Patent Publication Number: US-7216006-B2

Title: Control system for a plant including a slide mode controller

Description:
RELATED APPLICATIONS 
   This application is a divisional of U.S. patent application Ser. No. 10/312,108 (allowed), filed Dec. 20, 2002, now U.S. Pat. No. 7,050,864 which was a 35 U.S.C. §371 filing of International Application Number PCT/JP02/03895 filed Apr. 19, 2002, which claims priority to Japanese Patent Application No. 2001-123344 filed on Apr. 20, 2001, Japanese Patent Application No. 2001-125648 filed on Apr. 24, 2001, Japanese Patent Application No. 2001-146144 filed on May 16, 2001, Japanese Patent Application No. 2001-179926 filed on Jun. 14, 2001, Japanese Patent Application No. 2001-184540 filed on Jun. 19, 2001, Japanese Patent Application No. 2001-343998 filed on Nov. 9, 2001 in Japan. The contents of the aforementioned applications are hereby incorporated by reference. 

   TECHNICAL FIELD 
   The present invention relates to a control system for a plant, and more particularly to a control system for controlling a plant with a sliding mode controller based on a sliding mode control theory which is one of robust control theories. 
   BACKGROUND ART 
   One known control system based on a sliding mode control theory is disclosed in Japanese Patent Laid-open No. Hei 9-274504, for example. The publication proposes a method of setting a hyperplane in the sliding mode control theory according to the convergence state of a controlled state quantity. According to the proposed method, the convergence response and convergence stability of the sliding mode control is improved. 
   For controlling a plant, which is a controlled object, with the sliding mode controller, it is necessary to produce a model of the plant and determine model parameters representing the characteristics of the model of the plant (i.e. the controlled object). The model parameters may be set to predetermined constant values. However, the values of the model parameters usually change due to aging and disturbance. Therefore, it is desirable to use a model parameter identifier for identifying the model parameters on a real-time basis and carry out the sliding mode control using the model parameters that are identified by the model parameter identifier. 
   The model parameter identifier detects an identifying error which is a difference between the output of the plant which is calculated using the identified model parameters and the actual output of the plant, and corrects the model parameters in order to eliminate the identifying error. Therefore, regarding the model parameter identifier, the following problems may occur. 
   Due to nonlinear characteristics and disturbance whose average value is not “0”, the identifying error does not become “0” even though substantially optimum model parameters have actually been obtained. Therefore, the model parameters which do not need to be corrected are occasionally corrected. As a result, a drift occurs in which the values of the model parameters gradually shift from their optimum values to some other values to make the control performed by the sliding mode controller unstable. 
   Conventionally known is an air-fuel ratio control system in which a controlled object model is defined by modeling a plant, i.e., a controlled object, and model parameters of the controlled object model are identified by a parameter adjusting device (for example, Japanese Patent Laid-open No. Hei 11-73206). According to this system, the air-fuel ratio of an air-fuel mixture to be supplied to the internal combustion engine is feedback-controlled to a target air-fuel ratio by a self-tuning regulator using the identified model parameters. 
   In the above control system, each model parameter is calculated by adding an initial value of the model parameter and an updating component which is calculated according to the identifying error of the model parameters. 
   In the above control system, the initial values of the model parameter is determined irrespective of the state quantity of the plant at the time of calculating a control input to the plant. Therefore, the time period which is necessary for the model parameters to converge on their optimum values becomes longer when the characteristics of the plant change in a relatively short time period, which results in insufficient controlling performance. 
   One known control system based on the sliding mode control theory is disclosed in Japanese Patent Laid-open No. Hei 9-274504, for example. The publication shows that the stability of the sliding mode control is determined, and when the sliding mode control is determined to be unstable, the value calculated before the sliding mode control is determined to be unstable is used for a control input to the controlled object. 
   Another control system is known from Japanese Patent Laid-open No. Hei 11-93741. In this control system, the stability of the sliding mode control is determined, and when the sliding mode control is determined to be unstable, the control input to the controlled object is set to a predetermined value. 
   However, according to the above-described conventional control systems, when the sliding mode control is determined to be unstable, the control input to the plant is fixed, which makes it impossible to converge an output of the plant to the control target value. 
   In the control system shown in Japanese Patent Laid-open No. Hei 11-73206 described above, the model parameters are calculated by adding updating components, which are calculated according to the identifying error of the model parameters, to the initial values of the model parameters. 
   According to this control system shown in Japanese Patent Laid-open No. Hei 11-73206, a drift of the model parameters due to disturbances is prevented. However, since the model parameters calculated by the parameter adjusting device are used per se in the self tuning regulator, there remains a room for improvement in maintaining stability of the self tuning regulator. 
   A control system in which model parameters of a controlled object model which are obtained by modeling a plant, is identified on a real-time basis, and the plant is controlled using the identified model parameters, is shown in Japanese Patent Laid-open No. 2000-179384. 
   The control system shown in this publication includes an identifier for identifying the model parameters, an estimator for estimating an output of the plant, and a sliding mode controller which calculates a control input to the plant with the sliding mode control, using the model parameters identified by the identifier and the plant output estimated by the estimator. A difference between an output of the plant and a target value of the plant output, and a difference between an input of the plant and a reference value, are input to the identifier and the estimator. The reference value is variably set according to a manipulating quantity for manipulating the input of the plant. By setting the reference value variably according to the manipulating quantity, an effect of raising a converging speed of the control for converging the plant output to the target value, is obtained. 
   According to the control system shown in Japanese Patent Laid-open No. 2000-179384, a method for setting the reference value according to an adaptive control input which is calculated by the sliding mode controller, is adopted. Accordingly, this method for setting the reference value is applicable only to a control system using the adaptive sliding mode control. Therefore, a reference value setting method which is widely applicable is desired. 
   One known throttle valve opening control system for a vehicle is disclosed in Japanese Patent Laid-open No. Hei 8-261050. In this control system, an opening of a throttle valve which is actuated by a motor is controlled with a PID (Proportional, Integral, and Differential) control, and control constants of the PID control is set according to an operating condition of the vehicle. 
   Since the throttle valve actuating device, which is a controlled object, has non-linear characteristics, a sufficient control performance in control accuracy, control stability, and a follow-up characteristic (convergence characteristic) of the throttle valve opening to a target value, is not obtained by the conventional PID control. 
   DISCLOSURE OF INVENTION 
   It is therefore an object of the present invention to provide a control system for a plant, which can control the plant more stably when the model parameters of the controlled object model which are obtained by modeling the plant, which is a controlled object, are identified and the sliding mode control is performed using the identified model parameters. 
   Another object of the present invention is to provide a control system for a plant, which can promptly converge one or more model parameter of the controlled object model on its optimum value to thereby maintain good controllability, when the characteristics of the plant changes in a relatively short time period. 
   Another object of the present invention is to provide a control system for a plant, which can continue the control of converging an output of the plant to the control target value and execute a suitable process for stabilizing a sliding mode control, when the sliding mode control is determined to be unstable, to thereby improve the stability of the sliding mode control. 
   Another object of the present invention is to provide a control system for a plant, which identifies model parameters of a controlled object model which is obtained by modeling the plant (i.e. the controlled object), and improve stability of the control using the model parameters. 
   Another object of the present invention is to provide a control system for a plant, in which when using a controlled object model which is defined using a reference value of the input or output of the plant, the reference value is set by a widely applicable method, to thereby improve control performance. 
   Another object of the present invention is to provide a control system for a throttle valve actuating device, which can improve control accuracy, control stability, and a follow-up characteristic of the throttle valve opening to a target value. 
   To achieve the above object, the present invention provides a control system for a plant, comprising identifying means and a sliding mode controller. The identifying means identifies a model parameter vector (θ) of a controlled object model of a plant which is obtained by modeling the plant. The sliding mode controller controls the plant using the model parameter vector identified by the identifying means. The identifying means comprises an identifying error calculating means, an updating vector calculating means, and an updating vector correcting means. The identifying error calculating means calculates an identifying error (ide) of the model parameter vector. The updating vector calculating means calculates an updating vector (dθ) according to the identifying error. The updating vector correcting means corrects the updating vector by multiplying a past value of at least one element of the updating vector by a predetermined value (DELTAi, EPSi) which is greater than “0” and less than “1”. The identifying means calculates the model parameter vector by adding the corrected updating vector to a reference vector (θbase, θ(0)) of the model parameter vector. 
   With this configuration, the updating vector is calculated according to the identifying error of the model parameter vector, and corrected by multiplying the past value of at least one element of the updating vector by the predetermined value which is greater than “0” and less than “1”. The corrected updating vector is added to the reference vector of the model parameter vector to calculate the model parameter vector. Accordingly, values of the elements of the updating vector are limited, thus stabilizing the model parameter vector in the vicinity of the reference vector. As a result, the drift of the model parameters is prevented, to thereby improve the stability of the sliding mode control performed by the sliding mode controller. 
   Preferably, the updating vector correcting means does not multiply one of an element of the updating vector which is relevant to the input of the plant (an element relevant to the calculation of b 1 ) and an element of the updating vector which is irrelevant to the input and the output of the plant (an element relevant to the calculation of c 1 ), by the predetermined value. According to this configuration, the steady-state deviation which may be caused by correcting these elements can be prevented from occurring. 
   Preferably, the updating vector correcting multiplies at least one element of the reference vector (θ(0)) by the predetermined value (EPSi). Also in this case, the drift of the model parameters is prevented, to thereby improve the stability of the sliding mode control performed by the sliding mode controller. 
   The present invention further provides a control system for a plant, comprising an identifying means and a sliding mode controller. The identifying means identifies a model parameter vector of a controlled object model which is obtained by modeling the plant, based on an input and an output of the plant. The sliding mode controller controls the plant using the model parameter vector identified by the identifying means. The identifying means comprises identifying error calculating means and identifying error correcting means. The identifying error calculating means calculates an identifying error (ide) of the model parameter vector. The identifying error correcting means corrects the identifying error in a decreasing direction, if the identifying error is in a predetermined range (−EIDNRLMT≦ide≦EIDNRLMT). The identifying means calculates the model parameter vector using the identifying error (idenl) corrected by the identifying error correcting means. 
   With this configuration, if the identifying error is in the predetermined range, the identifying error is corrected in a decreasing direction, and the model parameter vector is calculated using the corrected identifying error. Therefore, the identifying error is suppressed to less accumulate in the model parameters, that is, the drift of the model parameters is prevented, to thereby improve the stability of the sliding mode control carried out by the sliding mode controller. 
   Preferably, the identifying error correcting means sets the identifying error to “0”, if the identifying error is in the predetermined range. With this configuration, the effect of the identifying error which is not to be reflected to the values of model parameters is eliminated, resulting in an increased effect of preventing the model parameters from drifting. 
   Preferably the predetermined range is set according to an amount (DDTHR) of change in a control target value or an amount (DTH) of change in the output of the plant. With this configuration, the range of the identifying error whose influence is to be reduced, becomes appropriate. Accordingly, it is prevented that the identifying error which is to be reflected to the values of model parameters is reduced or neglected as an unnecessary error. 
   Preferably, the identifying means identifies the model parameter vector according to a fixed gain algorithm. With this configuration, the amount of calculations for identifying the model parameters can be reduced. 
   Preferably, the identifying error calculating means performs a low-pass filtering of the identifying error and outputs the identifying error after the low-pass filtering. 
   With this configuration, the model parameter vector is identified using the identifying error which has been processed by the low-pass filtering. Accordingly, the frequency characteristics of the controlled object model are made closer to the actual frequency characteristics of the plant, to thereby improve the robustness and stability of the control carried out by the sliding mode controller. 
   Preferably, the control system further comprises predicting means for calculating a predicted value (PREDTH) of the output of the plant. With this configuration, the predicted value of the output of the plant is calculated by the predicting means. Therefore, the plant which has a dead time element can be controlled accurately. 
   Preferably, the predicting means calculates the predicted value using the model parameter vector identified by the identifying means. With this configuration, it is possible to calculate an accurate predicted value even when the dynamic characteristics of the plant changes with time or changes according to environmental conditions. 
   Preferably, a control input applied from the sliding mode controller to the plant includes an adaptive law input. With this configuration, good controllability can be obtained even in the presence of disturbance and/or a modeling error, which is a difference between the characteristics of the actual plant and the characteristics of the controlled object model. 
   Preferably, the plant includes a throttle valve actuating device ( 10 ) having a throttle valve ( 3 ) of an internal combustion engine and actuating means ( 6 ) for actuating the throttle valve, and the sliding mode controller calculates a parameter (DUT) for determining a control input to be applied to the throttle valve actuating device to make an opening (TH) of the throttle valve coincide with a target opening (THR). 
   With this configuration, the sliding mode controller performs the control to make the opening of the throttle valve coincide with the target opening, using stable model parameters identified by the identifying means. Consequently, the controllability of the throttle valve opening to the target opening can be improved, and the throttle valve opening can be controlled more stably. 
   Preferably, the plant includes an engine system ( 210 ) having an internal combustion engine ( 212 ) and fuel supply means ( 211 ) for supplying fuel to the engine, and the sliding mode controller calculates a parameter (DKAF) for determining a control input to be applied to the engine system to make an air-fuel ratio of an air-fuel mixture to be supplied to the engine coincide with a target air-fuel ratio. 
   With this configuration, the sliding mode controller performs the control to make the air-fuel ratio coincide with the target air-fuel ratio, using stable model parameters identified by the identifying means. Consequently, the controllability of the air-fuel ratio to the target air-fuel ratio can be improved, and the air-fuel ratio can be controlled more stably. 
   The present invention further provides a control system for a plant, comprising identifying means and control means. The identifying means identifies at least one model parameter (θ) of a controlled object model of a plant which is obtained by modeling the plant. The control means controls the plant using the at least one model parameter identified by the identifying means. The identifying means comprises a reference model parameter calculating means, an identifying error calculating means, and an updating component calculating means. The reference model parameter calculating means calculates at least one reference model parameter (θbase) according to a state quantity (DTH) of the plant. The identifying error calculating means calculates an identifying error (ide) of the at least one model parameter. The updating component calculating means calculates at least one updating component (dθ) according to the identifying error (ide). The identifying means calculates the at least one model parameter by adding the at least one updating component to the at least one reference model parameter. 
   With this configuration, the updating component(s) is/are calculated according to the identifying error, and the model parameter(s) is/are calculated by adding the updating component(s) to the reference model parameter(s) which is/are calculated according to the state quantity of the plant. Accordingly, the model parameter(s) is/are calculated by means of correcting the reference model parameter(s) corresponding to the always-changing state quantity of the plant, with the updating component(s), which makes it possible to converge the model parameter(s) more promptly compared with the conventional method. 
   Preferably, the state quantity of the plant is a parameter indicative of a change in the dynamic characteristic of the plant. The parameter indicative of a change in the dynamic characteristic of the plant may be an output (DTH, TH) of the plant, or a control target value (DTHR, THR) to which the output of the plant is controlled to converge. With this configuration, the reference model parameter(s) which reflect(s) the change in the dynamic characteristics of the plant can be obtained. This makes it possible to promptly converge the model parameter(s) especially when the plant includes a nonlinear element. 
   Preferably, the control means controls the plant with the sliding mode control. With this configuration, good controllability can be obtained even in the presence of disturbance, a modeling error (a difference between the characteristics of the actual plant and the characteristics of the controlled object model), and/or a dead time (a delay between the input and output of the controlled object). 
   Preferably the control input applied from the control means to the plant includes an adaptive law input. With this configuration, better controllability can be obtained even in the presence of disturbance and/or the modeling error. 
   Preferably, the plant includes a throttle valve actuating device having a throttle valve of an internal combustion engine and actuating means for actuating the throttle valve, and the sliding mode controller calculates a parameter for determining a control input to be applied to the throttle valve actuating device to make an opening of the throttle valve coincide with a target opening. 
   With this configuration, the control means performs the control to make the opening of the throttle valve coincide with the target opening, using model parameters identified by the identifying means. Consequently, the controllability of the throttle valve opening to the target opening can be improved. 
   The present invention further provides a control system for a plant, including a sliding mode controller for controlling the plant with the sliding mode control. The sliding mode controller includes switching function value calculating means, equivalent control input calculating means, reaching law input calculating means, control input calculating means, stability determining means, and stabilizing means. The switching function value calculating means calculates a value (σ) of a switching function defined as a linear function of a deviation between an output of the plant and a control target value. The equivalent control input calculating means calculates an equivalent control input (Ueq) that contributes to constraining a deviation state quantity on the switching straight line on which the value of the switching function becomes zero. The deviation state quantity is defined based on the deviation between the output of the plant and the control target value. The reaching law input calculating means calculates a reaching law input (Urch) that contributes to placing the deviation state quantity onto the switching straight line. The control input calculating means calculates a control input (Usl) to the plant by adding at least the equivalent control input (Ueq) and the reaching law input (Urch). The stability determining means determines whether or not the sliding mode control is stable. The stabilizing means reduces an absolute value of the equivalent control input (Ueq) when the sliding mode control is determined to be unstable. 
   With this configuration, the control input to the plant is calculated using the equivalent control input that contributes to constraining the deviation state quantity on the switching straight line, and the reaching law input that contributes to placing the deviation state quantity onto the switching straight line. When the sliding mode control is determined to be unstable, the absolute value of the equivalent control input is reduced. The equivalent control input may be most greatly influenced when the sliding mode control becomes unstable. Therefore, the reduction of the absolute value of the equivalent control input makes it possible to stabilize the sliding mode control while continuing the control of converging an output of the plant to the control target value. 
   The present invention provides another control system for a plant, including a sliding mode controller for controlling the plant with a sliding mode control. The sliding mode controller includes switching function value calculating means for calculating a value (σ) of a switching function defined as a linear function of a deviation between an output of the plant and a control target value, equivalent control input calculating means for calculating an equivalent control input (Ueq) that contributes to constraining a deviation state quantity which is defined based on the deviation between the output of the plant and the control target value, on a switching straight line on which the value of the switching function becomes zero, reaching law input calculating means for calculating a reaching law input (Urch) that contributes to placing the deviation state quantity onto the switching straight line, control input calculating means for calculating a control input to the plant by adding at least the equivalent control input (Ueq) and the reaching law input (Urch), stability determining means for determining whether or not the sliding mode control is stable, and stabilizing means for changing a control gain (F) that is used in calculating the reaching law input (Urch), to a value (XKRCHSTB) that makes the sliding mode control more stable, when the sliding mode control is determined to be unstable. 
   With this configuration, the control input to the plant is calculated using the equivalent control input that contributes to constraining the deviation state quantity on the switching straight line, and the reaching law input that contributes to placing the deviation state quantity onto the switching straight line. When the sliding mode control is determined to be unstable, the control gain that is used in calculating the reaching law input, to a value that makes the sliding mode control more stable. Accordingly, it is possible to stabilize the sliding mode control while continuing the control of converging an output of the plant to the control target value. 
   Preferably, the sliding mode controller further includes adaptive law input calculating means for calculating an adaptive law input (Uadp) that contributes to placing the deviation state quantity onto the switching straight line, and the control input calculating means calculates the control input (Usl) to the plant by adding the equivalent control input (Ueq), the reaching law input (Urch), and the adaptive law input (Uadp). 
   With this configuration, the control input applied to the plant is calculated by further adding the adaptive law input to the equivalent control input and the reaching law input. Accordingly, better controllability can be obtained even in the presence of disturbance and/or the modeling error (a difference between the characteristics of the actual plant and the characteristics of the controlled object model). 
   The present invention provides another control system for a plant, including a sliding mode controller for controlling the plant with a sliding mode control. The sliding mode controller includes switching function value calculating means for calculating a value (σ) of a switching function defined as a linear function of a deviation between an output of the plant and a control target value, equivalent control input calculating means for calculating an equivalent control input (Ueq) that contributes to constraining a deviation state quantity which is defined based on the deviation between the output of the plant and the control target value, on a switching straight line on which the value of the switching function becomes zero, reaching law input calculating means for calculating a reaching law input (Urch) that contributes to placing the deviation state quantity onto the switching straight line, adaptive law input calculating means for calculating an adaptive law input (Uadp) that contributes to placing the deviation state quantity onto the switching straight line, control input calculating means for calculating a control input to the plant by adding the equivalent control input (Ueq), the reaching law input (Urch), and the adaptive law input (Uadp), stability determining means for determining whether or not the sliding mode control is stable, and stabilizing means for changing a control gain (G) that is used in calculating the adaptive law input, to a value (XKADPSTB) that makes the sliding mode control more stable, when the sliding mode control is determined to be unstable. 
   With this configuration, the control input to the plant is calculated using the equivalent control input that contributes to constraining the deviation state quantity on the switching straight line, the reaching law input that contributes to placing the deviation state quantity onto the switching straight line, and the adaptive law input that contributes to placing the deviation state quantity onto the switching straight line. When the sliding mode control is determined to be unstable, the control gain that is used in calculating the adaptive law input, to a value that makes the sliding mode control more stable. Accordingly, it is possible to stabilize the sliding mode control while continuing the control of converging an output of the plant to the control target value. Using the adaptive law input makes it possible to obtain good controllability even in the presence of disturbance and/or the modeling error, thereby stabilizing the sliding mode control while continuing the control of converging an output of the plant to the control target value. 
   The present invention provides another control system for a plant, including a sliding mode controller for controlling the plant with a sliding mode control. The sliding mode controller includes switching function value calculating means for calculating a value (σ) of a switching function defined as a linear function of a deviation between an output of the plant and a control target value, stability determining means for determining whether or not the sliding mode control is stable, and stabilizing means for changing the switching function (VPOLE) so that a response speed of the sliding mode control may decrease, when the sliding mode control is determined to be unstable. 
   With this configuration, the switching function is changed so that a response speed of the sliding mode control may decrease, when the sliding mode control is determined to be unstable. Accordingly, it is possible to stabilize the sliding mode control while continuing the control of converging an output of the plant to the control target value. 
   Preferably, the stability determining means performs the stability determination, based on a value (σ) of the switching function. The tendency where the absolute value of the switching function increases indicates that the control becomes unstable. Therefore, by determining such tendency based on the value of the switching function, an unstable condition of the sliding mode control can easily be detected. 
   The present invention provides another control system for a plant, including identifying means for identifying at least one model parameter of a controlled object model that is obtained by modeling the plant, based on an input and an output of the plant, and a sliding mode controller for controlling the plant with a sliding mode control, using the at least one model parameter identified by the identifying means. The sliding mode controller includes stability determining means for determining whether or not the sliding mode control is stable, and stabilizing means for inhibiting the use of a predetermined model parameter (b 1 ), when the sliding mode control is determined to be unstable. 
   With this configuration, the use of the predetermined model parameter (b 1 ) is inhibited, when the sliding mode control is determined to be unstable. The sliding mode control becomes unstable when the identification by the identifying means becomes unstable. Therefore, inhibiting the use of the predetermined model parameter that gives great influence to the control, makes it possible to stabilize the sliding mode control while continuing the control of converging an output of the plant to the control target value. 
   Preferably, the plant includes a throttle valve actuating device having a throttle valve of an internal combustion engine and actuating means for actuating the throttle valve, and the sliding mode controller calculates a parameter for determining a control input to be applied to the throttle valve actuating device to make the opening of the throttle valve coincide with a target opening. 
   With this configuration, the control of making the opening of the throttle valve coincide with the target opening is performed with the sliding mode control. Accordingly, controllability of the throttle opening to the target opening is improved. In addition, more stable control can be accomplished by the stabilizing means. 
   The present invention provides another control system for a plant, including identifying means for identifying a model parameter vector of a controlled object model which is obtained by modeling the plant, based on an input and an output of the plant, and a sliding mode controller for controlling the plant using the model parameter vector identified by the identifying means. The identifying means includes identifying error calculating means for calculating an identifying error (ide) of the model parameter vector, updating vector calculating means for calculating an updating vector (dθ) according to the identifying error, updating vector correcting means for correcting the updating vector so that an influence of the past value of the identifying error may decrease, model parameter vector calculating means for calculating the model parameter vector by adding the corrected updating vector to a reference vector (θbase, θ(0)) of the model parameter vector, and limiting means for limiting a value of at least one element (a 1 , a 2 , b 1 , c 1 ) of the model parameter vector calculated by the model parameter vector calculating means, within a predetermined limit range. 
   With this configuration, the updating vector is calculated according to the identifying error of the model parameter vector, and corrected so that an influence of the past value of the identifying error may decrease. The corrected updating vector is added to the reference vector of the model parameter vector to calculate the model parameter vector. Further, the value of at least one element of the model parameter vector calculated by the model parameter vector calculating means is limited within the predetermined limit range. Accordingly, a drift of the model parameter(s) is prevented, and the control stability is further improved. 
   Preferably, the updating vector calculating means calculates the updating vector using a fixed gain algorithm. With this configuration, an amount of the calculation can be reduced. 
   Preferably, the updating vector correcting means corrects the updating vector by multiplying a past value of at least one element of the updating vector by a predetermined value (DELTAi, EPSi) which is greater than 0 and less than 1. With this configuration, an influence of the past value of the identifying error is reduced, and the drift of the model parameter(s) is prevented. 
   Preferably, the updating vector correcting means does not multiply one of an element (an element relating to the calculation of b 1 ) of the updating vector which is relevant to the input of the plant and an element (an element relating to the calculation of c 1 ) of the updating vector which is irrelevant to the input and the output of the plant, by the predetermined value (DELTAi, EPSi). According to this configuration, the steady-state deviation which may be caused by correcting these elements can be prevented from occurring. 
   Preferably, the updating vector correcting means multiplies at least one element of the reference vector (θ(0)) by the predetermined value. According to this configuration, the drift of the model parameters is prevented. 
   Preferably, the reference vector is calculated according to a parameter (DTH) indicative of a change in a dynamic characteristic of the plant. This makes it possible to promptly converge the model parameter(s) especially when the plant includes a nonlinear element. 
   The present invention provides another control system for a plant, including identifying means for identifying a model parameter vector of a controlled object model which is obtained by modeling the plant, based on an input and an output of the plant, and a sliding mode controller for controlling the plant using the model parameter vector identified by the identifying means. The identifying means includes identifying error calculating means for calculating an identifying error (ide) of the model parameter vector, identifying error correcting means for correcting the identifying error in a decreasing direction, if the identifying error is in a predetermined range (−EIDNRLMT≦ide≦EIDNRLMT), model parameter vector calculating means for calculating the model parameter vector using the identifying error (idenl) corrected by the identifying error correcting means, and limiting means for limiting a value of at least one element of the model parameter vector calculated by the model parameter vector calculating means, within a predetermined limit range. 
   With this configuration, if the identifying error is in the predetermined range, the identifying error is corrected in a decreasing direction, and the model parameter vector is calculated using the corrected identifying error. Further, the value of at least one element of the model parameter vector is limited within the predetermined limit range. Therefore, the drift of the model parameters is prevented, to thereby improve the control stability 
   Preferably, the limiting means limits values of a plurality of elements (a 1 , a 2 ) of the model parameter vector so that the plurality of elements satisfies a predetermined relation ( FIG. 18 ). 
   With this configuration, the values of a plurality of elements of the model parameter vector is limited so that the plurality of elements satisfies a predetermined relation. Accordingly, stability of the control using the model parameter vector can be improved. 
   Preferably, the identifying error correcting means sets the identifying error (ide) to “0”, if the identifying error (ide) is in the predetermined range (−EIDNRLMT≦ide≦EIDNRLMT). 
   Preferably, the plant includes a throttle valve actuating device having a throttle valve of an internal combustion engine and actuating means for actuating the throttle valve, and the sliding mode controller calculates a parameter for determining a control input to be applied to the throttle valve actuating device to make the opening of the throttle valve coincide with a target opening. 
   With this configuration, the control to make the opening of the throttle valve coincide with the target opening is performed using model parameters identified by the identifying means. Consequently, the controllability of the throttle valve opening to the target opening can be improved, and the throttle valve opening can be controlled more stably. 
   Further, the present invention provides another control system for a plant, including identifying means for identifying a plurality of model parameters of a controlled object model which is obtained by modeling the plant using a predetermined reference value (THDEF), the plurality of model parameters including a model parameter (c 1 ) which is irrelevant to an input and an output of the plant, control means for controlling the plant using the model parameters identified by the identifying means, and correcting means for correcting the predetermined reference value (THDEF) according to the model parameter (c 1 ) which is irrelevant to an input and an output of the plant. 
   With this configuration, the plurality of model parameters includes a model parameter which is irrelevant to an input and an output of the plant, and the predetermined reference value is corrected according to the model parameter which is irrelevant to an input and an output of the plant. Since the reference value is corrected by the model parameter itself of the controlled object model, the present invention is applicable to any control system which employs any control method. Further, the reference value of the controlled object model, which has a direct influence on the modeling error (a difference between the characteristics of the actual plant and the characteristics of the controlled object model), can be made coincide with an actual reference value of the plant. Accordingly, the modeling error can be reduced and the control performance can be improved. 
   Preferably, the correcting means calculates a correction value (thdefadp) by statistically processing the model parameter (c 1 ) which is irrelevant to an input and an output of the plant, and corrects the predetermined reference value (THDEF) with the correction value. 
   Preferably, the controlled object model is defined by at least one first model parameter (a 1 , a 2 ) which is relevant to an output of the plant, a second model parameter (b 1 ) which is relevant to a control input of the plant, and a third model parameter (c 1 ) which is irrelevant to the output and the control input of the plant. 
   Preferably, the plant includes a throttle valve actuating device ( 10 ) having a throttle valve ( 3 ) of an internal combustion engine and actuating means ( 6 ) for actuating the throttle valve, and the control means calculates a parameter (DUT) for determining a control input to be applied to the throttle valve actuating device ( 10 ) to make the opening (TH) of the throttle valve coincide with a target opening (THR). 
   Preferably, the throttle valve actuating device ( 10 ) includes first energizing means ( 4 ) for energizing the throttle valve in a closing direction, and second energizing means ( 5 ) for energizing the throttle valve in an opening direction, and maintains the throttle valve at a default opening (THDEF) by means of the first and second energizing means ( 4 ,  5 ), when the throttle valve is not actuated by the actuating means ( 6 ). The predetermined reference value is set to the default opening (THDEF). 
   The present invention provides another control system for a throttle valve actuating device including a throttle valve of an internal combustion engine and actuating means for actuating the throttle valve. The control system includes identifying means for identifying at least one model parameter of a controlled object model which is obtained by modeling the throttle valve actuating device, and control means for controlling an opening of the throttle valve to a target opening according to the at least one model parameter identified by the identifying means. The identifying means includes identifying error calculating means for calculating an identifying error (ide) of the at least one model parameter, and identifying error correcting means for correcting the identifying error in a decreasing direction, if the identifying error is in a predetermined range (−EIDNRLMT≦ide≦EIDNRLMT). The identifying means calculates the at least one model parameter using the identifying error corrected by the identifying error correcting means, and the predetermined range is set according to a value (DDTHRSQA) depending on the opening (TH) of the throttle valve or the target opening (THR). 
   With this configuration, the identifying error is corrected in a decreasing direction, if the identifying error is in a predetermined range, and the at least one model parameter is calculated using the corrected identifying error. Accordingly, an integration of the identifying error to the at least one model parameter is suppressed, which results in preventing a drift of the model parameter and improving a control stability. Further, the predetermined range is set according to the value depending on the opening of the throttle valve or the target opening. Accordingly, a range where a degree of influence of the identifying error should be reduced, becomes so suitable that it is prevented to reduce or neglect the identifying error to be reflected to a value of the model parameter, as an unnecessary error. 
   Preferably, the control system further includes predicting means for calculating a predicted value (PREDTH) of a future throttle valve opening, using the at least one model parameter identified by the identifying means. 
   With this configuration, the predicted value of a future throttle valve opening is predicted using the at least one model parameter identified by the identifying means. Accordingly, it is possible to calculate an accurate predicted value even when the dynamic characteristics of the plant changes with time or changes due to environmental conditions. 
   Preferably, the control means controls the throttle valve actuating device with a sliding mode control, using the at least one model parameter identified by the identifying means. According to this configuration, the controllability of the throttle valve opening to the target opening can be improved even in the presence of a modeling error, and the throttle valve opening can be controlled more stably. 

