US 20030009240 A1 Abstract A control system for a plant which controls the plant using an identifier and a sliding mode controller. The identifier identifies a model parameter vector of a controlled object model of the plant. The sliding mode controller controls the plant using the model parameter vector identified by the identifier. According to the identifier, an identifying error of the model parameter vector is calculated and an updating vector is calculated according to the identifying error. The updating vector is corrected by multiplying a past value of at least one element of the updating vector by a predetermined value which is greater than “0” and less than “1”. The model parameter vector is calculated by adding the corrected updating vector to a reference vector of the model parameter vector.
Claims(45) 1. A control system for a plant, comprising:
identifying means for identifying a model parameter vector of a controlled object model which is obtained by modeling said plant, based on an input and an output of said plant; and a sliding mode controller for controlling said plant using the model parameter vector identified by said identifying means; said identifying means comprising:
identifying error calculating means for calculating an identifying error of the model parameter vector;
updating vector calculating means for calculating an updating vector according to said identifying error; and
updating vector correcting means for correcting said updating vector by multiplying a past value of at least one element of said updating vector by a predetermined value which is greater than 0 and less than 1;
wherein said identifying means calculates the model parameter vector by adding the corrected updating vector to a reference vector of the model parameter vector.
2. A control system according to 3. A control system according to 4. A control system according to 5. A control system according to 6. A control system according to 7. A control system according to 8. A control system according to 9. A control system according to 10. A control system for a plant, comprising:
identifying means for identifying a model parameter vector of a controlled object model which is obtained by modeling said plant, based on an input and an output of said plant; and a sliding mode controller for controlling said plant using the model parameter vector identified by said identifying means; said identifying means comprising:
identifying error calculating means for calculating an identifying error of the model parameter vector; and
identifying error correcting means for correcting the identifying error in a decreasing direction, if the identifying error is in a predetermined range;
wherein said identifying means calculates the model parameter vector using the identifying error corrected by said identifying error correcting means.
11. A control system according to 12. A control system according to 13. A control system according to 14. A control system according to 15. A control system according to 16. A control method for a plant, comprising the steps of:
a) identifying a model parameter vector of a controlled object model which is obtained by modeling said plant, based on an input and an output of said plant; and b) controlling said plant with a sliding mode control using the identified model parameter vector; said step a) of identifying the model parameter vector, comprising the steps of:
i) calculating an identifying error of the model parameter vector;
ii) calculating an updating vector according to said identifying error; and
iii) correcting the updating vector by multiplying a past value of at least one element of the updating vector by a predetermined value which is greater than 0 and less than 1;
wherein the model parameter vector is calculated by adding the corrected updating vector to a reference vector of the model parameter vector.
17. A control method according to 18. A control method according to 19. A control method according to 20. A control method according to 21. A control method according to 22. A control method according to 23. A control method according to 24. A control method according to 25. A control method for a plant, comprising the steps of:
a) identifying a model parameter vector of a controlled object model which is obtained by modeling said plant, based on an input and an output of said plant; and b) controlling said plant with a sliding mode control using the model parameter vector identified by said identifying means; said step a) comprising the steps of:
i) calculating an identifying error of the model parameter vector; and
ii) correcting the identifying error in a decreasing direction, if the identifying error is in a predetermined range;
wherein the model parameter vector is calculated using the corrected identifying error.
26. A control method according to 27. A control method according to 28. A control method according to 29. A control method according to 30. A control method according to 31. A control system for a plant, comprising:
an identifying module for identifying a model parameter vector of a controlled object model which is obtained by modeling said plant, based on an input and an output of said plant; and a sliding mode controller for controlling said plant using the model parameter vector identified by said identifying module; said identifying module comprising:
an identifying error calculating module for calculating an identifying error of the model parameter vector;
an updating vector calculating module for calculating an updating vector according to said identifying error; and
an updating vector correcting module for correcting said updating vector by multiplying a past value of at least one element of said updating vector by a predetermined value which is greater than 0 and less than 1;
wherein said identifying module calculates the model parameter vector by adding the corrected updating vector to a reference vector of the model parameter vector.
