   +   d     )                     =                    -   1     s1b1     ·   F   ·     σ        (     k   +   d     )                       (   19   )                         
     Specifically, the reaching control law input Urch is determined in proportion to the value σ(k+d) of the switching function σ after the total dead time d, in view of the effect of the total dead time d. 
     The coefficient F in the equation (19) which determines the gain of the reaching control law is established to satisfy the condition expressed by the following equation (20): 
     
       
         0&lt;F&lt;2  (20) 
       
     
     The value of the switching function σ may possibly vary in an oscillating fashion (so-called chattering) with respect to the hyperplane σ=0. In order to suppress such chattering, it is preferable that the coefficient F relative to the reaching control law input Urch be established to further satisfy the condition of the following equation (21): 
      0&lt;F&lt;1  (21) 
     The adaptive control law input Uadp is basically determined according to the following equation (22) (ΔT in the equation (22) represents the period of the control cycles of the exhaust-side control unit  7   a ):                      Uadp        (   k   )       =                  -       (     S   ·   B     )       -   1         ·   G   ·       ∑     i   =   0       k   +   d            (         σ        (   i   )       ·   Δ                   T     )                     =                    -   1     s1b1     ·   G   ·       ∑     i   =   0       k   +   d            (         σ        (   i   )       ·   Δ                   T     )                       (   22   )                         
     The adaptive control law input Uadp is determined in proportion to an integrated value (which corresponds to an integral of the values of the switching function σ) over control cycles of the product of values of the switching function σ and the period ΔT of the exhaust-side control unit  7   a  until after the total dead time d, in view of the effect of the total dead time d. 
     The coefficient G (which determines the gain of the adaptive control law) in the equation (22) is established to satisfy the condition of the following equation (23):                G   =     J   ·       2   -   F       Δ                 T                
          (     0   &lt;   J   &lt;   2     )             (   23   )                         
     A specific process of deriving conditions for establishing the equations (16), (20), (21), (23) is described in detail in Japanese patent application No. 11-93741 and U.S. patent application Ser. No. 09/153,032, and will not be described in detail below. 
     In the present embodiment, the sliding mode controller  27  determines the sum (Ueq+Urch+Uadp) of the equivalent control input Ueq, the reaching control law input Urch, and the adaptive control law Uadp determined according to the respective equations (18), (19), (22) as the SLD manipulating input Usl to be applied to the object exhaust system E. However, the differential outputs VO 2 (K+d), VO 2 (k+d−1) of the O 2  sensor  6  and the value σ((k+d) of the switching function σ, etc. used in the equations (18), (19), (22) cannot directly be obtained as they are values in the future. 
     According to the present embodiment, therefore, the sliding mode controller  27  actually uses the estimated differential outputs VO 2 (k+d) bar, VO 2 (k+d−1) bar determined by the estimator  26 , instead of the differential outputs VO 2 (k+d), VO 2 (k+d−1) from the O 2  sensor  6  for determining the equivalent control input Ueq according to the equation (18), and calculates the equivalent control input Ueq in each control cycle according to the following equation (24):                      Ueq        (   k   )       =                    -   1     s1b1          {         [       s1   ·     (     a1   -   1     )       +   s2     ]     ·       VO2   _          (     k   +   d     )         +                                    (       s1   ·   a2     -   s2     )     ·       VO2   _          (     k   +   d   -   1     )         }                 (   24   )                         
     According to the present embodiment, furthermore, the sliding mode controller  27  actually uses time-series data of the estimated differential output VO 2  bar sequentially determined by the estimator  26  as described as a state quantity to be controlled, and defines a switching function σ bar according to the following equation (25) (the switching function σ bar corresponds to time-series data of the differential output VO 2  in the equation (15) which is replaced with time-series data of the estimated differential output VO 2  bar), in place of the switching function σ established according to the equation (25): 
     
       
         {overscore (σ( k ))}= s   1 ·{overscore (VO 2 )}( k )+ s   2 ·{overscore (VO 2 )}( k− 1)  (25) 
       
     
     The sliding mode controller  27  calculates the reaching control law input Urch in each control cycle according to the following equation (26), using the switching function σ bar represented by the equation (25), rather than the value of the switching function σ for determining the reaching control law input Urch according to the equation (19):                Urch        (   k   )       =         -   1       s1   ·   b1       ·   F   ·       σ   _          (     k   +   d     )                 (   26   )                         
     Similarly, the sliding mode controller  27  calculates the adaptive control law input Uadp in each control cycle according to the following equation (27), using the value of the switching function σ bar represented by the equation (25), rather than the value of the switching function σ for determining the adaptive control law input Uadp according to the equation (22):                Uadp        (   k   )       =         -   1       s1   ·   b1       ·   G   ·       ∑     i   =   0       k   +   d            (           σ   _          (   i   )       ·   Δ                   T     )                 (   27   )                         
     The latest identified gain coefficients a 1 (k) hat, a 2 (k) hat, b 1 (k) hat which have been determined by the identifier  25  are basically used as the gain coefficients a 1 , a 2 , b 1  that are required to calculate the equivalent control input Ueq, the reaching control law input Urch, and the adaptive control law input Uadp according to the equations (24), (26), (27). 
     The sliding mode controller  27  determines the sum of the equivalent control input Ueq, the reaching control law input Urch, and the adaptive control law input Uadp determined according to the equations (24), (26), (27), as the SLD manipulating input Usl to be applied to the object exhaust system E (see the equation (17)). The conditions for establishing the coefficients s 1 , s 2 , F, G used in the equations (24), (26), (27) are as described above. 
     The above process is a basic algorithm for determining the SLD manipulating input Usl (=target differential air-fuel ratio kcmd) to be applied to the object exhaust system E with the sliding mode controller  27 . According to the above algorithm, the SLD manipulating input Usl is determined to converge the estimated differential output VO 2  bar from the O 2  sensor  6  toward “0”, and as a result, to convert the output VO 2 /OUT from the O 2  sensor  6  toward the target value VO 2 /TARGET. 
     The sliding mode controller  27  eventually sequentially determines the target air-fuel ratio KCMD in each control cycle. The SLD manipulating input Usl determined as described above signifies a target value for the difference between the air-fuel ratio of the exhaust gas detected by the LAF sensor  5  and the reference value FLAF/BASE, i.e., the target differential air-fuel ratio kcmd. Consequently, the sliding mode controller  27  eventually determines the target air-fuel ratio KCMD by adding the reference value FLAF/BASE to the determined SLD manipulating input Usl in each control cycle according to the following equation (28):                      KCMD        (   k   )       =                  Usl        (   k   )       +     FLAF        /        BASE                   =                  Ueq        (   k   )       +     Urch        (   k   )       +     Uadp        (   k   )       +     FLAF        /        BASE                     (   28   )                         
     The above process is a basic algorithm for determining the target air-fuel ratio KCMD with the sliding mode controller  27  according to the present embodiment. 
     In the present embodiment, the stability of the adaptive sliding mode control process carried out by the sliding mode controller  27  is checked for limiting the value of the SLD manipulating input Usl. Details of such a checking process will be described later on. 
     A process carried out by the deteriorated state evaluating means  13   b  will be described below. 
     Various studies conducted by the inventors have revealed that when the target air-fuel ratio KCMD is determined by the sliding mode controller  27  and the fuel injection quantity of the internal combustion engine  1  is adjusted by the engine-side control unit  7   b  is adjusted in order converge the output KACT (the detected value of the air-fuel ratio of the internal combustion engine  1 ) of the LAF sensor to the target air-fuel ratio KCMD, the time-series data of the output VO 2 /OUT of the O 2  sensor  6  exhibits characteristic changes depending on the deteriorated state of the catalytic converter  3  with respect to the hyperplane σ=0. 
     Such characteristic changes will be described below with reference to FIGS. 5 through 7. FIGS. 5 through 7 show sampled data, represented by stippled dots, of the set of the time-series data VO 2 (k), VO 2 (k−1) of the differential output VO 2  of the O 2  sensor  6 , i.e., the state quantity X, which are obtained in respective control cycles of the exhaust-side control unit  7   a  when the fuel injection quantity of the internal combustion engine  1  is adjusted depending on the target air-fuel ratio KCMD, with respect to a brand-new catalytic converter  3 , a catalytic converter  3  that has been deteriorated to a relatively small degree, and a catalytic converter  3  that has been deteriorated to a relatively large degree. In each of FIGS. 5 through 7, the straight line represents the hyperplane σ=0. 
     As shown in FIG. 5, when the catalytic converter  3  is brand-new, the state quantities X tend to concentrate in the vicinity of the hyperplane σ=0. As the deterioration of the catalytic converter  3  progresses, as shown in FIGS. 6 and 7, the state quantities X tend to be distributed away from the hyperplane σ=0, i.e., vary in a larger range around the hyperplane σ=0. The state quantities X tend to be distributed more widely as the catalytic converter  3  is deteriorated to a larger degree, i.e., as the deterioration of the catalytic converter  3  progresses more greatly. Stated otherwise, as the deterioration of the catalytic converter  3  progresses, the switching function σ determined according to the equation (15) is likely to have a value more remote from “0”, and hence vary more widely from “0”. This appears to be due to the fact that as the deterioration of the catalytic converter  3  progresses, the exhaust system model according to the equation (1) tends to suffer an error, and hence the ability of the state quantity X to converge to the hyperplane σ=0 is lowered. 
     The above tendency is also exhibited by the value of the switching function σ bar determined by the equation (25) where the estimated differential output VO 2  bar determined by the estimator  26  is used as a variable component, i.e., the switching function used actually as the switching function for the sliding mode control process. Whereas the switching function σ bar employs the estimated value of the differential output VO 2  of the O 2  sensor  6 , the switching function according to the equation (15) employs the actual differential output VO 2  of the O 2  sensor  6 . Therefore, the latter appears to better reflect the actual deteriorated state of the catalytic converter  3 . 
     For the above reason, according to the present embodiment, the deteriorated state of the catalytic converter  3  is evaluated based on the value of the switching function σ according to the equation (15). As described above, the actual switching function for the sliding mode control process is the switching function σ bar defined according to the equation (25) where the estimated differential output VO 2  bar determined by the estimator  26  is used as a variable component. Strictly, the switching function σ according to the equation (15) is not the switching function for the sliding mode control process according to the present embodiment. The switching function σ according to the equation (15) will hereinafter be referred to as “deterioration evaluating linear function σ”. 
     An algorithm for evaluating the deteriorated state of the catalytic converter  3  based on the deterioration evaluating linear function σ with the deteriorated state evaluating means  13   b  is constructed as follows: 
     In view of the above tendency of the deterioration evaluating linear function σ to change as the deterioration of the catalytic converter  3  progresses, the deteriorated state evaluating means  13   b  sequentially determines the square σ 2  of the value of the deterioration evaluating linear function σ in each control cycle. 
     Then, the deteriorated state evaluating means  13   b  effects a low-pass filtering process on the square σ 2  to determine a central value of the square σ 2  (hereinafter represented by LSσ 2 ) as a deterioration evaluating parameter. 
     The above filtering process to determine the deterioration evaluating parameter LSσ 2  comprises a sequential statistic processing algorithm, and is represented by the following equation (29):                      LS                     σ   2          (   k   )         =                  LS                     σ   2          (     k   -   1     )         +         BP        (     k   -   1     )         1   +     BP        (     k   -   1     )           ·                                (         σ   2          (   k   )       -     LS                     σ   2          (     k   -   1     )           )                   (   29   )                         
     Thus, the deterioration evaluating parameter LSσ 2  is determined while being sequentially updated, in each control cycle of the exhaust-side control unit  7   a , from a previous value LSσ 2 (k−1) of the deterioration evaluating parameter LSσ 2 , a present value σ 2 (k) of the square σ 2 , and a gain parameter BP updated in each control cycle by a recursive formula expressed by the following equation (30):                BK        (   k   )       =       1     η                 1       ·     (     1   -       η                   2   ·     BP        (     k   -   1     )               η                 1     +     η                   2   ·     BP        (     k   -   1     )                 )     ·     BP        (     k   -   1     )                 (   30   )                         
     In the equation (30), η 1 , η 2  are set to values that satisfy the conditions: 0&lt;η 1 ≦1 and 0≦η 2 &lt;2. Depending on how the values of η 1 , η 2  are set, various specific algorithms including a fixed gain method, a degression method, a method of weighted least squares, a method of least squares, a fixed tracing method, etc. are constructed. According to the present embodiment, η 1  is set to a given positive value smaller than “1” (0&lt;η 1 &lt;1), and η 2 =1, and the algorithm of the method of weighted least squares is employed. 
     When the deterioration evaluating parameter LSσ 2  as the central value (the central value of the minimum square in the present embodiment) of the square σ 2  of the deterioration evaluating linear function σ is determined, the value of the deterioration evaluating parameter LSσ 2  exhibits a tendency shown in FIG. 8 with respect to the deteriorated state of the catalytic converter  3 . FIG. 8 shows the relationship between the deterioration evaluating parameter LSσ 2  determined as described above and the rate of flow of the exhaust gas through the catalytic converter  3  (hereinafter referred to as “exhaust gas volume”) at the respective deteriorated states of the catalytic converters  3  shown in FIGS. 5 through 7. 
     As shown in FIG. 8, the deterioration evaluating parameter LSσ 2  remains substantially constant irrespective of the exhaust gas volume at each of the deteriorated states of the catalytic converters  3 , and increases its value as the deterioration of the catalytic converters  3  progresses. Therefore, the deterioration evaluating parameter LSσ 2  represents the degree to which the catalytic converter  3  is deteriorated. 
     In the present embodiment, the deteriorated state of the catalytic converter  3  is evaluated to judge whether the catalytic converter  3  is in a state where it has been deteriorated to the extent that it needs to be replaced immediately or soon (such a deteriorated state will hereinafter be referred to as “deterioration-in-progress state”, or not (a state of the catalytic converter  3  which is not in the deterioration-in-progress state will hereinafter be referred to as “non-deteriorated state”). The deterioration-in-progress state is indicated by the deterioration indicator  29 . 
     As indicated by the broken line in FIG. 8, a threshold CATAGELMT is preset with respect to the deterioration evaluating parameter LSσ 2 . If the deterioration evaluating parameter LSσ 2  is equal to or greater than the threshold CATAGELMT, then the catalytic converter  3  is judged as being in the deterioration-in-progress state. If the deterioration evaluating parameter LSσ 2  is smaller than the threshold CATAGELMT, then the catalytic converter  3  is judged as being in the non-deteriorated state. 
     The algorithm described above is a basic algorithm for evaluating the deteriorated state of the catalytic converter  3  with the deteriorated state evaluating means  13   b . The deteriorated state evaluating means  13   b  also performs an additional process of recognizing how the exhaust gas volume changes upon evaluating the deteriorated state of the catalytic converter  3 . Such an additional process of recognizing how the exhaust gas volume changes will be described later on. 
     The general feedback controller  15  of the engine-side control unit  7   b , particularly, the adaptive controller  18 , will further be described below. 
     In FIG. 1, the general feedback controller  15  effects a feedback control process to converge the output KACT (the detected value of the air-fuel ratio) from the LAF sensor  5  toward the target air-fuel ratio KCMD as described above. If such a feedback control process were carried out under the known PID control only, it would be difficult keep stable controllability against dynamic behavioral changes including changes in the operating conditions of the internal combustion engine  1 , characteristic changes due to aging of the internal combustion engine  1 , etc. 
     The adaptive controller  18  is a recursive-type controller which makes it possible to carry out a feedback control process while compensating for dynamic behavioral changes of the internal combustion engine  1 . As shown in FIG. 9, the adaptive controller  18  comprises a parameter adjuster  30  for establishing a plurality of adaptive parameters using the parameter adjusting law proposed by I. D. Landau, et al., and a manipulated variable calculator  31  for calculating the feedback manipulated variable KSTR using the established adaptive parameters. 
     The parameter adjuster  30  will be described below. According to the parameter adjusting law proposed by I. D. Landau, et al., when polynomials of the denominator and the numerator of a transfer function B(Z −1 )/A(Z −1 ) of a discrete-system object to be controlled are generally expressed respectively by equations (31), (32), given below, an adaptive parameter θ hat (j) (j indicates the ordinal number of a control cycle) established by the parameter adjuster  30  is represented by a vector (transposed vector) according to the equation (33) given below. An input ξ(j) to the parameter adjuster  30  is expressed by the equation (34) given below. 
     In the present embodiment, it is assumed that the internal combustion engine  1 , which is an object to be controlled by the general feedback controller  15 , is considered to be a plant of a first-order system having a dead time dp corresponding to the time of three combustion cycles of the internal combustion engine  1 , and m=n=1, dp=3 in the equations (31)-(34), and five adaptive parameters s 0 , r 1 , r 2 , r 3 , b 0  are established (see FIG.  9 ). In the upper and middle expressions of the equation (34), us, ys generally represent an input (manipulated variable) to the object to be controlled and an output (controlled variable) from the object to be controlled. In the present embodiment, the input is the feedback manipulated variable KSTR and the output from the object (the internal combustion engine  1 ) is the output KACT (detected air-fuel ratio) from the LAF sensor  5 , and the input ζ(j) to the parameter adjuster  30  is expressed by the lower expression of the equation (34) (see FIG.  9 ). 
     
