Patent Publication Number: US-6990403-B2

Title: Control system for plant

Description:
BACKGROUND OF THE INVENTION 
   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. 
   A sliding mode controller is known from Japanese Patent Laid-open No. 2000-110636, for example. The sliding mode controller controls a plant with a sliding mode control using a controlled object model which is obtained by modeling the plant. 
   The sliding mode control is a high gain control which generally has an advantage of a high control speed. On the other hand, the sliding mode control has a characteristic that an overshoot with respect to a control target value may often occur. Therefore, when controlling a throttle valve actuating device which actuates a throttle valve of an internal combustion engine, using the sliding mode control, it is very important to suppress the overshoot. Specifically, if the overshoot occurs in the control of the throttle valve actuating device, this causes a collision of the throttle valve with a full-closed position stopper, or an inconvenience that an output of the internal combustion engine increases to a value which is greater than the required value. Accordingly, it is particularly important in the control of the throttle valve actuating device to reliably prevent such inconveniences. 
   However, according to the control system employing a conventional sliding mode controller, suppression of the overshoot is not sufficient, and further improvement is desired. 
   SUMMARY OF THE INVENTION 
   It is therefore an object of the present invention to provide a control system for a plant, which can reliably suppress an overshoot, which may often occur in the control by the sliding mode controller. 
   To achieve the above object, the present invention provides a control system for a plant, comprising identifying means ( 22 ) and a sliding mode controller ( 21 ). The identifying means ( 22 ) identifies a model parameter vector (θ) of a controlled object model which is obtained by modeling the plant. The sliding mode controller ( 21 ) controls the plant using the model parameter vector (θ) identified by the identifying means ( 22 ). The sliding mode controller ( 21 ) includes damping input calculating means for calculating a damping input (Udamp) according to a speed (DTH(k)−DTH(k−1)) of change in an output of the plant, and an element (b1) of the model parameter vector (θ). A control input (Usl) from the sliding mode controller (21) to the plant includes the damping input (Udamp) calculated by the damping input calculating means. 
   With this configuration, the damping input is calculated according to a speed of change in an output of the plant, and an element of the model parameter vector, and the plant is controlled with the control input including the damping input. The damping input exercises on the control input in such a direction that a change in the control input is suppressed. Accordingly, the overshoot can be suppressed by the damping input. In addition, by using the element of the identified model parameter vector, an appropriate damping effect corresponding to a change in the dynamic characteristic of the plant, can be obtained. As a result, the overshoot can reliably be suppressed. 
   Preferably, the element (b1) of the model parameter vector (θ) is an element which is relevant to the control input (DUT) to the plant. 
   Preferably, the element (b1) of the model parameter vector (θ) is multiplied by the control input (DUT) to the plant in an equation that defines the controlled object model. 
   Preferably, the damping input calculating means multiplies the speed (DTH(k)−DTH(k−1)) of change in the output of the plant by a control gain (Kdamp), and divides a value obtained by the multiplication by the element (b1) of the model parameter vector (θ), to calculate the damping input (Udamp). 
   Preferably, the control gain (Kdamp) is set according to the output (DTH) of the plant and an control target value (DTHR) of the output of the plant. 
   Preferably, the control gain (Kdamp) is calculated by multiplying a basic value (Kdampbs) depending on the output (DTH) of the plant by a correction coefficient (Kkdamp) depending on a moving average value (DDTHRAV) of an amount of change in the control target value (DTHR). 
   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 ( 3 ), and the sliding mode controller ( 21 ) 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 ( 3 ) coincide with a target opening (THR). 
   With this configuration, an overshoot in controlling the throttle valve opening to the target opening is reliably suppressed. Consequently, it is possible to reliably prevent a collision of the throttle valve with the full-closed position stopper, or an inconvenience that an output of the internal combustion engine increases to a value which is greater than the required value. 
   The above and other objects, features, and advantages of the present invention will become apparent from the following description when taken in conjunction with the accompanying drawings which illustrate embodiments of the present invention by way of example. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
       FIG. 1  is a schematic diagram showing a throttle valve actuating device and a control system for the throttle valve actuating device, according to a first embodiment of the present invention; 
       FIG. 2  is a functional block diagram showing functions realized by an electronic control unit (ECU) shown in  FIG. 1 ; 
       FIG. 3  is a diagram showing control characteristics of a sliding mode controller corresponding to a value of a switching function setting parameter (VPOLE); 
       FIG. 4  is a diagram showing a range for setting control gains (F, G) of the sliding mode controller; 
       FIGS. 5A and 5B  are diagram illustrating a drift of a model parameter; 
       FIGS. 6A and 6B  are diagrams illustrating a process of effecting low-pass filtering on an identifying error (ide); 
       FIG. 7  is a diagram illustrating frequency components of an output of a controlled object; 
       FIG. 8  is a diagram illustrating a sampling process using a short sampling period compared with a change rate of an output of the controlled object; 
       FIG. 9  is a diagram illustrating a manner in which a sampling frequency is set; 
       FIGS. 10A and 10B  are diagrams showing damping characteristics of a control deviation (e(k)); 
       FIG. 11  is a diagram showing a waveform representing how a throttle valve opening deviation (DTH) changes; 
       FIG. 12  is a diagram showing waveforms representing how a switching function value (σ) changes, the waveforms corresponding to the waveform shown in  FIG. 11 ; 
       FIGS. 13A ,  13 B, and  13 C are diagrams illustrating a method of setting control gains (F, G) of the sliding mode controller; 
       FIGS. 14A and 14B  are timing charts illustrating a problem that arises when the control gains (F, G) abruptly change; 
       FIGS. 15A through 15C  are timing charts illustrating a case in which a second period (ΔT 2 ) is used as a control period; 
       FIGS. 16A through 16D  are timing charts illustrating a case in which model parameters are calculated at intervals of a second period (ΔT 2 ) and a control period is set to a first period (ΔT 1 ); 
       FIG. 17  is a timing chart illustrating a moving-averaging calculation of model parameters; 
       FIG. 18  is a timing chart illustrating the manner in which a steady deviation is converged by an adaptive law input (Uadp); 
       FIGS. 19A and 19B  are timing charts illustrating a nonlinear input (Unl); 
       FIG. 20  is a diagram showing a table for calculating a nonlinear input gain (Knl); 
       FIG. 21  is a timing chart illustrating a change in a dither signal value (Fwave); 
       FIG. 22  is a diagram showing a relation between a frequency (fwave) of a forced vibration input and a resonant frequency (fr) of a controlled object; 
       FIGS. 23A through 23C  are timing charts illustrating reduction of an identifying error (ide), which is provided by a forced vibration input (Uwave); 
       FIGS. 24A and 24B  are timing charts illustrating an overshoot of the throttle valve opening deviation amount (DTH) and its improvement; 
       FIGS. 25A and 25B  are diagrams showing tables for setting a basic value (Kdampbs) and a correction coefficient (Kkdamp) of a damping control gain; 
       FIGS. 26A and 26B  are diagrams illustrating a limit process of model parameters (a 1 ″, a 2 ″); 
       FIG. 27  is a diagram illustrating a method of setting reference model parameters (a 1 base, a 2 base, b 1 base); 
       FIGS. 28A through 28C  are timing charts illustrating a problem with a conventional method of setting a reference model parameter (b 1 base); 
       FIGS. 29A through 29C  are timing charts illustrating a method of setting a reference model parameter (b 1 base) according to the first embodiment; 
       FIG. 30  is a flowchart showing a throttle valve opening control process; 
       FIG. 31  is a flowchart showing a process of setting a state variable executed in the process shown in  FIG. 30 ; 
       FIG. 32  is a flowchart showing a process of identifying model parameters executed in the process shown in  FIG. 30 ; 
       FIG. 33  is a flowchart showing a process of calculating an identifying error (ide) executed in the process shown in  FIG. 32 ; 
       FIG. 34  is a flowchart showing a first limit process executed in the process shown in  FIG. 30 ; 
       FIG. 35  is a flowcharht showing a limit process of model parameters (a 1 ″, a 2 ″) executed in the process shown in  FIG. 34 ; 
       FIG. 36  is a diagram illustrating the process shown in  FIG. 35 ; 
       FIG. 37  is a flowchart showing a limit process of a model parameter (b 1 ″) executed in the process shown in  FIG. 34 ; 
       FIG. 38  is a flowchart showing a limit process of a model parameter (c 1 ″) executed in the process shown in  FIG. 34 ; 
       FIG. 39  is a flowchart showing a second limit process executed in the process shown in  FIG. 30 ; 
       FIG. 40  is a flowchart showing a process of calculating a control input (Usl) executed in the process shown in  FIG. 30 ; 
       FIG. 41  is a flowchart showing a process of calculating a switching function value (σ) executed in the process shown in  FIG. 40 ; 
       FIG. 42  is a flowchart showing a process of calculating a switching function setting parameter (VPOLE) executed in the process shown in  FIG. 41 ; 
       FIG. 43  is a diagram showing a table used executed in the process shown in  FIG. 42 ; 
       FIG. 44  is a flowchart showing a process of calculating a reaching law input (Urch) executed in the process shown in  FIG. 40 ; 
       FIG. 45  is a flowchart showing a process of calculating an adaptive law input (Uadp) executed in the process shown in  FIG. 40 ; 
       FIG. 46  is a flowchart showing a process of calculating a nonlinear input (Unl) executed in the process shown in  FIG. 40 ; 
       FIG. 47  is a flowchart showing a process of calculating a forced vibration input (Uwave) executed in the process shown in  FIG. 40 ; 
       FIG. 48  is a diagram showing a table used executed in the process shown in  FIG. 47 ; 
       FIG. 49  is a flowchart showing a process of calculating a damping input (Udamp) executed in the process shown in  FIG. 40 ; 
       FIG. 50  is a flowchart showing a process of determining stability of the sliding mode controller executed in the process shown in  FIG. 30 ; 
       FIG. 51  is a schematic diagram of a hydraulic positioning apparatus according to a second embodiment of the present invention; and 
       FIG. 52  is a block diagram of a control system including the hydraulic positioning device shown in  FIG. 51 . 
   

   DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
   The preferred embodiments of the present invention will be described with reference to the following 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 . 
   A model defined by the equation (1) shown below is set as a controlled object model according to the frequency response characteristics of the throttle valve actuating device  10 . It has been confirmed that the frequency response characteristics of the model can be approximated 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 a discrete sampling time or a discrete control time which is digitized with a first period ΔT 1 , and DTH(k) is a throttle valve opening deviation amount defined by the equation (2) shown below.
 
 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 model parameters determining the characteristics of the controlled object model, and d is a dead time. The dead time d is a delay between the input and output of the controlled object model. For reducing the amount of calculations, it is effective to define a controlled object model by the equation (1a) shown below where the dead time d is set to “0”. A modeling error (a difference between the characteristics of the controlled object model and the characteristics of an actual controlled object (plant)) caused by setting the dead time d to “0”, is compensated by employing a sliding mode controller having robustness. “Robustness” of a control system means that control performance or control stability of the control system is not easily deteriorated even when the characteristics of the controlled object or disturbances change largely compared with an ordinary condition. 
                     DTH   ⁡     (     k   +   1     )       =       ⁢       a1   ×     DTH   ⁡     (   k   )         +     a2   ×     DTH   ⁡     (     k   -   1     )         +                     ⁢       b1   ×     DUT   ⁡     (   k   )         +   c1                   (     1   ⁢   a     )             
 
   In the equation (1a), the model parameter c 1  which is not relevant to the input and output of the controlled object, is employed in addition to the model parameters a 1  and a 2  which are relevant to the deviation DTH which is the output of the controlled object, and the model parameter b 1  which is relevant to the input duty ratio DUT which is the input of the controlled object. 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 described below. 
     FIG. 2  is a functional block diagram of the throttle valve control system which is realized by the ECU  7 . The throttle valve control system includes an adaptive sliding mode controller  21 , a model parameter identifier  22 , a model parameter scheduler  25 , 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, and subtractors  26  and  27 . 
   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 (specify) the response characteristics of the throttle valve opening TH to the target opening THR, using a specific parameter (a switching function setting parameter VPOLE to be described later). As a result, an optimum response characteristic can be specified according to the throttle valve opening TH. For example, 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  at the fully closed position. It is also possible to make the engine response corresponding to the operation of the accelerator pedal variable. Further, the sliding mode control makes it 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 first limit process, an oversampling and moving-averaging process, and a second limit process of the model parameter vector θ to calculate a corrected model parameter vector θL. The corrected model parameter vector θL is supplied 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. The first limit process, the oversampling and moving-averaging process, and the second limit process will be described later. 
   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, compensation for power supply voltage fluctuations, and adaptation to aging-dependent changes of hardware characteristics are possible. 
   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]) according to a target value DTHR which is defined as a deviation amount between a target opening THR(k) and the default opening THDEF by the following equation (3), the calculated reference model parameter vector θ base is supplied to the model parameter identifier  22 .
 