32. A control system according to 33. A control system according to 34. A control system according to 35. A control system according to 36. A control system according to 37. A control system according to 38. A control system according to 39. A control system according to 40. A control system for a plant, comprising:
an identifying module for identifying a model parameter vector of a controlled object model which is obtained by modeling said plant, based on an input and an output of said plant; and a sliding mode controller for controlling said plant using the model parameter vector identified by said identifying module; said identifying module comprising:
an identifying error calculating module for calculating an identifying error of the model parameter vector; and
an identifying error correcting module for correcting the identifying error in a decreasing direction, if the identifying error is in a predetermined range;
wherein said identifying module calculates the model parameter vector using the identifying error corrected by said identifying error correcting module.
41. A control system according to 42. A control system according to 43. A control system according to 44. A control system according to 45. A control system according to Description [0001] 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. [0002] One known control system based on a sliding mode control theory is disclosed in Japanese Patent Laid-open No. Hei 9-274504, for example. The publication proposes a method of setting a hyperplane in the sliding mode control theory according to the convergence state of a controlled state quantity. According to the proposed method, the convergence response and convergence stability of the sliding mode control is improved. [0003] For controlling a plant, which is a controlled object, with the sliding mode controller, it is necessary to produce a model of the plant and determine model parameters representing the characteristics of the model of the plant (i.e. the controlled object). The model parameters may be set to predetermined constant values. However, the values of the model parameters usually change due to aging and disturbance. Therefore, it is desirable to use a model parameter identifier for identifying the model parameters on a real-time basis and carry out the sliding mode control using the model parameters that are identified by the model parameter identifier. [0004] The model parameter identifier detects an identifying error which is a difference between the output of the plant which is calculated using the identified model parameters and the actual output of the plant, and corrects the model parameters in order to eliminate the identifying error. Therefore, regarding the model parameter identifier, the following problems may occur. [0005] Due to nonlinear characteristics and disturbance whose average value is not “0”, the identifying error does not become “0” even though substantially optimum model parameters have actually been obtained. Therefore, the model parameters which do not need to be corrected are occasionally corrected. As a result, a drift occurs in which the values of the model parameters gradually shift from their optimum values to some other values to make the control performed by the sliding mode controller unstable. [0006] It is therefore an object of the present invention to provide a control system for a plant, which can control the plant more stably when the model parameters of the controlled object model which are obtained by modeling the plant, which is a controlled object, are identified and the sliding mode control is performed using the identified model parameters. [0007] To achieve the above object, the present invention provides a control system for a plant, comprising identifying means and a sliding mode controller. The identifying means identifies a model parameter vector (θ) of a controlled object model of a plant which is obtained by modeling the plant. The sliding mode controller controls the plant using the model parameter vector identified by the identifying means. [0008] The identifying means comprises an identifying error calculating means, an updating vector calculating means, and an updating vector correcting means. The identifying error calculating means calculates an identifying error (ide) of the model parameter vector. The updating vector calculating means calculates an updating vector (dθ) according to the identifying error. The updating vector correcting means corrects the updating vector by multiplying a past value of at least one element of the updating vector by a predetermined value (DELTAi, EPSi) which is greater than “0” and less than “1”. The identifying means calculates the model parameter vector by adding the corrected updating vector to a reference vector (θbase, θ(0)) of the model parameter vector. [0009] With this configuration, the updating vector is calculated according to the identifying error of the model