       
           A ( Z   −1 )=1 +a   1   Z   −1   + . . . +anZ   −n   (31) 
       
     
     
       
           B ( Z   −1 )= b   0 + b   1   Z   −1   + . . . +bmZ   −m   (32) 
       
     
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             
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     The adaptive parameter θ hat expressed by the equation (33) is made up of a scalar quantity element b 0  hat (j) for determining the gain of the adaptive controller  18 , a control element BR hat (Z −1 ,j) expressed using a manipulated variable, and a control element S (Z −1 ,j) expressed using a controlled variable, which are expressed respectively by the following equations (35)-(37) (see the block of the manipulated variable calculator  31  shown in FIG.  9 ):                  b   ^          0     -   1            (   j   )       =     1   b0             (   35   )                         B   ^          R        (       Z     -   1       ,   j     )         =                  r1Z     -   1       +     r2Z     -   2       +   ⋯              +   rm   +   dp   -     1        Z     -     (     n   +   dp   -   1     )                         =                  r1Z     -   1       +     r2Z     -   2       +     r3Z     -   3                       (   36   )                         S   ^          (       Z     -   1       ,   j     )       =                s0   +     s1Z     -   1       +   ⋯   +   sn   -     1        Z     -     (     n   -   1     )                         =              s0                 (   37   )                         
     The parameter adjuster  30  establishes coefficients of the scalar quantity element and the control elements, described above, and supplies them as the adaptive parameter θ hat expressed by the equation (33) to the manipulated variable calculator  31 . The parameter adjuster  30  calculates the adaptive parameter θ hat so that the output KACT from the LAF sensor  5  will agree with the target air-fuel ratio KCMD, using time-series data of the feedback manipulated variable KSTR from the present to the past and the output KACT from the LAF sensor  5 . 
     Specifically, the parameter adjuster  30  calculates the adaptive parameter θ hat according to the following equation (38): 
     
       
         {circumflex over (θ)}( j )={circumflex over (θ)}( j −1)+Γ( j −1)·ζ( j−dp )· e *( j )  (38) 
       