 DTHR ( k )= THR ( k )− THDEF   (3)
 
   The subtractor  26  calculates a deviation amount between the default opening THDEF and the throttle valve opening TH as the throttle valve opening deviation amount DTH, and the subtractor  27  calculates a deviation amount between the default opening THDEF and the target opening THR as the target value DTHR (see the equations (2) and (3)). 
   Outline of the Adaptive Sliding Mode Controller 
   Principles of operation of the adaptive sliding mode controller  21  will be described below. 
   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 greater than “−1” and less than “1”. 
   On a phase plane defined by a vertical axis representing a deviation e(k) and a horizontal axis representing a 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. 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. 3 , by changing the value of the switching function setting parameter VPOLE in the equation (5), it is possible to change a damping characteristic of the deviation e(k), i.e., the follow-up characteristic of the throttle valve opening deviation amount DTH to follow up 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. Since the sliding mode controller is capable of specifying the damping characteristic of the deviation e(k) as a desired characteristic, the sliding mode controller is referred to as a response-specifying controller. 
   According to the sliding mode control, the converging speed 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 deviation amount DTH to obtain a response characteristic suitable for the operating condition of the throttle valve  3 . 
   As described above, according to the sliding mode control, the deviation e(k) is converged to “0” at an indicated 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 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 basically calculated as a sum of an equivalent control input Ueq(k), a reaching law input Urch(k), and an adaptive law input Uadp(k) by the following equation (6). 
                     DUT   ⁡     (   k   )       =     Usl   ⁡     (   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 the deviation state quantity onto the switching straight line. The adaptive law input Uadp(k) is an input for placing the 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 (8) shown below. The duty ratio DUT(k) calculated with the equation (8) 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 (9) and (10) shown below. 
                     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   )                     (   8   )                 Urch   ⁡     (   k   )       =         -   F     b1     ⁢     σ   ⁡     (   k   )                 (   9   )                 Uadp   ⁡     (   k   )       =         -   G     b1     ⁢       ∑     i   =   0     k     ⁢     Δ   ⁢           ⁢   T1   ⁢           ⁢   σ   ⁢           ⁢     (   i   )                   (   10   )             
 
where F and G represent respectively a reaching law control gain and an adaptive law control gain, which are set as described below, and ΔT 1  represents a control period. The control period is the first period ΔT 1  which is equal to a sampling period that is used to define the controlled object model.
 
   Then, 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) is 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 (11) through (13), in other words, the combination of the gains F and G should be located in a hatched region shown in FIG.  4 .
 
F&gt;0  (11)
 
G&gt;0  (12)
 
 F&lt; 2−(Δ T   1 / 2 ) G   (13)
 
   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 (8) through (10), and the duty ratio DUT(k) is calculated as a sum of those inputs. 
   Outline of the Model Parameter Identifier 
   Principles of operation of the model parameter identifier  22  will be described below. 
   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 (14).
 
θ( k )=θ( k− 1)+ KP ( k ) ide ( k )  (14)
 
θ( k ) T   =[a 1 ″, a 2 ″, b 1 ″, c 1″]  (15)
 
where a 1 ″, a 2 ″, b 1 ″, and c 1 ″ represent model parameters before a first limit process, described later, is carried out, ide(k) represents an identifying error defined by the equations (16), (17), and (18) 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 (19) shown below. In the equation (19), P(k) represents a quartic square matrix calculated by the equation (20) shown below.
 
 ide ( k )= DTH ( k )− DTHHAT ( k )  (16)
 
 DTHHAT ( k )=θ( k− 1) T ζ( k )  (17)
 
     ζ( k ) T   =[DTH ( k− 1),  DTH ( k− 2),  DUT ( k− 1), 1]  (18)               KP   ⁡     (   k   )       =         P   ⁡     (   k   )       ⁢     ζ   ⁡     (   k   )           1   +         ζ   T     ⁡     (   k   )       ⁢     P   ⁡     (   k   )       ⁢     ζ   ⁡     (   k   )                     (   19   )                   P   ⁡     (     k   +   1     )       =       1     λ   1       ⁢     (     E   -         λ   2     ⁢     P   ⁡     (   k   )       ⁢     ζ   ⁡     (   k   )       ⁢       ζ   T     ⁡     (   k   )             λ   1     +       λ   2     ⁢       ζ   T     ⁡     (   k   )       ⁢     P   ⁡     (   k   )       ⁢     ζ   ⁡     (   k   )               )     ⁢     P   ⁡     (   k   )           ⁢     
     ⁢     (     E   ⁢           ⁢   is   ⁢           ⁢   an   ⁢           ⁢   unit   ⁢           ⁢   matrix     )             (   20   )               
   In accordance with the setting of coefficients λ1 and λ2 in the equation (20), the identifying algorithm from the equations (14) through (20) becomes one of the following four identifying algorithm: 
   
     
       
         
             
             
             
           
             
                 
                 
             
           
          
             
                 
               λ 1 = 1, λ 2 = 0 
               Fixed gain algorithm 
             
             
                 
               λ 1 = 1, λ 2 = 1 
               Method-of-least-squares 
             
             
                 
                 
               algorithm 
             
             
                 
               λ 1 = 1, λ 2 = λ 
               Degressive gain algorithm 
             
             
                 
                 
               (λ is a given value other 
             
             
                 
                 
               than 0 or 1) 
             
             
                 
               λ 1 = λ, λ 2 = 1 
               Weighted Method-of-least- 
             
             
                 
                 
               squares algorithm 
             
             
                 
                 
               (λ is a given value other 
             
             
                 
                 
               than 0 or 1) 
             
             
                 
                 
             
          
         
       
     
   
   If the fixed gain algorithm is used to reduce the amount of calculations, then the equation (19) is simplified into the following equation (19a) where P represents a square matrix with constants as diagonal elements. 
               KP   ⁡     (   k   )       =       P   ⁢           ⁢     ζ   ⁡     (   k   )           1   +         ζ   T     ⁡     (   k   )       ⁢   P   ⁢           ⁢     ζ   ⁡     (   k   )                     (     19   ⁢   a     )             
 
   There are situations where model parameters calculated from the equations (14) through (18), (19a) gradually shifts from desired values. Specifically, as shown in  FIGS. 5A and 5B , if a residual identifying error 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 the residual identifying errors are accumulated, causing a drift in the model parameter. To prevent such a drift of the model parameters, the model parameter vector θ(k) is calculated by the following equation (14a) instead of the equation (14). 
                     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   )                         (     14   ⁢   a     )             
 
where DELTA represents a forgetting coefficient matrix in which the forgetting coefficient δ(i=1 through 3) and “1” are diagonal elements and other elements are all “0”, as shown by the following equation (21). 
             DELTA   =     [           δ   ⁢   1         0       0       0           0         δ   ⁢   2         0       0           0       0         δ   ⁢           ⁢   3         0           0       0       0       1         ]             (   21   )             
 
   The forgetting coefficient δi is set to a value between “0” and “1” (0&lt;δi&lt;1) and has a function to gradually reduce the effect of past identifying errors. In the equation (21), the coefficient which is relevant to the calculation of the model parameter c 1 ″ is set to “1”, holding the effect of past values. By setting one of the diagonal elements of the forgetting coefficient matrix DELTA, i.e., the coefficient which is relevant to the calculation of the model parameter c 1 ″, to “1”, it is possible to prevent a steady deviation between the target value DTHR and the throttle valve opening deviation amount DTH. The model parameters are prevented from drifting by setting other elements δ 1 , δ 2 , and δ 3  of the forgetting coefficient matrix DELTA to a value which is greater than “0” and less than 
   When the equation (14a) is rewritten into a recursive form, the following equations (14b) and (14c) are obtained. A process of calculating the model parameter vector θ(k) using the equations (14b) and (14c) rather than the equation (14) is hereinafter referred to as “δ correcting method”, and dδ(k) defined by the equation (14c) is referred to as “updating vector”.
 
θ( k )=θ(0)+ d θ( k )  (14b)
 
 d θ( k )= DELTA·d θ( k− 1)+ KP ( k ) ide ( k )  (14c)
 
   According to an algorithm using the δ correcting method, in addition to the drift preventing effect, a model parameter stabilizing effect can be obtained. Specifically, an initial vector θ(0) is maintained at all times, and values which can be taken by the elements of the updating vector dθ(k) are limited by the effect of the forgetting coefficient matrix DELTA. Therefore, the model parameters can be stabilized in the vicinity of their initial values. 
   Furthermore, since model parameters are calculated while adjusting the updating vector dθ(k) according to identifying process based on the input and output data of the actual controlled object, it is possible to calculate model parameters that match the actual controlled object. 
   It is preferable to calculate the model parameter vector θ(k) from the following equation (14d) which uses a reference model parameter vector θbase instead of the initial vector θ(0) in the equation (14b).
 
θ( k )=θbase+ d θ( k )  (14d)
 
   The reference model parameter vector θbase is set according to the target value DTHR 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. 
   Further, in the present embodiment, the identifying error ide(k) is subjected to a low-pass filtering. Specifically, when identifying the model parameters of a controlled object which has low-pass characteristics, the identifying weight of the identifying algorithm for the identifying error ide(k) has frequency characteristics as indicated by the solid line L 1  in  FIG. 6A . By the low-pass filtering of the identifying error ide(k), the frequency characteristics as indicated by the solid line L 1  are changed to 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 have frequency characteristics represented respectively by the solid lines L 3  and L 4  in  FIG. 6B . Specifically, if 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 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. 6A  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. 6B . 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. Accordingly, 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 (16), (17), and (18), 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−1). 
   When the identifying error which has been subjected to the low-pass filtering is represented by idef(k), then the updating vector dθ(k) is calculated from the following equation (14e) instead of the equation (14c).
 
 d θ( k )=DELTA× d θ( k− 1)+ KP ( k ) idef ( k )  (14e)
 