parameter vector, and corrected by multiplying the past value of at least one element of the updating vector by the predetermined value which is greater than “0” and less than “1”. The corrected updating vector is added to the reference vector of the model parameter vector to calculate the model parameter vector. Accordingly, values of the elements of the updating vector are limited, thus stabilizing the model parameter vector in the vicinity of the reference vector. As a result, the drift of the model parameters is prevented, to thereby improve the stability of the sliding mode control performed by the sliding mode controller. [0010] Preferably, the updating vector correcting means does not multiply one of an element of the updating vector which is relevant to the input of the plant (an element relevant to the calculation of b1) and an element of the updating vector which is irrelevant to the input and the output of the plant (an element related to the calculation of c1), by the predetermined value. The parameter b1 is an element of the model parameter vector which is multiplied to the input of the plant. The parameter c1 is an element of the model parameter vector which is irrelevant to the input and output of the plant. [0011] With this configuration, none of the element of the updating vector which is relevant to the input of the plant and the element of the updating vector which is irrelevant to the input and the output of the plant is multiplied by the predetermined value which is greater than “0” and less than “1”. The steady-state deviation which may be caused by correcting these elements can be prevented from occurring. [0012] Preferably, the updating vector correcting multiplies at least one element of the reference vector (θ(0)) by the predetermined value (EPSi). With this configuration, at least one element of the reference vector is multiplied by the predetermined value to calculate the model parameter vector. Also in this case, the drift of the model parameters is prevented, to thereby improve the stability of the sliding mode control performed by the sliding mode controller. [0013] The present invention further provides a control system for a plant, comprising an identifying means and a sliding mode controller. The identifying means identifies a model parameter vector of a controlled object model which is obtained by modeling the plant, based on an input and an output of the plant. The sliding mode controller controls the plant using the model parameter vector identified by said identifying means. The identifying means comprises identifying error calculating means and identifying error correcting means. The identifying error calculating means calculates an identifying error (ide) of the model parameter vector. The identifying error correcting means corrects the identifying error in a decreasing direction, if the identifying error is in a predetermined range (−EIDNRLM≦ide≦EIDNRLMT). The identifying means calculates the model parameter vector using the identifying error (idenl) corrected by the identifying error correcting means. [0014] With this configuration, if the identifying error is in the predetermined range, the identifying error is corrected in a decreasing direction, and the model parameter vector is calculated using the corrected identifying error. Therefore, the identifying error is suppressed to less accumulate in the model parameters, that is, the drift of the model parameters is prevented, to thereby improve the stability of the sliding mode control carried out by the sliding mode controller. [0015] Preferably, the identifying error correcting means sets the identifying error to “0”, if the identifying error is in the predetermined range. With this configuration, if the identifying error is in the predetermined range, the identifying error is set to “0”. Therefore, the effect of the identifying error which is not to be reflected to the values of model parameters is eliminated, resulting in an increased effect of preventing the model parameters from drifting. [0016] Preferably the predetermined range is set according to an amount (DDTHR) of change in a control target value or an amount (DTH) of change in the output of the plant. With this configuration, the predetermined range is set according to the amount of change in the control target value or the amount of change in the output of the plant. Therefore, it is prevented that the identifying error which is to be reflected to the values of model parameters is reduced or neglected as an unnecessary error. [0017] Preferably, the identifying means identifies the model parameter vector according to a fixed gain algorithm. With this configuration, the model parameter vector is calculated according to the fixed gain algorithm. Accordingly, the amount of calculations for identifying the model parameters can be reduced. [0018] Preferably, the identifying error calculating means performs a low-pass filtering of