     
     where Γ(j) represents a gain matrix (whose degree is indicated by m+n+dp) for determining a rate of establishing the adaptive parameter θ hat, and e*(j) an estimated error of the adaptive parameter θ hat. Γ(j) and e*(j) are expressed respectively by the following recursive formulas (39), (40):                      Γ        (   j   )       =                  1     λ                 1        (   j   )         ·                              [       Γ        (     j   -   1     )       -       λ                 2          (   j   )     ·     Γ        (     j   -   1     )       ·     ζ        (     j   -   dp     )       ·     ζ        (     j   -     dp   T       )       ·     Γ        (     j   -   1     )               λ                 1        (   j   )       +     λ                 2          (   j   )     ·       ζ   T          (     j   -   dp     )       ·     Γ        (     j   -   1     )       ·     ζ        (     j   -   dp     )                 ]                   (   39   )                         
     where 0&lt;λ 1 (j)≦1, 0≦λ 2 (j)&lt;2, Γ(0)&gt;0.                e   *     (   j   )       =           D        (     Z     -   1       )       ·     KACT        (   j   )         -           θ   ^     T          (     j   -   1     )       ·     ζ        (     j   -   dp     )             1   +         ζ   T          (     j   -   dp     )       ·     Γ        (     j   -   1     )       ·     ζ        (     j   -   dp     )                     (   40   )                         
     where D(Z −1 ) represents an asymptotically stable polynomial for adjusting the convergence. In the present embodiment, D(Z −1 )=1. 
     Various specific algorithms including the degressive gain algorithm, the variable gain algorithm, the fixed tracing algorithm, and the fixed gain algorithm are obtained depending on how λ 1 (j), λ 2 (j) in the equation (39) are selected. For a time-dependent plant such as a fuel injection process, an air-fuel ratio, or the like of the internal combustion engine  1 , either one of the degressive gain algorithm, the variable gain algorithm, the fixed gain algorithm, and the fixed tracing algorithm is suitable. 
     Using the adaptive parameter θ hat (s 0 , r 1 , r 2 , r 3 , b 0 ) established by the parameter adjuster  30  and the target air-fuel ratio KCMD determined by the deteriorated state evaluating means  13   a  of the exhaust-side main processor  13 , the manipulated variable calculator  31  determines the feedback manipulated variable KSTR according to a recursive formula expressed by the following equation (41):                      KSTR        (   j   )       =                  1   b0     ·     [       KCMD        (   j   )       -     s0   ·     KACT        (   j   )         -                                    r1   ·     KSTR        (     j   -   1     )         -     r2   ·                                    KSTR        (     j   -   2     )       -     r3   ·     KSTR        (     j   -   3     )           ]                   (   41   )                         
     The manipulated variable calculator  31  shown in FIG. 9 represents a block diagram of the calculations according to the equation (41). 
     The feedback manipulated variable KSTR determined according to the equation (41) becomes the target air-fuel ratio KCMD insofar as the output KACT of the LAF sensor  5  agrees with the target air-fuel ratio KCMD. Therefore, the feedback manipulated variable KSTR is divided by the target air-fuel ratio KCMD by the divider  19  for thereby determining the feedback manipulated variable kstr that can be used as the feedback correction coefficient KFB. 
     As is apparent from the foregoing description, the adaptive controller  18  thus constructed is a recursive-type controller taking into account dynamic behavioral changes of the engine  1  which is an object to be controlled. Stated otherwise, the adaptive controller  18  is a controller described in a recursive form to compensate for dynamic behavioral changes of the engine  1 , and more particularly a controller having a recursive-type adaptive parameter adjusting mechanism. 
     A recursive-type controller of this type may be constructed using an optimum regulator. In such a case, however, it generally has no parameter adjusting mechanism. The adaptive controller  18  constructed as described above is suitable for compensating for dynamic behavioral changes of the internal combustion engine  1 . 
     The details of the adaptive controller  18  have been described above. 
     The PID controller  17 , which is provided together with the adaptive controller  18  in the general feedback controller  15 , calculates a proportional term (P term), an integral term (I term), and a derivative term (D term) from the difference between the output KACT of the LAF sensor  5  and the target air-fuel ratio KCMD, and calculates the total of those terms as the feedback manipulated variable KLAF, as is the case with the general PID control process. In the present embodiment, the feedback manipulated variable KLAF is set to “1” when the output KACT of the LAF sensor  5  agrees with the target air-fuel ratio KCMD by setting an initial value of the integral term (I term) to “1”, so that the feedback manipulated variable KLAF can be used as the feedback correction coefficient KFB for directly correcting the fuel injection quantity. The gains of the proportional term, the integral term, and the derivative term are determined from the rotational speed and intake pressure of the internal combustion engine  1  using a predetermined map. 
     The switcher  20  of the general feedback controller  15  outputs the feedback manipulated variable KLAF determined by the PID controller  17  as the feedback correction coefficient KFB for correcting the fuel injection quantity if the combustion in the internal combustion engine  1  tends to be unstable as when the temperature of the coolant of the internal combustion engine  1  is low, the internal combustion engine  1  rotates at high speeds, or the intake pressure is low, or if the output KACT of the LAF sensor  5  is not reliable due to a response delay of the LAF sensor  5  as when the target air-fuel ratio KCMD changes largely or immediately after the air-fuel ratio feedback control process has started, or if the internal combustion engine  1  operates highly stably as when it is idling and hence no high-gain control process by the adaptive controller  18  is required. Otherwise, the switcher  20  outputs the feedback manipulated variable kstr which is produced by dividing the feedback manipulated variable KSTR determined by the adaptive controller  18  by the target air-fuel ration KCMD, as the feedback correction coefficient KFB for correcting the fuel injection quantity. This is because the adaptive controller  18  effects a high-gain control process and functions to converge the output KACT of the LAF sensor  5  quickly toward the target air-fuel ratio KCMD, and if the feedback manipulated variable KSTR determined by the adaptive controller  18  is used when the combustion in the internal combustion engine  1  is unstable or the output KACT of the LAF sensor  5  is not reliable, then the air-fuel ratio control process tends to be unstable. 
     Such operation of the switcher  20  is disclosed in detail in Japanese laid-open patent publication No. 8-105345 or U.S. Pat. No. 5,558,075, and will not be described in detail below. 
     Operation of the entire apparatus according to the present embodiment will be described below. 
     First, a process, carried out by the engine-side control unit  7   b , of calculating an output fuel injection quantity #nTout (n=1, 2, 3, 4) for each of the cylinders of the internal combustion engine  1  for controlling the air-fuel ratio of the internal combustion engine  1  will be described below with reference to FIG.  10 . The engine-side control unit  7   b  calculates an output fuel injection quantity #nTout (n=1, 2, 3, 4) for each of the cylinders in synchronism with a crankshaft angle period (TDC) of the internal combustion engine  1  as follows: 
     In FIG. 10, the engine-side control unit  7   b  reads outputs from various sensors including the LAF sensor  5  and the O 2  sensor  6  in STEPa. At this time, the output KACT of the LAF sensor  5  and the output VO 2 /OUT of the O 2  sensor  6 , including data obtained in the past, are stored in a time-series fashion in a memory (not shown). 
     Then, the basic fuel injection quantity calculator  8  corrects a fuel injection quantity corresponding to the rotational speed NE and intake pressure PB of the internal combustion engine  1  depending on the effective opening area of the throttle valve, thereby calculating a basic fuel injection quantity Tim in STEPb. The first correction coefficient calculator  9  calculates a first correction coefficient KTOTAL depending on the coolant temperature and the amount by which the canister is purged in STEPc. 
     The engine-side control unit  7   b  decides whether the operation mode of the internal combustion engine  1  is an operation mode (hereinafter referred to as “normal operation mode”) in which the fuel injection quantity is adjusted using the target air-fuel ratio KCMD generated by the target air-fuel ratio calculating means  13   a , and sets a value of a flag f/prism/on in STEPd. When the value of the flag f/prism/on is “1”, it means that the operation mode of the internal combustion engine  1  is the normal operation mode, and when the value of the flag f/prism/on is “0”, it means that the operation mode of the internal combustion engine  1  is not the normal operation mode. 
     The deciding subroutine of STEPd is shown in detail in FIG.  11 . As shown in FIG. 11, the engine-side control unit  7   b  decides whether the O 2  sensor  6  and the LAF sensor  5  are activated or not respectively in STEPd- 1 , STEPd- 2 . If neither one of the O 2  sensor  6  and the LAF sensor  5  is activated, since detected data from the O 2  sensor  6  and the LAF sensor  5  for use by the exhaust-side main processor  13  are not accurate enough, the operation mode of the internal combustion engine  1  is not the normal operation mode, and the value of the flag f/prism/on is set to “0” in STEPd- 10 . 
     Then, the engine-side control unit  7   b  decides whether the internal combustion engine  1  is operating with a lean air-fuel mixture or not in STEPd- 3 . The engine-side control unit  7   b  decides whether the ignition timing of the internal combustion engine  1  is retarded for early activation of the catalytic converter  3  immediately after the start of the internal combustion engine  1  or not in STEPd- 4 . The engine-side control unit  7   b  decides whether the throttle valve of the internal combustion engine  1  is substantially fully open or not in STEPd- 5 . The engine-side control unit  7   b  decides whether the supply of fuel to the internal combustion engine  1  is being stopped or not in STEPd- 6 . If either one of the conditions of these steps is satisfied, then since it is not preferable or not possible to control the supply of fuel to the internal combustion engine  1  using the target air-fuel ratio KCMD generated by the exhaust-side main processor  13 , the operation mode of the internal combustion engine  1  is not the normal operation mode, and the value of the flag f/prism/on is set to “0” in STEPd- 10 . 
     The engine-side control unit  7   b  then decides whether the rotational speed NE and the intake pressure PB of the internal combustion engine  1  fall within respective given ranges or not respectively in STEPd- 7 , STEPd- 8 . If either one of the rotational speed NE and the intake pressure PB does not fall within its given range, then since it is not preferable to control the supply of fuel to the internal combustion engine  1  using the target air-fuel ratio KCMD generated by the exhaust-side main processor  13 , the operation mode of the internal combustion engine  1  is not the normal operation mode, and the value of the flag f/prism/on is set to “0” in STEPd- 10 . 
     If the conditions of STEPd- 1 , STEPd- 2 , STEPd- 7 , STEPd- 8  are satisfied, and the conditions of STEPd- 3 , STEPd- 4 , STEPd- 5 , STEPd- 6  are not satisfied (at this time, the internal combustion engine  1  is in the normal operation mode), then the operation mode of the internal combustion engine  1  is judged as the normal operation mode, and the value of the flag f/prism/on is set to “1” in STEPd- 9 . 
     In FIG. 10, after the value of the flag f/prism/on has been set, the engine-side control unit  7   b  determines the value of the flag f/prism/on in STEPe. If f/prism/on=1, then the engine-side control unit  7   b  reads the target air-fuel ratio KCMD generated by the exhaust-side main processor  13  in STEPf. If f/prism/on=0, then the engine-side control unit  7   b  sets the target air-fuel ratio KCMD to a predetermined value in STEPg. The predetermined value to be established as the target air-fuel ratio KCMD is determined from the rotational speed NE and intake pressure PB of the internal combustion engine  1  using a predetermined map, for example. 
     In the local feedback controller  16 , the PID controllers  22  calculates respective feedback correction coefficients #nKLAF in order to eliminate variations between the cylinders, based on actual air-fuel ratios #nA/F (n=1, 2, 3, 4) of the respective cylinders which have been estimated from the output KACT of the LAF sensor  5  by the observer  21 , in STEPh. Then, the general feedback controller  15  calculates a feedback correction coefficient KFB in STEPi. 
     Depending on the operating conditions of the internal combustion engine  1 , the switcher  20  selects either the feedback manipulated variable KLAF determined by the PID controller  17  or the feedback manipulated variable kstr which has been produced by dividing the feedback manipulated variable KSTR determined by the adaptive controller  18  by the target air-fuel ratio KCMD (normally, the switcher  20  selects the feedback manipulated variable kstr). The switcher  20  then outputs the selected feedback manipulated variable KLAF or kstr as a feedback correction coefficient KFB. 
     When switching the feedback correction coefficient KFB from the feedback manipulated variable KLAF from the PID controller  17  to the feedback manipulated variable kstr from the adaptive controller  18 , the adaptive controller  18  determines a feedback manipulated variable KSTR in a manner to hold the correction coefficient KFB to the preceding correction coefficient KFB (=KLAF) as long as in the cycle time for the switching. When switching the feedback correction coefficient KFB from the feedback manipulated variable kstr from the adaptive controller  18  to the feedback manipulated variable KLAF from the PID controller  17 , the PID controller  17  calculates a present correction coefficient KLAF in a manner to regard the feedback manipulated variable KLAF determined by itself in the preceding cycle time as the preceding correction coefficient KFB (=kstr). 
     After the feedback correction coefficient KFB has been calculated, the second correction coefficient calculator  10  calculates in STEPj a second correction coefficient KCMDM depending on the target air-fuel ratio KCMD determined in STEPf or STEPg. 
     Then, the engine-side control unit  7   b  multiplies the basic fuel injection quantity Tim determined as described above, by the first correction coefficient KTOTAL, the second correction coefficient KCMDM, the feedback correction coefficient KFB, and the feedback correction coefficients #nKLAF of the respective cylinders, determining output fuel injection quantities #nTout of the respective cylinders in STEPk. The output fuel injection quantities #nTout are then corrected for accumulated fuel particles on intake pipe walls of the internal combustion engine  1  by the fuel accumulation corrector  23  in STEPm. The corrected output fuel injection quantities #nTout are applied to the non-illustrated fuel injectors of the internal combustion engine  1  in STEPn. 