   
     BRIEF DESCRIPTION OF DRAWINGS 
       FIG. 1  is a schematic view of a throttle valve control system according to a first embodiment of the present invention; 
       FIGS. 2A and 2B  are diagrams showing frequency characteristics of the throttle valve actuating device shown in  FIG. 1 ; 
       FIG. 3  is a functional block diagram showing functions realized by an electronic control unit (ECU) shown in  FIG. 1 ; 
       FIG. 4  is a diagram showing the relationship between control characteristics of a sliding mode controller and the value of a switching function setting parameter (VPOLE); 
       FIG. 5  is a diagram showing a range for setting control gains (F, G) of the sliding mode controller; 
       FIGS. 6A and 6B  are diagrams illustrative of a drift of model parameters; 
       FIGS. 7A through 7C  are diagrams showing functions for correcting an identifying error; 
       FIG. 8  is a diagram illustrating that a default opening deviation of a throttle valve is reflected to a model parameter (c 1 ′); 
       FIG. 9  is a flowchart showing a throttle valve opening control process; 
       FIG. 10  is a flowchart showing a process of setting state variables in the process shown in  FIG. 9 ; 
       FIG. 11  is a flowchart showing a process of performing calculations of a model parameter identifier in the process shown in  FIG. 9 ; 
       FIG. 12  is a flowchart showing a process of calculating an identifying error (ide) in the process shown in  FIG. 11 ; 
       FIGS. 13A and 13B  are diagrams illustrative of a process of low-pass filtering on the identifying error (ide); 
       FIG. 14  is a flowchart showing the dead zone process in the process shown in  FIG. 12 ; 
       FIG. 15  is a diagram showing a table used in the process shown in  FIG. 14 ; 
       FIG. 16  is a flowchart showing a process of stabilizing a model parameter vector (θ) in the process shown in  FIG. 11 ; 
       FIG. 17  is a flowchart showing a limit process of model parameters (a 1 ′, a 2 ′) in the process shown in  FIG. 16 ; 
       FIG. 18  is a diagram illustrative of the change in the values of the model parameters in the process shown in  FIG. 16 ; 
       FIG. 19  is a flowchart showing a limit process of a model parameter (b 1 ′) in the process shown in  FIG. 16 ; 
       FIG. 20  is a flowchart showing a limit process of a model parameter (c 1 ′) in the process shown in  FIG. 16 ; 
       FIG. 21  is a flowchart showing a process of performing calculations of a state predictor in the process shown in  FIG. 9 ; 
       FIG. 22  is a flowchart showing a process of calculating a control input (Usl) in the process shown in  FIG. 9 ; 
       FIG. 23  is a flowchart showing a process of calculating a predicted switching function value (σpre) in the process shown in  FIG. 22 ; 
       FIG. 24  is a flowchart showing a process of calculating the switching function setting parameter (VPOLE) in the process shown in  FIG. 23 ; 
       FIGS. 25A through 25C  are diagrams showing maps used in the process shown in  FIG. 24 ; 
       FIG. 26  is a flowchart showing a process of calculating an integrated value of the predicted switching function value (σpre) in the process shown in  FIG. 22 ; 
       FIG. 27  is a flowchart showing a process of calculating a reaching law input (Urch) in the process shown in  FIG. 22 ; 
       FIG. 28  is a flowchart showing a process of calculating an adaptive law input (Uadp) in the process shown in  FIG. 22 ; 
       FIG. 29  is a flowchart showing a process of determining the stability of the sliding mode controller in the process shown in  FIG. 9 ; 
       FIG. 30  is a flowchart showing a process of calculating a default opening deviation (thdefadp) in the process shown in  FIG. 9 ; 
       FIG. 31  is a functional block diagram showing functions realized by the electronic control unit (ECU) shown in  FIG. 1  according to a second embodiment of the present invention; 
       FIG. 32  is a flowchart showing a throttle valve opening control process according to the second embodiment; 
       FIG. 33  is a flowchart showing a process of performing calculations of a model parameter identifier in the process shown in FIG.  32 ; 
       FIG. 34  is a diagram showing a table used in the process shown in  FIG. 33 ; 
       FIG. 35  is a flowchart showing a process of calculating an identifying error (ide) in the process shown in  FIG. 33 ; 
       FIG. 36  is a flowchart showing a process of calculating a control input (Usl) in the process shown in  FIG. 32 ; 
       FIG. 37  is a flowchart showing a process of calculating a switching function value (σ) in the process shown in  FIG. 36 ; 
       FIG. 38  is a flowchart showing a process of calculating an integrated value of the switching function value (σ) in the process shown in  FIG. 36 ; 
       FIG. 39  is a flowchart showing a process of calculating a reaching law input (Urch) in the process shown in  FIG. 36 ; 
       FIG. 40  is a flowchart showing a process of calculating an adaptive law input (Uadp) in the process shown in  FIG. 36 ; 
       FIG. 41  is a flowchart showing a process of determining the stability of a sliding mode controller in the process shown in  FIG. 32 ; 
       FIG. 42  is a block diagram of a control system according to a third embodiment of the present invention; 
       FIG. 43  is a block diagram of a modification of the control system shown in  FIG. 42 ; 
       FIG. 44  is a block diagram of a control system according to a fourth embodiment of the present invention; 
       FIG. 45  is a block diagram of a modification of the control system shown in  FIG. 44 ; 
       FIG. 46  is a block diagram of a control system according to a fifth embodiment of the present invention; and 
       FIG. 47  is a block diagram of a control system according to a sixth embodiment of the present invention. 
   

   BEST MODE FOR CARRYING OUT THE INVENTION 
   The present invention will be explained below with reference to the drawings. 
   First Embodiment 
     FIG. 1  schematically shows a configuration of a throttle valve control system according to a first embodiment of the present invention. An internal combustion engine (hereinafter referred to as “engine”)  1  has an intake passage  2  with a throttle valve  3  disposed therein. The throttle valve  3  is provided with a return spring  4  as a first energizing means for energizing the throttle valve  3  in a closing direction, and a resilient member  5  as a second energizing means for energizing the throttle valve  3  in an opening direction. The throttle valve  3  can be actuated by a motor  6  as an actuating means through gears (not shown). When the actuating force from the motor  6  is not applied to the throttle valve  3 , an opening TH of the throttle valve  3  is maintained at a default opening THDEF (for example, 5 degrees) where the energizing force of the return spring  4  and the energizing force of the resilient member  5  are in equilibrium. 
   The motor  6  is connected to an electronic control unit (hereinafter referred to as “ECU”)  7 . The operation of the motor  6  is controlled by the ECU  7 . The throttle valve  3  is associated with a throttle valve opening sensor  8  for detecting the throttle valve opening TH. A detected signal from the throttle valve opening sensor  8  is supplied to the ECU  7 . 
   Further, the ECU  7  is connected to an acceleration sensor  9  for detecting a depression amount ACC of an accelerator pedal to detect an output demanded by the driver of the vehicle on which the engine  1  is mounted. A detected signal from the acceleration sensor  9  is supplied to the ECU  7 . 
   The ECU  7  has an input circuit, an A/D converter, a central processing unit (CPU), a memory circuit, and an output circuit. The input circuit is supplied with detected signals from the throttle valve opening sensor  8  and the acceleration sensor  9 . The A/D converter converts input signals into digital signals. The CPU carries out various process operations. The memory circuit has a ROM (read only memory) for storing processes executed by the CPU, and maps and tables that are referred to in the processes, a RAM for storing results of executing processes by the CPU. The output circuit supplies an energizing current to the motor  6 . The ECU  7  determines a target opening THR of the throttle valve  3  according to the depression amount ACC of the accelerator pedal, determines a control quantity DUT for the motor  6  in order to make the detected throttle valve opening TH coincide with the target opening THR, and supplies an electric signal according to the control quantity DUT to the motor  6 . 
   In the present embodiment, a throttle valve actuating device  10  that includes the throttle valve  3 , the return spring  4 , the resilient member  5 , and the motor  6  is a controlled object. An input to be applied to the controlled object is a duty ratio DUT of the electric signal applied to the motor  6 . An output from the controlled object is the throttle valve opening TH detected by the throttle valve opening sensor  8 . 
   When frequency response characteristics of the throttle valve actuating device  10  are measured, gain characteristics and phase characteristics indicated by the solid lines in  FIGS. 2A and 2B  are obtained. A model defined by the equation (1) shown below is set as a controlled object model. Frequency response characteristics of the model are indicated by the broken-line curves in  FIGS. 2A and 2B . It has been confirmed that the frequency response characteristics of the model are similar to the characteristics of the throttle valve actuating device  10 . 
                   DTH   ⁡     (     k   +   1     )       =       a1   ×     DTH   ⁡     (   k   )         +     a2   ×     DTH   ⁡     (     k   -   1     )         +     b1   ×     DUT   ⁡     (     k   -   d     )         +   c1             (   1   )               
where k is a parameter representing discrete time, and DTH(k) is a throttle valve opening deviation amount defined by the equation (2) shown below. DTH(k+1) is a throttle valve opening deviation amount at a discrete time (k+1).
   DTH ( k )= TH ( k )− THDEF   (2) 
where TH is a detected throttle valve opening, and THDEF is the default opening.
 