Review of the Sampling Period
 
   It has been confirmed by the inventors of the present invention that if the first period ΔT 1  which corresponds to the sampling period and control period of the controlled object model is set to a few milliseconds (e.g., 2 milliseconds), then the performance of suppressing disturbance becomes insufficient and the performance of adapting to variations and time-dependent changes of the hardware characteristics becomes insufficient. These problems will be described below in detail. 
   1) Insufficient Performance of Suppressing Disturbance 
   The equivalent control input Ueq which is calculated from the equation (8) is a feed-forward input for making the throttle valve opening deviation amount DTH follow the target value DTHR. Therefore, it is the reaching law input Urch(k) and the adaptive law input Uadp(k) calculated from the equations (9) and (10) that contributes to suppressing the effect of disturbances (e.g., changes in the friction force applied to a member which supports the valve body of the throttle valve  3 , or changes in the pressure applied to the valve body, i.e., the pressure applied to the valve body due to the difference between the pressures acting on both sides of the valve body). The reaching law input Urch(k) and the adaptive law input Uadp(k) are calculated based on the switching function value σ. 
   When setting the first period ΔT 1  to a value of about a few milliseconds, the present value e(k) and preceding value e(k−1) of the control deviation are substantially equal to each other, if a change rate of the throttle valve opening deviation amount DTH or the target value DTHR is low. Therefore, if the switching function setting parameter VPOLE in the equation (5) is set to a value close to “−1”, then the switching function value σ(k) becomes substantially “0”. As a result, the reaching law input Urch(k) and the adaptive law input Uadp(k) calculated from the equations (9) and (10) become substantially “0”, resulting in a large reduction in the disturbance suppressing performance of an adaptive sliding mode controller. That is, if a controlled object model is defined using a short sampling period compared with the change rate (change period) of the output of the controlled object model, then the disturbance suppressing performance of an adaptive sliding mode controller designed based on the controlled object model becomes greatly reduced. 
   2) Insufficient Performance of Adapting to Variations and Aging of the Hardware Characteristics 
   Adaptation to variations and aging of the hardware characteristics is carried out by sequentially identifying model parameters with the model parameter identifier  22 . 
   If the first period ΔT 1  is set to a value of about a few milliseconds, then the sampling frequency is a few hundreds Hz (e.g., about 500 Hz), and the Nyquist frequency fnyq is one-half the sampling frequency. Most of the frequency components of the throttle valve opening deviation amount DTH and the target value DTHR which are the output from the throttle valve actuating device  10  exist in a frequency range considerably lower than the Nyquist frequency fnyq as shown in  FIG. 7  (a frequency range below 5 Hz). In  FIG. 7 , Φth represents a power spectrum. Therefore, if a parameter which changes at a low rate is sampled at intervals of a relatively short period, as shown in  FIG. 8 , then an amount of change in the parameter cannot be observed. That is, the present value DTH(k) and the preceding value DTH(k−1) of the throttle valve opening deviation amount are substantially equal to each other. 
   When identifying model parameters using such detected data, a sum of the identified model parameters a1″ and a 2 ″ becomes substantially “1”, and each of the model parameters b 1 ″ and c 1 ″ becomes “0”. Thus, the identified model parameters do not accurately represent the dynamic characteristics of the controlled object. 
   As described above, if the model parameters are identified based on data sampled at intervals of a relatively short sampling period compared with the change rate (change period) of the output of the controlled object model, then the accuracy of the identified model parameters becomes greatly lowered, and the performance of adapting to variations and aging of the characteristics of the controlled object becomes insufficient. 
   If the sampling period is too long, then the Nyquist frequency fnyq apparently becomes too low, resulting in a reduction in controllability. However, it has been considered so far that no problem occurs due to a relatively short sampling period. The inventors of the present invention have made it clear that the controllability becomes reduced because of the short sampling period, if a control contemplating changes in the state of the controlled object is performed. 
   According to the present embodiment, the above problem is solved by making the sampling period of the controlled object longer according to the operating frequency range of the controlled object. On the other hand, it is empirically known that the controllability against nonlinear disturbances such as friction increases as the control period is shortened. Accordingly, the first period ΔT 1  set to about a few millimeters is employed as a control period of the adaptive sliding mode controller, and the sampling period that is used to define the controlled object model is set to a second period ΔT 2  which is longer than the first period ΔT 1 . 
   For example, if an upper-limit cut-off frequency of the operating frequency range of the controlled object operates is 1 Hz, then a minimum sampling frequency for observing motions of the controlled object is 2 Hz according to the sampling theorem. It has experimentally been confirmed that the highest sampling frequency for accurately identifying model parameters of a model which represents motions of the controlled object is about 20 Hz. Therefore, the sampling period that is used to define the controlled object model should preferably be set to a period corresponding to a frequency which is 3 times to 30 times the upper-limit cut-off frequency of the operating frequency range of the controlled object. 
   If Nyquist frequencies correspondifng to the first period ΔT 1  and the second period ΔT 2  are fnyq 1  and fnyq 2 , respectively, then their relationship is shown in  FIG. 9 . In  FIG. 9 , fsmp2 represents a sampling frequency corresponding to the second period ΔT 2 . 
   If the sampling frequency is set to a value which is shorter than a period corresponding to a frequency which is 30 times the upper-limit cut-off frequency, then the above-described problem occurs. If the sampling frequency is set to a value which is longer than a period corresponding to a frequency which is 3 times the upper-limit cut-off frequency, then the Nyquist frequency becomes too low for the operating frequency range of the controlled object, resulting in reduced controllability. 
   Further, in the present embodiment, the period of the identifying operation of the model parameter identifier is set to a period which is equal to the second period ΔT 2 . 
   If a discrete sampling time or s discrete control time which is digitized with the second period ΔT 2  is indicated by “n”, then the above-described equation (1a) for defining the controlled object model is rewritten to the equation (1b) shown below. Similarly, the above-described equations (3), (4), and (5) are rewritten to the equations (3a), (4a), and (5a) shown below. The controlled object model which is defined by the equation (1b) will hereinafter referred to as “ΔT 2  model”, and the controlled object model which is defined by the equation (1a) as “ΔT 1  model”. 
                     DTH   ⁡     (     n   +   1     )       =       ⁢       a1   ×     DTH   ⁡     (   n   )         +     a2   ×     DTH   ⁡     (     n   -   1     )         +                     ⁢       b1   ×     DUT   ⁡     (   n   )         +   c1                   (     1   ⁢   b     )             
 DTHR ( n )= THR ( n )− THDEF (3a) 
       e ( n )= DTH ( n )− DTHR ( n )  (4a)                     σ   ⁡     (   n   )       =       ⁢       e   ⁡     (   n   )       +     VPOLE   ×     e   ⁡     (     n   -   1     )                       =       ⁢       (       DTH   ⁡     (   n   )       -     DTHR   ⁡     (   n   )         )     +                     ⁢     VPOLE   ×     (       DTH   ⁡     (     n   -   1     )       -     DTHR   ⁡     (     n   -   1     )         )                     (     5   ⁢   a     )               
   The effect that lengthening the sampling period has on the switching function value σ will be described below. In order for the damping characteristic of the deviation e(k) in the ΔT 1  model and the damping characteristic of the deviation e(n) in the ΔT 2  model to be identical to each other on graphs whose horizontal axes represent time t as shown in  FIGS. 10A and 10B , the value of the switching function setting parameter VPOLE may be set as follows when the second period ΔT 2  is set to a value that is five times the first period ΔT 1 .
 
VPOLE of the Δ T   1  model=−0.9
 
VPOLE of the Δ T   2  model=−0.59
 
   If the switching function setting parameter VPOLE is thus set, and the throttle valve opening deviation amount DTH is vibrated by a low-frequency sine-wave disturbance as shown in  FIG. 11 , then the switching function values σ of the above two models change as shown in  FIG. 12 . Switching functions which are set so that the damping characteristics of the deviation e become identical, have different values with respect to the same disturbance. Specifically, the switching function value σ(n) of the ΔT 2  model is larger than the switching function value σ(k) of the ΔT 1  model. It is thus confirmed that the sensitivity of the switching function value σ to disturbance is increased by lowering the sampling frequency. Consequently, the performance of suppressing disturbance can be improved by using the switching function value σ(n) whose sensitivity to disturbance is increased. 
   Redesigning of Adaptive Sliding Mode Controller Based on ΔT 2  Model 
   The adaptive sliding mode controller is redesigned based on the ΔT 2  model. The output of the adaptive sliding mode controller is expressed by the following equation (6a). 
                     DUT   ⁡     (   n   )       =     Usl   ⁡     (   n   )                   =       Ueq   ⁡     (   n   )       +     Urch   ⁡     (   n   )       +     Uadp   ⁡     (   n   )                       (     6   ⁢   a     )             
 
   An equivalent control input Ueq(n) is obtained by replacing “k” with “n” in the equation (8). Since it is actually difficult to obtain a future value DTHR(n+1) of the target value, the equivalent control input Ueq(n) is calculated by the following equation (8a) from which the term relative to the target value DTHR is removed. It has experimentally been confirmed that the controller may become unstable if only the term of the future value DTHR(n+1) is removed and the present target value DTHR(n) and the preceding target value DTHR(n−1) are left. Therefore, the present target value DTHR(n) and the preceding target value DTHR(n−1) are also removed from the equation (8a). 
   The reaching law input Urch(n) and the adaptive law input Uadp(n) are calculated respectively from the equations (9a), (10a) shown below. 
                     Ueq   ⁡     (   n   )       =       ⁢       1   b1     ⁢     {         (     1   -   a1   -   VPOLE     )     ⁢     DTH   ⁡     (   n   )         +                         ⁢         (     VPOLE   -   a2     )     ⁢     DTH   ⁡     (     n   -   1     )         -   c1     }                 (     8   ⁢   a     )                 Urch   ⁡     (   n   )       =         -   F     b1     ⁢     σ   ⁡     (   n   )                 (     9   ⁢   a     )                 Uadp   ⁡     (   n   )       =         -   G     b1     ⁢       ∑     i   =   0     n     ⁢     Δ   ⁢           ⁢   T2   ⁢           ⁢     σ   ⁡     (   i   )                     (     10   ⁢   a     )             
 
   The gains F and G of the reaching law input Urch(n) and the adaptive law input Uadp(n) should preferably be set according to the switching function value σ(n) as shown in  FIG. 13A . By setting the gains F and G as shown in  FIG. 13A , the gains F and G decrease as the absolute value of the switching function value σ(n) increases. Accordingly, the throttle valve opening deviation amount DTH is prevented from overshooting with respect to the target value DTHR even when the target value DTHR abruptly changes. 
   Instead of setting the gains F and G as shown in  FIG. 13A , the gains F and G may be set according to the deviation e(n) or the throttle valve opening deviation amount DTH(n), as shown in  FIG. 13B  or  FIG. 13C . If the gains F and G are set according to the deviation e(n) as shown in  FIG. 13B , then the gains F and G decrease as the absolute value of the deviation e(n) increases. Accordingly, the throttle valve opening deviation amount DTH is prevented from overshooting with respect to the target value DTHR even when the target value DTHR abruptly changes. 
   If the gains F and G are set according to the throttle valve opening deviation amount DTH(n) as shown in  FIG. 13C , then the controllability can be improved when the throttle valve opening deviation amount DTH(n) is in the vicinity of “0”, i.e., when the throttle valve opening TH is in the vicinity of the default opening THDEF. 
   The gains F and G that are made variable raises the following problem: When the gain F or G changes stepwise due to a stepwise change in a parameter which determines the gain F or G as shown in  FIG. 14B , the reaching law input Urch or the adaptive law input Uadp abruptly changes as indicated by the broken line in  FIG. 14A , which may cause an abrupt change in the throttle valve opening TH. Therefore, the reaching law input Urch and the adaptive law input Uadp may be calculated respectively from the equations (9b) and (10b) instead of the equations (9a) and (10a). The reaching law input Urch and the adaptive law input Uadp thus calculated change gradually as indicated by the solid line in  FIG. 14A  even when the gains F and G abruptly change.
 
 Urch ( n )= Urch ( n− 1)−( F/b   1 )(σ( n )−σ( n− 1))  (9b)
 
 Uadp ( n )= Uadp ( n− 1)−( GΔT   2   /b   1 )×σ( n )  (10b)
 