the identifying error and outputs the identifying error after the low-pass filtering. With this configuration, the model parameter vector is identified using the identifying error which has been processed by the low-pass filtering. Accordingly, the frequency characteristics of the controlled object model are made closer to the actual frequency characteristics of the plant, to thereby improve the robustness and stability of the control carried out by the sliding mode controller. [0019] Preferably, the control system further comprises predicting means for calculating a predicted value (PREDTH) of the output of the plant. With this configuration, the predicted value of the output of the plant is calculated by the predicting means. Therefore, the plant which has a dead time element can be controlled accurately. [0020] Preferably, the predicting means calculates the predicted value using the model parameter vector identified by the identifying means. With this configuration, the predicted value is calculated using the model parameter vector identified by the identifying means. Therefore, it is possible to calculate an accurate predicted value even when the dynamic characteristics of the plant changes with time or changes according to environmental conditions. [0021] Preferably, a control input applied from the adaptive sliding mode controller to the plant includes an adaptive law input. With this configuration, the control input applied to the plant includes the adaptive law input. Accordingly, good controllability can be obtained even in the presence of disturbance and/or a modeling error, which is a difference between the characteristics of the actual plant and the characteristics of the controlled object model. [0022] More specifically, the plant includes a throttle valve actuating device having a throttle valve of an internal combustion engine and actuating means for actuating the throttle valve, and the sliding mode controller calculates a parameter for determining a control input to be applied to the throttle valve actuating device to make an opening of the throttle valve coincide with a target opening. With this configuration, the sliding mode controller performs the control to make the opening of the throttle valve coincide with the target opening, using stable model parameters identified by the identifying means. Consequently, the controllability of the throttle valve opening to the target opening can be improved, and the throttle valve opening can be controlled more stably. [0023] 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. [0024]FIG. 1 is a schematic view of a throttle valve control system according to a first embodiment of the present invention; [0025]FIGS. 2A and 2B are diagrams showing frequency characteristics of the throttle valve actuating device shown in FIG. 1; [0026]FIG. 3 is a functional block diagram showing functions realized by an electronic control unit (ECU) shown in FIG. 1; [0027]FIG. 4 is a diagram showing the relationship between control characteristics of a sliding mode controller and the value of a switching function setting parameter (VPOLE); [0028]FIG. 5 is a diagram showing a range for setting control gains (F, G) of the sliding mode controller; [0029]FIGS. 6A and 6B are diagrams illustrative of a drift of model parameters; [0030]FIGS. 7A through 7C are diagrams showing functions for correcting an identifying error; [0031]FIG. 8 is a diagram illustrating that a default opening deviation of a throttle valve is reflected to a model parameter (c1′); [0032]FIG. 9 is a flowchart showing a throttle valve opening control process; [0033]FIG. 10 is a flowchart showing a process of setting state variables in the process shown in FIG. 9; [0034]FIG. 11 is a flowchart showing a process of performing calculations of a model parameter identifier in the process shown in FIG. 9; [0035]FIG. 12 is a flowchart showing a process of calculating an identifying error (ide) in the process shown in FIG. 11; [0036]FIGS. 13A and 13B are diagrams illustrative of a process of low-pass filtering on the identifying error (ide); [0037]FIG. 14 is a flowchart showing the dead zone process in the process shown in FIG. 12; [0038]FIG. 15 is a diagram showing a table used in the process shown in FIG. 14; [0039]FIG. 16 is a flowchart showing a process of stabilizing a model parameter vector (θ) in the process shown in FIG. 11; [0040]FIG. 17 is a flowchart showing a limit process of model parameters (a1′, a2′) in the process shown in FIG. 16; [0041]FIG. 18 is a diagram illustrative of the change in the values of the model parameters in the process shown in FIG. 16; [0042]FIG. 19 is a flowchart showing a limit process of a model parameter (b1′) in the process shown in FIG. 16; [0043]FIG. 20 is a flowchart showing a limit process of a model parameter (c1′) in the process shown in FIG. 16; [0044]FIG. 21 is a flowchart showing a process of performing calculations of a state predictor