   In the equation (1), a 1 , a 2 , b 1 , and c 1  are parameters determining the characteristics of the controlled object model, and d is a dead time. The dead time is a delay between the input and output of the controlled object model. 
   The model defined by the equation (1) is a DARX model (delayed autoregressive model with exogeneous input) of a discrete time system, which is employed for facilitating the application of an adaptive control. 
   In the equation (1), the model parameter c 1  which is irrelevant to the input and output of the controlled object, in addition to the model parameters a 1  and a 2  which are relevant to the output deviation amount DTH and the model parameter b 1  which is relevant to the input duty ratio DUT. The model parameter c 1  is a parameter representing a deviation amount of the default opening THDEF and disturbance applied to the throttle valve actuating device  10 . In other words, the default opening deviation amount and the disturbance can be identified by identifying the model parameter c 1  simultaneously with the model parameters a 1 , a 2 , and b 1  by a model parameter identifier. 
     FIG. 3  is a functional block diagram of the throttle valve control system which is realized by the ECU  7 . The throttle valve control system as configured includes, an adaptive sliding mode controller  21 , a model parameter identifier  22 , a state predictor  23  for calculating a predicted throttle valve opening deviation amount (hereinafter referred to as “predicted deviation amount” or PREDTH(k)) where PREDTH(k) (=DTH(k+d)) after the dead time d has elapsed, and a target opening setting unit  24  for setting a target opening THR for the throttle valve  3  according to the accelerator pedal depression amount ACC. 
   The adaptive sliding mode controller  21  calculates a duty ratio DUT according to an adaptive sliding mode control in order to make the detected throttle valve opening TH coincide with the target opening THR, and outputs the calculated duty ratio DUT. 
   By using the adaptive sliding mode controller  21 , it is possible to change the response characteristics of the throttle valve opening TH to the target opening THR, using a specific parameter (VPOLE). As a result, it is possible to avoid shocks at the time the throttle valve  3  moves from an open position to a fully closed position, i.e., at the time the throttle valve  3  collides with a stopper for stopping the throttle valve  3  in the fully closed position. It is also possible to make the engine response corresponding to the operation of the accelerator pedal variable. Further, it is also possible to obtain a good stability against errors of the model parameters. 
   The model parameter identifier  22  calculates a corrected model parameter vector θL (θL T =[a 1 , a 2 , b 1 , c 1 ]) and supplies the calculated corrected model parameter vector θL to the adaptive sliding mode controller  21 . More specifically, the model parameter identifier  22  calculates a model parameter vector θ based on the throttle valve opening TH and the duty ratio DUT. The model parameter identifier  22  then carries out a limit process of the model parameter vector θ to calculate the corrected model parameter vector θL, and supplies the corrected model parameter vector θL to the adaptive sliding mode controller  21 . In this manner, the model parameters a 1 , a 2 , and b 1  which are optimum for making the throttle valve opening TH follow up the target opening THR are obtained, and also the model parameter c 1  indicative of disturbance and a deviation amount of the default opening THDEF is obtained. 
   By using the model parameter identifier  22  for identifying the model parameters on a real-time basis, adaptation to changes in engine operating conditions, compensation for hardware characteristics variations, compensatation for power supply voltage fluctuations, and adaptation to aging-dependent changes of hardware characteristics are possible. 
   The state predictor  23  calculates a throttle valve opening TH (predicted value) after the dead time d has elapsed, or more specifically a predicted deviation amount PREDTH, based on the throttle valve opening TH and the duty ratio DUT, and supplies the calculated deviation amount PREDTH to the adaptive sliding mode controller  21 . By using the predicted deviation amount PREDTH, the robustness of the control system against the dead time of the controlled object is ensured, and the controllability in the vicinity of the default opening THDEF where the dead time is large is improved. 
   Next, principles of operation of the adaptive sliding mode controller  21  will be hereinafter described. 
   First, a target value DTHR(k) is defined as a deviation amount between the target opening THR(k) and the default opening THDEF by the following equation (3).
 
 DTHR ( k )= THR ( k )− THDEF   (3)
 
   If a deviation e(k) between the throttle valve opening deviation amount DTH and the target value DTHR is defined by the following equation (4), then a switching function value σ(k) of the adaptive sliding mode controller is set by the following equation (5). 
                   e   ⁡     (   k   )       =       DTH   ⁡     (   k   )       -     DTHR   ⁡     (   k   )                 (   4   )                 σ   ⁡     (   k   )       =         e   ⁡     (   k   )       +     VPOLE   ×     e   ⁡     (     k   -   1     )           =       (       DTH   ⁡     (   k   )       -     DTHR   ⁡     (   k   )         )     +     VPOLE   ×     (       DTH   ⁡     (     k   -   1     )       -     DTHR   ⁡     (     k   -   1     )         )                   (   5   )               
where VPOLE is a switching function setting parameter that is set to a value which is greater than −1 and less than 1.
 
   On a phase plane defined by a vertical axis representing the deviation e(k) and a horizontal axis representing the preceding deviation e(k−1), a pair of the deviation e(k) and the preceding deviation e(k−1) satisfying the equation of “σ(k)=0” represents a straight line. The straight line is generally referred to as a switching straight line. A sliding mode control is a control contemplating the behavior of the deviation e(k) on the switching straight line. The sliding mode control is carried out so that the switching function value σ(k) becomes 0, i.e., the pair of the deviation e(k) and the preceding deviation e(k−1) exists on the switching straight line on the phase plane, to thereby achieve a robust control against disturbance and the modeling error (the difference between the characteristics of an actual plant and the characteristics of a controlled object model). As a result, the throttle valve opening deviation amount DTH is controlled with good robustness to follow up the target value DTHR. 
   As shown in  FIG. 4 , by changing the value of the switching function setting parameter VPOLE in the equation (5), it is possible to change damping characteristics of the deviation e(k), i.e., the follow-up characteristics of the throttle valve opening deviation amount DTH to follow the target value DTHR. Specifically, if VPOLE equals −1, then the throttle valve opening deviation amount DTH completely fails to follow up the target value DTHR. As the absolute value of the switching function setting parameter VPOLE is reduced, the speed at which the throttle valve opening deviation amount DTH follows up the target value DTHR increases. 
   The throttle valve control system is required to satisfy the following requirements A1 and A2: 
   A1) When the throttle valve  3  is shifted to the fully closed position, collision of the throttle valve  3  with the stopper for stopping the throttle valve  3  in the fully closed position should be avoided; and 
   A2) The controllability with respect to the nonlinear characteristics in the vicinity of the default opening THDEF (a change in the resiliency characteristics due to the equilibrium between the energizing force of the return spring  4  and the energizing force of the resilient member  5 , backlash of gears interposed between the motor  6  and the throttle valve  3 , and a dead zone where the throttle valve opening does not change even when the duty ratio DUT changes) should be improved. 
   Therefore, it is necessary to lower the speed at which the deviation e(k) converges, i.e., the converging speed of the deviation e(k), in the vicinity of the fully closed position of the throttle valve, and to increase the converging speed of the deviation e(k) in the vicinity of the default opening THDEF. 
   According to the sliding mode control, the converging speed of e(k) can easily be changed by changing the switching function setting parameter VPOLE. Therefore in the present embodiment, the switching function setting parameter VPOLE is set according to the throttle valve opening TH and an amount of change DDTHR (=DTHR(k)−DTHR(k−1)) of the target value DTHR, to thereby satisfy the requirements A1 and A2. 
   As described above, according to the sliding mode control, the deviation e(k) is converged to 0 at an indicated converging speed and robustly against disturbance and the modeling error by constraining the pair of the deviation e(k) and the preceding deviation e(k−1) on the switching straight line (the pair of e(k) and e(k−1) will be hereinafter referred to as “deviation state quantity”). Therefore, in the sliding mode control, it is important how to place the deviation state quantity onto the switching straight line and constrain the deviation state quantity on the switching straight line. 
   From the above standpoint, an input DUT(k) (also indicated as Usl(k)) to the controlled object (an output of the controller) is expressed as the sum of an equivalent control input Ueq(k), a reaching law input Urch(k), and an adaptive law input Uadp(k), as indicated by the following equation (6). 
   
     
       
         
           
             
               
                 
                   
                     
                       
                         DUT 
                         ⁡ 
                         
                           ( 
                           k 
                           ) 
                         
                       
                       = 
                         
                       ⁢ 
                       
                         Us1 
                         ⁡ 
                         
                           ( 
                           k 
                           ) 
                         
                       
                     
                   
                 
                 
                   
                     
                       = 
                         
                       ⁢ 
                       
                         
                           Ueq 
                           ⁡ 
                           
                             ( 
                             k 
                             ) 
                           
                         
                         + 
                         
                           Urch 
                           ⁡ 
                           
                             ( 
                             k 
                             ) 
                           
                         
                         + 
                         
                           Uadp 
                           ⁡ 
                           
                             ( 
                             k 
                             ) 
                           
                         
                       
                     
                   
                 
               
             
             
               
                 ( 
                 6 
                 ) 
               
             
           
         
       
     
   
   The equivalent control input Ueq(k) is an input for constraining the deviation state quantity on the switching straight line. The reaching law input Urch(k) is an input for placing deviation state quantity onto the switching straight line. The adaptive law input Uadp(k) is an input for placing deviation state quantity onto the switching straight line while reducing the modeling error and the effect of disturbance. Methods of calculating these inputs Ueq(k), Urch(k), and Uadp(k) will be described below. 
   Since the equivalent control input Ueq(k) is an input for constraining the deviation state quantity on the switching straight line, a condition to be satisfied is given by the following equation (7):
 
σ( k )=σ( k+ 1)  (7)
 
   Using the equations (1), (4), and (5), the duty ratio DUT(k) satisfying the equation (7) is determined by the equation (9) shown below. The duty ratio DUT(k) calculated with the equation (9) represents the equivalent control input Ueq(k). The reaching law input Urch(k) and the adaptive law input Uadp(k) are defined by the respective equations (10) and (11) shown below. 
                   DUT   ⁡     (   k   )       =         1   b1     ⁢     {         (     1   -   a1   -   VPOLE     )     ⁢     DTH   ⁡     (     k   +   d     )         +       (     VPOLE   -   a2     )     ⁢     DTH   ⁡     (     k   +   d   -   1     )         -   c1   +     DTHR   ⁡     (     k   +   d   +   1     )       +       (     VPOLE   -   1     )     ⁢     DTHR   ⁡     (     k   +   d     )         -     VPOLE   ×     DTHR   ⁡     (     k   +   d   -   1     )           }       =     Ueq   ⁡     (   k   )                 (   9   )                 Urch   ⁡     (   k   )       =         -   F     b1     ⁢     σ   ⁡     (     k   +   d     )                 (   10   )                 Uadp   ⁡     (   k   )       =         -   G     b1     ⁢       ∑     i   =   0       k   +   d       ⁢           ⁢     Δ   ⁢           ⁢   T   ⁢           ⁢     σ   ⁡     (   i   )                     (   11   )               
where F and G respectively represent a reaching law control gain and an adaptive law control gain, which are set as described below, and ΔT represents a control period.
 
   Calculating the equation (9) requires a throttle valve opening deviation amount DTH(k+d) after the elapse of the dead time d and a corresponding target value DTHR(k+d+1). Therefore, the predicted deviation amount PREDTH(k) calculated by the state predictor  23  is used as the throttle valve opening deviation amount DTH(k+d) after the elapse of the dead time d, and the latest target value DTHR is used as the target value DTHR(k+d+1). 
   Next, the reaching law control gain F and the adaptive law control gain G are determined so that the deviation state quantity can stably be placed onto the switching straight line by the reaching law input Urch and the adaptive law input Uadp. 
   Specifically, a disturbance V(k) is assumed, and a stability condition for keeping the switching function value σ(k) stable against the disturbance V(k) are determined to obtain a condition for setting the gains F and G. As a result, it has been obtained as the stability condition that the combination of the gains F and G satisfies the following equations (12) through (14), in other words, the combination of the gains F and G should be located in a hatched region shown in  FIG. 5 .
 
 F&gt; 0  (12)
 
 G&gt; 0  (13)
 
 F&lt; 2−(Δ T/ 2) G   (14)
 
   As described above, the equivalent control input Ueq(k), the reaching law input Urch(k), and the adaptive law input Uadp(k) are calculated from the equations (9) through (11), and the duty ratio DUT(k) is calculated as the sum of those inputs. 
   The model parameter identifier  22  calculates a model parameter vector of the controlled object model, based on the input (DUT(k)) and output (TH(k)) of the controlled object, as described above. Specifically, the model parameter identifier  22  calculates a model parameter vector θ(k) according to a sequential identifying algorithm (generalized sequential method-of-least-squares algorithm) represented by the following equation (15).
 
θ( k )=θ( k− 1)+ KP ( k ) ide ( k )  (15)
 
θ( k ) T   =[a 1 ′, a 2′,  b 1′,  c 1′]  (16)
 
where a 1 ′, a 2 ′, b 1 ζ, c 1 ′ represent model parameters before a limit process described later is carried out, ide(k) represents an identifying error defined by the equations (17), (18), and (19) shown below, where DTHHAT(k) represents an estimated value of the throttle valve opening deviation amount DTH(k) (hereinafter referred to as “estimated throttle valve opening deviation amount”) which is calculated using the latest model parameter vector θ(k−1), and KP(k) represents a gain coefficient vector defined by the equation (20) shown below. In the equation (20), P(k) represents a quartic square matrix calculated from the equation (21) shown below.
 
   
     
       
         
           
             
               
                 
                   i 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   d 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     e 
                     ⁡ 
                     
                       ( 
                       k 
                       ) 
                     
                   
                 
                 = 
                 
                   
                     DTH 
                     ⁡ 
                     
                       ( 
                       k 
                       ) 
                     
                   
                   - 
                   
                     DTHHAT 
                     ⁡ 
                     
                       ( 
                       k 
                       ) 
                     
                   
                 
               
             
             
               
                 ( 
                 17 
                 ) 
               
             
           
           
             
               
                 
                   DTHHAT 
                   ⁡ 
                   
                     ( 
                     k 
                     ) 
                   
                 
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                           - 
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                 ( 
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                                 ( 
                                 k 
                                 ) 
                               
                             
                             ⁢ 
                             
                               
                                 ζ 
                                 T 
                               
                               ⁡ 
                               
                                 ( 
                                 k 
                                 ) 
                               
                             
                           
                           
                             
                               λ 
                               1 
                             
                             + 
                             
                               
                                 λ 
                                 2 
                               
                               ⁢ 
                               
                                 
                                   ζ 
                                   T 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 P 
                                 ⁡ 
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 ζ 
                                 ⁡ 
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                             
                           
                         
                       
                       ) 
                     
                     ⁢ 
                     
                       P 
                       ⁡ 
                       
                         ( 
                         k 
                         ) 
                       
                     
                   
                 
                 ⁢ 
                 
                   
 
                 
                 ⁢ 
                 
                   ( 
                   
                     ⁢ 
                     
                       : 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     Identity 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     Matrix 
                   
                   ) 
                 
               
             
             
               
                 ( 
                 21 
                 ) 
               
             
           
         
       
     
   
   In accordance with the setting of coefficients λ1 and λ2 in the equation (21), the identifying algorithm from the equations (15) through (21) becomes one of the following four identifying algorithm: 
   
     
       
         
             
             
             
           
             
                 
                 
             
           
          
             
                 
               λ1 = 1, λ2 = 0 
               Fixed gain algorithm 
             
             
                 
               λ1 = 1, λ2 = 1 
               Method-of-least-squares algorithm 
             
             
                 
               λ1 = 1, λ2 = λ 
               Degressive gain algorithm 
             
             
                 
                 
               (λ is a given value other than 0, 1) 
             
             
                 
               λ1 = λ, λ2 = 1 
               Weighted Method-of-least- squares algorithm 
             
             
                 
                 
               (λ is a given value other than 0, 1) 
             
             
                 
                 
             
          
         
       
     
   
   In the present embodiment, it is required that the following requirements B1, B2, and B3 are satisfied: 
   B1) Adaptation to quasi-static dynamic characteristics changes and hardware characteristics variations 
   “Quasi-static dynamic characteristics changes” mean slow-rate characteristics changes such as power supply voltage fluctuations or hardware degradations due to aging. 
   B2) Adaptation to high-rate dynamic characteristics changes 
   Specifically, this means adaptation to dynamic characteristics changes depending on changes in the throttle valve opening TH. 
   B3) Prevention of a drift of model parameters 
   The drift, which is an excessive increase of the absolute values of model parameters, should be prevented. The drift of model parameters is caused by the effect of the identifying error, which should not be reflected to the model parameters, due to nonlinear characteristics of the controlled object. 
   In order to satisfy the requirements B1 and B2, the coefficients λ1 and λ2 are set respectively to a given value λ and “0” so that the weighted Method-of-least-squares algorithm is employed. 
   Next, the drift of model parameters will be described below. As shown in  FIG. 6A  and  FIG. 6B , if a residual identifying error, which is caused by nonlinear characteristics such as friction characteristics of the throttle valve, exists after the model parameters have been converged to a certain extent, or if a disturbance whose average value is not zero is steadily applied, then residual identifying errors are accumulated, causing a drift of model parameters. 
   Since such a residual identifying error should not be reflected to the values of model parameters, a dead zone process is carried out using a dead zone function Fn 1  as shown in  FIG. 7A . Specifically, a corrected identifying error idenl(k) is calculated from the following equation (23), and a model parameter vector θ(k) is calculated using the corrected identifying error idenl(k). That is, the following equation (15a) is used instead of the above equation (15). In this manner, the requirement B3) can be satisfied.
 
 idenl ( k )= Fnl ( ide ( k ))  (23)
 
θ( k )=θ( k− 1)+ KP ( k ) idenl ( k )  (15a)
 
   The dead zone function Fn 1  is not limited to the function shown in  FIG. 7A . A discontinuous dead zone function as shown in  FIG. 7B  or an incomplete dead zone function as shown in  FIG. 7C  may be used as the dead zone function Fnl. However, it is impossible to completely prevent the drift if the incomplete dead zone function is used. 
   The amplitude of the residual identifying error changes according to an amount of change in the throttle valve opening TH. In the present embodiment, a dead zone width parameter EIDNRLMT which defines the width of the dead zone shown in  FIGS. 7A through 7C  is set according to the square average value DDTHRSQA of an amount of change in the target throttle valve opening THR. Specifically, the dead zone width parameter EIDNRLMT is set such that it increases as the square average value DDTHRSQA increases. According to such setting of the dead zone width parameter EIDNRLMT, it is prevented to neglect an identifying error to be reflected to the values of the model parameters as the residual identifying error. In the following equation (24), DDTHR represents an amount of change in the target throttle valve opening THR, which is calculated from the following equation (25): 
   
     
       
         
           
             
               
                 
                   DDTHRSQA 
                   ⁡ 
                   
                     ( 
                     k 
                     ) 
                   
                 
                 = 
                 
                   
                     1 
                     
                       n 
                       + 
                       1 
                     
                   
                   ⁢ 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         0 
                       
                       n 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         DDTHR 
                         ⁡ 
                         
                           ( 
                           i 
                           ) 
                         
                       
                       2 
                     
                   
                 
               
             
             
               
                 ( 
                 24 
                 ) 
               
             
           
           
             
               
                 
                   
                     
                       
                         DDTHR 
                         ⁡ 
                         
                           ( 
                           k 
                           ) 
                         
                       
                       = 
                         
                       ⁢ 
                       
                         
                           DTHR 
                           ⁡ 
                           
                             ( 
                             k 
                             ) 
                           
                         
                         - 
                         
                           DTHR 
                           ⁡ 
                           
                             ( 
                             
                               k 
                               - 
                               1 
                             
                             ) 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       = 
                         
                       ⁢ 
                       
                         
                           THR 
                           ⁡ 
                           
                             ( 
                             k 
                             ) 
                           
                         
                         - 
                         
                           THR 
                           ⁡ 
                           
                             ( 
                             
                               k 
                               - 
                               1 
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
             
             
               
                 ( 
                 25 
                 ) 
               
             
           
         
       
     
   
   Since the throttle valve opening deviation amount DTH is controlled to the target value DTHR by the adaptive sliding mode controller  21 , the target value DTHR in the equation (25) may be changed to the throttle valve opening deviation amount DTH. In this case, an amount of change DDTH in the throttle valve opening deviation amount DTH may be calculated, and the dead zone width parameter EIDNRLMT may be set according to the square average value DDTHRSQA obtained by replacing DDTHR in the equation (24) with DDTH. 
   For further improving the robustness of the control system, it is effective to further stabilize the adaptive sliding mode controller  21 . In the present embodiment, the elements a 1 ′, a 2 ′, b 1 ′, and c 1 ′ of the model parameter vector θ(k) calculated from the equation (15) are subjected to the limit process so that a corrected model parameter vector θL(k) (θL(k) T =[a 1 , a 2 , b 1 , c1]) is calculated. The adaptive sliding mode controller  21  performs a sliding mode control using the corrected model parameter vector θL(k). The limit process will be described later in detail referring to the flowcharts. 
   Next, a method for calculating the predicted deviation amount PREDTH in the state predictor  23  will be described below. 
   First, matrixes A, B and vectors X(k), U(k) are defined according to the following equations (26) through (29). 
   