Review of the Calculation Period
 
   If the second period ΔT 2  is used as a sampling period for the controlled object model, then, as shown in  FIGS. 15A through 15C , the control period is usually also set to the second period ΔT 2  that is longer than the first period ΔT 1 . The longer control period, however, causes the following problems: 
   1) A better controllability is obtained by detecting and correcting as soon as possible an error of the output with respect to the target value, when the error is generated by a nonlinear disturbance such as friction of the actuating mechanism of the throttle valve. If the sampling period is made longer, then the detection of the error is delayed, resulting in low controllability. 
   2) When making the control period longer, the period of inputting the target value into the controller becomes longer. Therefore, the dead time in making the output follow up a change in the target value also becomes longer. Changes in the target value at a high frequency (high speed) cannot be reflected to the output. 
   Therefore, in the present embodiment, the adaptive sliding mode controller  21 , the model parameter identifier  22 , and the model parameter scheduler  25  are constructed based on a model which is defined using the second period ΔT 2  as a sampling period. The adaptive sliding mode controller  21  calculates a control input at intervals of the first period ΔT 1 , the model parameter identifier  22  identifies a model parameter vector θ at intervals of the second period ΔT 2 , and the model parameter scheduler  25  calculates a reference model parameter vector θ base at intervals of the second period ΔT 2 . 
     FIGS. 16A through 16D  are timing charts illustrating calculation timings of the parameters described above, when the second period ΔT 2  is set to a value five times the first period ΔT 2  (ΔT 2 =5ΔT 1 ). In  FIGS. 16A through 16D , a model parameter vector θ(n−1) is calculated based on throttle valve opening deviation amounts DTH at time (n−1) (=time (k−5)) and time (n−2) (=time (k−10)), a control input DUT at time (n−1), and a target value DTHR at time (n−1), using a reference model parameter vector θbase(n−1) at time (n−1). A control input DUT(k−5) is calculated using target values DTHR(k−5) and DTHR(k−10), throttle valve opening deviation amounts DTH(k−5) and DTH(k−10), and the model parameter vector θ(n−1). A control input DUT(k−4) is calculated using target values DTHR(k−4) and DTHR(k−9), throttle valve opening deviation amounts DTH(k−4) and DTH(k−9), and the model parameter vector θ(n−1). A control input DUT(k−3) is calculated using target values DTHR(k−3) and DTHR(k−8), throttle valve opening deviation amounts DTH(k−3) and DTH(k−8), and the model parameter vector θ(n−1). 
   When employing the above calculation timings, the period of updating model parameters which are used to calculate the control input DUT becomes longer than the period of updating the control input DUT by the controller  21 . As a result, the period of updating model parameters affects the control input DUT, which may possibly cause resonance in the control system. 
   Therefore, in the present embodiment, such resonance in the control system is prevented by sampling (oversampling) model parameters which are identified intervals of the second period ΔT 2 , at intervals of the first period ΔT 1  which is the control period, storing the sampled data in a ring buffer, and using values obtained by effecting a moving-averaging process on the data stored in the ring buffer as model parameters for the control. 
     FIG. 17  is a timing chart illustrating the above calculation sequence.  FIG. 17 , similar to  FIGS. 16A through 16D , shows a case where ΔT 2  equals 5ΔT 1 . In the illustrated example, the latest nine oversampled data are averaged. Specifically, model parameters obtained by averaging three model parameter vectors θ(n−2), five model parameter vectors θ(n−1), and one model parameter vector θ(n), are used in a calculation carried out by the sliding mode controller at time k. At another time, e.g., at time (k−3), model parameters obtained by averaging one model parameter vector θ(n−3), five model parameter vectors θ(n−2), and three model parameter vectors 0 (n−1), are used in a calculation carried out by the sliding mode controller. 
   A model parameter vector θ′ shown in  FIG. 17  represents a model parameter vector which has been subjected to a first limit process and an oversampling and moving-averaging process to be described later. 
   Details of the Adaptive Sliding Mode Controller 
   Details of the adaptive sliding mode controller  21  will be described below. The controlled object model is a model which is defined using the second period ΔT 2 . As a calculation period of the adaptive sliding mode controller  21 , the first period ΔT 1  rather than the second period ΔT 2  is employed as described above. Accordingly, time “k” rather than time “n” is used as a discrete time. 
   In the present embodiment, a control input DUT(k) is calculated from the equation (6b) instead of the equation (6a) in order to improve the response to small changes in the target value DTHR and reduce the overshooting of the throttle valve opening deviation amount DTH with respect to the target value DTHR. In the equation (6b), the control input DUT(k) is calculated using a nonlinear input Unl(k), a forced vibration input Uwave(k), and a damping input Udamp(k) in addition to the equivalent control input Ueq(k), the reaching law input Urch(k), and the adaptive law input Uadp(k). 
                     DUT   ⁡     (   k   )       =       ⁢     Usl   ⁡     (   k   )                   =       ⁢       Ueq   ⁡     (   k   )       +     Urch   ⁡     (   k   )       +     Uadp   ⁡     (   k   )       +                     ⁢       Unl   ⁡     (   k   )       +     Uwave   ⁡     (   k   )       +     Udamp   ⁡     (   k   )                       (     6   ⁢   b     )             
 
   In the equation (6b), the equivalent control input Ueq(k), the reaching law input Urch(k), and the adaptive law input Uadp(k) are calculated from the following equations (8b), (9), and (10c), and the switching function value σ(k) is calculated from the following equation (5b). 
                     Ueq   ⁡     (   k   )       =       ⁢       1   b1     ⁢     {         (     1   -   a1   -   VPOLE     )     ⁢     DTH   ⁡     (   k   )         +                         ⁢         (     VPOLE   -   a2     )     ⁢     DTH   ⁡     (     k   -   k0     )         -   c1     }                 (     8   ⁢   b     )                 Urch   ⁡     (   k   )       =         -   F     b1     ⁢     σ   ⁡     (   k   )                 (   9   )                 Uadp   ⁡     (   k   )       =       Uadp   ⁡     (     k   -   1     )       -       G   b1     ⁢   Δ   ⁢           ⁢   T1   ×     σ   ⁡     (   k   )                   (     10   ⁢   c     )                       σ   ⁡     (   k   )       =       ⁢       e   ⁡     (   k   )       +     VPOLE   ×     e   ⁡     (     k   -   k0     )                       =       ⁢       DTH   ⁡     (   k   )       -     DTHR   ⁡     (   k   )       +                     ⁢     VPOLE   ×     (       DTH   ⁡     (     k   -   k0     )       -     DTHR   ⁡     (     k   -   k0     )         )                     (     5   ⁢   b     )             
 
   In the equations (5b) and (8b), k 0  represents a parameter corresponding to a sampling time interval of the deviation e(k) involved in the calculation of the switching function value σ. In the present embodiment, the parameter k 0  is set to (ΔT 2 /ΔT 1 ) (e.g., “5”) corresponding to the second period ΔT 2 . By setting the sampling time interval of the deviation e(k) involved in the calculation of the switching function value σ to the second period ΔT 2 , it is possible to calculate a switching function value suitable for a frequency range in which the characteristics of the controlled object model and the characteristics of the plant substantially coincide with each other. As a result, the performance of suppressing disturbances and the modeling error can be further improved. 
   Since the sampling period for the modeling is set to the second period ΔT 2  and the control period is set to the first period ΔT 1 , the equations (5b), (8b), and (10c) are different from the above-described equations (5), (8a), and (lob). 
   The nonlinear input Unl is an input for suppressing a nonlinear modeling error due to backlash of speed reduction gears for actuating the valve body of the throttle valve  3 , and placing the deviation state quantity onto the switching straight line. The forced vibration input Uwave is an input for suppressing nonlinear characteristics due to friction of the actuating mechanism of the throttle valve  3 . The damping input Udamp is an input for preventing the throttle valve opening deviation amount DTH from overshooting with respect to the target value DTHR. 
   First, the nonlinear input Unl will be described below. 
   In a throttle valve actuating device of the type which actuates a valve body through speed reduction gears, a steady deviation due to backlash of the speed reduction gears as shown in  FIG. 18  is generated when the target value DTHR is slightly changing, and a certain time period is required to eliminate the steady deviation. Particularly, such a tendency grows after the direction of change in the target value DTHR and the throttle valve opening deviation amount DTH is reversed. 
   According to a controller using the equation (6a) which does not include the nonlinear input Unl, the above steady deviation is converged to “0” by the adaptive law input Uadp and the model parameter c 1  which are included in the equation (8) for calculating the equivalent control input Ueq. However, since the converging rate of the steady deviation is low, no sufficient controllability is obtained.  FIG. 18  shows the manner in which the adaptive law input Uadp changes, and the steady deviation is converged to “0”. According to a control process using the equation (6a), the steady deviation can be reduced to “0” by using at least one of the adaptive law input Uadp and the model parameter c1. 
   In the present embodiment, a nonlinear input Unl(k) calculated from the following equation (22) is used in order to solve the above problem.
 
 Unl ( k )=− Knl×sgn (σ( k )) b 1  (22)
 
where sgn(σ(k)) represents a sign function whose value equals “1” when σ (k) has a positive value, and equals “−1” when σ(k) has a negative value. Knl is a nonlinear input gain.
 
   When the nonlinear input Unl(k) is used, the response to the target value DTHR which is slightly changing is as shown in  FIG. 19A , and the nonlinear input Unl(k) changes as shown in  FIG. 19B . That is, the convergence of the steady deviation is prevented from being delayed as shown in  FIG. 18 . 
   However, as understood from  FIGS. 19A and 19B , a chattering phenomenon is caused by adding the nonlinear input Unl. This chattering phenomenon, which may be sometimes caused by the sliding mode controller, is not caused when using the equation (6a). In the present embodiment, by using the adaptive law input Uadp and the model parameter c 1  and using the forced vibration input Uwave, a modeling error to be compensated by the nonlinear input Unl is minimized, and hence the amplitude of the nonlinear input Unl, i.e., the amplitude of chattering, is minimized. 
   Further, in the present embodiment, the nonlinear input gain Knl is set according to the throttle valve opening deviation amount DTH as shown in  FIG. 20 . When the throttle valve opening deviation amount DTH is near “0”, i.e., when the throttle valve opening TH is near the default opening THDEF, a steady deviation is suppressed by increasing the nonlinear input gain Knl. 
   The forced vibration input Uwave will be described below. 
   In a controlled object, such as the throttle valve actuating device  10 , the controllability with respect to minute changes of the target value may be lowered by the friction characteristics of sliding members for actuating the valve body of the throttle valve  3 . 
   For compensating for the friction characteristics, there is known a method of adding a dither input to the control input at intervals of a predetermined period. In the present embodiment, the forced vibration input Uwave is calculated as the dither input from the following equation (23).
 
 Uwave ( k )= Kwave×Fwave ( k )×| ide ( n )|/ b   1   (23)
 
where Kwave is a dither input basic gain, Fwave(k) is a dither signal value, and ide(n) is an identifying error of model parameters.
 
   As a dither signal for obtaining the dither signal value Fwave, a series of a basic waveform shown in  FIG. 21  is employed, and the repetitive frequency thereof is set to a frequency which is not in the vicinity of the resonant frequency of the controlled object, as shown in  FIG. 22 , in order to avoid resonance of the control system. In  FIG. 22 , fr represents the resonant frequency of the control system, and fwave represents the frequency of the dither signal. 
   In a frequency range lower than the resonant frequency fr, the nonlinear input Unl exhibits the same effect as the forced vibration input Uwave. Therefore, the dither signal frequency fwave is set to a frequency higher than the resonant frequency fr. More specifically, the dither signal frequency fwave is set to a frequency within a rejection frequency band (outside a pass frequency band) of the controlled object which has a low-pass characteristic (a characteristic which attenuates high-frequency components). 
   The forced vibration input Uwave, similar to the nonlinear input Unl, may become a cause of the chattering. Therefore, an amplitude of the forced vibration input Uwave should be set according to the friction characteristics of the controlled object. However, the friction characteristics of the throttle valve actuating device vary depending on the characteristic variations and aging of hardware arrangements, and the pressure acting on the valve body. Therefore, it is not appropriate to set the forced vibration input Uwave according to the throttle valve opening (throttle valve opening deviation amount), like the nonlinear input Unl. 
   According to the present embodiment, in view of the fact that since the controlled object model is a linear model, the nonlinear characteristics such as friction characteristics are not reflected in the model parameters, but appear as the identifying error ide, the amplitude of the forced vibration input Uwave is set according to the absolute value of the identifying error ide, as indicated by the equation (23). In this manner, it is possible to set the amplitude according to changes in the friction characteristics. 
     FIGS. 23A through 23C  are timing charts illustrating an effect of the forced vibration input (Uwave). At the time an excessive friction region starts (t 1 ) and at the time the excessive friction region ends (t 2 ), the identifying error ide increases and hence the forced vibration input Uwave increases. Accordingly, a control error of the throttle valve opening deviation amount DTH is prevented from increasing. 
   The damping input Udamp will be described below. 
   In controlling the throttle valve actuating device, it is important to avoid a collision with a stopper when the valve body of the throttle valve moves to a fully closed position. It is also important to prevent the engine drive power from increasing over a level which is greater than the driver&#39;s demand. The sliding mode control generally has a high-speed response characteristic, but has a tendency to often cause an overshoot with respect to the target value. 
   Therefore, in the present embodiment, the damping input Udamp is used as a control input for suppressing the overshoot. 
   It is considered that the damping input Udamp for suppressing the overshoot may be defined by the following three equations.
 
 Udamp   1 ( k )=− Kdamp   1 ( e ( k )− e ( k− 1))/ b   1   (24)
 
 Udamp   2 ( k )=− Kdamp   2 (σ( k )−σ( k− 1))/ b   1   (25)
 
 Udamp   3 ( k )=− Kdamp   3 ( DTH ( k )− DTH ( k− 1))/ b   1   (26)
 
where Kdamp 1 , Kdamp 2 , and Kdamp 3  represent damping control gains.
 