in the process shown in FIG. 9; [0045]FIG. 22 is a flowchart showing a process of calculating a control input (Usl) in the process shown in FIG. 9; [0046]FIG. 23 is a flowchart showing a process of calculating a predicted switching function value (σpre) in the process shown in FIG. 22; [0047]FIG. 24 is a flowchart showing a process of calculating the switching function setting parameter (VPOLE) in the process shown in FIG. 23; [0048]FIGS. 25A through 25C are diagrams showing maps used in the process shown in FIG. 24; [0049]FIG. 26 is a flowchart showing a process of calculating an integrated value of the predicted switching function value (σpre) in the process shown in FIG. 22; [0050]FIG. 27 is a flowchart showing a process of calculating a reaching law input (Urch) in the process shown in FIG. 22; [0051]FIG. 28 is a flowchart showing a process of calculating an adaptive law input (Uadp) in the process shown in FIG. 22; [0052]FIG. 29 is a flowchart showing a process of determining the stability of the sliding mode controller in the process shown in FIG. 9; [0053]FIG. 30 is a flowchart showing a process of calculating a default opening deviation (thdefadp) in the process shown in FIG. 9; [0054]FIG. 31 is a functional block diagram showing functions realized by the electronic control unit (ECU) shown in FIG. 1 according to a second embodiment of the present invention; [0055]FIG. 32 is a flowchart showing a throttle valve opening control process according to the second embodiment; [0056]FIG. 33 is a flowchart showing a process of performing calculations of a model parameter identifier in the process shown in FIG. 32; [0057]FIG. 34 is a diagram showing a table used in the process shown in FIG. 33; [0058]FIG. 35 is a flowchart showing a process of calculating an identifying error (ide) in the process shown in FIG. 33; [0059]FIG. 36 is a flowchart showing a process of calculating a control input (Usl) in the process shown in FIG. 32; [0060]FIG. 37 is a flowchart showing a process of calculating a switching function value (σ) in the process shown in FIG. 36; [0061]FIG. 38 is a flowchart showing a process of calculating an integrated value of the switching function value (σ) in the process shown in FIG. 36; [0062]FIG. 39 is a flowchart showing a process of calculating a reaching law input (Urch) in the process shown in FIG. 36; [0063]FIG. 40 is a flowchart showing a process of calculating an adaptive law input (Uadp) in the process shown in FIG. 36; [0064]FIG. 41 is a flowchart showing a process of determining the stability of a sliding mode controller in the process shown in FIG. 32; [0065]FIG. 42 is a block diagram of a control system according to a third embodiment of the present invention; [0066]FIG. 43 is a block diagram of a modification of the control system shown in FIG. 42; [0067]FIG. 44 is a block diagram of a control system according to a fourth embodiment of the present invention; and [0068]FIG. 45 is a block diagram of a modification of the control system shown in FIG. 44. [0069] First Embodiment [0070]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”) [0071] The motor [0072] Further, the ECU [0073] The ECU [0074] In the present embodiment, a throttle valve actuating device [0075] When frequency response characteristics of the throttle valve actuating device [0076] where k is a parameter representing discrete time, and DTH(k) is a throttle valve opening deviation amount defined by the equation (2) shown below. DTH(k+1) is a throttle valve opening deviation amount at a discrete time (k+1). [0077] where TH is a detected throttle valve opening, and THDEF is the default opening. [0078] In the equation (1), a1, a2, b1, and c1 are parameters determining the characteristics of the controlled object model, and d is a dead time. The dead time is a delay between the input and output of the controlled object model. [0079] The model defined by the equation (1) is a DARX model (delayed autoregressive model with exogeneous input) of a discrete time system, which is employed for facilitating the application of an adaptive control. [0080] In the equation (1), the model parameter c1 which is irrelevant to the input and output of the controlled object, in addition to the model parameters a1 and a2 which are relevant to the output deviation amount DTH and the model parameter b1 which is relevant to the input duty ratio DUT. The model parameter c1 is a parameter representing a deviation amount of the default opening THDEF and disturbance applied to the throttle valve actuating device [0081]FIG. 3 is a functional block diagram of the throttle valve control system which is realized by the ECU [0082] The adaptive sliding mode controller [0083] By using the adaptive sliding mode controller [0084] The model parameter identifier [0085] By using the model parameter identifier [0086] The state predictor [0087] Next, principles of operation of the adaptive sliding mode controller [0088] First, a target