     
       
         
           
             
               
                 A 
                 = 
                 
                   [ 
                   
                     
                       
                         a1 
                       
                       
                         a2 
                       
                     
                     
                       
                         1 
                       
                       
                         0 
                       
                     
                   
                   ] 
                 
               
             
             
               
                 ( 
                 26 
                 ) 
               
             
           
           
             
               
                 B 
                 = 
                 
                   [ 
                   
                     
                       
                         b1 
                       
                       
                         c1 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                     
                   
                   ] 
                 
               
             
             
               
                 ( 
                 27 
                 ) 
               
             
           
           
             
               
                 
                   X 
                   ⁡ 
                   
                     ( 
                     k 
                     ) 
                   
                 
                 = 
                 
                   [ 
                   
                     
                       
                         
                           DTH 
                           ⁡ 
                           
                             ( 
                             k 
                             ) 
                           
                         
                       
                     
                     
                       
                         
                           DTH 
                           ⁡ 
                           
                             ( 
                             
                               k 
                               - 
                               1 
                             
                             ) 
                           
                         
                       
                     
                   
                   ] 
                 
               
             
             
               
                 ( 
                 28 
                 ) 
               
             
           
           
             
               
                 
                   U 
                   ⁡ 
                   
                     ( 
                     k 
                     ) 
                   
                 
                 = 
                 
                   [ 
                   
                     
                       
                         
                           DUT 
                           ⁡ 
                           
                             ( 
                             k 
                             ) 
                           
                         
                       
                     
                     
                       
                         1 
                       
                     
                   
                   ] 
                 
               
             
             
               
                 ( 
                 29 
                 ) 
               
             
           
         
       
     
   
   By rewriting the equation (1) which defines the controlled object model, using the matrixes A, B and the vectors X(k), U(k), the following equation (30) is obtained.
 
 X ( k+ 1)= AX ( k )+ BU ( k−d )  (30)
 
   Determining X(k+d) from the equation (30), the following equation (31) is obtained. 
   
     
       
         
           
             
               
                 
                   X 
                   ⁡ 
                   
                     ( 
                     
                       k 
                       + 
                       d 
                     
                     ) 
                   
                 
                 = 
                 
                   
                     
                       A 
                       d 
                     
                     ⁢ 
                     
                       X 
                       ⁡ 
                       
                         ( 
                         k 
                         ) 
                       
                     
                   
                   + 
                   
                     
                       [ 
                       
                         
                           A 
                           
                             d 
                             - 
                             1 
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         B 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           A 
                           
                             d 
                             - 
                             2 
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         B 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         ⋯ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         AB 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         B 
                       
                       ] 
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             
                               U 
                               ⁡ 
                               
                                 ( 
                                 
                                   k 
                                   - 
                                   1 
                                 
                                 ) 
                               
                             
                           
                         
                         
                           
                             
                               U 
                               ⁡ 
                               
                                 ( 
                                 
                                   k 
                                   - 
                                   2 
                                 
                                 ) 
                               
                             
                           
                         
                         
                           
                             ⋮ 
                           
                         
                         
                           
                             
                               U 
                               ⁡ 
                               
                                 ( 
                                 
                                   k 
                                   - 
                                   d 
                                 
                                 ) 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
             
             
               
                 ( 
                 31 
                 ) 
               
             
           
         
       
     
   
   If matrixes A′ and B′ are defined by the following equations (32), (33), using the model parameters a 1 ′, a 2 ′, b 1 ′, and c 1 ′ which are not subjected to the limit process, a predicted vector XHAT(k+d) is given by the following equation (34). 
   
     
       
         
           
             
               
                 
                   A 
                   ′ 
                 
                 = 
                 
                   [ 
                   
                     
                       
                         
                           a1 
                           ′ 
                         
                       
                       
                         
                           a2 
                           ′ 
                         
                       
                     
                     
                       
                         1 
                       
                       
                         0 
                       
                     
                   
                   ] 
                 
               
             
             
               
                 ( 
                 32 
                 ) 
               
             
           
           
             
               
                 
                   B 
                   ′ 
                 
                 = 
                 
                   [ 
                   
                     
                       
                         
                           b1 
                           ′ 
                         
                       
                       
                         
                           c2 
                           ′ 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                     
                   
                   ] 
                 
               
             
             
               
                 ( 
                 33 
                 ) 
               
             
           
           
             
               
                 
                   XHAT 
                   ⁡ 
                   
                     ( 
                     
                       k 
                       + 
                       d 
                     
                     ) 
                   
                 
                 = 
                 
                   
                     
                       A 
                       
                         ′ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         d 
                       
                     
                     ⁢ 
                     
                       X 
                       ⁡ 
                       
                         ( 
                         k 
                         ) 
                       
                     
                   
                   + 
                   
                     
                       [ 
                       
                         
                           A 
                           
                             
                               ′ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               d 
                             
                             - 
                             1 
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           B 
                           ′ 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           A 
                           
                             
                               ′ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               d 
                             
                             - 
                             2 
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         ⋯ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           A 
                           ′ 
                         
                         ⁢ 
                         
                           B 
                           ′ 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           B 
                           ′ 
                         
                       
                       ] 
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             
                               U 
                               ⁡ 
                               
                                 ( 
                                 
                                   k 
                                   - 
                                   1 
                                 
                                 ) 
                               
                             
                           
                         
                         
                           
                             
                               U 
                               ⁡ 
                               
                                 ( 
                                 
                                   k 
                                   - 
                                   2 
                                 
                                 ) 
                               
                             
                           
                         
                         
                           
                             ⋮ 
                           
                         
                         
                           
                             
                               U 
                               ⁡ 
                               
                                 ( 
                                 
                                   k 
                                   - 
                                   d 
                                 
                                 ) 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
             
             
               
                 ( 
                 34 
                 ) 
               
             
           
         
       
     
   
   The first-row element DTHHAT(k+d) of the predicted vector XHAT(k+d) corresponds to the predicted deviation amount PREDTH(k), and is given by the following equation (35). 
                   PREDTH   ⁡     (   k   )       =       DTHHAT   ⁡     (     k   +   d     )       =       α   ⁢           ⁢   1   ×     DTH   ⁡     (   k   )         +     α   ⁢           ⁢   2   ×     DTH   ⁡     (     k   -   1     )         +     β   ⁢           ⁢   1   ×     DUT   ⁡     (     k   -   1     )         +     β   ⁢           ⁢   2   ×     DUT   ⁡     (     k   -   2     )         +   ⋯   +     β   ⁢           ⁢   d   ×     DUT   ⁡     (     k   -   d     )         +     γ   ⁢           ⁢   1     +   γ2   +   ⋯   +     γ   ⁢           ⁢   d                 (   35   )               
where α1 represents a first-row, first-column element of the matrix A′ d , α2 represents a first-row, second-column element of the matrix A′ d , βi represents a first-row, first-column element of the matrix A′ d−i B′, and γi represents a first-row, second-column element of the matrix A′ d−i B′.
 
   By applying the predicted deviation amount PREDTH(k) calculated from the equation (35) to the equation (9), and replacing the target values DTHR(k+d+1), DTHR(k+d), and DTHR(k+d−1) respectively with DTHR(k), DTHR(k−1), and DTHR(k−2), the following equation (9a) is obtained. From the equation (9a), the equivalent control input Ueq(k) is calculated. 
   
     
       
         
           
             
               
                 
                   DUT 
                   ⁡ 
                   
                     ( 
                     k 
                     ) 
                   
                 
                 = 
                 
                   
                     1 
                     b1 
                   
                   ⁢ 
                   
                     { 
                     
                       
                         
                           
                             ( 
                             
                               1 
                               - 
                               a1 
                               - 
                               VPOLE 
                             
                             ) 
                           
                           ⁢ 
                           
                             PREDTH 
                             ⁡ 
                             
                               ( 
                               k 
                               ) 
                             
                           
                         
                         + 
                         
                           
                             ( 
                             
                               VPOLE 
                               - 
                               a2 
                             
                             ) 
                           
                           ⁢ 
                           PREDTH 
                           ⁢ 
                           
                               
                             
                               
                                 ( 
                                 
                                   k 
                                   - 
                                   1 
                                 
                                 ) 
                               
                               - 
                               c1 
                               + 
                               
                                 DTHR 
                                 ⁡ 
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                               + 
                               
                                 
                                   ( 
                                   
                                     VPOLE 
                                     - 
                                     1 
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   DTHR 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       k 
                                       - 
                                       1 
                                     
                                     ) 
                                   
                                 
                               
                               - 
                               
                                 VPOLE 
                                 × 
                                 
                                   DTHR 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       k 
                                       - 
                                       2 
                                     
                                     ) 
                                   
                                 
                               
                             
                             } 
                           
                         
                       
                       = 
                       
                         Ueq 
                         ⁡ 
                         
                           ( 
                           k 
                           ) 
                         
                       
                     
                   
                 
               
             
             
               
                 ( 
                 
                   9 
                   ⁢ 
                   a 
                 
                 ) 
               
             
           
         
       
     
   
   Using the predicted deviation amount PREDTH(k) calculated from the equation (35), a predicted switching function value σpre(k) is defined by the following equation (36). The reaching law input Urch(k) and the adaptive law input Uadp(k) are calculated respectively from the following equations (10a) and (11a). 
   
     
       
         
           
             
               
                 
                   σ 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     pre 
                     ⁡ 
                     
                       ( 
                       k 
                       ) 
                     
                   
                 
                 = 
                 
                   
                     ( 
                     
                       
                         PREDTH 
                         ⁡ 
                         
                           ( 
                           k 
                           ) 
                         
                       
                       - 
                       
                         DTHR 
                         ⁡ 
                         
                           ( 
                           
                             k 
                             - 
                             1 
                           
                           ) 
                         
                       
                     
                     ) 
                   
                   + 
                   
                     VPOLE 
                     ⁡ 
                     
                       ( 
                       
                         
                           PREDTH 
                           ⁡ 
                           
                             ( 
                             
                               k 
                               - 
                               1 
                             
                             ) 
                           
                         
                         - 
                         
                           DTHR 
                           ⁡ 
                           
                             ( 
                             
                               k 
                               - 
                               2 
                             
                             ) 
                           
                         
                       
                       ) 
                     
                   
                 
               
             
             
               
                 ( 
                 36 
                 ) 
               
             
           
           
             
               
                 
                   Urch 
                   ⁡ 
                   
                     ( 
                     k 
                     ) 
                   
                 
                 = 
                 
                   
                     
                       - 
                       F 
                     
                     b1 
                   
                   ⁢ 
                   σ 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     pre 
                     ⁡ 
                     
                       ( 
                       k 
                       ) 
                     
                   
                 
               
             
             
               
                 ( 
                 
                   10 
                   ⁢ 
                   a 
                 
                 ) 
               
             
           
           
             
               
                 
                   Uadp 
                   ⁡ 
                   
                     ( 
                     k 
                     ) 
                   
                 
                 = 
                 
                   
                     
                       - 
                       G 
                     
                     b1 
                   
                   ⁢ 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         0 
                       
                       k 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       Δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       T 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       σ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         pre 
                         ⁡ 
                         
                           ( 
                           i 
                           ) 
                         
                       
                     
                   
                 
               
             
             
               
                 ( 
                 
                   11 
                   ⁢ 
                   a 
                 
                 ) 
               
             
           
         
       
     
   
   The model parameter c 1 ′ is a parameter representing a deviation of the default opening THDEF and disturbance. Therefore, as shown in  FIG. 8 , the model parameter c 1 ′ changes with disturbance, but can be regarded as substantially constant in a relatively short period. In the present embodiment, the model parameter c 1 ′ is statistically processed, and the central value of its variations is calculated as a default opening deviation thdefadp. The default opening deviation thdefadp is used for calculating the throttle valve opening deviation amount DTH and the target value DTHR. 
   Generally, the method of least squares is known as a method of the statistic process. In the statistic process according to the method of least squares, all data, i.e., all identified parameters c 1 ′, obtained in a certain period are stored in a memory and the stored data is subjected to a batch calculation of the statistic process at a certain timing. However, the batch calculation requires a memory having a large storage capacity for storing all data, and an increased amount of calculations are necessary because inverse matrix calculations are required. 
   Therefore, according to the present embodiment, the sequential method-of-least-squares algorithm for adaptive control which is indicated by the equations (15) through (21) is applied to the statistic process, and the central value of the least squares of the model parameter c 1  is calculated as the default opening deviation thdefadp. 
   Specifically, in the equations (15) through (21), by replacing θ(k) and θ(k) T  with thdefadp, replacing ζ(k) and ζ(k) T  with “1”, replacing ide(k) with ecl(k), replacing KP(k) with KPTH(k), replacing P(k) with PTH(k), and replacing λ1 and λ2 respectively with λ1′ and λ2′, the following equations (37) through (40) are obtained. 
   
     
       
         
           
             
               
                 
                   thdefadp 
                   ⁡ 
                   
                     ( 
                     
                       k 
                       + 
                       1 
                     
                     ) 
                   
                 
                 = 
                 
                   
                     thdefadp 
                     ⁡ 
                     
                       ( 
                       k 
                       ) 
                     
                   
                   + 
                   
                     
                       KPTH 
                       ⁡ 
                       
                         ( 
                         k 
                         ) 
                       
                     
                     ⁢ 
                     
                       ec1 
                       ⁡ 
                       
                         ( 
                         k 
                         ) 
                       
                     
                   
                 
               
             
             
               
                 ( 
                 37 
                 ) 
               
             
           
           
             
               
                 
                   KPTH 
                   ⁡ 
                   
                     ( 
                     k 
                     ) 
                   
                 
                 = 
                 
                   
                     PTH 
                     ⁡ 
                     
                       ( 
                       k 
                       ) 
                     
                   
                   
                     1 
                     + 
                     
                       PTH 
                       ⁡ 
                       
                         ( 
                         k 
                         ) 
                       
                     
                   
                 
               
             
             
               
                 ( 
                 38 
                 ) 
               
             
           
           
             
               
                 
                   PTH 
                   ⁡ 
                   
                     ( 
                     
                       k 
                       + 
                       1 
                     
                     ) 
                   
                 
                 = 
                 
                   
                     1 
                     
                       λ 
                       1 
                       ′ 
                     
                   
                   ⁢ 
                   
                     ( 
                     
                       1 
                       - 
                       
                         
                           
                             λ 
                             2 
                             ′ 
                           
                           ⁢ 
                           
                             PTH 
                             ⁡ 
                             
                               ( 
                               k 
                               ) 
                             
                           
                         
                         
                           
                             λ 
                             1 
                             ′ 
                           
                           + 
                           
                             
                               λ 
                               2 
                               ′ 
                             
                             ⁢ 
                             
                               PTH 
                               ⁡ 
                               
                                 ( 
                                 k 
                                 ) 
                               
                             
                           
                         
                       
                     
                     ) 
                   
                   ⁢ 
                   
                     PTH 
                     ⁡ 
                     
                       ( 
                       k 
                       ) 
                     
                   
                 
               
             
             
               
                 ( 
                 39 
                 ) 
               
             
           
           
             
               
                 
                   ec1 
                   ⁡ 
                   
                     ( 
                     k 
                     ) 
                   
                 
                 = 
                 
                   
                     
                       c1 
                       ′ 
                     
                     ⁡ 
                     
                       ( 
                       k 
                       ) 
                     
                   
                   - 
                   
                     thdefadp 
                     ⁡ 
                     
                       ( 
                       k 
                       ) 
                     
                   
                 
               
             
             
               
                 ( 
                 40 
                 ) 
               
             
           
         
       
     
   
   One of the four algorithms described above can be selected according to the setting of the coefficients λ1′ and λ2′. In the equation (39), the coefficient λ1′ is set to a given value other than 0 or 1, and the coefficient λ2′ is set to 1, thus employing the weighted method of least squares. 
   For the calculations of the equations (37) through (40), the values to be stored are thdefadp(k+1) and PTh(k+1) only, and no inverse matrix calculations are required. Therefore, by employing the sequential method-of-least-squares algorithm, the model parameter c1 can be statistically processed according to the method of least squares while overcoming the shortcomings of a general method of least squares. 
   The default opening deviation thdefadp obtained as a result of the statistic process is applied to the equations (2) and (3), and the throttle valve opening deviation amount DTH(k) and the target value DTHR(k) are calculated from the following equations (41) and (42) instead of the equations (2) and (3).
 