   The change rates of the deviation e(k) and the switching function value σ(k) in the equations (24) and (25) become high either when the change rate of the throttle valve opening deviation amount DTH is high, or when the change rate of the target value DTHR is high. Therefore, the absolute value of the damping input Udamp increases in the both cases. The damping input Udamp has a function for suppressing other control inputs for converging the throttle valve opening deviation amount DTH to the target value DTHR. Therefore, if the damping input Udamp 1  or Udamp 2  defined by the equation (24) or (25) is employed, then control inputs for following up the target value DTHR are suppressed when the target value DTHR varies largely. As a result, the response speed becomes lower. 
   On the other hand, an absolute value of the damping input Udamp 3  defined by the equation (26) increases to suppress other control inputs only when the change rate of the throttle valve opening increases. In other words, the damping input Udamp 3  does not suppress other control inputs when the target value DTHR varies greatly. Therefore, the damping input Udamp 3  is capable of achieving both overshoot suppression and a high response speed, which cannot be achieved by the damping input Udamp 1  or Udamp 2  defined by the equation (24) or (25). 
   Accordingly, in the present embodiment, the damping input Udamp is calculated form the following equation (27).
 
 Udamp=−Kdamp ( DTH ( k )− DTH ( k− 1))/ b   1   (27)
 
     FIGS. 24A and 24B  are timing charts illustrating an overshoot suppressing effect of the damping input Udamp, and show response characteristics of the throttle valve opening deviation amount DTH when the target value DTHR is changed stepwise as indicated by the broken lines. The overshoot shown in  FIG. 24A  is suppressed by the damping input Udamp as shown in  FIG. 24B . 
   Since the equation (27) includes the model parameter b 1 , an overshoot can appropriately be suppressed even when the dynamic characteristics of the throttle valve actuating device 10 have changed. 
   With respect to the damping control gain Kdamp in the equation (27), the controllability can further be improved by changing the damping control gain Kdamp according to the throttle valve opening deviation amount DTH and the target value DTHR. Therefore, in the present embodiment, a basic value Kdampbs is set according to the throttle valve opening deviation amount DTH as shown in  FIG. 25A , and a correction coefficient Kkdamp is calculated according to a moving average value DDTHRAV of amounts of change in the target value DTHR as shown in  FIG. 25B . Further, the damping control gain Kdamp is calculated from the equation (28) shown below. Since the basic value Kdampbs is set to a small value when the throttle valve opening TH is in the vicinity of the default opening (DTH≈0), the damping effect is lowered, and a high response speed is obtained. When the moving average value DDTHRAV is equal to or greater than a predetermined positive value, the correction coefficient Kkdamp is set to a value greater than “1”. This is because an overshoot is prone to occur when the throttle valve opening TH increases.
 
 Kdamp=Kdampbs×Kkdamp   (28)
 
   The moving average value DDTHRAV is calculated by the following equation (29): 
               DDTHRAV   ⁡     (   k   )       =       ∑     i   =   0     iAV     ⁢     (       DTHR   ⁡     (     k   -   i     )       -       DTHR   ⁡     (     k   -   i   -   1     )       /     (     iAV   +   1     )                     (   29   )             
 
where iAV represents a number that is set to “50”, for example.
 
Details of the Model Parameter Identifier
 
   Since the identifying process is carried out by the model parameter identifier  22  at intervals of the second period ΔT 2 , equations obtained by changing “k” in the equations shown in the description of the outline of the model parameter identifier to “n” are given below. LF( ) in the equation (30) below represents the low-pass filtering of the identifying error in the form of a function.
 
θ( n )=θbase+ d θ( n )  (14f)
 
 d θ( n )= DELTA·d θ( n− 1)+ KP ( n ) idef ( n )  (14g)
 
 KP ( n )= P ζ( n )/(1+ζ T ( n ) P ζ( n ))  (19b)
 
 idef ( n )= LF ( ide ( n ))  (30)
 
 ide ( n )= DTH ( n )− DTHHAT ( n )  (16a)
 
 DTHHAT ( n )=θ( n− 1) T ζ( n )  (17a)
 
θ( n ) T   =[a   1 ″( n ),  a   2 ″( n ),  b   1 ″( n ),  c   1 ″( n )]  (15a)
 
ζ( n ) T   =[DTH ( n− 1),  DTH ( n− 2),  DUT (n−1), 1]  (18a)
 
             DELTA   =     [           δ   ⁢   1         0       0       0           0         δ   ⁢   2         0       0           0       0         δ   ⁢   3         0           0       0       0       1         ]             (   21   )             
 
   The elements a 1 ″, a 2 ″, b 1 ″, and c 1 ″ of the model parameter vector θ(n) calculated by the equation (14f) are subjected to a limit process described below in order to improve robustness of the control system. 
     FIGS. 26A and 26B  are diagrams illustrating a limit process of the model parameters a 1 ″ and a 2 ″.  FIGS. 26A and 26B  show a plane defined by the horizontal axis of the model parameter a 1 ″ and the vertical axis of the model parameter a 2 ″. If the model parameters a 1 ″ and a 2 ″ are located outside a stable region which is indicated as a hatched region, then a limit process is performed to change them to values corresponding to an outer edge of the stable region. 
   If the model parameter b 1 ″ falls outside a range between an upper limit value XIDB 1 H and a lower limit value XIDB 1 L, then a limit process is performed to change the model parameter b 1 ″ to the upper limit value XIDB 1 H or the lower limit value XIDB 1 L. If the model parameter c 1 ″ falls outside of a range between an upper limit value XIDC 1 H and a lower limit value XIDC 1 L, then a limit process is performed to change the model parameter c 1 ″ to the upper limit value XIDC 1 H or the lower limit value XIDC 1 L. 
   A set of the above limit processes (first limit process) is expressed by the equation (31) shown below. θ*(n) represents the limited model parameter vector, whose elements are expressed by the equation (32) shown below.
 
θ*( n )= LMT (θ( n ))  (31)
 
θ*( n ) T   =[a   1 *( n ),  a   2 *( n ),  b   1 *( n ),  c   1 *( n )]  (32)
 
   In a control system which was formerly proposed by the inventers of the present invention, the preceding updating vector dθ(n−1) which is used to calculate the updating vector dθ(n) from the equation (14g) and the preceding model parameter vector θ(n−1) which is used to calculate the estimated throttle valve opening deviation amount DTHHAT(k) includes model parameters that are not subjected to the limit process. In the present embodiment, a vector calculated by the equation (33) shown below is used as the preceding updating vector dθ(n−1), and a limited model parameter vector θ*(n−1) is used as the preceding model parameter vector which is used to calculate the estimated throttle valve opening deviation amount DTHHAT(k), as shown by the following equation (17b).
 
 d θ( n− 1)=θ*( n− 1)−θbase( n− 1)  (33)
 
 DTHHAT ( n )=θ*( n− 1) T ζ( n )  (17b)
 
   The reasons for the above process are described below. 
   If a point corresponding to coordinates determined by the model parameters a 1 ″ and a 2 ″ (hereinafter referred to as “model parameter coordinates”) is located at a point PA 1  shown in  FIG. 26B , then a limit process is performed to move a point corresponding to the model parameter coordinates to a point PAL positioned on an outer edge of the stable region. If the throttle valve opening deviation amount DTH changes and a point corresponding to the model parameter coordinates to which the model parameters a 1 ″ and a 2 ″ are to be converged, changes to a point PA 2 , then the movement from the point PA 1  to the point PA 2  is slower than the movement from the point PAL to the point PA 2 . That is, when the control process carried out by the adaptive sliding mode controller  21  is adapted to the dynamic characteristics of the controlled object, a dead time is produced, which may lower the controllability. 
   Therefore, in the present embodiment, the limited model parameter vector θ*(n−1) is applied to the equations (33) and (17b) to calculate the present model parameter vector θ(n). 
   A model parameter vector θ*(k) obtained at time k by oversampling the model parameter vector θ*(n) after the first limit process at the time k is expressed by the following equation (32a).
 
θ*( k ) T   =[a   1 *( k ),  a   2 *( k ),  b   1 *( k ),  c   1 *( k )]  (32a)
 
   When a model parameter vector θ′(k) obtained by moving-averaging of the oversampled model parameter vector θ*(k) is expressed by the following equation (32b), then elements a 1 ′(k), a 2 ′(k), b 1 ′(k), and c 1 ′(k) of the model parameter vector θ′(k) are calculated by the following equations (34) through (37). 
     θ′( k ) T   =[a   1 ′( k ),  a   2 ′( k ),  b   1 ′( k ),  c   1 ′( k )]  (32b)                 a1   ′     ⁡     (   k   )       =       ∑     i   =   0     m     ⁢         a1   *     ⁡     (     k   -   i     )       /     (     m   +   1     )                 (   34   )                   a2   ′     ⁡     (   k   )       =       ∑     i   =   0     m     ⁢         a2   *     ⁡     (     k   -   i     )       /     (     m   +   1     )                 (   35   )                   b1   ′     ⁡     (   k   )       =       ∑     i   =   0     m     ⁢         b1   *     ⁡     (     k   -   i     )       /     (     m   +   1     )                 (   36   )                   c1   ′     ⁡     (   k   )       =       ∑     i   =   0     m     ⁢         c1   *     ⁡     (     k   -   i     )       /     (     m   +   1     )                 (   37   )               
   where (m+1) represents the number of data which are subjected to the moving-averaging, and “m” is set to “4”, for example. 
   Then, as shown by the equation (38) described below, the model parameter vector θ′(k) is subjected to a limit process (second limit process) similar to the above limit process, thus calculating a corrected model parameter vector θL(k) expressed by the equation (39) shown below, because the model parameter a 1 ′ and/or the model parameter a 2 ′ may change so that a point corresponding to the model parameters a 1 ′ and a 2 ′ moves out of the stable region shown in  FIGS. 26A and 26B  due to the moving-averaging calculations. The model parameters b 1 ′ and c 1 ′ are not actually limited because they do not change out of the limited range by the moving-averaging calculations.
 
θ L ( k )= LMT (θ′( k ))  (38)
 
θ L ( k ) T   =[a   1   , a   2   , b   1   , c   1 ]  (39)
 