value DTHR(k) is defined as a deviation amount between the target opening THR(k) and the default opening THDEF by the following equation (3). [0089] 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). σ( [0090] where VPOLE is a switching function setting parameter that is set to a value which is greater than −1 and less than 1. [0091] On a phase plane defined by a vertical axis representing the deviation e(k) and a horizontal axis representing the preceding deviation e(k−1), a pair of the deviation e(k) and the preceding deviation e(k−1) satisfying the equation of “σ(k)=0” represents a straight line. The straight line is generally referred to as a switching straight line. A sliding mode control is a control contemplating the behavior of the deviation e(k) on the switching straight line. The sliding mode control is carried out so that the switching function value σ(k) becomes 0, i.e., the pair of the deviation e(k) and the preceding deviation e(k−1) exists on the switching straight line on the phase plane, to thereby achieve a robust control against disturbance and the modeling error (the difference between the characteristics of an actual plant and the characteristics of a controlled object model). As a result, the throttle valve opening deviation amount DTH is controlled with good robustness to follow up the target value DTHR. [0092] As shown in FIG. 4, by changing the value of the switching function setting parameter VPOLE in the equation (5), it is possible to change damping characteristics of the deviation e(k), i.e., the follow-up characteristics of the throttle valve opening deviation amount DTH to follow the target value DTHR. Specifically, if VPOLE equals −1, then the throttle valve opening deviation amount DTH completely fails to follow up the target value DTHR. As the absolute value of the switching function setting parameter VPOLE is reduced, the speed at which the throttle valve opening deviation amount DTH follows up the target value DTHR increases. [0093] The throttle valve control system is required to satisfy the following requirements A1 and A2: [0094] A1) When the throttle valve [0095] A2) The controllability with respect to the nonlinear characteristics in the vicinity of the default opening THDEF (a change in the resiliency characteristics due to the equilibrium between the energizing force of the return spring [0096] Therefore, it is necessary to lower the speed at which the deviation e(k) converges, i.e., the converging speed of the deviation e(k), in the vicinity of the fully closed position of the throttle valve, and to increase the converging speed of the deviation e(k) in the vicinity of the default opening THDEF. [0097] According to the sliding mode control, the converging speed of e(k) can easily be changed by changing the switching function setting parameter VPOLE. Therefore in the present embodiment, the switching function setting parameter VPOLE is set according to the throttle valve opening TH and an amount of change DDTHR (=DTHR(k)−DTHR(k−1)) of the target value DTHR, to thereby satisfy the requirements A1 and A2. [0098] As described above, according to the sliding mode control, the deviation e(k) is converged to 0 at an indicated converging speed and robustly against disturbance and the modeling error by constraining the pair of the deviation e(k) and the preceding deviation e(k−1) on the switching straight line (the pair of e(k) and e(k−1) will be hereinafter referred to as “deviation state quantity”). Therefore, in the sliding mode control, it is important how to place the deviation state quantity onto the switching straight line and constrain the deviation state quantity on the switching straight line. [0099] From the above standpoint, an input DUT(k) (also indicated as Usl(k)) to the controlled object (an output of the controller) is expressed as the sum of an equivalent control input Ueq(k), a reaching law input Urch(k), and an adaptive law input Uadp(k), as indicated by the following equation (6). [0100] The equivalent control input Ueq(k) is an input for constraining the deviation state quantity on the switching straight line. The reaching law input Urch(k) is an input for placing deviation state quantity onto the switching straight line. The adaptive law input Uadp(k) is an input for placing deviation state quantity onto the switching straight line while reducing the modeling error and the effect of disturbance. Methods of calculating these inputs Ueq(k), Urch(k), and Uadp(k) will be described below. [0101] 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( [0102] Using the equations (1), (4), and (5), the duty ratio DUT(k) satisfying the equation (7) is determined by the equation (9) shown below. The duty ratio DUT(k) calculated with the equation (9) represents the equivalent control input Ueq(k). The reaching law input Urch(k) and the adaptive law input Uadp(k) are defined by the respective equations (10) and (11) shown below.