 DTH ( k )= TH ( k )− THDEF+thdefadp   (41)
 
 DTHR ( k )= THR ( k )− THDEF+thdefadp   (42)
 
   Using the equations (41) and (42), even when the default opening THDEF is shifted from its designed value due to characteristic variations or aging of the hardware, the shift can be compensated to perform an accurate control process. 
   Operation processes executed by the CPU in the ECU  7  for realizing the functions of the adaptive sliding mode controller  21 , the model parameter identifier  22 , and the state predictor  23  will be described below. 
     FIG. 9  is a flowchart showing a process of the throttle valve opening control. The process is executed by the CPU in the ECU  7  in every predetermined period of time (e.g., 2 msec). 
   In step S 11 , a process of setting a state variable shown in  FIG. 10  is performed. Calculations of the equations (41) and (42) are executed to determine the throttle valve opening deviation amount DTH(k) and the target value DTHR(k) (steps S 21  and S 22  in  FIG. 10 ). The symbol (k) representing a current value may sometimes be omitted as shown in  FIG. 10 . 
   In step S 12 , a process of performing calculations of the model parameter identifier as shown in  FIG. 11 , i.e., a process of calculating the model parameter vector θ(k) from the equation (15a), is carried out. Further, the model parameter vector θ(k) is subjected to the limit process so that the corrected model parameter vector θL(k) is calculated. 
   In step S 13 , a process of performing calculations of the state predictor as shown in  FIG. 21  is carried out to calculate the predicted deviation amount PREDTH(k). 
   Next, using the corrected model parameter vector θL(k) calculated in step S 12 , a process of calculating the control input Usl(k) as shown in  FIG. 22  is carried out in step S 14 . Specifically, the equivalent control input Ueq, the reaching law input Urch(k), and the adaptive law input Uadp(k) are calculated, and the control input Usl(k) (=duty ratio DUT(k)) is calculated as a sum of these inputs Ueq(k), Urch(k), and Uadp(k). 
   In step S 16 , a process of stability determination of the sliding mode controller as shown in  FIG. 29  is carried out. Specifically, the stability is determined based on a differential value of the Lyapunov function, and a stability determination flag FSMCSTAB is set. When the stability determination flag FSMCSTAB is set to “1”, this indicates that the adaptive sliding mode controller  21  is unstable. 
   If the stability determination flag FSMCSTAB is set to “1”, indicating that the adaptive sliding mode controller  21  is unstable, the switching function setting parameter VPOLE is set to a predetermined stabilizing value XPOLESTB (see steps S 231  and S 232  in  FIG. 24 ), and the equivalent control input Ueq is set to “0”. That is, the control process by the adaptive sliding mode controller  21  is switched to a control process based on only the reaching law input Urch and the adaptive law input Uadp, to thereby stabilize the control (see steps S 206  and S 208  in  FIG. 22 ). 
   Further, when the adaptive sliding mode controller  21  has become unstable, the equations for calculating the reaching law input Urch and the adaptive law input Uadp are changed. Specifically, the values of the reaching law control gain F and the adaptive law control gain G are changed to values for stabilizing the adaptive sliding mode controller  21 , and the reaching law input Urch and the adaptive law input Uadp are calculated without using the model parameter b 1  (see  FIGS. 27 and 28 ). According to the above stabilizing process, it is possible to quickly terminate the unstable state of the adaptive sliding mode controller  21 , and to bring the adaptive sliding mode controller  21  back to its stable state. 
   In step S 17 , a process of calculating the default opening deviation thdefadp as shown in  FIG. 30  is performed to calculate the default opening deviation thdefadp. 
     FIG. 11  is a flowchart showing a process of performing calculations of the model parameter identifier  22 . 
   In step S 31 , the gain coefficient vector KP(k) is calculated from the equation (20). Then, the estimated throttle valve opening deviation amount DTHHAT(k) is calculated from the equation (18) in step S 32 . In step S 33 , a process of calculating the identifying error idenl(k) as shown in  FIG. 12  is carried out. The estimated throttle valve opening deviation amount DTHHAT(k) calculated in step S 32  is applied to the equation (17) to calculate the identifying error ide(k). Further in step S 32 , the dead zone process using the function shown in  FIG. 7A  is carried out to calculate the corrected identifying error idenl. 
   In step S 34 , the model parameter vector θ(k) is calculated from the equation (15a). Then, the model parameter vector θ(k) is subjected to the stabilization process in step S 35 . That is, each of the model parameters is subjected to the limit process to calculate the corrected model parameter vector θL(k). 
     FIG. 12  is a flowchart showing a process of calculating the identifying error idenl(k) which is carried out in step S 33  shown in  FIG. 11 . 
   In step S 51 , the identifying error ide(k) is calculated from the equation (17). Then, it is determined whether or not the value of a counter CNTIDST which is incremented in step S 53  is greater than a predetermined value XCNTIDST that is set according to the dead time d of the controlled object (step S 52 ). The predetermined value XCNTIDST is set, for example, to “3” according to a dead time d=2. Since the counter CNTIDST has an initial value of “0”, the process first goes to step S 53 , in which the counter CNTIDST is incremented by “1”. Then, the identifying error ide(k) is set to “0” in step S 54 , after which the process goes to step S 55 . Immediately after starting identifying the model parameter vector θ(k), no correct identifying error can be obtained by the equation (17). Therefore, the identifying error ide(k) is set to “0” according to steps S 52  through S 54 , instead of using the calculated result of the equation (17). 
   If the answer to the step S 52  is affirmative (YES), then the process immediately proceeds to step S 55 . 
   In step S 55 , the identifying error ide(k) is subjected to a low-pass filtering. Specifically, when identifying the model parameters of an controlled object which has low-pass characteristics, the identifying weight of the method-of-least-squares algorithm for the identifying error ide(k) has frequency characteristics as indicated by the solid line L 1  in  FIG. 13A . By the low-pass filtering of the identifying error ide(k), the frequency characteristics as indicated by the solid line L 1  is changed to a frequency characteristics as indicated by the broken line L 2  where the high-frequency components are attenuated. The reason for executing the low-pass filtering will be described below. 
   The frequency characteristics of the actual controlled object having low-pass characteristics and the controlled object model thereof are represented respectively by the solid lines L 3  and L 4  in  FIG. 13B . Specifically, if the model parameters are identified by the model parameter identifier  22  with respect to the controlled object which has low-pass characteristics (characteristics of attenuating high-frequency components), the identified model parameters are largely affected by the high-frequency-rejection characteristics so that the gain of the controlled object model becomes lower than the actual characteristics in a low-frequency range. As a result, the sliding mode controller  21  excessively corrects the control input. 
   By changing the frequency characteristics of the weighting of the identifying algorithm to the characteristics indicated by the broken line L 2  in  FIG. 13A  according to the low-pass filtering, the frequency characteristics of the controlled object are changed to frequency characteristics indicated by the broken line L 5  in  FIG. 13B . As a result, the frequency characteristics of the controlled object model is made to coincide with the actual frequency characteristics, or the low frequency gain of the controlled object model is corrected to a level which is slightly higher than the actual gain. Accordinly, it is possible to prevent the control input from being excessively corrected by the sliding mode controller  21 , to thereby improve the robustness of the control system and further stabilize the control system. 
   The low-pass filtering is carried out by storing past values ide(k−i) of the identifying error (e.g., 10 past values for i=1 through 10) in a ring buffer, multiplying the past values by weighting coefficients, and adding the products of the past values and the weighting coefficients. 
   Since the identifying error ide(k) is calculated from the equations (17), (18), and (19), the same effect as described above can be obtained by performing the same low-pass filtering on the throttle valve opening deviation amount DTH(k) and the estimated throttle valve opening deviation amount DTHHAT(k), or by performing the same low-pass filtering on the throttle valve opening deviation amounts DTH(k−1), DTH(k−2) and the duty ratio DUT(k−d−1). 
   Referring back to  FIG. 12 , the dead zone process as shown in  FIG. 14  is carried out in step S 56 . In step S 61  shown in  FIG. 14 , “n” in the equation (24) is set, for example, to “5” to calculate the square average value DDTHRSQA of an amount of change of the target throttle valve opening THR. Then, an EIDNRLMT table shown in  FIG. 15  is retrieved according to the square average value DDTHRSQA to calculate the dead zone width parameter EIDNRLMT (step S 62 ). 
   In step S 63 , it is determined whether or not the identifying error ide(k) is greater than the dead zone width parameter EIDNRLMT. If ide(k) is greater than EIDNRLMT, the corrected identifying error idenl(k) is calculated from the following equation (43) in step S 67 .
 
 idenl ( k )= ide ( k )− EIDNRLMT   (43)
 
   If the answer to step S 63  is negative (NO), it is determined whether or not the identifying error ide(k) is greater than the negative value of the dead zone width parameter EIDNRLMT with a minus sign (step S 64 ). 
   If ide(k) is less than −EIDNRLMT, the corrected identifying error idenl(k) is calculated from the following equation (44) in step S 65 .
 
 idenl ( k )= ide ( k )+ EIDNRLMT   (44)
 
   If the identifying error ide(k) is in the range between +EIDNRLMT and −EIDNRLMT, the corrected identifying error idenl(k) is set to “0” in step S 66 . 
     FIG. 16  is a flowchart showing a process of stabilizing the model parameter vector θ(k), which is carried out in step S 35  shown in  FIG. 11 . 
   In step S 71  shown in  FIG. 16 , flags FA 1 STAB, FA 2 STAB, FB 1 LMT, and FC 1 LMT used in this process are initialized to be set to “0”. In step S 72 , the limit process of the model parameters a 1 ′ and a 2 ′ shown in  FIG. 17  is executed. In step S 73 , the limit process of the model parameter b 1 ′ shown in  FIG. 19  is executed. In step S 74 , the limit process of the model parameter c 1 ′ shown in  FIG. 20  is executed. 
     FIG. 17  is a flowchart showing the limit process of the model parameters a 1 ′ and a 2 ′, which is carried out in the step S 72  shown in  FIG. 16 .  FIG. 18  is a diagram illustrative of the process shown in  FIG. 17 , and will be referred to with  FIG. 17 . 
   In  FIG. 18 , combinations of the model parameters a 1 ′ and a 2 ′ which are required to be limited are indicated by “x” symbols, and the range of combinations of the model parameters a 1 ′ and a 2 ′ which are stable are indicated by a hatched region (hereinafter referred to as “stable region”). The limit process shown in  FIG. 17  is a process of moving the combinations of the model parameters a 1 ′ and a 2 ′ which are in the outside of the stable region into the stable region at positions indicated by “◯” symbols. 
   In step S 81 , it is determined whether or not the model parameter a 2 ′ is greater than or equal to a predetermined a 2  lower limit value XIDA 2 L. The predetermined a 2  lower limit value XIDA 2 L is set to a negative value greater than “−1”. Stable corrected model parameters a 1  and a 2  are obtained when setting the predetermined a 2  lower limit value XIDA 2 L to “−1”. However, the predetermined a 2  lower limit value XIDA 2 L is set to a negative value greater than “−1” because the matrix A defined by the equation (26) to the “n”th power may occasionally become unstable (which means that the model parameters a 1 ′ and a 2 ′ do not diverge, but oscillate). 
   If a 2 ′ is less than XIDA 2 L in step S 81 , the corrected model parameter a 2  is set to the lower limit value XIDA 2 L, and an a 2  stabilizing flag FA 2 STAB is set to “1”. When the a 2  stabilizing flag FA 2 STAB is set to “1”, this indicates that the corrected model parameter a 2  is set to the lower limit value XIDA 2 L. In  FIG. 18 , the correction of the model parameter in a limit process P 1  of steps S 81  and S 82  is indicated by the arrow lines with “P 1 ”. 
   If the answer to the step S 81  is affirmative (YES), i.e., if a 2 ′ is greater than or equal to XIDA 2 L, the corrected model parameter a 2  is set to the model parameter a 2 ′ in step S 83 . 
   In steps S 84  and S 85 , it is determined whether or not the model parameter a 1 ′ is in a range defined by a predetermined a 1  lower limit value XIDA 1 L and a predetermined a 1  upper limit value XIDA 1 H. The predetermined a 1  lower limit value XIDA 1 L is set to a value which is equal to or greater than “−2” and lower than “0”, and the predetermined a 1  upper limit value XIDA 1 H is set to “2”, for example. 
   If the answers to steps S 84  and S 85  are affirmative (YES), i.e., if a 1 ′ is greater than or equal to XIDA 1 L and less than or equal to XIDA 1 H, the corrected model parameter a 1  is set to the model parameter a 1 ′ in step S 88 . 
   If a 1 ′ is less than XIDA 1 L in step S 84 , the corrected model parameter a 1  is set to the lower limit value XIDA 1 L and an a 1  stabilizing flag FA 1 STAB is set to “1” in step S 86 . If a 1 ′ is greater than XIDA 1 H in step S 85 , the corrected model parameter a 1  is set to the upper limit value XIDA 1 H and the a 1  stabilizing flag FA 1 STAB is set to “1” in step S 87 . When the a 1  stabilizing flag FA 1 STAB is set to “1”, this indicates that the corrected model parameter a 1  is set to the lower limit value XIDA 1 L or the upper limit value XIDA 1 H. In  FIG. 18 , the correction of the model parameter in a limit process P 2  of steps S 84  through S 87  is indicated by the arrow lines with “P 2 ”. 
   In step S 90 , it is determined whether or not the sum of the absolute value of the corrected model parameter a 1  and the corrected model parameter a 2  is less than or equal to a predetermined stability determination value XA 2 STAB. The predetermined stability determination value XA 2 STAB is set to a value close to “1” but less than “1” (e.g., “0.99”). 
   Straight lines L 1  and L 2  shown in  FIG. 18  satisfy the following equation (45).
 
 a 2 +|a 1 |XA 2 STAB   (45)
 
   Therefore, in step S 90 , it is determined whether or not the combination of the corrected model parameters a 1  and a 2  is placed at a position on or lower than the straight lines L 1  and L 2  shown in  FIG. 18 . If the answer to step S 90  is affirmative (YES), the limit process immediately ends, since the combination of the corrected model parameters a 1  and a 2  is in the stable region shown in  FIG. 18 . 
   If the answer to step S 90  is negative (NO), it is determined whether or not the corrected model parameter a 1  is less than or equal to a value obtained by subtracting the predetermined a 2  lower limit value XIDA 2 L from the predetermined stability determination value XA 2 STAB in step S 91  (since XIDA 2 L is less than “0”, XA 2 STAB−XIDA 2 L is greater than XA 2 STAB). If the corrected model parameter a 1  is equal to or less than (XA 2 STAB−XIDA 2 L), the corrected model parameter a 2  is set to (XA 2 STAB−|a 1 |) and the a 2  stabilizing flag FA 2 STAB is set to “1” in step S 92 . 
   If the corrected model parameter a 1  is greater than (XA 2 STAB−XIDA 2 L) in step S 91 , the corrected model parameter a 1  is set to (XA 2 STAB−XIDA 2 L) in step S 93 . Further in step S 93 , the corrected model parameter a 2  is set to the predetermined a 2  lower limit value XIDA 2 L, and the a 1  stabilizing flag FA 1 STAB and the a 2  stabilizing flag FA 2 STAB are set to “1” in step S 93 . 
   In  FIG. 18 , the correction of the model parameter in a limit process P 3  of steps S 91  and S 92  is indicated by the arrow lines with “P 3 ”, and the correction of the model parameter in a limit process P 4  in steps S 91  and S 93  is indicated by the arrow lines with “P 4 ”. 
   As described above, the limit process shown in  FIG. 17  is carried out to bring the model parameters a 1 ′ and a 2 ′ into the stable region shown in  FIG. 18 , thus calculating the corrected model parameters a1 and a2. 
     FIG. 19  is a flowchart showing a limit process of the model parameter b 1 ′, which is carried out in step S 73  shown in  FIG. 16 . 
   In steps S 101  and S 102 , it is determined whether or not the model parameter b 1 ′ is in a range defined by a predetermined b 1  lower limit value XIDB 1 L and a predetermined b 1  upper limit value XIDB 1 H. The predetermined b 1  lower limit value XIDB 1 L is set to a positive value (e.g., “0.1”), and the predetermined b 1  upper limit value XIDB 1 H is set to “1”, for example. 
   If the answers to steps S 101  and S 102  are affirmative (YES), i.e., if b 1 ′ is greater than or equal to XIDB 1 L and less than or equal to XIDB 1 H, the corrected model parameter b 1  is set to the model parameter b 1 ′ in step S 105 . 
   If b 1 ′ is less than XIDB 1 L in step S 101 , the corrected model parameter b 1  is set to the lower limit value XIDB 1 L, and a b 1  limiting flag FB 1 LMT is set to “1” in step S 104 . If b 1 ′ is greater than XIDB 1 H in step S 102 , then the corrected model parameter b 1  is set to the upper limit value XIDB 1 H, and the b 1  limiting flag FB 1 LMT is set to “1” in step S 103 . When the b 1  limiting flag FB 1 LMT is set to “1”, this indicates that the corrected model parameter b 1  is set to the lower limit value XIDB 1 L or the upper limit value XIDB 1 H. 
     FIG. 20  is a flowchart showing a limit process of the model parameter c 1 ′, which is carried out in step S 74  shown in  FIG. 16 . 
   In steps S 111  and S 112 , it is determined whether or not the model parameters c1′ is in a range defined by a predetermined c 1  lower limit value XIDC 1 L and a predetermined c 1  upper limit value XIDC 1 H. The predetermined c 1  lower limit value XIDC 1 L is set to “−60”, for example, and the predetermined c 1  upper limit value XIDC 1 H is set to “60”, for example. 
   If the answers to steps S 111  and S 112  are affirmative (YES), i.e., if c 1 ′ is greater than or equal to XIDC 1 L and less than or equal to XIDC 1 H, the corrected model parameter c 1  is set to the model parameter c 1 ′ in step S 115 . 
   If c 1 ′ is less than XIDC 1 L in step S 111 , the corrected model parameter c 1  is set to the lower limit value XIDC 1 L, and a c 1  limiting flag FC 1 LMT is set to “1” in step S 114 . If c 1 ′ is greater than XIDC 1 H in step S 112 , the corrected model parameter c 1  is set to the upper limit value XIDC 1 H, and the c 1  limiting flag FC 1 LMT is set to “1” in step S 113 . When the c 1  limiting flag FC 1 LMT is set to “1”, this indicates that the corrected model parameter c 1  is set to the lower limit value XIDC 1 L or the upper limit value XIDC 1 H. 
     FIG. 21  is a flowchart showing a process of calculations of the state predictor, which is carried out in step S 13  shown in  FIG. 9 . 
   In step S 121 , the matrix calculations are executed to calculate the matrix elements α 1 , α 2 , β 1 , β 2 , γ 1  through γd in the equation (35). 
   In step S 122 , the predicted deviation amount PREDTH(k) is calculated from the equation (35). 
     FIG. 22  is a flowchart showing a process of calculation of the control input Usl (=DUT) applied to the throttle valve actuating device  10 , which is carried out in step S 14  shown in  FIG. 9 . 
   In step S 201 , a process of calculation of the predicted switching function value σpre, which is shown in  FIG. 23 , is executed. In step S 202 , a process of calculation of the integrated value of the predicted switching function value σpre, which is shown in  FIG. 26 , is executed. In step S 203 , the equivalent control input Ueq is calculated from the equation (9). In step S 204 , a process of calculation of the reaching law input Urch, which is shown in  FIG. 27 , is executed. In step S 205 , a process of calculation of the adaptive law input Uadp, which is shown in  FIG. 28 , is executed. 
   In step S 206 , it is determined whether or not the stability determination flag FSMCSTAB set in the process shown in  FIG. 29  is “1”. When the stability determination flag FSMCSTAB is set to “1”, this indicates that the adaptive sliding mode controller  21  is unstable. 
   If FSMCSTAB is “0” in step S 206 , indicating that the adaptive sliding mode controller  21  is stable, the control inputs Ueq, Urch, and Uadp which are calculated in steps S 203  through S 205  are added, thereby calculating the control input Usl in step S 207 . 
   If FSMCSTAB is “1” in step S 206 , indicating that the adaptive sliding mode controller  21  is unstable, the sum of the reaching law input Urch and the adaptive law input Uadp is calculated as the control input Usl. In other words, the equivalent control input Ueq is not used for calculating the control input Usl, thus preventing the control system from becoming unstable. 
   In steps S 209  and S 210 , it is determined whether or not the calculated control input Usl is in a range defined between a predetermined upper limit value XUSLH and a predetermined lower limit value XUSLL. If the control input Usl is in the range between XUSLH and XUSLL, the process immediately ends. If the control input Usl is less than or equal to the predetermined lower limit value XUSLL in step S 209 , the control input Usl is set to the predetermined lower limit value XUSLL in step S 212 . If the control input Usl is greater than or equal to the predetermined upper limit value XUSLH in step S 210 , the control input Usl is set to the predetermined upper limit value XUSLH in step S 211 . 
     FIG. 23  is a flowchart showing a process of calculating the predicted switching function value σpre, which is carried out in step S 201  shown in  FIG. 22 . 
   In step S 221 , the process of calculating the switching function setting parameter VPOLE, which is shown in  FIG. 24  is executed. Then, the predicted switching function value σpre(k) is calculated from the equation (36) in step S 222 . 
   In steps S 223  and S 224 , it is determined whether or not the calculated predicted switching function value σpre(k) is in a range defined between a predetermined upper value XSGMH and a predetermined lower limit value XSGML. If the calculated predicted switching function value σpre(k) is in the range between XSGMH and XSGML, the process shown in  FIG. 23  immediately ends. If the calculated predicted switching function value σpre(k) is less than or equal to the predetermined lower limit value XSGML in step S 223 , the calculated predicted switching function value σpre(k) is set to the predetermined lower limit value XSGML in step S 225 . If the calculated predicted switching function value σpre(k) is greater than or equal to the predetermined upper limit value XSGMH in step S 224 , the calculated predicted switching function value σpre(k) is set to the predetermined upper limit value XSGMH in step S 226 . 
     FIG. 24  is a flowchart showing a process of calculating the switching function setting parameter VPOLE, which is carried out in step S 221  shown in  FIG. 23 . 
   In step S 231 , it is determined whether or not the stability determination flag FSMCSTAB is “1”. If FSMCSTAB is “1” in step S 231 , indicating that the adaptive sliding mode controller  21  is unstable, the switching function setting parameter VPOLE is set to a predetermined stabilizing value XPOLESTB in step S 232 . The predetermined stabilizing value XPOLESTB is set to a value which is greater than “−1” but very close to “−1” (e.g., “−0.999”). 
   If FSMCSTAB is “0”, indicating that the adaptive sliding mode controller  21  is stable, an amount of change DDTHR(k) in the target value DTHR(k) is calculated from the following equation (46) in step S 233 .
 