Details of the Model Parameter Scheduler
 
   The reference model parameters a 1 base, a 2 base, b 1 base, and c 1 base are set by the model parameter scheduler  25 . The reference model parameters a 1 base and a 2 base are set according to the target value DTHR as shown in  FIG. 27 . By setting the reference model parameters a 1 base and a 2 base set according to the target value DTHR, a higher controllability, particularly, a quicker response can be obtained, compared with the case where the reference model parameters a 1 base and a 2 base is set according to the throttle valve opening deviation amount DTH. 
   The reference model parameter c 1 base is always set to “0”, because The reference model parameter c 1 base does not depend on the operating condition of the throttle valve actuating device (the target value DTHR or the throttle valve opening deviation amount DTH). The reference model parameter b 1 base which is relevant to the control input DUT is always set to the lower limit value XIDB 1 L of the model parameter b 1  irrespective of the operating condition of the throttle valve actuating device. 
   The reference model parameter b 1 base is always set to the lower limit value XIDB 1 L because of the following reason. 
   As shown in  FIG. 28B , in the case where the model parameter b 1  used by the adaptive sliding mode controller  21  prior to time tS is corrected by a b 1  component db 1  (see  FIG. 28C ) of the updating vector dθ and the model parameter b 1  is less than the reference model parameter b 1 base, if the target value DTHR changes stepwise from DTHR 1  to DTHR 2  at time tS as shown in  FIG. 28A , it is assumed that the target value DTHR is equal to the value DTHR 2  and a value to be taken by the model parameter b 1  is b 1 s shown in  FIG. 28B . 
   In this example, since several steps are required for the model parameter identifier  22  to correct the reference model parameter b 1 base, several steps are also required for the updating component db 1  which has corrected the reference model parameter b 1 base in a negative direction prior to time tS to become an appropriate value after time tS. Therefore, during the period of those several steps, the model parameter b 1  takes a value which is much less than a desired value b 1 s. As a result, the adaptive sliding mode controller  21  calculates a control input DUT which performs excessive correction, and an overshoot of the throttle valve opening deviation amount DTH may be caused as shown in  FIG. 28A . 
   Therefore, in the present embodiment, the reference model parameter b 1 base is always set to the lower limit value XDB 1 L to avoid the drawbacks shown in  FIGS. 28A through 28C . By setting the reference model parameter b 1 base to the lower limit value XDB 1 L, the updated component db 1  always takes a positive value as shown in  FIG. 29C . Therefore, even in the presence of the identification delay, for example, it is prevented that the model parameter b 1  takes a value which is much less than the desired value b 1 s (see  FIG. 29B ), and the adaptive sliding mode controller  21  is prevented from performing excessive correction due to the identification delay. As a result, as shown in  FIG. 29A , the overshoot of the throttle valve opening deviation amount DTH can be suppressed. 
   Processes Executed by the CPU of the ECU  7   
   Processes executed by the CPU of the ECU  7  for realizing the above functions of the controller  21 , the model parameter identifier  22 , and the model parameter scheduler  25  will be described below. 
     FIG. 30  is a flowchart showing a throttle valve opening control process, which is executed by the CPU of the ECU  7  at intervals of a predetermined period, e.g., 2 msec. 
   In step S 11 , a process of setting a state variable shown in  FIG. 31  is carried out. Specifically, calculations of the equations (2) and (3) are carried out to determine the throttle valve opening deviation amount DTH(k) and the target value DTHR(k) in steps S 31  and S 32  in  FIG. 31 . The symbol (k) or (n) representing a current value may occasionally be omitted. 
   In step S 12 , it is determined whether or not the value of a counter IDCOUNT is “0”. Since the counter IDCOUNT is initially set to “0”, the process proceeds from step S 12  to step S 14 , in which a process of identifying a model parameter is carried out, i.e., a process of calculating a model parameter vector θ(n) is carried out. Then, a first limit process shown in  FIG. 34  is carried out to calculate a model parameter vector θ(n) in step S 15 . Specifically, the limit process of the model parameter vector θ(n) is executed to calculate the model parameter vector θ*(n). Elements a 1 *(n), a 2 *(n), b 1 *(n), and c 1 *(n) of the calculated model parameter vector θ*(n) are stored in a ring buffer for the oversampling process. Specifically, a predetermined number N of each elements, i.e., elements of θ(k), θ*(k+1), . . . , θ*(k+N−1) are stored in the ring buffer. The predetermined number N represents a ratio of the second period ΔT 2  to the first period ΔT 1  (ΔT 2 /ΔT 1 ), and is set to “5”, for example. 
   In step S 16 , the counter IDCOUNT is set to the predetermined number N. Therefore, in the next execution of this process, the answer to step S 12  becomes negative (NO), and the value of the counter IDCOUNT is decremented by “1” in step S 13 . Thereafter, the process proceeds to step S 17 . Therefore, steps from S 14  to S 16  are carried out once in every N times. 
   In step S 17 , a model parameter vector θ′(k) is calculated by the moving-averaging of the limited model parameter vector θ*(n). Specifically, the model parameter stored in the ring buffer is applied to the equations (34) through (37) to calculate model parameters a 1 ′(k), a 2 ′(k), b 1 ′(k), and c 1 ′(k). 
   In step S 18 , a second limit process shown in  FIG. 39  is carried out. Specifically, the limit process of the model parameters a 1 ′(k) and a 2 ′(k) calculated in step S 17  is carried out to calculate a corrected model parameter vector θL(k). The model parameters b 1 ′(k) and c 1 ′(k) are directly applied to elements b 1 (k) and c 1 (k), respectively, of the corrected model parameter vector θL(k). 
   In step S 19 , a process of calculating a control input Usl(k) shown in  FIG. 40  is carried out. Specifically, an equivalent control input Ueq(k), a reaching law input Urch(k), an adaptive law input Uadp(k), a nonlinear input Unl(k), a forced vibration input Uwave, and a damping input Udamp(k) are calculated, and the calculated inputs are summed up to a control input Usl(k) (=duty ratio DUT(k)). 
   In step S 20 , a process of stability determination of the sliding mode controller shown in  FIG. 50  is carried out. Specifically, the stability of the sliding mode controller is determined based on the differential of a Lyapunov function, and a stability determination flag FSMCSTAB is set. The stability determination flag FSMCSTAB is referred to when performing the calculation of the control input Usl(k). 
     FIG. 32  is a flowchart showing the process of identifying model parameters in step S 14  shown in  FIG. 30 . 
   In step S 41 , the gain coefficient vector KP(n) is calculated from the equation (19b). Then, the estimated throttle valve opening deviation amount DTHHAT(n) is calculated from the equation (17b) in step S 42 . 
   In step S 43 , a process of calculating ide(n) shown in  FIG. 33  is carried out to calculate the identifying error ide(n). In step S 44 , the updating vector dθ(n) is calculated from the equations (14g), (33). A θbase table shown in  FIG. 27  is retrieved according to the target value DTHR to calculate the reference model parameter vector θbase in step S 45 . In the θbase table, values of the reference model parameters a 1 base and a 2 base are actually set. The reference model parameter b 1 base is set to the minimum value XIDB 1 L of the model parameter b 1 . The reference model parameter c 1 base is set to “0”. 
   In step S 46 , the model parameter vector θ(n) is calculated from the equation (14f). Thereafter, the process shown in  FIG. 32  ends. 
     FIG. 33  is a flowchart showing a process of calculating an identifying error ide(n) in step S 43  shown in  FIG. 32 . 
   In step S 51 , the identifying error ide(n) is calculated from the equation (16a). 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 ). XCNTIDST is set to “2”, since the dead time d is approximated to “0” in the present embodiment. Since the counter CNTIDST has an initial value of “0”, the process first proceeds to step S 53 , in which the counter CNTIDST is incremented by “1”. Next, the identifying error ide(n) is set to “0” in step S 54 , and the process proceeds to step S 55 . Immediately after the identification of the model parameter vector θ(n) starts, no correct identifying error is obtained by the calculation of the equation (16a). Therefore, the identifying error ide(n) is set to “0” by steps S 52  through S 54 , without using the calculated result of the equation (16a). 
   If the answer to the step S 52  is affirmative (YES), the process immediately proceeds to step S 55 . 
   In step S 55 , the identifying error ide(n) is subjected to a low-pass filtering process. Specifically, a process of correcting the frequency characteristics of the controlled object as described above with reference to  FIGS. 6A and 6B , is carried out. 
     FIG. 34  is a flowchart showing the first limit process carried out in step S 15  shown in  FIG. 30 . 
   In step S 71 , flags FA 1 STAB, FA 2 STAB, FB 1 LMT, and FC1LMT used in this process are initialized by setting each flag to “0”. In step S 72 , the limit process of the model parameters a 1 ″ and a 2 ″ shown in  FIG. 35  is executed. In step S 73 , the limit process of the model parameter b 1 ″ shown in  FIG. 37  is executed. In step S 74 , the limit process of the model parameter c 1 ″ shown in  FIG. 38  is executed. 
     FIG. 35  is a flowchart showing the limit process of the model parameters a 1 ″ and a 2 ″ which is carried out in step S 72  shown in  FIG. 34 .  FIG. 36  is a diagram illustrating the process shown in  FIG. 35 , and will be referred to with  FIG. 35 . 
   In  FIG. 36 , 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 is indicated by a hatched region (hereinafter referred to as “stable region”). The process shown in  FIG. 35  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 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 (40) 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). 
             A   =     [           a1   *           a2   *             1       0         ]             (   40   )             
 
   If a 2 ″ is less than XIDA 2 L in step S 81 , then the 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” in step S 82 . When the a 2  stabilizing flag FA 2 STAB is set to “1”, this indicates that the model parameter a 2 * is set to the lower limit value XIDA 2 L. In  FIG. 36 , 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 step S 81  is affirmative (YES), i.e., if a 2 ″ is greater than or equal to XIDA 2 L, then the 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 al 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 less 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, then the 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 , then the model parameter a 1 * is set to the lower limit value XIDA 1 L and an al* 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 , then the model parameter a 1  is set to the upper limit value XIDA 1 H and the al stabilizing flag FAISTAB is set to “1” in step S 87 . When the al stabilizing flag FA 1 STAB is set to “1”, this indicates that the model parameter a 1 * is set to the lower limit value XIDA 1 L or the upper limit value XIDA 1 H. In  FIG. 36 , the correction of the model parameters 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 model parameter a 1 * and the model parameter a 2 * is equal to or less than a predetermined stability determining value XA 2 STAB. The predetermined stability determining 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. 37  satisfy the following equation (41).
 
 a   2   *+|a   1 *|=XA 2 STAB  (41)
 