[0103] where F and G respectively represent a reaching law control gain and an adaptive law control gain, which are set as described below, and ΔT represents a control period. [0104] Calculating the equation (9) requires a throttle valve opening deviation amount DTH(k+d) after the elapse of the dead time d and a corresponding target value DTHR(k+d+1). Therefore, the predicted deviation amount PREDTH(k) calculated by the state predictor [0105] Next, the reaching law control gain F and the adaptive law control gain G are determined so that the deviation state quantity can stably be placed onto the switching straight line by the reaching law input Urch and the adaptive law input Uadp. [0106] Specifically, a disturbance V(k) is assumed, and a stability condition for keeping the switching function value σ(k) stable against the disturbance V(k) are determined to obtain a condition for setting the gains F and G. As a result, it has been obtained as the stability condition that the combination of the gains F and G satisfies the following equations (12) through (14), in other words, the combination of the gains F and G should be located in a hatched region shown in FIG. 5. F>0 (12) G>0 (13) [0107] As described above, the equivalent control input Ueq(k), the reaching law input Urch(k), and the adaptive law input Uadp(k) are calculated from the equations (9) through (11), and the duty ratio DUT(k) is calculated as the sum of those inputs. [0108] The model parameter identifier θ( σ( [0109] where a1′, a2′, b1′, c1′ represent model parameters before a limit process described later is carried out, ide(k) represents an identifying error defined by the equations (17), (18), and (19) shown below, where DTHHAT(k) represents an estimated value of the throttle valve opening deviation amount DTH(k) (hereinafter referred to as “estimated throttle valve opening deviation amount”) which is calculated using the latest model parameter vector θ(k−1), and KP(k) represents a gain coefficient vector defined by the equation (20) shown below. In the equation (20), P(k) represents a quartic square matrix calculated from the equation (21) shown below. ζ( [0110] In accordance with the setting of coefficients λ1 and λ2 in the equation (21), the identifying algorithm from the equations (15) through (21) becomes one of the following four identifying algorithm: [0111] λ1=1, λ2=0 [0112] Fixed gain algorithm [0113] λ1=1, λ2=1 [0114] Method-of-least-squares algorithm [0115] λ1=1, λ2=λ [0116] Degressive gain algorithm [0117] (λ is a given value other than 0, 1) [0118] λ1=λ, λ2=1 [0119] Weighted Method-of-least-squares algorithm [0120] (λ is a given value other than 0, 1) [0121] In the present embodiment, it is required that the following requirements B1, B2, and B3 are satisfied: [0122] B1) Adaptation to Quasi-Static Dynamic Characteristics Changes and Hardware Characteristics Variations [0123] “Quasi-static dynamic characteristics changes” mean slow-rate characteristics changes such as power supply voltage fluctuations or hardware degradations due to aging. [0124] B2) Adaptation to High-Rate Dynamic Characteristics Changes [0125] Specifically, this means adaptation to dynamic characteristics changes depending on changes in the throttle valve opening TH. [0126] B3) Prevention of a Drift of Model Parameters [0127] The drift, which is an excessive increase of the absolute values of model parameters, should be prevented. The drift of model parameters is caused by the effect of the identifying error, which should not be reflected to the model parameters, due to nonlinear characteristics of the controlled object. [0128] In order to satisfy the requirements B1 and B2, the coefficients λ1 and λ2 are set respectively to a given value λ and “0” so that the weighted Method-of-least-squares algorithm is employed. [0129] Next, the drift of model parameters will be described below. As shown in FIG. 6A and FIG. 6B, if a residual identifying error, which is caused by nonlinear characteristics such as friction characteristics of the throttle valve, exists after the model parameters have been converged to a certain extent, or if a disturbance whose average value is not zero is steadily applied, then residual identifying errors are accumulated, causing a drift of model parameters. [0130] Since such a residual identifying error should not be reflected to the values of model parameters, a dead zone process