     In the internal combustion engine  1 , the fuel injectors inject fuel into the respective cylinders according to the respective output fuel injection quantities #nTout. 
     The above calculation of the output fuel injection quantities #nTout and the fuel injection of the internal combustion engine  1  are carried out in successive cycle times synchronous with the crankshaft angle period of the internal combustion engine  1  for controlling the air-fuel ratio of the internal combustion engine  1  in order to converge the output KACT of the LAF sensor  5  (the detected air-fuel ratio) toward the target air-fuel ratio KCMD. While the feedback manipulated variable kstr from the adaptive controller  18  is being used as the feedback correction coefficient KFB, the output KACT of the LAF sensor  5  is quickly converged toward the target air-fuel ratio KCMD with high stability against behavioral changes such as changes in the operating conditions of the internal combustion engine  1  or characteristic changes thereof. A response delay of the internal combustion engine  1  is also appropriately compensated for. 
     Concurrent with the above fuel control for the internal combustion engine  1 , the exhaust-side control unit  7   a  executes a main routine shown in FIG. 12 in control cycles of a constant period. 
     As shown in FIG. 12, the exhaust-side control unit  7   a  decides whether the processing of the exhaust-side main processor  13  is to be executed or not, and sets a value of a flag f/prism/cal indicative of whether the processing is to be executed or not in STEP 1 . When the value of the flag f/prism/cal is “0”, it means that the processing of the exhaust-side main processor  13  is not to be executed, and when the value of the flag f/prism/cal is “1”, it means that the processing of the exhaust-side main processor  13  is to be executed. 
     The deciding subroutine in STEP 1  is shown in detail in FIG.  13 . As shown in FIG. 13, the exhaust-side control unit  7   a  decides whether the O 2  sensor  6  and the LAF sensor  5  are activated or not respectively in STEP 1 - 1 , STEP 1 - 2 . If neither one of the O 2  sensor  6  and the LAF sensor  5  is activated, since detected data from the O 2  sensor  6  and the LAF sensor  5  for use by the exhaust-side main processor  13  are not accurate enough, the value of the flag f/prism/cal is set to “0” in STEP 1 - 6 . Then, in order to initialize the identifier  25  as described later on, the value of a flag f/id/reset indicative of whether the identifier  25  is to be initialized or not is set to “1” in STEP 1 - 7 . When the value of the flag f/id/reset is “1”, it means that the identifier  25  is to be initialized, and when the value of the flag f/id/reset is “0”, it means that the identifier  25  is not to be initialized. 
     The exhaust-side control unit  7   a  decides whether the internal combustion engine  1  is operating with a lean air-fuel mixture or not in STEP 1 - 3 . The exhaust-side control unit  7   a  decides whether the ignition timing of the internal combustion engine  1  is retarded for early activation of the catalytic converter  3  immediately after the start of the internal combustion engine  1  or not in STEP 1 - 4 . If the conditions of these steps are satisfied, then since the target air-fuel ratio KCMD calculated to adjust the output VO 2 /OUT of the O 2  sensor  6  to the target value VO 2 /TARGET is not used for the fuel control for the internal combustion engine  1 , the value of the flag f/prism/cal is set to “0” in STEP 1 - 6 , and the value of the flag f/id/reset is set to “1” in order to initialize the identifier  25  in STEP 1 - 7 . 
     In FIG. 12, after the above deciding subroutine, the exhaust-side control unit  7   a  decides whether a process of identifying (updating) the gain coefficients a 1 , a 2 , b 1  with the identifier  25  is to be executed or not, and sets a value of a flag f/id/cal indicative of whether the process of identifying (updating) the gain coefficients a 1 , a 2 , b 1  is to be executed or not in STEP 2 . When the value of the flag f/id/cal is “0”, it means that the process of identifying (updating) the gain coefficients a 1 , a 2 , b 1  is not to be executed, and when the value of the flag f/id/cal is “1”, it means that the process of identifying (updating) the gain coefficients a 1 , a 2 , b 1  is to be executed. 
     In the deciding process of STEP 2 , the exhaust-side control unit  7   a  decides whether the throttle valve of the internal combustion engine  1  is substantially fully open or not, and also decides whether the supply of fuel to the internal combustion engine  1  is being stopped or not. If either one of these conditions is satisfied, then since it is difficult to adjust the gain coefficients a 1 , a 2 , b 1  appropriately, the value of the flag f/id/cal is set to “0”. If neither one of these conditions is satisfied, then the value of the flag f/id/cal is set to “1” to identify (update) the gain coefficients a 1 , a 2 , b 1  with the identifier  25 . 
     Referring back to FIG. 12, the exhaust-side control unit  7   a  calculates the latest differential outputs kact(k) (=KACT(k)−FLAF/BASE), VO 2 (k) (=VO 2 /OUT(k)−VO 2 /TARGET) respectively from the subtractors  11 ,  12  in STEP 3 . Specifically, the subtractors  11 ,  12  select latest ones of the time-series data read and stored in the non-illustrated memory in STEPa shown in FIG. 10, and calculate the differential outputs kact(k), VO 2 (k). The differential outputs kact(k), VO 2 (k), as well as data given in the past, are stored in a Lime-series manner in a memory (not shown) in the exhaust-side control unit  7   a.    
     Then, in STEP 4 , the exhaust-side control unit  7   a  determines the value of the flag f/prism/cal set in STEP 1 . If the value of the flag f/prism/cal is “0”, i.e., if the processing of the exhaust-side main processor  13  is not to be executed, then the exhaust-side control unit  7   a  forcibly sets the SLD manipulating input Usl (the target differential air-fuel ratio kcmd) to be determined by the sliding mode controller  27 , to a predetermined value in STEP 13 . The predetermined value may be a fixed value (e.g., “0”) or the value of the SLD manipulating input Usl determined in a preceding control cycle. 
     After the SLD manipulating input Usl is set to the predetermined value in STEP 12 , the exhaust-side control unit  7   a  adds the reference value FLAF/BASE to the SLD manipulating input Usl for thereby determining a target air-fuel ratio KCMD in the present control cycle in STEP  14 . Then, the processing in the present control cycle is finished. 
     If the value of the flag f/prism/cal is “1” in STEP 4 , i.e., if the processing of the exhaust-side main processor  13  is to be executed, then the exhaust-side control unit  7   a  effects the processing of the identifier  25  in STEP 5 . 
     The processing subroutine of STEP 5  is shown in detail in FIG.  14 . 
     The identifier  25  determines the value of the flag f/id/cal set in STEP 2  in STEP 5 - 1 . If the value of the flag f/id/cal is “0”, then since the process of identifying the gain coefficients a 1 , a 2 , b 1  with the identifier  25  is not carried out, control immediately goes back to the main routine shown in FIG.  12 . 
     If the value of the flag f/id/cal is “1”, then the identifier  25  determines the value of the flag f/id/reset set in STEP 1  with respect to the initialization of the identifier  25  in STEP 5 - 2 . If the value of the flag f/id/reset is “1”, the identifier  25  is initialized in STEP 5 - 3 . When the identifier  25  is initialized, the identified gain coefficients a 1  hat, a 2  hat, b 1  hat are set to predetermined initial values (the identified gain coefficient vector Θ according to the equation (4) is initialized), and the elements of the matrix P (diagonal matrix) according to the equation (9) are set to predetermined initial values. The value of the flag f/id/reset is reset to “0”. 
     Then, the identifier  25  calculates the identified differential output VO 2 (k) hat using the present identified gain coefficients a 1 (k−1) hat, a 2 (k−1) hat, b 1 (k−1) hat and the past data VO 2 (k−1), VO 2 (k−2), kact(k−d−1) of the differential outputs VO 2 , kact calculated in each control cycle in STEP 3 , according to the equation (3) in STEP 5 - 4 . 
     The identifier  25  then calculates the vector Kθ(k) to be used in determining the new identified gain coefficients a 1  hat, a 2  hat, b 1  hat according to the equation (9) in STEP 5 - 5 . Thereafter, the identifier  25  calculates the identified error id/e(k), i.e., the difference between the identified differential output VO 2  hat and the actual differential output VO 2  (see the equation (7)), in STEP 5 - 6 . 
     The identified error id/e(k) obtained in STEP 5 - 6  may basically be calculated according to the equation (7). In the present embodiment, however, a value (=VO 2 (k)−VO 2 (k) hat) calculated according to the equation (7) from the differential output VO 2  acquired in each control cycle in STEP 3  (see FIG.  12 ), and the identified differential output VO 2  hat calculated in each control cycle in STEP 5 - 4  is filtered with low-pass characteristics to calculate the identified error id/e(k). 
     This is because since the object exhaust system E including the catalytic converter  3  generally has low-pass characteristics, it is preferable to attach importance to the low-frequency behavior of the object exhaust system E in appropriately identifying the main coefficients a 1 , a 2 , b 1  of the exhaust system model. 
     Both the differential output VO 2  and the identified differential output VO 2  hat may be filtered with the same low-pass characteristics. For example, after the differential output VO 2  and the identified differential output VO 2  hat have separately been filtered, the equation (7) may be calculated to determine the identified error id/e(k). The above filtering is carried out by a moving average process which is a digital filtering process. 
     Thereafter, the identifier  25  calculates a new identified gain coefficient vector Θ(k), i.e., new identified gain coefficients a 1 (k) hat, a 2 (k) hat, b 1 (k) hat, according to the equation (8) using the identified error id/e(k) determined in STEP 5 - 6  and KO(k) calculated in SETP 5 - 5  in STEP 5 - 7 . 
     After having calculated the new identified gain coefficients a 1 (k) hat, a 2 (k) hat, b 1 (k) hat, the identifier  25  further limits the values of the gain coefficients a 1  hat, a 2  hat, b 1  hat (elements of the identified gain coefficient vector Θ) to meet predetermined conditions in STEP 5 - 8 . Then, the identifier  25  updates the matrix P(k) according to the equation (10) for the processing of a next control cycle in STEP 5 - 9 , after which control returns to the main routine shown in FIG.  12 . 
     The process of limiting the identified gain coefficients a 1  hat, a 2  hat, b 1  hat in STEP 5 - 8  comprises a process of limiting the values of the identified gain coefficients a 1  hat, a 2  hat, b 1  hat to a certain combination, i.e., a process of limiting a point (a 1  hat, a 2  hat) to a given region on a coordinate plane which has the identified gain coefficients a 1  hat, a 2  hat as its component, and a process of limiting the value of the identified gain coefficient b 1  hat within a given range. According to the former process, if a point (a 1 (k) hat, a 2 (k) hat) on the coordinate plane which is determined by the identified gain coefficients a 1 (k) hat, a 2 (k) hat calculated in STEP 5 - 7  deviates from the given region on the coordinate plane, then the values of the identified gain coefficients a 1 (k) hat, a 2 (k) hat are forcibly limited to the values of the point in the given region. According to the latter process, if the value of the identified gain coefficient b 1 (k) hat calculated in STEP 5 - 7  exceeds the upper or lower limit of the given range, then the value of the identified gain coefficient b 1 (k) hat is forcibly limited to the upper or lower limit of the given range. 
     The above process of limiting the identified gain coefficients a 1  hat, a 2  hat, b 1  hat serves to maintain stability of the SLD manipulating input Usl (the target differential air-fuel ratio kcmd) calculated by the sliding mode controller  27 , and hence the target air-fuel ratio KCMD. 
     Specific details of the process of limiting the identified gain coefficients a 1  hat, a 2  hat, b 1  hat are described in Japanese laid-open patent publication No. 11-153051 and U.S. patent application Ser. No. 09/153,300, and will not be described herein. 
     The preceding values a 1 (k−1) hat, a 2 (k−1) hat, b 1 (k−1) hat of the identified gain coefficients used to determine the new identified gain coefficients a 1 (k) hat, a 2 (k) hat, b 1 (k) hat in STEP 5 - 7  are the values of the identified gain coefficients after which have been limited in STEP 5 - 8  in the preceding control cycle. 
     Details of the processing of the identifier  25  in STEP 5  shown in FIG. 12 have been described above. 
     In FIG. 12, after the processing of the identifier  25  has been carried out, the exhaust-side control unit  7   a  determines the values of the gain coefficients a 1 , a 2 , b 1  in STEP 6 . Specifically, if the value of the flag f/id/cal set in STEP 2  is “1”, i.e., if the gain coefficients a 1 , a 2 , b 1  have been identified by the identifier  25 , then-the gain coefficients a 1 , a 2 , b 1  are set to the latest identified gain coefficients a 1 (k) hat, a 2 (k) hat, b 1 (k) hat determined by the identifier  25  in STEP 5  (limited in STEP 5 - 8 ). If the value of the flag f/id/cal is “0”, i.e., if the gain coefficients a 1 , a 2 , b 1  have not been identified by the identifier  25 , then the gain coefficients a 1 , a 2 , b 1  are set to predetermined values, respectively. 
     Then, the exhaust-side control unit  7   a  effects a processing operation of the estimator  26 , i.e., calculates the estimated differential output VO 2  bar, in STEP 7 . 
     The estimator  26  calculates the coefficients α1, α2, βj (j=1, 2, . . . , d) to be used in the equation (13), using the gain coefficients a 1 , a 2 , b 1  determined in STEP 6  (these values are basically the identified gain coefficients a 1  hat, a 2  hat, b 1  hat) according to the equation (12). 
     Then, the estimator  26  calculates the estimated differential output VO 2 (k+d) bar (estimated value of the differential output VO 2  after the total dead time d from the time of the present control cycle) according to the equation (13), using the time-series data VO 2 (k), VO 2 (k−1), from before the present control cycle, of the differential output VO 2  of the O 2  sensor calculated in each control cycle in STEP 3 , the time-series data kact(k−j) (j=0, . . . , d 1 ), from before the present control cycle, of the differential output kact of the LAF sensor  5 , the time-series data kcmd(k−j) (=Usl(k−j), j=1, . . . , d 2 −1), from before the preceding control cycle, of the target differential air-fuel ratio kcmd (=the SLD manipulating input Usl) given in each control cycle from the sliding mode controller  27 , and the coefficients α1, α2, βj calculated as described above. 
     Then, the exhaust-side control unit  7   a  calculates the SLD manipulating input Usl (=the target differential air-fuel ratio kcmd) with the sliding mode controller  27  in STEP 8 . 
     Specifically, the sliding mode controller  27  calculates a value σ(k+d) bar (corresponding to an estimated value, after the total dead time d, of the linear function σ defined according to the equation (15)), after the total dead time d from the present control cycle, of the switching function σ bar defined according to the equation (25), using the time-series data VO 2 (k+d) bar, VO 2 (k+d−1) bar of the estimated differential output VO 2  bar determined by the estimator  26  in STEP 7 . 