 DDTHR ( k )= DTHR ( k )− DTHR ( k− 1)  (46)
 
   In step S 234 , a VPOLE map is retrieved according to the throttle valve opening deviation amount DTH and the amount of change DDTHR calculated in step S 233  to calculate the switching function setting parameter VPOLE. As shown in  FIG. 25A , the VPOLE map is set so that the switching function setting parameter VPOLE increases when the throttle valve opening deviation amount DTH has a value in the vicinity of “0”, i.e., when the throttle valve opening TH is in the vicinity of the default opening THDEF, and the switching function setting parameter VPOLE has a substantially constant value regardless of changes of the throttle valve opening deviation amount DTH, when the throttle valve opening deviation amount DTH has values which are not in the vicinity of “0”. The VPOLE map is also set so that the switching function setting parameter VPOLE increases as the amount of change DDTHR in target value increases as indicated by the solid line in  FIG. 25B , and the switching function setting parameter VPOLE increases as the amount of change DDTHR in the target value has a value in the vicinity of “0” as indicated by the broken line in  FIG. 25B , when the throttle valve opening deviation amount DTH has a value in the vicinity of “0”. 
   Specifically, when the target value DTHR for the throttle valve opening changes greatly in the decreasing direction, the switching function setting parameter VPOLE is set to a relatively small value. This makes it possible to prevent the throttle valve  3  from colliding with the stopper for stopping the throttle valve  3  in the fully closed position. In the vicinity of the default opening THDEF, the switching function setting parameter VPOLE is set to a relatively large value, which improves the controllability in the vicinity of the default opening THDEF. 
   As shown in  FIG. 25C , the VPOLE map may be set so that the switching function setting parameter VPOLE decreases when the throttle valve opening TH is in the vicinity of the fully closed opening or the fully open opening. Therefore, when the throttle valve opening TH is in the vicinity of the fully closed opening or the fully open opening, the speed for the throttle valve opening TH to follow up the target opening THR is reduced. As a result, collision of the throttle valve  3  with the stopper can more positively be avoided (the stopper also stops the throttle valve  3  in the fully open position). 
   In steps S 235  and S 236 , it is determined whether or not the calculated switching function setting parameter VPOLE is in a range defined between a predetermined upper limit value XPOLEH and a predetermined lower limit XPOLEL. If the switching function setting parameter VPOLE is in the range between XPOLEH and XPOLEL, the process shown immediately ends. If the switching function setting parameter VPOLE is less than or equal to the predetermined lower limit value XPOLEL in step S 236 , the switching function setting parameter VPOLE is set to the predetermined lower limit value XPOLEL in step S 238 . If the switching function setting parameter VPOLE is greater than or equal to the predetermined upper limit value XPOLEH in step S 235 , the switching function setting parameter VPOLE is set to the predetermined upper limit value XPOLEH in step S 237 . 
     FIG. 26  is a flowchart showing a process of calculating an integrated value of σpre, SUMSIGMA, of the predicted switching function value σpre. This process is carried out in step S 202  shown in  FIG. 22 . The integrated value SUMSIGMA is used for calculating the adaptive law input Uadp in the process shown in  FIG. 28  which will be described later (see the equation (11a)). 
   In step S 241 , the integrated value SUMSIGMA is calculated from the following equation (47) where ΔT represents a calculation period.
 
SUMSIGMA( k )=SUMSIGMA( k− 1)+σ pre×ΔT   (47)
 
   In steps S 242  and S 243 , it is determined whether or not the calculated integrated value SUMSIGMA is in a range defined between a predetermined upper limit value XSUMSH and a predetermined lower limit value XSUMSL. If the integrated value SUMSIGMA is in the range between XSUMSH and XSUMSL, the process immediately ends. If the integrated value SUMSIGMA is less than or equal to the predetermined lower limit value XSUMSL in step S 242 , the integrated value SUMSIGMA is set to the predetermined lower limit value XSUMSL in step S 244 . If the integrated value SUMSIGMA is greater than or equal to the predetermined upper limit value XSUMSH in step S 243 , the integrated value SUMSIGMA is set to the predetermined upper limit value XSUMSH in step S 245 . 
     FIG. 27  is a flowchart showing a process of calculating the reaching law input Urch, which is carried out in step S 204  shown in  FIG. 22 . 
   In step S 261 , it is determined whether or not the stability determination flag FSMCSTAB is “1”. If the stability determination flag FSMCSTAB is “0”, indicating that the adaptive sliding mode controller  21  is stable, the control gain F is set to a normal gain XKRCH in step S 262 , and the reaching law input Urch is calculated from the following equation (48), which is the same as the equation ( 10   a ), in step S 263 .
 
 Urch=−F×σpre/b 1  (48)
 
   If the stability determination flag FSMCSTAB is “1”, indicating that the adaptive sliding mode controller  21  is unstable, the control gain F is set to a predetermined stabilizing gain XKRCHSTB in step S 264 , and the reaching law input Urch is calculated according to the following equation (49), which does not include the model parameter b 1 , in step S 265 .
 
 Urch=−F×σpre   (49)
 
   In steps S 266  and S 267 , it is determined whether the calculated reaching law input Urch is in a range defined between a predetermined upper limit value XURCHH and a predetermined lower limit value XURCHL. If the reaching law input Urch is in the range between XURCHH and XURCHL, the process immediately ends. If the reaching law input Urch is less than or equal to the predetermined lower limit value XURCHL in step S 266 , the reaching law input Urch is set to the predetermined lower limit value XURCHL in step S 268 . If the reaching law input Urch is greater than or equal to the predetermined upper limit value XURCHH in step S 267 , the reaching law input Urch is set to the predetermined upper limit value XURCHH in step S 269 . 
   As described above, when the adaptive sliding mode controller  21  becomes unstable, the control gain F is set to the predetermined stabilizing gain XKRCHSTB, and the reaching law input Urch is calculated without using the model parameter b 1 , which brings the adaptive sliding mode controller  21  back to its stable state. When the identifying process carried out by the model parameter identifier  22  becomes unstable, the adaptive sliding mode controller  21  becomes unstable. Therefore, by using the equation (49) that does not include the model parameter b 1  which has become unstable, the adaptive sliding mode controller  21  can be stabilized. 
     FIG. 28  is a flowchart showing a process of calculating the adaptive law input Uadp, which is carried out in step S 205  shown in  FIG. 22 . 
   In step S 271 , it is determined whether or not the stability determination flag FSMCSTAB is “1”. If the stability determination flag FSMCSTAB is “0”, indicating that the adaptive sliding mode controller  21  is stable, the control gain G is set to a normal gain XKADP in step S 272 , and the adaptive law input Uadp is calculated from the following equation (50), which corresponds to the equation (11a), in step S 273 .
 
 Uadp=−G× SUMSIGMA/b1  (50)
 
   If the stability determination flag FSMCSTAB is “1”, indicating that the adaptive sliding mode controller  21  is unstable, the control gain G is set to a predetermined stabilizing gain XKADPSTB in step S 274 , and the adaptive law input Uadp is calculated according to the following equation (51), which does not include the model parameter b 1 , in step S 275 .
 
 Uadp=−G ×SUMSIGMA  (51)
 
   As described above, when the adaptive sliding mode controller  21  becomes unstable, the control gain G is set to the predetermined stabilizing gain XKADPSTB, and the adaptive law input Uadp is calculated without using the model parameter b 1 , which brings the adaptive sliding mode controller  21  back to its stable state. 
     FIG. 29  is a flowchart showing a process of determining the stability of the sliding mode controller, which is carried out in step S 16  shown in  FIG. 9 . In this process, the stability is determined based on a differential value of the Lyapunov function, and the stability determination flag FSMCSTAB is set according to the result of the stability determination. 
   In step S 281 , a switching function change amount Dσpre is calculated from the following equation (52). Then, a stability determining parameter SGMSTAB is calculated from the following equation (53) in step S 282 .
 
 Dσpre=σpre ( k )−σ pre ( k− 1)  (52)
 
 SGMSTAB=Dσpre×σpre ( k )  (53)
 
   In step S 283 , it is determined whether or not the stability determination parameter SGMSTAB is less than or equal to a stability determining threshold XSGMSTAB. If SGMSTAB is greater than XSGMSTAB, it is determined that the adaptive sliding mode controller  21  may possibly be unstable, and an unstability detecting counter CNTSMCST is incremented by “1” in step S 285 . If SGMSTAB is less than or equal to XSGMSTAB, the adaptive sliding mode controller  21  is determined to be stable, and the count of the unstability detecting counter CNTSMCST is not incremented but maintained in step S 284 . 
   In step S 286 , it is determined whether or not the value of the unstability detecting counter CNTSMCST is less than or equal to a predetermined count XSSTAB. If CNTSMCST is less than or equal to XSSTAB, the adaptive sliding mode controller  21  is determined to be stable, and a first determination flag FSMCSTAB 1  is set to “0” in step S 287 . If CNTSMCST is greater than XSSTAB, the adaptive sliding mode controller  21  is determined to be unstable, and the first determination flag FSMCSTAB 1  is set to “1” in step S 288 . The value of the unstability detecting counter CNTSMCST is initialized to “0”, when the ignition switch is turned on. 
   In step S 289 , a stability determining period counter CNTJUDST is decremented by “1”. It is determined whether or not the value of the stability determining period counter CNTJUDST is “0” in step S 290 . The value of the stability determining period counter CNTJUDST is initialized to a predetermined determining count XCJUDST, when the ignition switch is turned on. Initially, therefore, the answer to step S 290  is negative (NO), and the process immediately goes to step S 295 . 
   If the count of the stability determining period counter CNTJUDST subsequently becomes “0”, the process goes from step S 290  to step S 291 , in which it is determined whether or not the first determination flag FSMCSTAB 1  is “1”. If the first determination flag FSMCSTAB 1  is “0”, a second determination flag FSMCSTAB 2  is set to “0” in step S 293 . If the first determination flag FSMCSTAB 1  is “1”, the second determination flag FSMCSTAB 2  is set to “1” in step S 292 . 
   In step S 294 , the value of the stability determining period counter CNTJUDST is set to the predetermined determining count XCJUDST, and the unstability detecting counter CNTSMCST is set to “0”. Thereafter, the process goes to step S 295 . 
   In step S 295 , the stability determination flag FSMCSTAB is set to the logical sum of the first determination flag FSMCSTAB 1  and the second determination flag FSMCSTAB 2 . The second determination flag FSMCSTAB 2  is maintained at “1” until the value of the stability determining period counter CNTJUDST becomes “0”, even if the answer to step S 286  becomes affirmative (YES) and the first determination flag FSMCSTAB 1  is set to “0”. Therefore, the stability determination flag FSMCSTAB is also maintained at “1” until the value of the stability determining period counter CNTJUDST becomes “0”. 
     FIG. 30  is a flowchart showing a process of calculating the default opening deviation thdefadp, which is carried out in step S 17  shown in  FIG. 9 . 
   In step S 251  shown  FIG. 30 , a gain coefficient KPTH(k) is calculated according to the following equation (54).
 
 KPTH ( k )= PTH ( k− 1)/(1 +PTH ( k− 1))  (54)
 
where PTH(k−1) represents a gain parameter calculated in step S 253  when the present process was carried out in the preceding cycle.
 
   In step S 252 , the model parameter c 1 ′ calculated in the process of calculations of the model parameter identifier as shown in  FIG. 11  and the gain coefficient KPTH(k) calculated in step S 251  are applied to the following equation (55) to calculate a default opening deviation thdefadp(k). 
   
     
       
         
           
             
               
                 
                   thdefadp 
                   ⁡ 
                   
                     ( 
                     k 
                     ) 
                   
                 
                 = 
                 
                   
                     thdefadp 
                     ⁡ 
                     
                       ( 
                       
                         k 
                         - 
                         1 
                       
                       ) 
                     
                   
                   + 
                   
                     
                       KPTH 
                       ⁡ 
                       
                         ( 
                         k 
                         ) 
                       
                     
                     × 
                     
                       ( 
                       
                         
                           c1 
                           ′ 
                         
                         - 
                         
                           thdefadp 
                           ⁡ 
                           
                             ( 
                             
                               k 
                               - 
                               1 
                             
                             ) 
                           
                         
                       
                       ) 
                     
                   
                 
               
             
             
               
                 ( 
                 55 
                 ) 
               
             
           
         
       
     
   
   In step S 253 , a gain parameter PTH(k) is calculated from the following equation (56): 
   
     
       
         
           
             
               
                 
                   
                     
                       PTH 
                       ⁡ 
                       
                         ( 
                         k 
                         ) 
                       
                     
                     = 
                     
                       
                         { 
                         
                           1 
                           - 
                           
                             PTH 
                             ⁡ 
                             
                               ( 
                               
                                 k 
                                 - 
                                 1 
                               
                               ) 
                             
                           
                         
                         } 
                       
                       / 
                       
                         ( 
                         
                           XDEFADPW 
                           + 
                           
                             PTH 
                             ⁡ 
                             
                               ( 
                               
                                 k 
                                 - 
                                 1 
                               
                               ) 
                             
                           
                         
                         ) 
                       
                     
                   
                   } 
                 
                 × 
                 
                   
                     PTH 
                     ⁡ 
                     
                       ( 
                       
                         k 
                         - 
                         1 
                       
                       ) 
                     
                   
                   / 
                   XDEFADPW 
                 
               
             
             
               
                 ( 
                 56 
                 ) 
               
             
           
         
       
     
   
   The equation (56) is obtained by setting λ1′ and λ2′ in the equation (39) respectively to a predetermined value XDEFADP and “1”. 
   According to the process shown in  FIG. 30 , the model parameter c1′ is statistically processed by the sequential method-of-weighted-least-squares algorithm to calculate the default opening deviation thdefadp. 
   In the present embodiment, the throttle valve actuating device  10  and a portion of the ECU  7  (i.e., an output circuit for supplying an energizing current to the motor  6 ) correspond to a plant. The process shown in  FIG. 22  corresponds to a sliding mode controller. The process shown in  FIG. 11  corresponds to an identifying means. The process shown in  FIG. 12  corresponds to an identifying error calculating means. The process shown in  FIG. 14  corresponds to an identifying error correcting means. The process shown in  FIG. 21  corresponds to a predicting means. 
   Second Embodiment 
   In the first embodiment described above, the controlled object model is defined by the equation (1) including the dead time d, and the predicted deviation amount PREDTH after the elapse of the dead time d is calculated with the state predictor  23 , to thereby control the controlled object model which includes the dead time. Accordingly, it is necessary to execute calculations corresponding to the state predictor  23  in the CPU, and the amount of calculations executed by the CPU becomes large. In the second embodiment, in order to reduce the calculation load on the CPU, the controlled object model is defined by the following equation (1a) where the dead time d is set to “0”, and the modeling error caused by setting the dead time d to “0” is compensated by the robustness of the adaptive sliding mode control. 
   
     
       
         
           
             
               
                 
                   DTH 
                   ⁡ 
                   
                     ( 
                     
                       k 
                       + 
                       1 
                     
                     ) 
                   
                 
                 = 
                 
                   
                     a1 
                     × 
                     
                       DTH 
                       ⁡ 
                       
                         ( 
                         k 
                         ) 
                       
                     
                   
                   + 
                   
                     a2 
                     × 
                     
                       DTH 
                       ⁡ 
                       
                         ( 
                         
                           k 
                           - 
                           1 
                         
                         ) 
                       
                     
                   
                   + 
                   
                     b1 
                     × 
                     
                       DUT 
                       ⁡ 
                       
                         ( 
                         k 
                         ) 
                       
                     
                   
                   + 
                   c1 
                 
               
             
             
               
                 ( 
                 
                   1 
                   ⁢ 
                   a 
                 
                 ) 
               
             
           
         
       
     
   
   In order to further reduce the calculation load on the CPU, the fixed gain algorithm is employed as the algorithm for identifying the model parameters. 
   For further stabilizing the control, another process instead of the dead zone process is employed as the process for preventing the drift of the model parameters. 
   The second embodiment will be described below primarily with respect to details which are different from those of the first embodiment. Other details except for what will be described below are identical to those of the first embodiment. 
     FIG. 31  is a functional block diagram of a throttle valve control system which is realized by the ECU  7 . The throttle valve control system comprises an adaptive sliding mode controller  21   a , a model parameter identifier  22   a , a model parameter scheduler  25 , and a target opening setting unit  24  for setting a target opening THR for the throttle valve  3  according to the accelerator pedal depression amount ACC. 
   The adaptive sliding mode controller  21   a  is supplied with a detected throttle valve opening TH instead of the predicted deviation amount PREDTH. The adaptive sliding mode controller  21   a  calculates a duty ratio DUT in order to make the throttle valve opening TH coincide with a target opening THR. 
   Using the adaptive sliding mode controller  21   a  offers the same advantages as described above in the first embodiment, and achieves the robustness of the control system against the dead time of the controlled object. Therefore, it is possible to compensate for the modeling error that is caused by setting the dead time d to “0”. 
   The model parameter identifier  22   a  calculates a corrected model parameter vector θL (θL T =[a 1 , a 2 , b 1 , c 1 ]) according to a method which is different from the method in the first embodiment, and supplies the calculated corrected model parameter vector θL to the adaptive sliding mode controller  21   a . More specifically, the model parameter identifier  22   a  calculates a model parameter vector θ by correcting a reference model parameter vector θbase supplied from the model parameter scheduler  25 , based on the throttle valve opening TH and the duty ratio DUT. The model parameter identifier  22   a  then performs the limit process of the model parameter vector θ to calculate a corrected model parameter vector θL, and supplies the corrected model parameter vector θL to the adaptive sliding mode controller  21   a . In this manner, model parameters a 1 , a 2 , and b 1  which are optimum for making the throttle valve opening TH follow up the target opening THR are obtained, and a model parameter c 1  indicative of disturbance and a deviation of the default opening THDEF is obtained. 
   The model parameter scheduler  25  calculates a reference model parameter vector θbase (θbase T =[a 1 base, a 2 base, b 1 base, c 1 base]) based on the throttle valve opening TH, and supplies the calculated reference model parameter vector θbase to the model parameter identifier  22   a.    
   In the present embodiment, since the controlled object model is defined by the above equation (1a), the adaptive sliding mode controller  21   a  calculates the equivalent control input Ueq(k), the reaching law input Urch(k), and the adaptive law input Uadp(k) from the following equations (9b), (10b), and (11b) instead of the above equations (9a), (10a), and (11a). 
   
     
       
         
           
             
               
                 
                   DUT 
                   ⁡ 
                   
                     ( 
                     k 
                     ) 
                   
                 
                 = 
                 
                   
                     
                       1 
                       b1 
                     
                     ⁢ 
                     
                       { 
                       
                         
                           
                             ( 
                             
                               1 
                               - 
                               a1 
                               - 
                               VPOLE 
                             
                             ) 
                           
                           ⁢ 
                           
                             DTH 
                             ⁡ 
                             
                               ( 
                               k 
                               ) 
                             
                           
                         
                         + 
                         
                           
                             ( 
                             
                               VPOLE 
                               - 
                               a2 
                             
                             ) 
                           
                           ⁢ 
                           
                             DTH 
                             ⁡ 
                             
                               ( 
                               
                                 k 
                                 - 
                                 1 
                               
                               ) 
                             
                           
                         
                         - 
                         c1 
                         + 
                         
                           DTHR 
                           ⁡ 
                           
                             ( 
                             
                               k 
                               + 
                               1 
                             
                             ) 
                           
                         
                         + 
                         
                           
                             ( 
                             
                               VPOLE 
                               - 
                               1 
                             
                             ) 
                           
                           ⁢ 
                           
                             DTHR 
                             ⁡ 
                             
                               ( 
                               k 
                               ) 
                             
                           
                         
                         - 
                         
                           VPOLE 
                           × 
                           
                             DTHR 
                             ⁡ 
                             
                               ( 
                               
                                 k 
                                 - 
                                 1 
                               
                               ) 
                             
                           
                         
                       
                       } 
                     
                   
                   = 
                   
                     Ueq 
                     ⁡ 
                     
                       ( 
                       k 
                       ) 
                     
                   
                 
               
             
             
               
                 ( 
                 
                   9 
                   ⁢ 
                   b 
                 
                 ) 
               
             
           
           
             
               
                 
                   Urch 
                   ⁡ 
                   
                     ( 
                     k 
                     ) 
                   
                 
                 = 
                 
                   
                     
                       - 
                       F 
                     
                     b1 
                   
                   ⁢ 
                   
                     σ 
                     ⁡ 
                     
                       ( 
                       k 
                       ) 
                     
                   
                 
               
             
             
               
                 ( 
                 
                   10 
                   ⁢ 
                   b 
                 
                 ) 
               
             
           
           
             
               
                 
                   Uadp 
                   ⁡ 
                   
                     ( 
                     k 
                     ) 
                   
                 
                 = 
                 
                   
                     
                       - 
                       G 
                     
                     b1 
                   
                   ⁢ 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         0 
                       
                       k 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       Δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       T 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         σ 
                         ⁡ 
                         
                           ( 
                           i 
                           ) 
                         
                       
                     
                   
                 
               
             
             
               
                 ( 
                 
                   11 
                   ⁢ 
                   b 
                 
                 ) 
               
             
           
         
       
     
   
   The equations (9b), (10b), and (11b) are obtained by setting the dead time d to “0” in the equations (9), (10), and (11). 
   The model parameter identifier  22   a  calculates a model parameter vector of the controlled object model based on the input (DUT(k)) and output (TH(k)) of the controlled object, as described above. Specifically, the model parameter identifier  22   a  calculates a model parameter vector θ(k) from the following equation (15).
 