   Therefore, in step S 90 , it is determined whether or not the combination of the 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. 36 . If the answer to step S 90  is affirmative (YES), then the limit process immediately ends, since the combination of the model parameters a 1 * and a 2 * is in the stable region shown in  FIG. 36 . 
   If the answer to step S 90  is negative (NO), then it is determined whether or not the model parameter a 1 * is less than a value obtained by subtracting the predetermined a 2  lower limit value XIDA 2 L from the predetermined stability determining 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 model parameter a 1 * is equal to or less than (XA 2 STAB−XIDA 2 L), then the 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 model parameter a 1 * is greater than (XA 2 STAB−XIDA 2 L) in step S 91 , then the model parameter a 1 * is set to (XA 2 STAB−XIDA 2 L) in step S 93 . Further in step S 93 , the model parameter a 2 * is set to the predetermined a 2  lower limit value XIDA 2 L, and the al stabilizing flag FA 1 STAB and the a 2  stabilizing flag FA 2 STAB are set to “1”. 
   In  FIG. 36 , the correction of the model parameters 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 parameters in a limit process P 4  of steps S 91  and S 93  is indicated by the arrow lines with “P 4 ”. 
   As described above, the limit process shown in  FIG. 35  is carried out to bring the model parameters a 1 ″ and a 2 ″ into the stable region shown in  FIG. 36 , thus calculating the model parameters a 1 * and a 2 *. 
     FIG. 37  is a flowchart showing a limit process of the model parameters b 1 ″, which is carried out in step S 73  shown in  FIG. 34 . 
   In steps S 101  and S 102  shown in  FIG. 37 , it is determined whether or not the model parameters 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 predetermined 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 answer to steps S 101  and S 102  is 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, then the model parameter b 1 * is set to the model parameter b 1 ″ in step S 105 . 
   If b 1 ″ is less than XIDBIL in step S 101 , then the model parameter b 1 * is set to the lower limit value XIDBIL, 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 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 model parameter b 1 * is set to the lower limit value XIDB 1 L or the upper limit value XIDB 1 H. 
     FIG. 38  is a flowchart showing a limit process of the model parameter c 1 ″, which is carried out in step S 74  shown in  FIG. 34 . 
   In steps S 111  and S 112  shown in  FIG. 38 , it is determined whether or not the model parameters c 1 ″ 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 answer to steps S 111  and S 112  is 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, then the 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 , then the 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 , then the model parameter c 1 * is set to the upper limit value XIDC 1 H, and the c 1  limiting flag FCLLMT 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. 39  is a flowchart showing the second limit process carried out in step S 18  shown in  FIG. 30 . The second limit process is essentially the same as the first limit process shown in  FIG. 35  except that the model parameters a 1 ″ and a 2 ″ in the limit process shown in  FIG. 35  are replaced respectively with the model parameters a 1 ′ and a 2 ′, and the model parameters a 1 * and a 2 * in the limit process shown in  FIG. 35  are replaced respectively with the model parameters a 1 ″ and a 2 ″. Specifically, the moving-averaged model parameters a 1 ′ and a 2 ′ are subjected to a limit process of steps S 121  through S 133 , which is similar to the limit process shown in  FIG. 35 , thereby calculating corrected model parameters a 1  and a 2 . 
     FIG. 40  is a flowchart showing a process of calculating a control input Usl, which is carried out in step S 19  shown in  FIG. 30 . 
   In step S 201 , a process of calculating a switching function value σ shown in  FIG. 41  is carried out. In step S 202 , an equivalent control input Ueq is calculated from the equation (8b). In step S 203 , a process of calculating a reaching law input Urch shown in  FIG. 44  is carried out. In step S 204 , a process of calculating an adaptive law input Uadp shown in  FIG. 45  is carried out. In step S 205 , a process of calculating a nonlinear input Unl shown in  FIG. 46  is carried out. In step S 206 , a process of calculating a forced vibration input Uwave shown in  FIG. 47  is carried out. In step S 207 , a process of calculating a damping input Udamp shown in  FIG. 49  is carried out. 
   In step S 208 , it is determined whether or not the stability determination flag FSMCSTAB set in a process shown in  FIG. 50  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 equal to “0” in step S 208 , indicating that the adaptive sliding mode controller  21  is stable, then the control inputs Ueq, Urch, Uadp, Unl, Uwave, and Udamp calculated in steps S 202  through S 207  are added, thereby calculating the control input Usl in step S 209 . 
   If FSMCSTAB is equal to “1” in step S 208 , indicating that the adaptive sliding mode controller  21  is unstable, then 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, the nonlinear input Unl, the forced vibration input Uwave, and the damping input Udamp are not used for calculating the control input Usl, which prevents the control system from becoming unstable. 
   In steps S 211  and S 212 , it is determined whether or not the calculated control input Usl is in a range defined by a predetermined upper limit value XUSLH and a predetermined lower limit value XUSLL. If the control input Usl is in this range, then the process shown in  FIG. 40  immediately ends. If the control input Usl is equal to or less than the predetermined lower limit value XUSLL in step S 211 , then the control input Usl is set to the predetermined lower limit value XUSLL in step S 214 . If the control input Usl is equal to or greater than the predetermined upper limit value XUSLH in step S 212 , then the control input Usl is set to the predetermined upper limit value XUSLH in step S 213 . 
     FIG. 41  is a flowchart showing a process of calculating the switching function value σ which is carried out in step S 201  shown in  FIG. 40 . 
   In step S 221 , a VPOLE calculation process shown in  FIG. 42  is carried out to calculate the switching function setting parameter VPOLE. Then, the switching function value σ(k) is calculated from the equation (5b) in step S 222 . 
   In steps S 223  and  224 , it is determined whether or not the calculated switching function value σ(k) is in a range defined by a predetermined upper limit value XSGMH and a predetermined lower limit value XSGML. If the calculated switching function value σ(k) is in this range, then the process shown in  FIG. 41  immediately ends. If the calculated switching function value σ(k) is equal to or less than the predetermined lower limit value XSGML in step S 223 , then the calculated switching function value σ(k) is set to the predetermined lower limit value XSGML in step S 225 . If the calculated switching function value σ(k) is equal to or greater than the predetermined upper limit value XSGMH in step S 224 , then the calculated switching function value σ(k) is set to the predetermined upper limit value XSGMH in step S 226 . 
     FIG. 42  is a flowchart showing the VPOLE calculation process which is carried out in step S 221  shown in  FIG. 41 . 
   In step S 231  shown in  FIG. 42 , it is determined whether or not the stability determination flag FSMCSTAB is “1”. If FSMCSTAB is equal to “1” in step S 231 , indicating that the adaptive sliding mode controller  21  is unstable, then 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 greater than −1” but very close to “−1” (e.g., “−0.999”). 
   If FSMCSTAB is equal to “0”, indicating that the adaptive sliding mode controller  21  is stable, then a VPOLE table shown in  FIG. 43  is retrieved according to the throttle valve opening deviation amount DTH to calculate a switching function setting parameter VPOLE in step S 234 . The VPOLE table is set so that the switching function setting parameter VPOLE increases when the throttle valve opening deviation amount DTH takes a value in vicinity of “0”, i.e., when the throttle valve opening TH takes a value in vicinity of the default opening THDEF, and the switching function setting parameter VPOLE is substantially constant regardless of changes in the throttle valve opening deviation amount DTH when the throttle valve opening deviation amount DTH takes a value which is not in the vicinity of “0”. Therefore, when the throttle valve opening TH is in 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. 
   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 by a predetermined upper limit value XPOLEH and a predetermined lower limit value XPOLEL. If the switching function setting parameter VPOLE is in this range, then the process shown in  FIG. 42  immediately ends. If the switching function setting parameter VPOLE is equal to or less than the predetermined lower limit value XPOLEL in step S 236 , then 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 equal to or greater than the predetermined upper limit value XPOLEH in step S 235 , then the switching function setting parameter VPOLE is set to the predetermined upper limit value XPOLEH in step S 237 . 
     FIG. 44  is a flowchart showing a process of calculating the reaching law input Urch, which is carried out in step S 203  shown in  FIG. 40 . 
   In step S 261  shown in  FIG. 44 , 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, then the control gain F is set according to the switching function value σ as shown in  FIG. 13A  (Step S 262 ). 
   The reaching law input Urch is calculated according to the following equation (42), which is the same as the equation (9), in step S 263 .
 
 Urch=−Fx×σ/b 1  (42)
 
   If the stability determination flag FSMCSTAB is “1”, indicating that the adaptive sliding mode controller  21  is unstable, then 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 (43), which does not include the model parameter b 1 , in step S 265 .
 
 Urch=−F×σ   (43)
 
   In steps S 266  and S 267 , it is determined whether or not the calculated reaching law input Urch is in a range defined by a predetermined upper limit value XURCHH and a predetermined lower limit value XURCHL. If the reaching law input Urch is in this range, then the process shown in  FIG. 44  is immediately put to an end. If the reaching law input Urch is equal to or less than the predetermined lower limit value XURCHL in step S 266 , then 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 equal to or greater than the predetermined upper limit value XURCHH in step S 267 , then 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 (43) that does not include the model parameter b 1  which has become unstable, the adaptive sliding mode controller  21  can be stabilized. 
     FIG. 45  is a flowchart showing a process of calculating an adaptive law input Uadp, which is carried out in step S 204  shown in  FIG. 40 . 
   In step S 271 , it is determined whether or not the switching function value σ is equal to or less than a predetermined lower limit value−XSGMSL. If a is less than or equal to −XSGMSL, then a switching function parameter SGMS is set to the predetermined lower limit value −XSGMSL in step S 272 . If σ is greater than −XSGMSL, it is determined whether or not the switching function value σ is equal or greater than a predetermined upper limit value XSGMSL in step S 273 . If σ is greater than or equal to XSGMSL, then the switching function parameter SGMS is set to the predetermined upper limit value XSGMSL in step S 274 . If the switching function value σ falls between the predetermined lower limit value −XSGMSL and the predetermined upper limit value XSGMSL, then the switching function parameter SGMS is set to the switching function value σ in step S 275 . 
   The switching function value σ used in calculating the adaptive law input Uadp is limited in steps S 271  through S 275 . The switching function parameter SGMS is a parameter corresponding to the limited switching function value σ. The limit process makes it possible to prevent the throttle valve opening deviation amount DTH from overshooting with respect to the target value DTHR when the target value DTHR changes abruptly. 
   In step S 276 , it is determined whether or not the stability determination flag FSMCSTAB is “1”. If FSMCSTAB is equal to “0”, indicating that the adaptive sliding mode controller  21  is stable, then the control gain G is set according to the switching function value σ as shown in  FIG. 13A  in step S 279 . 
   Then, the switching function parameter SGMS and the control gain G are applied to the equation (44) shown below to calculate an adaptive law input Uadp(k) in step S 280 . The equation (44) is similar to the equation (10c) except that the switching function value σ in the equation (10c) is replaced with the switching function parameter SGMS.
 
 Uadp ( k )= Uadp ( k− 1)− G×SGMS×ΔT   1   /b 1     (44)
 
   If FSMCSTAB is equal to “1” in step S 276 , indicating that the adaptive sliding mode controller  21  is unstable, then the control gain G is set to a predetermining stabilized gain SKADPSTB in step S 277 , and an adaptive law input Uadp(k) is calculated from the equation (45) in step S 278 . The equation (45) is an equation obtained by removing the model parameter b 1  from the equation (44).
 
 Uadp ( k )= Uadp (k−1)− G×SGMS×ΔT   1   (45)
 
   In steps S 281  and  282 , it is determined whether or not the calculated adaptive law input Uadp is in a range defined by a predetermined upper limit value XUADPH and a predetermined lower limit value XUADPL. If the adaptive law input Uadp is in this range, then the process shown in  FIG. 45  immediately ends. If the adaptive law input Uadp is equal to or less than the predetermined lower limit value XUADPL in step S 282 , then the adaptive law input Uadp is set to the predetermined lower limit value XUADPL in step S 284 . If the adaptive law input Uadp is equal to or greater than the predetermined upper limit value XUADPH in step S 281 , then the adaptive law input Uadp is set to the predetermined upper limit value XUADPH in step S 283 . 
     FIG. 46  is a flowchart showing a process of calculating a nonlinear input Unl, which is carried out in step S 205  shown in  FIG. 40 . 
   In step S 301 , a nonlinear input gain Knl is calculated according to the throttle valve opening deviation amount DTH (see  FIG. 20 ). In step S 302 , it is determined whether or not the switching function value σ is equal to or less than a predetermined lower limit value −XNLTH. If σ is greater than −XNLTH, then it is determined whether the switching function value σ is equal to or greater than a predetermined upper limit value XNLTH in step S 304 . If the switching function value σ falls between the predetermined upper limit value XNLTH and the predetermined lower limit value −XNLTH, then the switching function value σ is set to a nonlinear input parameter SNL (step S 306 ). 
   If the switching function value σ is equal to or less than the predetermined lower limit value −XNLTH, then the nonlinear input parameter SNL is set to “−1” in step S 303 . If the switching function value σ is equal to or greater than the predetermined upper limit value XNLTH, then the nonlinear input parameter SNL is set to “1” in step S 305 . 
   In step S 307 , a nonlinear input Unl(k) is calculated according to the following equation (46).
 
 Unl ( k )=− Knl×SNL/b   1   (46)
 
   In the process shown in  FIG. 46 , the nonlinear input parameter SNL is used in place of the sign function sgn(σ(k)) in the equation (22), and the switching function value σ is directly applied in a predetermined range where the absolute value of the switching function value σ is small. This makes it possible to suppress the chattering due to the nonlinear input Unl. 
     FIG. 47  is a flowchart showing a process of calculating a forced vibration input Uwave which is carried out in step S 206  shown in  FIG. 40 . 
   In step S 311 , a time parameter twave(k) is calculated from the following equation (47).
 
 twave ( k )= twave ( k− 1)+ XTWAVEINC   (47)
 
where XTWAVEINC represents an elapsed time period which is set to the execution period of this process.
 
   In step S 312 , it is determined whether or not the time parameter twave(k) is equal to or greater than a predetermined period TPERIOD (e.g., 1 second). If twave(k) is less than TPERIOD, then the process proceeds to step S 314 . If twave(k) is greater than or equal to TPERIOD, then the time parameter twave(k) is reset to “0” in step S 313 . Thereafter, the process proceeds to step S 314 . 
   In step S 314 , an Fwave table shown in  FIG. 48  is retrieved according to the time parameter twave(k) to calculate a dither signal value Fwave. The waveform shown in  FIG. 48  is slightly different from the waveform shown in  FIG. 21 . The Fwave table may be set according to the waveform shown in  FIG. 21 . 
   In step S 315 , the dither input basic gain Kwave and the identifying error ide(k) are applied to the following equation (48) to calculate a dither input gain KWID (see the equation (23)).
 
 KWID=Kwave×|ide ( k )|  (48)
 
   In step S 316 , it is determined whether or not the dither input gain KWID is less than a predetermined upper limit value XKWIDL. If KWID is less than XKWIDL, the process proceeds to step S 320 . If the dither input gain KWID is equal to or greater than the predetermined upper limit value XKWIDL, then the dither input gain KWID is set to the predetermined upper limit value XKWIDL in step S 318 . 
   In step S 320 , a forced vibration input Uwave(k) is calculated according to the following equation (49), which is substantially the same as the equation (23).
 