     At this time, the sliding mode controller  27  keeps the value of the switching function σ bar within a predetermined allowable range. If the value σ(k+d) bar determined as described above exceeds the upper or lower limit of the allowable range, then the sliding mode controller  27  forcibly limits the value σ(k+d) bar to the upper or lower limit of the allowable range. This is because if the value of the switching function σ bar were excessive, the reaching control law input Urch would be excessive, and the adaptive control law Uadp would change abruptly, tending to impair the stability of the process of converging the output VO 2 /OUT of the O 2  sensor  6  to the target value VO 2 /TARGET. 
     Then, the sliding mode controller  27  accumulatively adds values σ(k+d) bar·ΔT, produced by multiplying the value σ(k+d) bar of the switching function σ bar by the period ΔT (constant period) of the control cycles of the exhaust-side control unit  7   a . That is, the sliding mode controller  27  adds the product σ(k+d) bar·ΔT of the value σ(k+d) bar and the period ΔT calculated in the present control cycle to the sum determined in the preceding control cycle, thus calculating an integrated value a bar (hereinafter represented by “Σσ bar”) which is the calculated result of the term Σ(σ bar·ΔT) of the equation (27). 
     In the present embodiment, the sliding mode controller  27  keeps the integrated value Σσ bar in a predetermined allowable range. If the integrated value Σσ bar exceeds the upper or lower limit of the allowable range, then the sliding mode controller  27  forcibly limits the integrated value Σσ bar to the upper or lower limit of the allowable range. This is because if the integrated value Σσ bar were excessive, the adaptive control law Uadp determined according to the equation (27) would be excessive, tending to impair the stability of the process of converging the output VO 2 /OUT of the O 2  sensor  6  to the target value VO 2 /TARGET. 
     Then, the sliding mode controller  27  calculates the equivalent control input Ueq, the reaching control law input Urch, and the adaptive control law Uadp according to the respective equations (24), (26), (27), using the time-series data VO 2 (k+d)bar, VO 2 (k+d−1) bar of the present and past values of the estimated differential output VO 2  bar determined by the estimator  26  in STEP 7 , the value σ(k+d) bar of the switching function σ bar and its integrated value Σσ bar which are determined as described above, and the gain coefficients a 1 , a 2 , b 1  determined in STEP  6  (which are basically the gain coefficients a 1 (k) hat, a 2 (k) hat, b 1 (k) hat). 
     The sliding mode controller  27  then adds the equivalent control input Ueq, the reaching control law input Urch, and the adaptive control law Uadp to calculate the SLD manipulating input Usl, i.e., the input (=the target differential air-fuel ratio kcmd) to be applied to the object exhaust system E for converging the output signal VO 2 /OUT of the O 2  sensor  6  toward the target value VO 2 /TARGET. 
     After the SLD manipulating input Usl has been calculated, the exhaust-side control unit  7   a  determines the stability of the adaptive sliding mode control process carried out by the sliding mode controller  27 , or more specifically, the ability of the controlled state of the output VO 2 /OUT of the O 2  sensor  6  based on the adaptive sliding mode control process (hereinafter referred to as “SLD controlled state”), and sets a value of a flag f/sld/stb indicative of whether the SLD controlled state is stable or not in STEP 9 . 
     The determining subroutine of STEP 9  is shown in detail in FIG.  15 . 
     As shown in FIG. 15, the exhaust-side control unit  7   a  calculates a difference Δσ bar (corresponding to a rate of change of the switching function σ bar) between the present value σ(k+d) bar of the switching function σ bar calculated in STEP 8  and a preceding value σ(k+d−1) bar thereof in STEP 9 - 1 . 
     Then, the exhaust-side control unit  7   a  decides whether or not a product Δσ bar ·σ(k+d) bar (corresponding to the time-differentiated function of a Lyapunov function σ bar 2 /2 relative to the σ bar) of the difference Δσ bar and the present value σ(k+d) bar is equal to or smaller than a predetermined value ε(≦0) in STEP 9 - 2 . 
     The difference Δσ bar ·σ(k+d) bar (hereinafter referred to as “stability determining parameter Pstb”) will be described below. If the stability determining parameter Pstb is greater than 0 (Pstb&gt;0), then the value of the switching function σ bar is basically changing away from “0”. If the stability determining parameter Pstb is equal to or smaller than 0 (Pstb≧0), then the value of the switching function σ bar is basically converged or converging to “0”. Generally, in order to converge a controlled variable to its target value according to the sliding mode control process, it is necessary that the value of the switching function be stably converged to “0”. Basically, therefore, it is possible to determine whether the SLD controlled state is stable or unstable depending on whether or not the value of the stability determining parameter Pstb is equal to or smaller than 0. 
     If, however, the stability of the SLD controlled state is determined by comparing the value of the stability determining parameter Pstb with “0”, then the determined result of the stability is affected even by slight noise contained in the value of the switching function σ bar. According to the present invention, therefore, the predetermined value ε with which the stability determining parameter Pstb is to be compared in STEP 9 - 2  is of a positive value slightly greater than “0”. 
     If Pstb&gt;ε in STEP 9 - 2 , then the SLD controlled state is judged as being unstable, and the value of a timer counter tm (count-down timer) is set to a predetermined initial value T M  (the timer counter tm is started) in order to inhibit the determination of the target air-fuel ratio KCMD using the SLD manipulating input Usl calculated in STEP 8  for a predetermined time in STEP 9 - 4 . Thereafter, the value of the flag f/sld/stb is set to “0” in STEP 9 - 5 , after which control returns to the main routine shown in FIG.  12 . 
     If Pstb≦ε in STEP 9 - 2 , then the exhaust-side control unit  7   a  decides whether the present value σ(k+d) bar of the switching function σ bar falls within a predetermined range or not in STEP 9 - 3 . 
     If the present value σ(k+d) bar of the switching function σ bar does not fall within the predetermined range, then since the present value σ(k+d) bar spaced widely apart from “0”, the SLD controlled state is considered to be unstable. Therefore, if the present value σ(k+d) bar of the switching function σ bar does not fall within the predetermined range in STEP 9 - 3 , then the SLD controlled state is judged as being unstable, and the processing of STEP 9 - 4  through STEP 9 - 5  is executed to start the timer counter tm and set the value of the flag f/sld/stb to “0”. 
     In the present embodiment, since the value of the switching function σ bar is limited within the allowable range in STEP 8 , the decision processing in STEP 9 - 3  may be dispensed with. 
     If the present value σ(k+d) bar of the switching function σ bar falls within the predetermined range in STEP 9 - 3 , then the exhaust-side control unit  7   a  counts down the timer counter tm for a predetermined time Δtm in STEP 9 - 6 . The exhaust-side control unit  7   a  then decides whether or not the value of the timer counter tm is equal to or smaller than “0”, i.e., whether a time corresponding to the initial value T M  has elapsed from the start of the timer counter tm or not, in STEP 9 - 7 . 
     If tm&gt;0, i.e., if the timer counter tm is still measuring time and its set time has not yet elapsed, then since no substantial time has elapsed after the SLD controlled state is judged as unstable in STEP 9 - 2  or STEP 9 - 3 , the SLD controlled state tends to become unstable. Therefore, if tm&gt;0 in STEP 9 - 7 , then the value of the flag f/sld/stb is set to “0” in STEP 9 - 5 . 
     If tm≦0 in STEP 9 - 7 , i.e., if the set time of the timer counter tm has elapsed, then the SLD controlled stage is judged as being stable, and the value of the flag f/sld/stb is set to “1” in STEP 9 - 8 . 
     According to the above processing, if the SLD controlled state is judged as being unstable, then the value of the flag f/sld/stb is set to “0”, and if the SLD controlled state is judged as being stable, then the value of the flag f/sld/stb is set to “1”. 
     In the present embodiment, the above process of determining the stability of the SLD controlled state is by way of illustrative example only. The stability of the SLD controlled state may be determined by any of various other processes. For example, in each given period longer than the control cycle, the frequency with which the value of the stability determining parameter Pstb in the period is greater than the predetermined value ε is counted. If the frequency is in excess of a predetermined value, then the SLD controlled state is judged as unstable. Otherwise, the SLD controlled state is judged as stable. 
     Referring back to FIG. 12, after a value of the flag f/sld/stb indicative of the stability of the SLD controlled state has been set, the exhaust-side control unit  7   a  determines the value of the flag f/sld/stb in STEP 10 . If the value of the flag f/sld/stb is “1”, i.e., if the SLD controlled state is judged as being stable, then the sliding mode controller  27  limits the SLD manipulating input Usl calculated in STEP  8  in STEP 11 . Specifically, the sliding mode controller  27  determines whether the present value of the SLD manipulating input Usl calculated in STEP 8  falls in a predetermined allowable range or not. If the present value of the SLD manipulating input Usl exceeds the upper or lower limit of the allowable range, then the sliding mode controller  27  forcibly limits the present value Usl(k) of the SLD manipulating input Usl to the upper or lower limit of the allowable range. 
     The SLD manipulating input Usl (=the target differential air-fuel ratio kcmd) limited in STEP 11  is stored in a memory (not shown) in a time-series fashion, and will be used in the processing operation of the estimator  26 . 
     Then, the deteriorated state evaluating means  13   b  of the exhaust-side main processor  13  performs the process of evaluating the deteriorated state of the catalytic converter  3  (described later on) in STEP 12 . The sliding mode controller  27  adds the reference value FLAF/BASE to the SLD manipulating input Usl limited in STEP 11 , thus calculating the target air-fuel ratio KCMD in STEP 14 . The processing in the present control cycle is now put to an end. 
     If f/sld/stb=0 in STEP 10 , i.e., if the SLD controlled state is judged as unstable, then the exhaust-side control unit  7   a  forcibly sets the SLD manipulating input Usl in the present control cycle to a predetermined value (the fixed value or the preceding value of the SLD manipulating input Usl) in STEP 13 . The exhaust-side control unit  7   a  calculates the target air-fuel ratio KCMD according to the equation (28) in STEP  14 . Then, the processing in the present control cycle is finished. 
     The target air-fuel ratio KCMD finally determined in STEP 14  is stored in a memory (not shown) in a time-series fashion in each control cycle. When the general feedback controller  15  is to use the target air-fuel ratio KCMD determined by the exhaust-side control unit  7   a  (see STEPf in FIG.  10 ), the latest one of the time-series data of the target air-fuel ratio KCMD thus stored is selected. 
     The process of evaluating the deteriorated state of the catalytic converter  3  in STEP 12  will be described below with reference to FIG.  16 . 
     The deteriorated state evaluating means  13   b  calculates the value of the deterioration evaluating linear function σ determined according to the equation (15), from the time-series data VO 2 (k), VO 2 (k−1) of the differential output VO 2  of the O 2  sensor  6  which is calculated in STEP 3  shown in FIG. 12, i.e., the present value of the differential output VO 2  and the past value thereof in the preceding control cycle, in STEP 12 - 1 . 
     The values of the coefficients s 1 , s 2  used to calculate the value of the deterioration evaluating linear function σ are identical to the values of the coefficients s 1 , s 2  used by the sliding mode controller  27  to determine the value of the switching function σ bar. 
     Then, the deteriorated state evaluating means  13   b  determines the value of a flag F/DONE in STEP 12 - 2 . When the value of the flag F/DONE is “1”, then it indicates that the evaluation of the deteriorated state of the catalytic converter  3  is completed during the present operation of the internal combustion engine  1 , and when the value of the flag F/DONE is “0”, then it indicates that the evaluation of the deteriorated state of the catalytic converter  3  is not completed during the present operation of the internal combustion engine  1 . When the internal combustion engine  1  starts to operate, the value of the flag F/DONE is initialized to “0”. 
     If F/DONE=0, i.e., if the evaluation of the deteriorated state of the catalytic converter  3  is not completed, then the deteriorated state evaluating means  13   b  performs a process of determining a varying state of the exhaust gas volume (the rate of flow of the exhaust gas through the exhaust pipe  2 ) in STEP 12 - 3 . More specifically, the deteriorated state evaluating means  13   b  determines whether the exhaust gas volume is kept at a substantially constant level, i.e., in a cruise state, or not, and sets the value of a flag F/CRS. When the value of the flag F/CRS is “1”, then it indicates that the exhaust gas volume is in the cruise state, and when the value of the flag F/CRS is “0”, then it indicates that the exhaust gas volume is not in the cruise state. The process of determining a varying state of the exhaust gas volume is carried out in a period of 1 second, for example (hereinafter referred to as “exhaust gas volume variation determining period”) longer than the period (30-100 ms) of the control cycles of the exhaust-side control unit  7   a , and is shown in detail in FIG.  17 . 
     As shown in FIG. 17, the deteriorated state evaluating means  13   b  calculates an estimated value ABSV of the present exhaust gas volume (hereinafter referred to as “estimated exhaust gas volume”) from the detected data of the present rotational speed NE and intake pressure PB of the internal combustion engine  1  according to the following equation (42) in STEP 12 - 3 - 1 :              ABSV   =       NE   1500     ·   PB   ·   SVPRA             (   42   )                         
     In the present embodiment, the exhaust gas volume when the rotational speed of the internal combustion engine  1  is 1500 rpm is used as a reference. Therefore, the detected value of the rotational speed NE is divided by “1500” in the above equation (42). In the equation (42), SVPRA represents a predetermined constant depending on the displacement of the internal combustion engine  1 . 
     Instead of estimating the exhaust gas volume as described above, the exhaust gas volume may be estimated from the fuel supply quantity and intake air quantity of the internal combustion engine  1  or may be directly detected using flow sensor. 
     Then, the deteriorated state evaluating means  13   b  effects a predetermined filtering process on the estimated exhaust gas volume calculated in STEP 12 - 3 - 1  in each exhaust gas volume variation determining period for thereby determining an exhaust gas volume variation parameter SVMA that represents the varying state of the exhaust gas volume in STEP 12 - 3 - 2 . 
     The above filtering process is expressed by the following equation (43): 
     