θ( k )=θ( k− 1)+ KP ( k ) ide ( k )  (15)
 
   The identifying error ide(k) in the equation (15) is defined by the following equations (17), (18), and (19a). The gain coefficient vector KP(k) is defined by the following equation (20), and the square matrix P(k) in the equation (20) is calculated from the following equation (21). 
   
     
       
         
           
             
               
                 
                   i 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   d 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     e 
                     ⁡ 
                     
                       ( 
                       k 
                       ) 
                     
                   
                 
                 = 
                 
                   
                     DTH 
                     ⁡ 
                     
                       ( 
                       k 
                       ) 
                     
                   
                   - 
                   
                     DTHHAT 
                     ⁡ 
                     
                       ( 
                       k 
                       ) 
                     
                   
                 
               
             
             
               
                 ( 
                 17 
                 ) 
               
             
           
           
             
               
                 
                   DTHHAT 
                   ⁡ 
                   
                     ( 
                     k 
                     ) 
                   
                 
                 = 
                 
                   
                     
                       θ 
                       ⁡ 
                       
                         ( 
                         
                           k 
                           - 
                           1 
                         
                         ) 
                       
                     
                     T 
                   
                   ⁢ 
                   
                     ζ 
                     ⁡ 
                     
                       ( 
                       k 
                       ) 
                     
                   
                 
               
             
             
               
                 ( 
                 18 
                 ) 
               
             
           
           
             
               
                 
                   
                     ζ 
                     ⁡ 
                     
                       ( 
                       k 
                       ) 
                     
                   
                   T 
                 
                 = 
                 
                   [ 
                   
                     
                       DTH 
                       ⁡ 
                       
                         ( 
                         
                           k 
                           - 
                           1 
                         
                         ) 
                       
                     
                     , 
                     
                       DTH 
                       ⁡ 
                       
                         ( 
                         
                           k 
                           - 
                           2 
                         
                         ) 
                       
                     
                     , 
                     
                       DUT 
                       ⁡ 
                       
                         ( 
                         
                           k 
                           - 
                           1 
                         
                         ) 
                       
                     
                     , 
                     1 
                   
                   ] 
                 
               
             
             
               
                 ( 
                 
                   19 
                   ⁢ 
                   a 
                 
                 ) 
               
             
           
           
             
               
                 
                   KP 
                   ⁡ 
                   
                     ( 
                     k 
                     ) 
                   
                 
                 = 
                 
                   
                     
                       P 
                       ⁡ 
                       
                         ( 
                         k 
                         ) 
                       
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       ζ 
                       ⁡ 
                       
                         ( 
                         k 
                         ) 
                       
                     
                   
                   
                     1 
                     + 
                     
                       
                         
                           ζ 
                           T 
                         
                         ⁡ 
                         
                           ( 
                           k 
                           ) 
                         
                       
                       ⁢ 
                       
                         P 
                         ⁡ 
                         
                           ( 
                           k 
                           ) 
                         
                       
                       ⁢ 
                       
                         ζ 
                         ⁡ 
                         
                           ( 
                           k 
                           ) 
                         
                       
                     
                   
                 
               
             
             
               
                 ( 
                 20 
                 ) 
               
             
           
           
             
               
                 
                   
                     P 
                     ⁡ 
                     
                       ( 
                       
                         k 
                         + 
                         1 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       1 
                       
                         λ 
                         1 
                       
                     
                     ⁢ 
                     
                       ( 
                       
                         - 
                         
                           
                             
                               λ 
                               2 
                             
                             ⁢ 
                             
                               P 
                               ⁡ 
                               
                                 ( 
                                 k 
                                 ) 
                               
                             
                             ⁢ 
                             
                               ζ 
                               ⁡ 
                               
                                 ( 
                                 k 
                                 ) 
                               
                             
                             ⁢ 
                             
                               
                                 ζ 
                                 T 
                               
                               ⁡ 
                               
                                 ( 
                                 k 
                                 ) 
                               
                             
                           
                           
                             
                               λ 
                               1 
                             
                             + 
                             
                               
                                 λ 
                                 2 
                               
                               ⁢ 
                               
                                 
                                   ζ 
                                   T 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 P 
                                 ⁡ 
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 ζ 
                                 ⁡ 
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                             
                           
                         
                       
                       ) 
                     
                     ⁢ 
                     
                       P 
                       ⁡ 
                       
                         ( 
                         k 
                         ) 
                       
                     
                   
                 
                 ⁢ 
                 
                   
 
                 
                 ⁢ 
                 
                   ( 
                   
                     ⁢ 
                     
                       : 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     Identity 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     Matrix 
                   
                   ) 
                 
               
             
             
               
                 ( 
                 21 
                 ) 
               
             
           
         
       
     
   
   In the present embodiment, the following requirements B4 and B5 are required to be satisfied, in addition to the requirements B1 through B3 which should be satisfied like the first embodiment should be satisfied. 
   B1) Adaptation to quasi-static dynamic characteristic changes and hardware characteristics variations 
   “Quasi-static dynamic characteristics changes” mean slow-rate characteristics changes such as power supply voltage fluctuations or hardware degradations due to aging. 
   B2) Adaptation to high-rate dynamic characteristics changes 
   Specifically, this means adaptation to dynamic characteristics changes depending on changes in the throttle valve opening TH. 
   B3) Prevention of a drift of model parameters 
   The drift, which is an excessive increase of the absolute values of model parameters, should be prevented. The drift of model parameters is caused by the effect of the identifying error, which should not be reflected to the model parameters, due to nonlinear characteristics of the controlled object. 
   B4) Matching with the calculating capability of the ECU 
   Specifically, the amount of calculations is required to be further reduced. 
   B5) The stabilization of model parameters (control performance) 
   Specifically, variations of identified model parameters should be minimized. 
   In order to satisfy the requirement B4, the coefficients λ1 and λ2 are set respectively to “1” and “0”, to thereby employ the fixed gain algorithm. Accordingly, the square matrix P(k) is made constant, and the calculation of the equation (21) can be omitted. As a result, the amount of calculations can greatly be reduced. 
   Specifically, when the fixed gain algorithm is employed, the equation (20) is simplified into the following equation (20a) where P represents a square matrix having constants as diagonal elements. 
   
     
       
         
           
             
               
                 
                   KP 
                   ⁡ 
                   
                     ( 
                     k 
                     ) 
                   
                 
                 = 
                 
                   
                     P 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       ζ 
                       ⁡ 
                       
                         ( 
                         k 
                         ) 
                       
                     
                   
                   
                     1 
                     + 
                     
                       
                         
                           ζ 
                           T 
                         
                         ⁡ 
                         
                           ( 
                           k 
                           ) 
                         
                       
                       ⁢ 
                       P 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ζ 
                         ⁡ 
                         
                           ( 
                           k 
                           ) 
                         
                       
                     
                   
                 
               
             
             
               
                 ( 
                 
                   20 
                   ⁢ 
                   a 
                 
                 ) 
               
             
           
         
       
     
   
   According to the algorithm thus simplified, the amount of calculations can be reduced. However, the identifying ability is also slightly reduced. Further, the equation (15) for calculating the model parameter vector θ(k) can be rewritten to the following equation (15b) and has an integral structure of the identifying error ide(k). Therefore, the identifying error ide(k) is likely to be integrated to the model parameters to cause the drift of the model parameters. 
                   θ   ⁡     (   k   )       =       θ   ⁡     (   0   )       +       KP   ⁡     (   1   )       ⁢     ide   ⁡     (   1   )         +       KP   ⁡     (   2   )       ⁢     ide   ⁡     (   2   )         +   ⋯   +       KP   ⁡     (   k   )       ⁢     ide   ⁡     (   k   )                   (     15   ⁢   b     )               
where θ(0) represents an initial vector having elements of initial values of the model parameters.
 
   In the present embodiment, in order to prevent the drift of the model parameters, the model parameter vector θ(k) is calculated from the following equation (15c) instead of the above equation (15b). 
                         θ   ⁡     (   k   )       =       ⁢       θ   ⁡     (   0   )       +       DELTA     k   -   1       ×     KP   ⁡     (   1   )       ⁢     ide   ⁡     (   1   )         +                     ⁢         DELTA     k   -   2       ×     KP   ⁡     (   2   )       ⁢     ide   ⁡     (   2   )         +   ⋯   +                     ⁢       DELTA   ×     KP   ⁡     (     k   -   1     )       ⁢     ide   ⁡     (     k   -   1     )         +                     ⁢       KP   ⁡     (   k   )       ⁢     ide   ⁡     (   k   )                       (     15   ⁢   c     )               
where DELTA represents a forgetting coefficient vector having forgetting coefficients DELTAi (i=1 through 4) as elements, as indicated by the following equation.
 
   DELTA=[DELTA 1 , DELTA 2 , DELTA 3 , DELTA 4 ] 
   The forgetting coefficients DELTAi are set to a value between 0 and 1 (0&lt;DELTAi&lt;1) and have a function to gradually reduce the influence of past identifying errors. One of the coefficient DELTA 3  which is relevant to the calculation of the model parameter b 1 , and the coefficient DELTA 4  which is relevant to the calculation of the model parameter c 1  is set to “1”. In other words, one of the forgetting coefficients DELTA 3  and DELTA 4  is made noneffective. By setting one or more of the elements of the forgetting coefficient vector DELTA to “1”, it is possible to prevent a steady-state deviation between the target value DTHR and the throttle valve opening deviation amount DTH from occurring. If both of the coefficients DELTA 3  and DELTA 4  are set to “1”, the effect of preventing the model parameters from drifting becomes insufficient. Accordingly, it is preferable to set only one of the coefficients DELTA 3  or DELTA 4  to “1”. 
   If the equation (15) is rewritten into a recursive form, the following equations (15d) and (15e) are obtained. A process of calculating the model parameter vector θ(k) from the equations (15d) and (15e) instead of the equation (15) is hereinafter referred to as a δ correcting method, and dθ(k) defined by the equation (15e) is referred to as “updating vector”.
 
θ( k )=θ(0)+ d θ( k )  (15d)
 
 d θ( k )=DELTA× d θ( k− 1)+ KP ( k ) ide ( k )  (15e)
 
   According to the algorithm using the δ correcting method, the drift preventing effect which satisfies the requirement B3 and the model parameter stabilizing effect which satisfies the requirement B5 are obtained. Specifically, the initial vector θ(0) is maintained at all times, and the values which can be taken by the elements of the updating vector dθ(k) are limited by the effect of the coefficient vector DELTA. Therefore, the model parameters can be stabilized in the vicinity of their initial values. 
   Furthermore, since the model parameters are calculated while adjusting the updating vector dθ(k) according to the identifying process based on the input and output data of the actual controlled object, the model parameters matching the actual controlled object can be obtained, and hence the above requirement B1 is satisfied. 
   In order to satisfy the requirement B2, the model parameter vector θ(k) is calculated from the following equation (15f) which uses the reference model parameter vector θbase, instead of the initial vector θ(0) in the equation (15d).
 
θ( k )=θbase+ d θ( k )  (15f)
 
   The reference model parameter vector θbase is set according to the throttle valve opening deviation amount DTH (TH−THDEF) by the model parameter scheduler  25 . Consequently, the reference model parameter vector θbase can be adapted to changes in the dynamic characteristics which correspond to changes in the throttle valve opening TH, and hence the above requirement B2 is satisfied. 
   As described above, according to the present embodiment, the amount of calculations by the ECU is reduced by employing the fixed gain algorithm (the requirement B4). The adaptation to quasi-static dynamic characteristics changes and hardware characteristics variations (the requirement B1), the stabilization of the model parameters (control performance) (the requirement B5), and the prevention of the drift of the model parameter (the requirement B3) are achieved by employing the algorithm using the δ correcting method. Further, the adaptation to dynamic characteristics changes corresponding to changes in the throttle opening TH is achieved by employing the model parameter scheduler  25  (the requirement B2). 
   The elements a 1 ′, a 2 ′, b 1 ′, and c 1 ′ of the model parameter vector θ( k ) calculated from the equation (15f) are subjected to the limit process so that the corrected model parameter vector θL(k) (θL(k) T =[a 1 , a 2 , b 1 , c 1 ]) is calculated like the first embodiment. 
   In addition, the model parameter c 1 ′ is statistically processed, and the central value of its variations is calculated as the default opening deviation thdefadp and used to calculate the throttle valve opening deviation amount DTH and the target value DTHR from the equations (41) and (42) like the first embodiment.
 
 DTH ( k )= TH ( k )− THDEF+thdefadp   (41)
 
 DTHR ( k )= THR ( k )− THDEF+thdefadp   (42)
 
   Operations of the CPU of the ECU  7  for realizing the functions of the adaptive sliding mode controller  21   a , the model parameter identifier  22   a , and the model parameter scheduler  25  will be described below. 
     FIG. 32  is a flowchart showing a throttle valve opening control process. The throttle valve opening control process differs from the throttle valve opening control process shown in  FIG. 9  in that step S 13  (the calculations of the state predictor) in the latter process is omitted, and steps S 12 , S 14 , and S 16  in the latter process are changed respectively to steps S 12   a , S 14   a , and S 16   a.    
   In step S 12   a , a process of performing calculations of the model parameter identifier as shown in  FIG. 33 . Specifically, a process of calculating the model parameter vector θ(k) from the equation (15f) is carried out, and the model parameter vector θ(k) is subjected to the limit process so that the corrected model parameter vector θL(k) is calculated. 
   In step S 14   a , the control input Usl(k) shown in  FIG. 36  is calculated using the corrected model parameter vector θL(k). Specifically, the equivalent control input Ueq, the reaching law input Urch(k), and the adaptive law input Uadp(k) are calculated, and the sum of these inputs is calculated as the control input Usl(k) (duty ratio DUT(k)). 
   In step S 16   a , a process of determining the stability of the sliding mode controller as shown in  FIG. 41  is carried out. Specifically, the stability of the sliding mode controller is determined using the switching function value σ instead of the predicted switching function value σpre, and the stability determination flag FSMCSTAB is set. The process performed when the stability determination flag FSMCSTAB is set to “1” is the same as that the first embodiment. 
     FIG. 33  is a flowchart showing a process of calculations of the model parameter identifier  22   a . The process shown in  FIG. 33  is different from the process of calculations of the model parameter identifier as shown in  FIG. 11  in that steps S 31  through S 34  in the process shown in  FIG. 11  are changed to steps S 31   a  through S 34   a , and steps S 33   b  and S 33   c  are added. 
   In step S 31   a , the gain coefficient vector KP(k) is calculated from the equation (20a). Then, the estimated throttle valve opening deviation amount DTHHAT(k) is calculated from the equations (18) and ( 19   a ) in step S 32   a.    
   In step S 33   a , a process of calculating ide(k) as shown in  FIG. 35  is carried out to calculate the identifying error ide(k). In step S 33   b , the updating vector dθ(k) is calculated from the equation (15e). Then, a θbase table shown in  FIG. 34  is retrieved according to the throttle valve opening deviation amount DTH to calculate the reference model parameter vector θbase in step S 33   c . In the θbase table, reference model parameters a 1 base, a 2 base, and b 1 base are set. When the throttle valve opening deviation amount DTH has a value close to “0” (i.e., when the throttle opening TH is in the vicinity of the default opening THDEF), the reference model parameters a 1 base and b 1 base decrease and the reference model parameter a 2 base increases. The reference model parameter c 1 base is set to “0”. 
   In step S 34   a , the model parameter vector θ(k) is calculated from the equation (15f). Then, the process of stabilizing the model parameter vector θ(k) is executed like the first embodiment (step S 35 ). That is, the model parameters are subjected to the limit process so that the corrected model parameter vector θL(k) is calculated. 
     FIG. 35  is a flowchart showing a process of calculating the identifying error ide(k) which is carried out in step S 33   a  shown in  FIG. 33 . The process shown in  FIG. 35  is different from the process shown in  FIG. 12  in that the dead zone process in step S 56  shown in  FIG. 12  is omitted and step S 51  shown in  FIG. 12  is changed to step S 51   a . In the present embodiment, the dead zone process is not performed, since the drift of the model parameters is prevented by the δ correcting method. 
   In step S 51   a , the estimated throttle valve opening deviation amount DTHHAT(k) is calculated from the equations (18) and ( 19   a ), and the identifying error ide(k) is calculated using the estimated throttle valve opening deviation amount DTHHAT(k). 
   In the present embodiment, the predetermined value XCNTIDST in step S 52  is set to “2”, for example, because the dead time d of the controlled object model is set to “0”. 
     FIG. 36  is a flowchart showing a process of calculating the control input Usl (DUT) applied to the throttle valve actuating device  10 , which is carried out in step S 14   a  shown in  FIG. 32 . The process shown in  FIG. 36  is different from the process shown in  FIG. 22  in that steps S 201  through S 205  shown in  FIG. 22  are changed respectively to steps S 201   a  through S 205   a.    
   In step S 201   a , the switching function value σ is calculated as shown in  FIG. 37 . In step S 202   a , the integrated value of the switching function value σ is calculated as shown in  FIG. 38 . In step S 203   a , the equivalent control input Ueq is calculated from the equation (9b). In step S 204   a , the reaching law input Urch is calculated as shown in FIG.  39 . In step S 205   a , the adaptive law input Uadp is calculated as shown in  FIG. 40 . 
     FIG. 37  is a flowchart showing a process of calculating the switching function value σ which is carried out in step S 201   a  shown in  FIG. 36 . The process shown in  FIG. 37  is different from the process shown in  FIG. 23  in that steps S 222  through S 226  shown in  FIG. 23  are changed respectively to steps S 222   a  through S 226   a.    
   In step S 222   a , the switching function value σ(k) is calculated from the equation (5). Steps S 223   a  through S 226   a  are the steps obtained by replacing “σpre” in steps S 223  through S 226  shown in  FIG. 23  with “σ”. Accordingly, the switching function value σ(k) is subjected to the limit process in the same manner as the process shown in  FIG. 23 . 
     FIG. 38  is a flowchart showing a process of calculating an integrated value SUMSIGMAa of the switching function value a which is carried out in step S 202   a  shown in  FIG. 36 . The process shown in  FIG. 38  is different from the process shown in  FIG. 26  in that steps S 241  through S 245  shown in  FIG. 26  are changed respectively to steps S 241   a  through S 245   a . The integrated value SUMSIGMAa is used for the calculation of the adaptive law input Uadp in a process shown in  FIG. 40  described below (see the equation (11b)). 
   In step S 241   a , the integrated value SUMSIGMAa is calculated according to the following equation (47a):
 
SUMSIGMA a ( k )=SUMSIGMA a ( k− 1)+σ×Δ T   (47a)
 
   In steps S 242   a  through S 245   a , the calculated integrated value SUMSIGMAa is subjected to the limit process in the same manner as the process shown in  FIG. 26 . 
     FIG. 39  is a flowchart showing a process of calculating the reaching law input Urch, which is carried out in step S 204   a  shown in  FIG. 36 . The process shown in  FIG. 39  is different from the process shown in  FIG. 27  in that steps S 263  and S 265  shown in  FIG. 27  are changed respectively to steps S 263   a  and S 265   a.    
   In the present embodiment, using the switching function value σ instead of the predicted switching function value σpre, the reaching law input Urch is calculated in step S 263   a , when the adaptive sliding mode controller  21   a  is stable, and the reaching law input Urch is calculated in step S 265   a  when the adaptive sliding mode controller  21   a  is unstable. 
     FIG. 40  is a flowchart showing a process of calculating the adaptive law input Uadp which is carried out in step S 205   a  shown in  FIG. 36 . The process shown in  FIG. 40  is different from the process shown in  FIG. 28  in that steps S 273  and S 275  shown in  FIG. 28  are changed respectively to steps S 273   a  and S 275   a.    
   In the present embodiment, using the integrated value SUMSIGMAa of the switching function value σ, the adaptive law input Uadp is calculated in step S 273   a , when the adaptive sliding mode controller  21   a  is stable, and the adaptive law input Uadp is calculated in step S 275   a  when the adaptive sliding mode controller  21   a  is unstable. 
     FIG. 41  is a flowchart showing a process of determining the stability of the sliding mode controller which is carried out in step S 16   a  shown in  FIG. 32 . The process shown in  FIG. 41  is different from the process shown in  FIG. 29  in that steps S 281  and S 282  shown in  FIG. 29  are changed respectively to steps S 281   a  and S 282   a.    
   In step S 281   a , the switching function change amount Dσ is calculated according to the following equation (52a). The stability determining parameter SGMSTAB is calculated according to the following equation (53a) in step S 282   a . That is, the stability is determined on the basis of the switching function value σ instead of the predicted switching function value σpre.
 