 Uwave ( k )= KWID×Fwave/b   1   (49)
 
     FIG. 49  is a flowchart showing a process of calculating a damping input Udamp which is carried out in step S 207  shown in  FIG. 40 . 
   In step S 331 , a moving average value DTHRAV of an amount of change in the target value DTHR is calculated according to the above-described equation (29). In step S 332 , a basic value Kdampbs of a damping control gain is calculated according to the throttle valve opening deviation amount DTH (see  FIG. 25A ). In step S 333 , a correction coefficient Kkdamp of a damping control gain is calculated according to the moving average value DDTHRAV in step S 333  (see  FIG. 25B ). 
   In step S 334 , a damping control gain Kdamp is calculated by multiplying the basic value Kdampbs by the correction coefficient Kkdamp. Then, a damping input Udamp(k) is calculated according to the following equation (27) (shown again).
 
 Udamp ( k )=− Kdamp ×( DTH ( k )− DTH ( k− 1))/ b   1   (27)
 
     FIG. 50  is a flowchart showing a process of stability determination of the sliding mode controller, which is carried out in step S 20  shown in  FIG. 30 . In this process, the stability is determined based on the differential of a Lyapunov function, and the stability determination flag FSMCSTAB is set according to the result of the stability determination. 
   In step S 351 , a switching function change amount Dσ is calculated from the following equation (50). A stability determining parameter SGMSTAB is calculated from the following equation (51) in step S 352 .
 
 Dσ =σ( k )−σ( k−k 0 )   (50)
 
 SGMSTAB=D σ×σ( k )  (51)
 
   In step S 353 , it is determined whether or not the stability determining parameter SGMSTAB is equal to or less than a stability determining threshold XSGMSTAB. If SGMSTAB is greater than XSGMSTAB, then 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 355 . If SGMSTAB is less than or equal to XSGMSTAB, then 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 354 . 
   In step S 356 , it is determined whether or not the value of the unstability detecting counter CNTSMCST is equal to or less than a predetermined count XSSTAB. If CNTSMCST is less than or equal to XSSTAB, then 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 357 . If CNTSMCST is greater than XSSTAB, then 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 358 . The value of the unstability detecting counter CNTSMCST is initialized to “0” when the ignition switch is turned on. 
   In step S 359 , a stability determining period counter CNTJUDST is decremented by “1”. It is then determined whether or not the value of the stability determining period counter CNTJUDST is “0” in step S 360 . 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 360  is negative (NO), and the process immediately goes to step S 365 . 
   If the value of the stability determining period counter CNTJUDST subsequently becomes “0”, then the process goes from step S 360  to step S 361 , 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”, then a second determination flag FSMCSTAB 2  is set to “0” in step S 363 . If the first determination flag FSMCSTAB 1  is “1”, then the second determination flag FSMCSTAB 2  is set to “1” in step S 362 . 
   In step S 364 , 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 365 . 
   In step S 365 , the stability determination flag FSMCSTAB is set to the logical sum of the first determination flag FSMCSTAB 1  and the second determination flag FSMCSTAB2. 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 356  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”. 
   In the present embodiment, the throttle valve actuating device  10  and a portion of the ECU  7 , i.e., the output circuit for supplying an energizing current to the motor  6 , correspond to a plant, and the ECU  7  constitutes a sliding mode controller, an identifying means, and a damping input calculating means. More specifically, step S 19  shown in  FIG. 30 , i.e., the process shown in  FIG. 40 , corresponds to the sliding mode controller. Steps S 12  through S 18  shown in  FIG. 30  correspond to the identifying means. The process shown in  FIG. 49  corresponds to the damping input calculating means. 
   Second Embodiment 
     FIG. 51  is a diagram showing the configuration of a hydraulic positioning device and its control system, which is a control system for a plant according to a second embodiment of the present invention. Such a hydraulic positioning device can be used for a continuously variable valve timing mechanism for continuously varying the valve timing of the intake and exhaust valves. The continuously variable valve timing mechanism changes rotational phases of the cams for driving the intake and exhaust valves to shift the opening/closing timing of the intake and exhaust valves, which improves the charging efficiency of the engine and reduces the pumping loss of the engine. 
   The hydraulic positioning device includes a piston  64 , a hydraulic cylinder  61  in which the piston  64  is fitted, an electrically-driven spool valve  67 , a hydraulic pump  65 , an oil pressure supply line  66  for supplying an oil pressure from the hydraulic pump  65  to the electrically-driven spool valve  67 , a first oil passage  68  for supplying a first oil pressure P 1  to a first oil pressure chamber  62  of the hydraulic cylinder  61 , a second oil passage  69  for supplying a second oil pressure P 2  to a second oil pressure chamber  63  of the hydraulic cylinder  61 , and an oil pressure release line  70  for returning hydraulic oil discharged from the electrically-driven spool valve  67  to an oil pan (not shown). 
   A potentiometer  71  is provided for detecting a position PACT of the piston  64 , and a signal indicating the detected position PACT is supplied to an electronic control unit (ECU)  72 . 
   A target position PCMD is input to the ECU  72 . The ECU  72  calculates a control quantity DUT so that the detected position PACT coincides with the target position PCMD, and supplies an electrical signal according to the control quantity DUT to the electrically-driven spool valve  67 . 
   The electrically-driven spool valve  67  moves the position of a valve element (not shown) according to the control quantity DUT, and outputs the first and second oil pressure P 1  and P 2  according to the position of the valve element. When the pressure difference DP (=P 1 −P 2 ) between the first and second oil pressures P 1  and P 2  is a positive value, the piston  64  moves to the right as viewed in  FIG. 51 . When the pressure difference DP is a negative value, the piston  64  moves to the left as viewed in  FIG. 51 . In the condition where the detected position PACT coincides with the target position PCMD, the pressure difference DP is maintained at “0”. 
     FIG. 52  is a block diagram showing a control system for controlling the hydraulic positioning device shown in  FIG. 51  with an adaptive sliding mode controller. 
   The control system  80  includes an identifier  81 , an adaptive sliding mode controller  82 , a scheduler  83 , and subtractors  85 ,  86 . The control system  80  is realized by processes which are carried out by a CPU included in the ECU  72 . 
   The subtractor  85  subtracts a reference value PBASE from the detected position PACT to calculate a detected position deviation amount DPACT. The subtractor  86  subtracts the reference value PBASE from the target position PCMD to calculate a target value DPCMD. The reference value PBASE is preset to an optimum value based on the operating characteristics of the hydraulic positioning device. 
   The detected position PACT and the detected position deviation amount DPACT in the present embodiment correspond respectively to the throttle opening TH and the throttle valve opening deviation amount DTH in the first embodiment. The target position PCMD and the target value DPCMD in the present embodiment correspond respectively to the target opening THR and the target value DTHR in the first embodiment. 
   The scheduler  83 , similarly to the model parameter scheduler  25  in the first embodiment, calculates a reference model parameter vector θbase according to the target value DPCMD, and supplies the reference model parameter vector θbase to the identifier  81 . 
   The identifier  81 , similarly to the model parameter identifier  22  in the first embodiment, calculates a corrected model parameter vector θL(k) according to the control quantity DUT as a control input and the detected position deviation amount DPACT as a control output. Specifically, the identifier  81  calculates an identifying error ide(n) from the equations (52) and (53) shown below. An input/output parameter vector ζ(n) is defined from the equation (54) shown below.
 
 ide ( n )= DPACT ( n )( k )− DPACTHAT ( n )  (52)
 
 DPACTHAT ( n )=θ*( n− 1) T ζ( n )  (53)
 
ζ T ( n )=[ DPACT ( n− 1),  DPACT ( n− 2),  DUT ( n− 1),  1 ]  (54)
 
   The identifying error ide(n) is applied to the equation (30), and the equations (14f), (14g), (19b), and (33) are used to calculate a model parameter vector θ(n). The calculated model parameter vector θ(n) is subjected to a first limit process, which is similar to the first limit process in the first embodiment, to calculate a model parameter vector θ*(n). The model parameter vector θ*(n) is oversampled and moving-averaged to calculate a model parameter vector θ′(k). The model parameter vector θ′(k) is subjected to a second limit process, which is similar to the second limit process in the first embodiment, to calculate a corrected model parameter vector θL(k). 
   The adaptive sliding mode controller  82 , similarly to the adaptive sliding mode controller  21  in the first embodiment, applies the detected position deviation amount DPACT to the equation (55) shown below to calculate an equivalent control input Ueq(k). The adaptive sliding mode controller  82  calculates a switching function value σ(k) from the equation (56) shown below, and applies the switching function value σ(k) to the equations (9) and (10c) described above to calculate a reaching law input Urch(k) and an adaptive law input Uadp(k). A switching function setting parameter VPOLE and control gains F and G are set to values suitable for the controlled object in the present embodiment, i.e., the hydraulic positioning device. 
                     Ueq   ⁡     (   k   )       =       ⁢       (     1   /   b1     )     ⁢     {         (     1   -   a1   -   VPOLE     )     ⁢     DPACT   ⁡     (   k   )         +                         ⁢         (     VPOLE   -   a2     )     ⁢     DPACT   ⁡     (     k   -   k0     )         -   c1     }                 (   55   )                       σ   ⁡     (   k   )       =       ⁢       (       DPACT   ⁡     (   k   )       -     DPCMD   ⁡     (   k   )         )     +                     ⁢     VPOLE   ⁡     (       DPACT   ⁡     (     k   -   k0     )       -     DPCMD   ⁡     (     k   -   k0     )         )                     (   56   )             
 
   The adaptive sliding mode controller  82  applies the switching function value σ(k) calculated from the equation (56) to the above equation (22) to calculate a nonlinear input Unl(k). A nonlinear input gain Knl is set a value suitable for the controlled object in the present embodiment. 
   The adaptive sliding mode controller 82 applies the identifying error ide(n) calculated from the equation (52) to the above-described equation (23) to calculate a forced vibration input Uwave. A dither input basic gain Kwave and a dither signal value Fwave are set to values suitable for the controlled object in the present embodiment. 
   The adaptive sliding mode controller  82  calculates a damping input Udamp(k) from the equation (57) shown below. A damping control gain Kdamp is set to a value suitable for the controlled object in the present embodiment.
 
 Udamp ( k )=− Kdamp ×( DACT ( k )− DACT ( k− 1))/ b   1   (57)
 
   The adaptive sliding mode controller  82  adds the equivalent control input Ueq(k), the reaching law input Urch(k), the adaptive law input Uadp(k), the nonlinear input Unl(k), the forced vibration input Uwave(k), and the damping input Udamp(k) to thereby calculate a control input Usl (=DUT). 
   Since the control system  80  performs a control process in which the control output TH and the target opening THR in the first embodiment are replaced respectively by the control output PACT and the target position PCMD, the control output PACT is controlled to follow up the target position PCMD with good robustness as similar to the first embodiment. 
   According to the present embodiment, the hydraulic positioning device shown in  FIG. 52  corresponds to a plant, and the ECU  72  constitutes a sliding mode controller, an identifying means, and a damping input calculating means in the claimed invention. 
   The present invention is not limited to the above embodiments, but various modifications may be made. For example, while the hydraulic positioning device is shown in the second embodiment, the control process carried out by the control system  80  in the second embodiment may be applied to a pneumatic positioning device which uses pneumatic pressure instead of hydraulic pressure. 
   The response-specifying controller that performs a feedback control to make an output of a controlled object coincide with a target value and specifies the damping characteristic of a control deviation of the feedback control process, is not limited to an adaptive sliding mode controller. A controller for performing a back stepping control which realizes control results similar to those of the sliding mode control, may be used as a response-specifying controller. 
   In the above embodiments, the period of the calculation for identifying model parameters is set to a period which is equal to the second period ΔT 2 . However, the period of the calculation for identifying model parameters may not necessarily be set to the same period as the second period ΔT 2 , but may be set to a period between the first period ΔT 1  and the second period ΔT 2 , or a period which is longer than the second period ΔT 2 . 
   In the above embodiments, the parameter k 0  indicative of the sampling time interval for the deviation e(k) involved in the calculation of the switching function value σ is set to ΔT 2 /ΔT 1  which is a discrete time corresponding to the second period ΔT 2 . Alternatively, the parameter k 0  may be set to another integer which is greater than “1”. 
   The present invention may be embodied in other specific forms without departing from the spirit or essential characteristics thereof. The presently disclosed embodiments are therefore to be considered in all respects as illustrative and not restrictive, the scope of the invention being indicated by the appended claims, rather than the foregoing description, and all changes which come within the meaning and range of equivalency of the claims are, therefore, to be embraced therein.