       
         SVMA=(ABSV( n )−ABSV( n −1))+(ABSV( n −2)−ABSV( n −3))+(ABSV( n −4)−ABSV( n −5))  (43) 
       
     
     Specifically, the exhaust gas volume variation parameter SVMA is calculated by determining a moving average of changes of the estimated exhaust gas volume ABSV over a plurality of exhaust gas volume variation determining periods (three exhaust gas volume variation determining periods in the present embodiment). In the equation (43), “n” represents the ordinal number of the cycle of the exhaust gas volume variation determining period. 
     The exhaust gas volume variation parameter SVMA thus calculated represents a rate of change in the estimated exhaust gas volume ABSV. Consequently, as the value of the exhaust gas volume variation parameter SVMA is closer to “0”, the time-dependent change of the estimated exhaust gas volume ABSV is smaller, i.e., the estimated exhaust gas volume ABSV is substantially constant. 
     Then, the deteriorated state evaluating means  13   b  compares the square of the exhaust gas volume variation parameter SVMA, i.e., the square SVMA 2 , with a predetermined value δ in STEP 12 - 3 - 3 . The predetermined value δ is a positive value near “0”. 
     If SVMA 2 ≦δ, i.e., if the present exhaust gas volume suffers a relatively large variation, then the deteriorated state evaluating means  13   b  sets the value of a timer counter (count-down timer) TMCRSJUD to a predetermined initial value X/TMCRSJST in STEP 12 - 3 - 4 . As the exhaust gas volume is not in the cruise state, i.e., the exhaust gas volume is not kept at a substantially constant level, the deteriorated state evaluating means  13   b  sets the flag F/CRS to “0” in STEP 12 - 3 - 5 , after which control returns to the main routine shown in FIG.  16 . 
     If SVMA 2 &lt;δ in STEP 12 - 3 - 3 , i.e., if the present exhaust gas volume suffers a relatively small variation, then the deteriorated state evaluating means  13   b  counts down the value of the timer counter TMCRSJUD by a predetermined value in each exhaust gas volume variation determining period as long as the present exhaust gas volume suffers a relatively small variation, in STEP 12 - 3 - 6 . Then, the deteriorated state evaluating means  13   b  determines whether or not the value of the timer counter TMCRSJUD becomes “0” or smaller, i.e., whether the set time of timer counter TMCRSJUD has elapsed or not, in STEP 12 - 3 - 7 . 
     If TMCRSJUD≦0, i.e., if the set time of the timer counter TMCRSJUD has elapsed, then the deteriorated state evaluating means  13   b  decides that the exhaust gas volume is in the cruise state, and holds the value of the timer counter TMCRSJUD to “0” in STEP 12 - 3 - 8 . Then, the deteriorated state evaluating means  13   b  sets the value of the flag F/CRS to “1” in STEP 12 - 3 - 9 , after which control returns to the main routine shown in FIG.  16 . 
     If TMCRSJUD&gt;0 in STEP 12 - 3 - 7 , i.e., if the set time of the timer counter TMCRSJUD has not elapsed, then the deteriorated state evaluating means  13   b  sets the value of the flag F/CRS to “0” in STEP 12 - 3 - 5 , after which control returns to the main routine shown in FIG.  16 . 
     The processing sequence described above with reference to FIG. 17 represents the processing in STEP 12 - 3  shown in FIG.  16 . According to the processing in STEP 12 - 3 , if the square SVMA 2  of the exhaust gas volume variation parameter SVMA is SVMA 2 &lt;δ, i.e., the variation of the exhaust gas volume is small, continuously for a time, e.g., 10 to 15 seconds, corresponding to the initial value X/TMCRSJST of the timer counter TMCRSJUD, the deteriorated state evaluating means  13   b  decides that the exhaust gas volume is in the cruise state, and sets the value of the flag F/CRS to “1”. Otherwise, the deteriorated state evaluating means  13   b  decides that the exhaust gas volume is not in the cruise state, and sets the value of the flag F/CRS to “0”. 
     The processing in STEP 12 - 3  allows a proper recognition of the state in which the exhaust gas volume is maintained at a substantially constant level. In each control cycle of the exhaust-side control unit  7   a  in one exhaust gas volume variation determining period, the value of the flag F/CRS is kept constant. 
     Referring back to FIG. 16, the deteriorated state evaluating means  13   b  performs a process of calculating the deterioration evaluating parameter LSσ 2  in STEP 12 - 4 . The process of calculating the deterioration evaluating parameter LSσ 2  will be described below with reference to FIG.  18 . 
     The deteriorated state evaluating means  13   b  determines whether certain conditions for calculating the deterioration evaluating parameter LSσ 2  are satisfied or not in STEP 12 - 4 - 1 . The conditions include the value of the flag F/CRS set in STEP 12 - 3  and the value of the flag f/prism/on set by the engine-side control unit  7   b  in STEPd shown in FIG.  10 . 
     If F/CRS=1, i.e., if the exhaust gas volume is in the cruise state, then the deteriorated state evaluating means  13   b  decides that the condition for calculating the deterioration evaluating parameter LSσ 2  (hereinafter referred to as “deterioration evaluating condition”) is not satisfied. Therefore, without calculating the deterioration evaluating parameter LSσ 2 , control goes back to the main routine shown in FIG.  16 . 
     While the exhaust gas volume is in the cruise state, i.e., while exhaust gas volume is maintained at a substantially constant level, the deterioration evaluating parameter LSσ 2  is not calculated for the following reason: In the cruise state, the output VO 2 /OUT of the O 2  sensor  6  is likely to be held stably to the target value VO 2 /TARGET, and hence the value of the deterioration evaluating linear function σ is less apt to change even when the deterioration of the catalytic converter  3  has progressed. In the cruise state, therefore, the value of the deterioration evaluating linear function σ does not tend to have a tendency depending on the deteriorated state of the catalytic converter  3  described above with reference to FIG.  7 . In the present embodiment, therefore, the deterioration evaluating parameter LSσ 2  is not calculated in the cruise state. 
     If f/prism/on=0 in STEP 12 - 4 - 1 , i.e., if the operation mode of the internal combustion engine  1  is other than the normal operation mode in which the fuel supply of the internal combustion engine  1  is controlled depending on the target air-fuel ratio KCMD that is determined by the sliding mode controller  27  of the exhaust-side control unit  7   a , then the deteriorated state evaluating means  13   b  also decides that the deterioration evaluating condition is not satisfied, and does not calculate the deterioration evaluating parameter LSσ 2 , and control returns to the main routine shown in FIG.  16 . This is because for appropriately evaluating the deteriorated state of the catalytic converter  3  with the deterioration evaluating parameter LSσ 2 , it is preferable to determine the deterioration evaluating parameter LSσ 2  using the data of the differential output VO 2  of the O 2  sensor  6  that is obtained while the air-fuel ratio of the internal combustion engine  1  is being controlled depending on the target air-fuel ratio KCMD generated by the sliding mode controller  27  according to the adaptive sliding mode control process. 
     In STEP 12 - 4 - 1 , the deteriorated state evaluating means  13   b  also determines whether the speed of the vehicle with the internal combustion engine  1  mounted thereon is in a predetermined range or not, whether a certain time has elapsed after the startup of the internal combustion engine  1  or not, and whether the catalytic converter  3  has been activated or not. If these conditions are not satisfied, then the deteriorated state evaluating means  13   b  determines that the deterioration evaluating condition is not satisfied. Therefore, without calculating the deterioration evaluating parameter LSσ 2 , control goes back to the main routine shown in FIG.  16 . 
     If the deterioration evaluating condition is satisfied in STEP 12 - 4 - 1  (at this time, F/CRS=0 and f/prism/on=1), then the deteriorated state evaluating means  13   b  calculates the square σ 2  of the deterioration evaluating linear function σ determined in each control cycle of the exhaust-side control unit  7   a  in STEP 12 - 1  shown in FIG. 16 in STEP 12 - 4 - 2 . 
     The deteriorated state evaluating means  13   b  calculates a new deterioration evaluating parameter LSσ 2 (k) from the present value σ 2 (k) of the square σ 2 , the present value LSσ 2 (k−1) of the deterioration evaluating parameter LSσ 2 , and the present value BP(k−1) of the gain parameter BP determined by the recursive formula expressed by the equation (30), according to the equation (29) in STEP 12 - 4 - 3 . 
     After updating the value of the gain parameter BP according to the equation (30) in STEP 12 - 4 - 4 , the deteriorated state evaluating means  13   b  increments, by “1”, the value of a counter CB 1 P which counts the number of times that the deterioration evaluating parameter LSσ 2  and the gain parameter BP are updated, which number corresponds to the number of values of the deterioration evaluating linear function σ used to determine the deterioration evaluating parameter LSσ 2 , in STEP 12 - 4 - 5 . Thereafter, control returns to the main routine shown in FIG.  16 . 
     The values of the deterioration evaluating parameter LSσ 2  and the gain parameter BP which are determined respectively in STEP 12 - 4 - 3  and STEP 12 - 4 - 4  are stored in a nonvolatile memory such as an EEPROM or the like (not shown) when the internal combustion engine  1  is shut off, so that those values will not be lost when the internal combustion engine  1  is not operating. When the internal combustion engine  1  operates next time, the stored values of the deterioration evaluating parameter LSσ 2  and the gain parameter BP are used as their initial values. The initial values of the deterioration evaluating parameter LSσ 2  and the gain parameter BP at the time the internal combustion engine  1  operates for the first time are “0” and “1”, respectively. The value of the counter CB 1 P is initialized to “0” at the time of the startup of the internal combustion engine  1 . 
     In FIG. 16, after calculating (updating) the value of the deterioration evaluating parameter LSσ 2  as described above, the deteriorated state evaluating means  13   b  evaluates the deteriorated state of the catalytic converter  3  based on the deterioration evaluating parameter LSσ 2  in STEP 12 - 5 . The process of evaluating the deteriorated state of the catalytic converter  3  will be described below with reference to FIG.  19 . 
     The deteriorated state evaluating means  13   b  determines whether the present value BP(k) of the gain parameter BP and the preceding value BP(k−1) thereof are substantially equal to each other or not, i.e., whether the gain parameter BP has substantially been converged or not, in STEP 12 - 5 - 1 , and then determines whether or not the value of the counter CB 1 P is equal to or greater than a predetermined value CB 1 CAT, i.e., whether the number of values of the deterioration evaluating linear function σ used to determine the deterioration evaluating parameter LSσ 2  has reached the predetermined value CB 1 CAT or not, in STEP 12 - 5 - 2 . 
     In the present embodiment, if the data of the deterioration evaluating parameter LSσ 2  and the gain parameter BP are not held, i.e., if the values thereof are initialized to “0”, as when the battery of the vehicle (not shown) is temporarily removed before the internal combustion engine  1  is started or as when the internal combustion engine  1  operates for the first time, then the predetermined value to be compared with the value of the counter CB 1 P in STEP 12 - 5 - 2  is set to a value greater than if the data of the deterioration evaluating parameter LSσ 2  and the gain parameter BP are held. 
     If either of the conditions in STEP 12 - 5 - 1  and STEP 12 - 5 - 2  is not satisfied, then the deterioration evaluating parameter LSσ 2  determined in STEP 12 - 4  in the present control cycle is considered to be not sufficiently converged to the central value of the square σ 2  of the deterioration evaluating linear function σ. Therefore, the processing in STEP 12 - 5  is finished without evaluating the deteriorated state of the catalytic converter  3  based on the deterioration evaluating parameter LSσ 2 . 
     If either of the conditions in STEP 12 - 5 - 1  and STEP 12 - 5 - 2  is satisfied, then since the deterioration evaluating parameter LSσ 2  determined in STEP 12 - 4  in the present control cycle is representative of the central value of the square σ 2  of the deterioration evaluating linear function σ, the deteriorated state evaluating means  13   b  compares the deterioration evaluating parameter LSσ 2  with the threshold CATAGELMT shown in FIG. 8 in STEP 12 - 5 - 3 . 
     If LSσ 2 ≧CATAGELMT, then the deteriorated state evaluating means  13   b  decides that the deteriorated state of the catalytic converter  3  is in the deterioration-in-progress state in which it needs to be replaced immediately or soon. The deteriorated state evaluating means  13   b  controls the deterioration indicator  29  to indicate the deteriorated state of the catalytic converter  3  in STEP 12 - 5 - 4 . After setting the value of the flag F/DONE to “1”, indicating that the evaluation of the deteriorated state of the catalytic converter  3  is completed, in STEP 12 - 5 - 5 . The processing in STEP 12 - 5  is now finished. 
     If LSσ 2 &lt;CATAGELMT in STEP 12 - 5 - 3 , since the catalytic converter  3  is in the non-deteriorated state, the deteriorated state evaluating means  13   b  does not control the deterioration indicator  29 , but sets the value of the flag F/DONE to “1” in STEP 12 - 5 - 5 . The processing in STEP 12 - 5  is now finished. 
     The above processing represents the process that is carried out by the deteriorated state evaluating means  13   b  in STEP 12  shown in FIG.  12 . 
     In the apparatus according to the above embodiment, the target air-fuel ratio calculating means  13   a  of the exhaust-side main processor  13  sequentially determines a target air-fuel ratio for the internal combustion engine  1 , i.e., a target value for the air-fuel ratio of the exhaust gas entering the catalytic converter  3 , according to the adaptive sliding mode control process in order to converge (settle) the output VO 2 /OUT from the O 2  sensor  6  downstream of the catalytic converter  3  to the target value VO 2 /TARGET. The target air-fuel ratio calculating means  13   a  adjusts the amount of fuel injected into the internal combustion engine  1  in order to converge the output KACT of the LAF sensor  5  to the target air-fuel ratio KCMD, for thereby feedback-controlling the air-fuel ratio of the internal combustion engine  1  at the target air-fuel ratio KCMD. In this manner, the output signal VO 2 /OUT of the O 2  sensor  6  is converted to the target value VO 2 /TARGET, and the catalytic converter  3  can maintain its optimum exhaust gas purifying performance without being affected by its own aging. 
     Concurrent with the above control of the air-fuel ratio of the internal combustion engine  1 , the deteriorated state evaluating means  13   b  of the exhaust-side main processor  13  sequentially determines a deterioration evaluating linear function σ from the time-series data of the differential output VO 2  of the O 2  sensor  6 . The deteriorated state evaluating means  13   b  determines a deterioration evaluating parameter LSσ 2  as the central value (the central value of the minimum square in the present embodiment) of the square σ 2  of the deterioration evaluating linear function σ, according to the sequential statistic processing algorithm (the algorithm of the method of weighted least squares in the present embodiment). The deteriorated state evaluating means  13   b  then compares the deterioration evaluating parameter LSσ 2  with the predetermined threshold CATAGELMT thereby to evaluate the deteriorated state of the catalytic converter  3 . 
     In this fashion, it is possible to evaluate the deteriorated state of the catalytic converter  3  while maintaining the optimum purifying performance of the catalytic converter  3 . Because the deterioration evaluating parameter LSσ 2  is the central value of the square σ 2  of the deterioration evaluating linear function σ, its correlation to the deteriorated state of the catalytic converter  3  is high, so that the deteriorated state of the catalytic converter  3  can appropriately be evaluated based on the deterioration evaluating parameter LSσ 2 . 
     In the present embodiment, in situations where the exhaust gas volume is maintained at a substantially constant level, i.e., in the cruise state, i.e., variations of the exhaust gas volume are small and the value of the deterioration evaluating linear function σ is unlikely to change, the deterioration evaluating parameter LSσ 2  is not calculated. In other situations, the deterioration evaluating parameter LSσ 2  is calculated to evaluate the deteriorated state of the catalytic converter  3 . Therefore, the deterioration evaluating parameter LSσ 2  representative of the deteriorated state of the catalytic converter  3  is highly reliable, allowing the deteriorated state of the catalytic converter  3  to be evaluated accurately. 
     The sliding mode controller  27  calculates a target air-fuel ratio KCMD to converge the estimated differential output VO 2  bar of the O 2  sensor  6  determined by the estimator  26  to “0” and, as a result, to converge the output VO 2 /OUT of the O 2  sensor  6  to the target value VO 2 /TARGET. It is thus possible to compensate for the effect of the dead time d 1  of the object exhaust system E and the effect of the dead time t 2  of the air-fuel ratio manipulating system which is made up of the internal combustion engine  1  and the engine-side control unit  7   b , and to increase the stability of the control process of converging the output VO 2 /OUT of the O 2  sensor  6  to the target value VO 2 /TARGET. Inasmuch as the gain coefficients a 1 , a 2 , b 1  which are parameters of the exhaust system model that is used by the sliding mode controller  27  and the estimator  26  in their processes are sequentially identified by the identifier  25 , any effect that behavioral changes of the object exhaust system E have on the control process of converging the output VO 2 /OUT of the O 2  sensor  6  to the target value VO 2 /TARGET is held to a minimum. As a consequence, the control process of converging the output VO 2 /OUT of the O 2  sensor  6  to the target value VO 2 /TARGET can stably be carried out. 
     In the apparatus according to the present embodiment, therefore, the deteriorated state of the catalytic converter  3  can be evaluated highly reliably while the desired purifying performance of the catalytic converter  3  is reliably maintained. 
     The present invention is not limited to the above first embodiment, but may be modified as follows: 
     In the first embodiment, the central value of the minimum square σ 2  of the deterioration evaluating linear function σ is used as the deterioration evaluating parameter LSσ 2 . However, the central value of the minimum square of the absolute value of the deterioration evaluating linear function σ may be determined as the deterioration evaluating parameter. According to such a modification, in STEP 12 - 4  shown in FIG. 16, the absolute value of the deterioration evaluating Linear function σ is determined instead of the square σ 2  the deterioration evaluating linear function σ, and “σ 2 ” in the equation (29) is replaced with the determined absolute value to obtain a deterioration evaluating parameter that exhibits the same tendency as the deterioration evaluating parameter LSσ 2  with respect to the deteriorated state of the catalytic converter  3 . By comparing the obtained deterioration evaluating parameter with a predetermined value, the deteriorated state of the catalytic converter  3  can be evaluated in the same manner as with the first embodiment. 