 D σ=σ( k )−σ( k− 1)  (52a)
 
 SGMSTAB=Dσ×σ ( k )  (53a)
 
   In the present embodiment, the throttle valve actuating device  10  and a portion of the ECU  7 , i.e., an output circuit for supplying an energizing current to the motor  6 , correspond to a plant. The process shown in  FIG. 36  corresponds to a sliding mode controller. The process shown in  FIG. 33  corresponds to an identifying means. The process shown in  FIG. 35  corresponds to an identifying error calculating means. The process shown in  FIG. 35  corresponds to an identifying error correcting means. Step S 33   b  shown  FIG. 33  corresponds to an updating vector calculating means and an updating vector correcting means. 
   Third Embodiment 
     FIG. 42  is a block diagram showing a control system according to the third embodiment of the present invention. 
   As shown in  FIG. 42 , the control system comprises a plant  101  as a controlled object, a pH sensor  102  for detecting a pH of a mixed liquid which is an output from the plant, a subtractor  103  for subtracting a first reference value V 1 BASE from an output V 1 OUT from the pH sensor  102 , a target value generator  104  for generating a control target value V 1 TARGET, a control quantity determining unit  105  for determining a first control quantity U 1 , and an adder  106  for adding the first control quantity U 1  and a second reference value V 2 BASE and outputting a second control quantity U 2 . 
   The subtractor  103 , the target value generator  104 , the control quantity determining unit  105 , and the adder  106  are specifically implemented by an electronic control unit having a CPU, a memory, and an input/output circuit. 
   The plant  101  includes a flow rate control valve  111  for controlling the flow rate of an alkaline liquid according to the second control quantity U 2 , and a stirrer  112  for stirring an alkaline liquid supplied via the flow rate control valve  111  and an acid liquid. The plant  101  stirs the alkaline liquid and the acid liquid and outputs a mixed liquid having a desired pH value. 
   The control quantity determining unit  105  comprises an identifier  121  for identifying a model parameter vector of a controlled object model which is obtained by modeling the plant  101 , an adaptive sliding mode controller  122 , and a predictor  123 . The identifier  121 , the adaptive sliding mode controller  122 , and the predictor  123  correspond respectively to the model parameter identifier  22 , the adaptive sliding mode controller  21 , and the state predictor  23  according to the first embodiment, and have the same functions as those of the model parameter identifier  22 , the adaptive sliding mode controller  21 , and the state predictor  23 . 
   The relations between the components and parameters of the third embodiment and the components and parameters of the first embodiment will be described below. 
   The pH sensor  102  corresponds to the throttle valve opening sensor  8 , and the output VIOUT of the pH sensor  102  corresponds to the throttle valve opening TH. The first target value V 1 BASE corresponds to the default opening THDEF. In the present preferred embodiment, the first target value V 1 BASE is set to a pH value corresponding to the neutral value. Therefore, a deviation amount DV 1  corresponds to the throttle valve opening deviation amount DTH. The target value generator  104  corresponds to the target opening setting unit  24 , and the control target value V 1 TARGET corresponds to the target value DTHR of the throttle valve opening deviation amount. In the first embodiment, the function of the subtractor  103  is included in the model parameter identifier  22  and the state predictor  23 . 
   The second reference value V 2 BASE is added to bias the central value of the first control quantity U 1  which is the output of the adaptive sliding mode controller  122 . In the first embodiment, there is no component corresponding to the adder  106 , and hence the second reference value V 2 BASE substantially equals “0” (i.e., U 1 =U 2 =Usl). In the present embodiment, the second reference value V 2 BASE is set to such a value that the opening of the flow rate control valve  111  is 50%, for example. 
   The flow rate control valve  111  corresponds to a switching element (which is included in the output circuit of the ECU  7 , but not shown in the drawings and not explained the description) that is turned on and off by a pulse signal having the duty ratio DUT. The alkaline liquid corresponds to the supply voltage. An output flow rate V 2  of the flow rate control valve  111  corresponds to an energizing current of the motor  6 . The stirrer  112  corresponds to the motor  6  and the valve body of the throttle valve  3 . The acid liquid corresponds to the intake pipe negative pressure acting on the valve body of the throttle valve  3  and the energizing forces of the return spring  4  and the resilient member  5 . A pH value V 1  of the mixed liquid outputted from the stirrer  112  corresponds to the actual throttle valve opening. 
   Because of the above relations, the plant  101  can be modeled in the same manner as the first embodiment, and the same control process as the control process according to the first embodiment can be applied to the plant  101 . Specifically, the identifier  121  performs the same operation as that of the first embodiment to calculate the corrected model parameter vector θL, based on the first control quantity U 1  and the deviation amount DV 1 . The predictor  123  performs the same operation as that of the first embodiment to calculate the predicted deviation PREDV 1 , based on the first control quantity U 1 , the deviation amount DV 1 , and the corrected model parameter vector θL. The adaptive sliding mode controller  122  performs the same operation as that of the first embodiment to calculate the first control quantity U 1  in order to make the predicted deviation PREDV 1  coincide with the control target value V 1 TARGET, based on the predicted deviation amount PREDV 1  and the corrected model parameter vector θL. Therefore, the output V 1  of the plant  101  can be made to coincide with a desired pH value by setting the control target value V 1 TARGET to a desired relative pH value (a desired deviation amount from the first reference value V 1 BASE). 
   In the present embodiment, the identifier  121  corresponds to an identifying means, and includes an identifying error calculating means and an identifying error correcting means. The predictor  123  corresponds to a predicting means. 
   Modification of Third Embodiment 
     FIG. 43  shows a modification of the control system shown in  FIG. 42 . In the modification, a plant  101   a , instead of the plant  101  shown in  FIG. 42 , is a controlled object. The plant  101   a  is constituted by adding a flow rate sensor  113  and a feedback controller  114  to the plant  101  shown in  FIG. 42 . The flow rate sensor  113  detects an output flow rate V 1  of the flow control valve  111 , and the feedback controller  114  controls the flow control valve  111  to make an output V 2 OUT of the flow rate sensor  113  coincide with a flow rate corresponding to the second control quantity U 2 . 
   The modeling and the control process which are the same as those of the third embodiment are also applicable to the plant including the local feedback loop as shown in  FIG. 43 . 
   Since the circuit for energizing the motor in the first embodiment is already known, this circuit has not been described in detail. The circuit for energizing the motor may include a current sensor for detecting an output current of the switching element that is turned on and off, and a feedback control process may be carried out to make a detected current value ID coincide with a current value IR corresponding to the control quantity Usl. The present modification corresponds to a structure where the above circuit arrangement is applied to the first embodiment. 
   Fourth Embodiment 
     FIG. 44  is a block diagram of a control system according to the fourth embodiment of the present invention. The control system shown in  FIG. 44  corresponds to the control system according to the second embodiment, and is similar to the control system shown in  FIG. 42  except that the control quantity determining unit  105  shown in  FIG. 42  is replaced with a control quantity determining unit  105   a . The details of the control system shown in  FIG. 44  which will not be described below are identical to those of the control system according to the third embodiment. 
   The control quantity determining unit  105   a  comprises an identifier  121   a , an adaptive sliding mode controller  122   a , and a parameter scheduler  124 . 
   The identifier  121   a , the adaptive sliding mode controller  122   a , and the parameter scheduler  124  correspond respectively to, and have the same functions as, the model parameter identifier  22   a , the adaptive sliding mode controller  21   a , and the model parameter scheduler  25  according to the second embodiment. 
   The parameter scheduler  124  performs the same operation as that of the second embodiment to calculate the reference model parameter vector θbase, based on the deviation amount DV 1 . The identifier  121   a  performs the same operation as that of the second embodiment to calculate the corrected model parameter vector θL based on the first control quantity U 1 , the deviation amount DV 1 , and the reference model parameter vector θbase. The adaptive sliding mode controller  122   a  performs the same operation as that of the second embodiment to calculate the first control quantity U 1  based on the deviation amount DV 1  and the corrected model parameter vector θL, to make the deviation amount DV 1  coincide with the control target value V 1 TARGET. Therefore, the output V 1  of the plant  101  can be made to coincide with a desired pH value by setting the control target value V 1 TARGET to a desired relative pH value (a deviation amount from the first reference value V 1 BASE). 
   In the present embodiment, the identifier  121   a  corresponds to an identifying means, and includes an identifying error calculating means, an updating vector calculating means, and an updating vector correcting means. 
   Modification of Fourth Embodiment 
     FIG. 45  shows a modification of the control system shown in  FIG. 44 . In the modification, a plant  101   a , instead of the plant  101  shown in  FIG. 44 , is a controlled object. The plant  101   a  is identical to the plant  101   a  shown in  FIG. 43 . 
   The modeling and the control process which are the same as those of the fourth embodiment are also applicable to the plant including the local feedback loop as shown in  FIG. 45 . 
   Fifth Embodiment 
     FIG. 46  is a block diagram showing a control system according to the fifth embodiment of the present invention. 
   As shown in  FIG. 46 , the control system comprises an engine system  201  as a controlled object, which includes an internal combustion engine  212 , an air-fuel ratio sensor  202  for detecting an air-fuel ratio of an air-fuel mixture to be supplied to the engine  212 , a subtractor  203  for subtracting a reference value KBS from an output KACT (having a value converted to an equivalent ratio) from the air-fuel ratio sensor  202 , a target value generator  204  for generating a control target value DKCMD, a control quantity determining unit  205  for determining a control quantity DKAF, an adder  206  for adding the control quantity DKAF and the reference value KBS and outputting a correction quantity KAF, and a fuel amount calculating unit  207  for calculating a fuel amount TOUT using the correction quantity KAF and other feedfoward control terms. 
   The subtractor  203 , the target value generator  204 , the control quantity determining unit  205 , the adder  206 , and the fuel amount calculating unit  207  are specifically implemented by an electronic control unit having a CPU, a memory, and an input/output circuit. 
   The engine system  201  includes a fuel injection valve  211  for injecting fuel according to the fuel amount TOUT, and an internal combustion engine  212  in which an air-fuel mixture of the fuel supplied via the fuel injection valve  211  and the intake air is combusted. The engine system  201  generates a rotation torque by combusting the fuel, and emits combusted gases (exhaust gases). 
   The control quantity determining unit  205  comprises an identifier  221  for identifying a model parameter vector of a controlled object model which is obtained by modeling the engine system  201 , an adaptive sliding mode controller  222 , and a predictor  223 . The identifier  221 , the adaptive sliding mode controller  222 , and the predictor  223  correspond respectively to the model parameter identifier  22 , the adaptive sliding mode controller  21 , and the state predictor  23  according to the first embodiment, and have the same functions as those of the model parameter identifier  22 , the adaptive sliding mode controller  21 , and the state predictor  23 . 
   The relations between the components and parameters of the fifth embodiment and the components and parameters of the first embodiment will be described below. 
   The air-fuel ratio sensor  202  corresponds to the throttle valve opening sensor  8 , and the output KACT of the air-fuel ratio sensor  202  corresponds to the throttle valve opening TH. The reference value KBS corresponds to the default opening THDEF. In the present embodiment, the reference value KBS is set to a value (1.0) corresponding to the stoichiometric air-fuel ratio. Therefore, a deviation amount DK corresponds to the throttle valve opening deviation amount DTH. The target value generator  204  corresponds to the target opening setting unit  24 , and the control target value DKCMD corresponds to the target value DTHR of the throttle valve opening deviation amount. In the first embodiment, the function of the subtractor  203  is included in the model parameter identifier  22  and the state predictor  23 . 
   The reference value KBS is added to bias the central value of the control quantity DKAF which is the output of the adaptive sliding mode controller  222 . 
   The fuel injection valve  211  corresponds to a switching element (which is included in the output circuit of the ECU  7 , but not shown in the drawings and not explained the description) that is turned on and off by a pulse signal having the duty ratio DUT. The fuel corresponds to the supply voltage. A fuel amount FA injected from the injection valve  211  corresponds to an energizing current of the motor  6 . The engine  212  corresponds to the motor  6  and the valve body of the throttle valve  3 . The intake air corresponds to the intake pipe negative pressure acting on the valve body of the throttle valve  3  and the energizing forces of the return spring  4  and the resilient member  5 . An oxygen concentration VO 2  in the exhaust gases output from the engine  212  corresponds to the actual throttle valve opening. 
   Because of the above relations, the engine system  201  can be modeled in the same manner as the first embodiment, and the same control process as the control process according to the first embodiment can be applied to the engine system  201 . Specifically, the identifier  221  performs the same operation as that of the first embodiment to calculate the corrected model parameter vector θL, based on the control quantity DKAF and the deviation amount DK. The predictor  223  performs the same operation as that of the first embodiment to calculate the predicted deviation PREDK, based on the control quantity DKAF, the deviation amount DK, and the corrected model parameter vector θL. The adaptive sliding mode controller  222  performs the same operation as that of the first embodiment to calculate the control quantity DKAF in order to make the predicted deviation PREDK coincide with the control target value DKCMD, based on the predicted deviation amount PREDK and the corrected model parameter vector θL. Therefore, the air-fuel ratio of the air-fuel mixture to be supplied to the engine  212  can be made to coincide with a desired air-fuel ratio by setting the control target value DKCMD to a desired air-fuel ratio correction value (a desired deviation amount from the reference value KBS). 
   In the present embodiment, the identifier  221  corresponds to an identifying means, and includes an identifying error calculating means and an identifying error correcting means. The predictor  223  corresponds to a predicting means. 
   Sixth Embodiment 
     FIG. 47  is a block diagram of a control system according to the sixth embodiment of the present invention. The control system shown in  FIG. 47  corresponds to the control system according to the second embodiment, and is similar to the control system shown in  FIG. 46  except that the control quantity determining unit  205  shown in  FIG. 46  is replaced with a control quantity determining unit  205   a . The details of the control system shown in  FIG. 47  which will not be described below are identical to those of the control system according to the fifth embodiment. 
   The control quantity determining unit  205   a  comprises an identifier  221   a , an adaptive sliding mode controller  222   a , and a parameter scheduler  224 . 
   The identifier  221   a , the adaptive sliding mode controller  222   a , and the parameter scheduler  224  correspond respectively to, and have the same functions as, the model parameter identifier  22   a , the adaptive sliding mode controller  21   a , and the model parameter scheduler  25  according to the second embodiment. 
   The parameter scheduler  224  performs the same operation as that of the second embodiment to calculate the reference model parameter vector θbase, based on the deviation amount DK. The identifier  221   a  performs the same operation as that of the second embodiment to calculate the corrected model parameter vector θL based on the control quantity DKAF, the deviation amount DK, and the reference model parameter vector θbase. The adaptive sliding mode controller  222   a  performs the same operation as that of the second embodiment to calculate the control quantity DKAF based on the deviation amount DK and the corrected model parameter vector θL, to make the deviation amount DK coincide with the control target value DKCMD. Therefore, the air-fuel ratio of the air-fuel mixture to be supplied to the engine  212  can be made to coincide with a desired air-fuel ratio by setting the control target value DKCMD to a desired air-fuel ratio correction value (a desired deviation amount from the reference value KBS). 
   In the present embodiment, the identifier  221   a  corresponds to an identifying means, and includes an identifying error calculating means, an updating vector calculating means, and an updating vector correcting means. 
   Other Embodiments 
   The δ correcting method may be replaced with a ε correcting method described below, as a method of calculating the identifying error ide(k) of the model parameters. Specifically, the following equation (15g) may be used instead of the equation (15c) to calculate the model parameter vector θ(k). 
                         θ   ⁡     (   k   )       =       ⁢         EPS   k     ⁢     θ   ⁡     (   0   )         +                     ⁢         EPS     k   -   1       ×     KP   ⁡     (   1   )       ⁢     ide   ⁡     (   1   )         +                     ⁢         EPS     k   -   2       ×     KP   ⁡     (   2   )       ⁢     ide   ⁡     (   2   )         +   ⋯   +                     ⁢       EPS   ×     KP   ⁡     (     k   -   1     )       ⁢     ide   ⁡     (     k   -   1     )         +       KP   ⁡     (   k   )       ⁢     ide   ⁡     (   k   )                         (     15   ⁢   g     )               
where EPS represents a forgetting coefficient vector having forgetting coefficients EPSi (i=1 through 4) as its elements, as indicated by the following equation.
 EPS=[EPS1, EPS2, EPS3, EPS4] 
   Like the forgetting coefficients DELTAi, the forgetting coefficients EPS1, EPS2, and EPS4 are set to a value between “0” and “1” (0&lt;EPSi&lt;1) and have a function to gradually reduce the influence of past identifying errors. 
   In the ε correcting method, the coefficient EPS3 which is relevant to the calculation of the model parameter b 1  must be set to “1” for the following reasons. In the ε correcting method, the all values of the model parameters becomes closer to zero, as the identifying error ide(k) becomes less. Since the model parameter b 1  is applied to the denominator of the equations (9b), (10b), and (11b), the input Usl applied to the controlled object diverges as the model parameter b 1  becomes closer to “0”. 
   The equation (15g) is different from the equation (15c) in that the initial vector θ(0) is also multiplied by the forgetting coefficient vector EPS. 
   If the equation (15g) is rewritten into a recursive form, the following equation (15h) is obtained. The method of calculating the model parameter vector θ(k) from the equation (15h) instead of the equation (15) is referred to as the ε correcting method.
 
θ( k )= EPS ×θ( k −1)+ KP ( k ) ide ( k )  (15h)
 
   The ε correcting method also has a function to gradually reduce the influence of past identifying errors eid. Accordingly, the drift of the model parameters are prevented by the ε correcting method. 
   In the second embodiment, the drift of the model parameters is prevented by the δ correcting method. However, like the first embodiment, the corrected identifying error idenl(k) may be calculated according to the dead zone process ( FIG. 14 ), and the model parameter vector θ(k) may be calculated using the corrected identifying error idenl(k). 
   In the first embodiment, the dead zone process may be replaced with the δ correcting method or the ε correcting method. If the δ correcting method is employed in the first embodiment, it is preferable to employ the model parameter scheduler like the second embodiment, and calculate the model parameter vector θ by adding the updating vector and the reference vector θbase calculated by the model parameter scheduler. 
   INDUSTRIAL APPLICABILITY 
   According to the control system for a plant of the present invention, one or more model parameters of a controlled object model which is obtained by modeling the plant as a controlled object, are identified, and the sliding mode control using the identified model parameters can be stabilized. Specifically, the present invention is applicable to the control of an actuating device of a throttle valve that controls an amount of air supplied to an internal combustion engine, a combustion system including an internal combustion engine, a chemical plant, or the like. The present invention contributes to improve stability of control when controlling the above controlled object with the sliding mode control. Further, the present invention is also applicable to the control of an engine having an crank shaft mounted vertically, such as an outboard engine for driving a ship.