     Rather than the square σ 2  of the deterioration evaluating linear function σ or the central value of the minimum square of the absolute value, the central value of an average value of the square σ 2  or the absolute value may be determined as the deterioration evaluating parameter. Alternatively, a variance of the value of the deterioration evaluating linear function σ, or more accurately a variation with respect to “0” and an average value of the square σ 2  of the value of the deterioration evaluating linear function σ, or a standard deviation (the square root of a variance) may be determined as the deterioration evaluating parameter. The deterioration evaluating parameter thus determined exhibits the same tendency as the deterioration evaluating parameter LSσ 2  with respect to the deteriorated state of the catalytic converter  3 . Therefore, by comparing the obtained deterioration evaluating parameter with a predetermined value, the deteriorated state of the catalytic converter  3  can be evaluated. 
     In the first embodiment, the deterioration evaluating linear function σ is determined according to the equation (15) whose variable components are represented by two time-series data of the differential output VO 2  of the O 2  sensor  6 . However, the deterioration evaluating linear function may be defined by a linear function whose variable components are represented by more time-series data of the differential output VO 2 . According to such a modification, the switching function of the sliding mode control process is preferably defined by a linear function where the time-series data of the differential output VO 2  included in the deterioration evaluating linear function is replaced with the time-series data of the estimated differential output VO 2  bar. 
     The deterioration evaluating linear function may alternatively be determined by an equation similar to the equation (15) where the differential outputs VO 2 (k), VO 2 (k−1) of the equation (15) are replaced with the outputs VO 2 /OUT(k), VO 2 /OUT(k−1) of the O 2  sensor  6 . According to this modification, the central value of the deterioration evaluating linear function is basically represented by “(s 1 +s 2 )·VO 2 /TARGET”. If a parameter representing the degree to which the value of the deterioration evaluating linear function varies with respect to the central value (s 1 +s 2 )·VO 2 /TARGET, such as the square of the difference between the central value (s 1 +s 2 )·VO 2 /TARGET and the value of the deterioration evaluating linear function, or the central value of the minimum square of the absolute value, is determined as the deterioration evaluating parameter, then the deteriorated state of the catalytic converter  3  can be evaluated in the same manner as with the first embodiment. 
     Furthermore, a linear function whose variable components are represented by time-series data of the switching function σ bar according to the equation (6), i.e., time-series data of the estimated differential output VO 2  bar of the O 2  sensor  6 , may be used as the deterioration evaluating linear function. It is preferable for the purpose of increasing the reliability of the evaluated result to use the deterioration evaluating linear function σ according to the equation (15) which employs the actual differential output VO 2  of the O 2  sensor  6  as a variable component, rather than the switching function σ bar which employs the estimated differential output VO 2  bar that is an estimated value after the total dead time d of the differential output VO 2  of the O 2  sensor  6 , because the deterioration evaluating linear function σ better reflects the actual state of the catalytic converter  3 . 
     In the first embodiment, the square  2  of the deterioration evaluating linear function σ is used to evaluate the deteriorated state of the catalytic converter  3 . However, it is possible to use the product of the value of the linear function σ and its rate of change, which represents the stability determining parameter Pstb used in STEP 9  to determine the stability of the SLD controlled state, for evaluating the deteriorated state of the catalytic converter  3 . In such a modification, if a variance of the product, or more generally a value representing the degree to which the value of the product varies, is determined as the deterioration evaluating parameter, then it is possible to evaluate the deteriorated state of the catalytic converter  3  based on the deterioration evaluating parameter thus determined. 
     In the first embodiment, the deteriorated state of the catalytic converter  3  is evaluated as one of the two states, i.e., the deterioration-in-progress state and the non-deteriorated state. However, if an increased number of thresholds are used for comparison with the deterioration evaluating parameter LSσ 2 , then the deteriorated state of the catalytic converter  3  may be evaluated as three or more deteriorated states. In this case, different evaluations may be indicated depending on those three or more deteriorated states. 
     In the first embodiment, the algorithm of the sliding mode control process is constructed on the basis of an exhaust system model expressed as a discrete time system. However, the algorithm of the sliding mode control process may be constructed on the basis of a model which expresses the object exhaust system E as a continuous time system. In this modification, the switching function for the sliding mode control process may be expressed by a linear function whose variable components are represented by the differential output VO 2  of the O 2  sensor  6  and its rate of change, for example. 
     In the first embodiment, the adaptive sliding mode control process is employed to calculate the target air-fuel ratio KCMD. However, the sliding mode control process which does not use the adaptive control law (adaptive algorithm) may be employed. In this modification, the target air-fuel ratio KCMD may be determined according to an equation that is similar to the equation (28) except that the term of the adaptive control law input Uadp is removed therefrom. 
     In the first embodiment, the effect of the total dead time d is compensated for by the estimator  26  in calculating the target air-fuel ratio KCMD. If the dead time of the air-fuel ratio manipulating system is negligibly small, then only the dead time d 1  of the object exhaust system E may be compensated for. In this modification, the estimator  26  sequentially determines in each control cycle the estimated value VO 2 (k+d 1 ) after the dead time d 1  of the differential output VO 2  of the O 2  sensor  6 , according to the following equation (44) which is similar to the equation (12) except that “kcmd” and “d” are replaced respectively with “kact” and “d 1 ”:                        VO2   _          (     k   +   d1     )       =                  α                   1   ·     VO2        (   k   )           +     α                   2   ·     VO2        (     k   -   1     )           +                                ∑     j   =   1     d1          β                   j   ·     kact        (     k   -   j     )                           (   44   )                         
     where 
     α1=the first-row, first-column element of A d1 , 
     α2=the first-row, second-column element of A d1 , 
     βj=the first-row elements of A j−1 ·B        A   =     [         a1       a2           1       0         ]             B   =     [         b1           0         ]                     
     In this modification, the sliding mode controller  27  determines in each control cycle the equivalent control input Ueq, the reaching control law input Urch, and the adaptive control law input Uadp according to equations which are similar to the equations (24)-(27) except that “d” is replaced with “d 1 ”, and adds the equivalent control input Ueq, the reaching control law input Urch, and the adaptive control law input Uadp to determine the target differential air-fuel ratio kcmd for thereby determining the target air-fuel ratio KCMD which has been compensated for the effect of the dead time d 1  of the object exhaust system E. 
     According to the above modification, the processing of the identifier  25 , the deteriorated state evaluating means  13   b , and the engine-side control unit  7   b  may be the same as the processing thereof in the first embodiment. 
     If the dead time d 1  of the object exhaust system E as well as the dead time d 1  of the air-fuel ratio manipulating system is negligibly small, then the estimator  26  may be dispensed with. In this modification, the processing operation of the sliding mode controller  27  and the identifier  25  may be performed with d=d 1 =0. 
     In the first embodiment, the identifier  25  is employed. However, the gain coefficients a 1 , a 2 , b 1  of the exhaust system model may be of predetermined fixed values, or may be set to suitable values from the rotational speed and intake pressure of the internal combustion engine  1  using a map. 
     In the first embodiment, the O 2  sensor  6  is used as the exhaust gas sensor downstream of the catalytic converter  3 . However, for maintaining the desired purifying performance of the catalytic converter  3 , any of various other sensors may be employed insofar as they can detect the concentration of a certain component of the exhaust gas downstream of the catalytic converter to be controlled. For example, a CO sensor is employed if the carbon monoxide (CO) in the exhaust gas downstream of the catalytic converter is controlled, an NOx sensor is employed if the nitrogen oxide (NOx) in the exhaust gas downstream of the catalytic converter is controlled, and an HC sensor is employed if the hydrocarbon (HC) in the exhaust gas downstream of the catalytic converter is controlled. If a three-way catalytic converter is employed, then it can be controlled to maximize its purifying performance irrespective of which of the above gas components is detected for its concentration. If a reducing catalytic converter or an oxidizing catalytic converter is employed, then its purifying performance can be increased by directly detecting a gas component to be purified. 
     In the first embodiment, the sliding mode control process is employed as the feedback control process for converging the output VO 2 /OUT of the O 2  sensor  6  to the target value VO 2 /TARGET. However, it is possible to employ another feedback control process to evaluate the deteriorated state of the catalytic converter  3  while converging the output VO 2 /OUT of the O 2  sensor  6  to the target value VO 2 /TARGET. A second embodiment of the present invention which is directed to such an arrangement will be described below with reference to FIGS. 20 through 22. 
     The second embodiment differs from the first embodiment only as to the functional structure and processing of the exhaust-side control unit  7   a . Those structural and processing details of the second embodiment which are identical to those of the first embodiment are shown in identical figures and denoted by identical reference characters, and will not be described in detail below. 
     FIG. 20 shows in block form a functional structure of an engine-side control unit  7   a  according to the second embodiment. As with the first embodiment, the engine-side control unit  7   a  according to the second embodiment performs, in given control cycles, a process of sequentially generating a target air-fuel ratio KCMD (a target value for the air-fuel ratio detected by the LAF sensor  5 ) to converge the output VO 2 /OUT of the O 2  sensor  6  downstream of the catalytic converter  3  (see FIG. 1) to the target value VO 2 /TARGET, and a process of evaluating the deteriorated state of the catalytic converter  3 . The control cycles of the engine-side control unit  7   a  are of a constant period as with the first embodiment. 
     For performing the above processes, the engine-side control unit  7   a  has the subtractor  12  for sequentially calculating a difference (=VO 2 /OUT−VO 2 /TARGET) between the output VO 2 /OUT from the O 2  sensor  6  and the target value VO 2 /TARGET therefor, i.e., the differential output VO 2 , and the deteriorated state evaluating means  13   b  for evaluating the deteriorated state of the catalytic converter  3  using the time-series data of the differential output VO 2  and controlling the deterioration indicator  29  for its operation, as with the first embodiment. The engine-side control unit  7   a  also has a target air-fuel ratio calculating means  30  as an air-fuel ratio manipulated variable determining means for sequentially calculating the target air-fuel ratio KCMD from the data of the differential output VO 2  according to a PID (proportional plug integral plus derivative) control process which serves as one feedback control process. 
     The processing of the subtractor  12  and the deteriorated state evaluating means  13   b  is identical to that of the first embodiment. In the second embodiment, the values of the coefficients s 1 , s 2  of the deterioration evaluating linear function σ required for the processing of the deteriorated state evaluating means  13   b  (see STEP 12 - 1  shown in FIG. 16) may be identical to those used in the first embodiment. Basically, while the air-fuel ratio of the internal combustion engine is being controlled depending on the target air-fuel ratio calculated by the target air-fuel ratio calculating means  30  as described later on, the values of the coefficients s 1 , s 2  for causing the value of the deterioration evaluating linear function σ to clearly exhibit the tendency shown in FIGS. 5 through 7 with respect to the deteriorated state of the catalytic converter  3  may be established through experimentation or the like. 
     The target air-fuel ratio calculating means  30  comprises a PID controller  31  for sequentially generating an air-fuel ratio manipulated variable Upid required to converge the differential output VO 2  of the O 2  sensor  6  to “0” according to the PID control process (described in detail later on), and an adder  32  for adding a predetermined air-fuel ratio reference value KBS to the air-fuel ratio manipulated variable Upid thereby to calculate a target air-fuel ratio KCMD. 
     The air-fuel ratio manipulated variable Upid is signified as a corrective quantity of the target air-fuel ratio KCMD with respect to the air-fuel ratio reference value KBS, and corresponds to the SLD manipulating input Usl (=the target differential air-fuel ratio kcmd) in the first embodiment. The air-fuel ratio reference value KBS added to the air-fuel ratio manipulated variable Upid is a central air-fuel ratio for the target air-fuel ratio KCMD, and corresponds to the reference value FLAF/BASE in the first embodiment. In the present embodiment, the air-fuel ratio reference value KBS is a value close to a stoichiometric air-fuel ratio which is determined from the detected values of the rotational speed NE and intake pressure PB of the internal combustion engine  1  using a predetermined map. 
     Other details than the above exhaust-side control unit  7   a , i.e., the functional arrangement of the engine-side control unit  7   b  and the arrangement of the exhaust system of the internal combustion engine  1 , are exactly the same as those of the first embodiment. 
     Operation of the apparatus according to the second embodiment, including detailed processing of the target air-fuel ratio calculating means  30 , will be described below. 
     The process performed by the engine-side control unit  7   b  is identical to the process in the first embodiment, and the processing sequence shown in FIGS. 10 and 11, i.e., the process of adjusting the fuel injection quantity of the internal combustion engine  1 , is sequentially executed in control cycles in synchronism with the TDC (crankshaft angle period) by the engine-side control unit  7   b . However, the target air-fuel ratio KCMD read by the engine-side control unit  7   b  in STEPf shown in FIG. 10 is the latest target air-fuel ratio KCMD that is calculated by the target air-fuel ratio calculating means  30  of the exhaust-side control unit  7   a.    
     In the present embodiment, the exhaust-side control unit  7   a  executes a main routine shown in FIG. 21 in given control cycles concurrent with the processing of the engine-side control unit  7   b.    
     Specifically, the exhaust-side control unit  7   a  determines the air-fuel ratio reference value KBS from the present rotational speed NE and intake pressure PB of the internal combustion engine  1  using a predetermined map in STEP 21 . 
     Then, the exhaust-side control unit  7   a  determines the value of the flag f/prism/on which is set by the engine-side control unit  7   b  in STEPd shown in FIG. 10 in STEP 22 . If f/prism/on=0, i.e., if the operation mode of the internal combustion engine  1  is not the normal operation mode in which the air-fuel ratio of the internal combustion engine  1  is manipulated to converge the output VO 2 /OUT of the O 2  sensor  6  to the target value VO 2 /TARGET, then the exhaust-side control unit  7   a  sets the target air-fuel ratio KCMD(k) in the present control cycle to the air-fuel ratio reference value KBS determined in STEP 21  in STEP 30 . Thereafter, the processing in the present control cycle is put to an end. 
     If f/prism/on=1 in STEP 22 , i.e., if the operation mode of the internal combustion engine  1  is the normal operation mode, then the subtractor  12  calculates the latest differential output VO 2 (k) (=VO 2 /OUT−VO 2 /TARGET) of the O 2  sensor  6  in STEP 23 . Specifically, the subtractor  12  selects the latest one of the time-series data of the output VO 2 /OUT of the O 2  sensor  6  which have been read and stored in the non-illustrated memory in STEPa shown in FIG. 10, and calculates the differential output VO 2 (k). The differential output VO 2 (k), including data calculated in the past (i.e., the differential output VO 2 (k−1) calculated in the preceding control cycle), is stored in a memory (not shown). 
     Then, the exhaust-side control unit  7   a  performs the processing of the target air-fuel ratio calculating means  30  in STEP 24 -STEP 27 . 
     The PID controller  31  of the target air-fuel ratio calculating means  30  determines the values of respective gain coefficients KVP, KVI, KVD of proportional, integral, and derivative terms relative to the PID control process for converging the differential output VO 2  of the O 2  sensor  6  to “0”, from the present rotational speed NE and intake pressure PB of the internal combustion engine  1  using a predetermined map in STEP 24 . 
     Then, the PID controller  31  calculates the equations (45)-(47), shown below, using present and preceding values VO 2 (k), VO 2 (k−1) of the differential output VO 2  of the O 2  sensor  6  determined in STEP 23 , and the latest gain coefficients KVP, KVI, KVD determined in STEP 24 , to determine present values VREFP(k), VREFI(k), VREFD(k) of the proportional, integral, and derivative terms, respectively. The PID controller  31  also adds the present values VREFP(k), VREFI(k), VREFD(k) of the proportional, integral, and derivative terms according to the equation (48), shown below, to determine a basic manipulated variable VREF as a basis for the air-fuel ratio manipulated variable Usl in STEP 25 . 
     
       
         VREFP( k )=VO 2 ( k )·KVP  (45) 
       
     
     
       
         VREFI( k )=VREFI( k −1)+VO 2 ( k )·KVI  (46) 
       
     
     
       
         VREFD( k )=(VO 2 ( k )−VO 2 ( k −1))·KVD  (47) 
       
     
     
       
         VREF=VREFP( k )+VREFI( k )+VREFD( k )  (48) 
       
     
     Furthermore, the PID controller  31  performs a limiting process for limiting the basic manipulated variable VREF in STEP 26 . In the limiting process, if the basic manipulated variable VREF determined in STEP 25  exceeds a predetermined upper or lower limit, the PID controller  31  forcibly limits the basic manipulated variable VREF to the upper or lower limit. 
     Then, the PID controller  31  determines an air-fuel ratio manipulated variable Upid from the limited basic manipulated variable VREF using a predetermined data table shown in FIG.  22 . 
     The data table shown in FIG. 22 is basically arranged such that as the basic manipulated variable VREF is larger, the air-fuel ratio manipulated variable Upid is larger. In a range of values of the basic manipulated variable VREF, i.e., a range S in FIG. 22, that are determined by the PID controller  31  while the output VO 2 /OUT of the O 2  sensor  6  is converged substantially closely to the target value VO 2 /TARGET, the air-fuel ratio manipulated variable Upid changes to a small degree as the basic manipulated variable VREF changes, and is held substantially closely to “0”. This is because in the vicinity of the target value VO 2 /TARGET, the output VO 2 /OUT of the O 2  sensor  6  greatly changes even when the air-fuel ratio changes slightly, as indicated by the solid-line curve a in FIG.  2 . 
     After determining the air-fuel ratio manipulated variable Upid, the adder  32  of the target air-fuel ratio calculating means  30  adds the air-fuel ratio reference value KBS to the air-fuel ratio manipulated variable Upid, thus determining the target air-fuel ratio KCMD(k) in the present control cycle in STEP 28 . 
     The target air-fuel ratio KCMD thus determined is stored in a memory (not shown) in a time-series fashion in each control cycle of the exhaust-side control unit  7   a . When the engine-side control unit  7   b  reads the target air-fuel ratio KCMD in STEPf shown in FIG. 10, it selects the latest data of the target air-fuel ratio KCMD thus stored in the memory. 
     After the target air-fuel ratio calculating means  30  has determined the target air-fuel ratio KCMD as described above, the deteriorated state evaluating means  13   b  performs a process of evaluating the deteriorated state of the catalytic converter  3  in STEP 29 . The process of evaluating the deteriorated state of the catalytic converter  3  performed by the deteriorated state evaluating means  13   b  in STEP 29  is exactly the same as the process in the first embodiment. Specifically, the deteriorated state evaluating means  13   b  carries out the processing sequence shown in FIGS. 16 through 19 as described above, using the time-series data of the differential output VO 2  of the O 2  sensor  6  determined in each control cycle in STEP 23 . The deteriorated state evaluating means  13   b  evaluates the deteriorated state of the catalytic converter  3  as the deterioration-in-progress state or the non-deteriorated state. If the deteriorated state of the catalytic converter  3  is the deterioration-in-progress state, then the deteriorated state is indicated by the deterioration indicator  29 . 
     In the apparatus according to the second embodiment, as with the first embodiment, the deteriorated state of the catalytic converter  3  is evaluated while the air-fuel ratio of the internal combustion engine  1  is being manipulated to converge the output VO 2 /OUT of the O 2  sensor  6  downstream of the catalytic converter  3  to the target value VO 2 /TARGET. Therefore, the deteriorated state of the catalytic converter  3  can be evaluated while maintaining the appropriate purifying performance of the catalytic converter  3 . 
     Since the deteriorated state of the catalytic converter  3  is evaluated in exactly the same manner as with the first embodiment, the deteriorated state of the catalytic converter  3  can be evaluated appropriately based on the deterioration evaluating parameter LSσ 2  whose correlation to the deteriorated state of the catalytic converter  3  is high and which is highly reliable. 
     In the present embodiment, the exhaust-side control unit  7   a  performs its processing in control cycles of a constant period. However, the exhaust-side control unit  7   a  may perform its processing in synchronism with the TDC with the engine-side control unit  7   b , or in control cycles of a period which is several times (plural times) one TDC. 
     The deterioration evaluating parameter and the evaluation of the deteriorated state of the catalytic converter  3  based thereon may be modified in the same fashion as with the first embodiment as described above. 
     INDUSTRIAL APPLICABILITY 
     As is clear from the above description, according to the present invention, the deteriorated state of a catalytic converter such as a three-way catalytic converter, which is disposed in the exhaust system of the internal combustion engine loaded in an automobile or a hybrid vehicle, can be evaluated automatically and appropriately, thus the present invention is effective for use, e.g., in reporting the evaluation results.