is carried out using a dead zone function Fn θ( [0131] The dead zone function Fn [0132] The amplitude of the residual identifying error changes according to an amount of change in the throttle valve opening TH. In the present embodiment, a dead zone width parameter EIDNRLMT which defines the width of the dead zone shown in FIGS. 7A through 7C is set according to the square average value DDTHRSQA of an amount of change in the target throttle valve opening THR. Specifically, the dead zone width parameter EIDNRLMT is set such that it increases as the square average value DDTHRSQA increases. According to such setting of the dead zone width parameter EIDNRLMT, it is prevented to neglect an identifying error to be reflected to the values of the model parameters as the residual identifying error. In the following equation (24), DDTHR represents an amount of change in the target throttle valve opening THR, which is calculated from the following equation (25):
[0133] Since the throttle valve opening deviation amount DTH is controlled to the target value DTHR by the adaptive sliding mode controller [0134] For further improving the robustness of the control system, it is effective to further stabilize the adaptive sliding mode controller [0135] Next, a method for calculating the predicted deviation amount PREDTH in the state predictor [0136] First, matrixes A, B and vectors X(k), U(k) are defined according to the following equations (26) through (29).
[0137] By rewriting the equation (1) which defines the controlled object model, using the matrixes A, B and the vectors X(k), U(k), the following equation (30) is obtained. [0138] Determining X(k+d) from the equation (30), the following equation (31) is obtained.
[0139] If matrixes A′ and B′ are defined by the following equations (32), (33), using the model parameters a1′, a2′, b1′, and c1′ which are not subjected to the limit process, a predicted vector XHAT(k+d) is given by the following equation (34).
[0140] The first-row element DTHHAT(k+d) of the predicted vector XHAT(k+d) corresponds to the predicted deviation amount PREDTH(k), and is given by the following equation (35).
[0141] where α1 represents a first-row, first-column element of the matrix A′ [0142] By applying the predicted deviation amount PREDTH(k) calculated from the equation (35) to the equation (9), and replacing the target values DTHR(k+d+1), DTHR(k+d), and DTHR(k+d−1) respectively with DTHR(k), DTHR(k−1), and DTHR(k−2), the following equation (9a) is obtained. From the equation (9a), the equivalent control input Ueq(k) is calculated.
[0143] Using the predicted deviation amount PREDTH(k) calculated from the equation (35), a predicted switching function value σpre(k) is defined by the following equation (36). The reaching law input Urch(k) and the adaptive law input Uadp(k) are calculated respectively from the following equations (10a) and (11a). σ [0144] The model parameter c1′ is a parameter representing a deviation of the default opening THDEF and disturbance. Therefore, as shown in FIG. 8, the model parameter c1′ changes with disturbance, but can be regarded as substantially constant in a relatively short period. In the present embodiment, the model parameter c1′ is statistically processed, and the central value of its variations is calculated as a default opening deviation thdefadp. The default opening deviation thdefadp is used for calculating the throttle valve opening deviation amount DTH and the target value DTHR. [0145] Generally, the method of least squares is known as a method of the statistic process. In the statistic process according to the method of least squares, all data, i.e., all identified parameters c1′, obtained in a certain period are stored in a memory and the stored data is subjected to a batch calculation of the statistic process at a certain timing. However, the batch calculation requires a memory having a large storage capacity for storing all data, and an increased amount of calculations are necessary because inverse matrix calculations are required. [0146] Therefore, according to the present embodiment, the sequential method-of-least-squares algorithm for adaptive control which is indicated by the equations (15) through (21) is applied to the statistic process, and the central value of the least squares of the model parameter c1 is calculated as the default opening deviation thdefadp. [0147] Specifically, in the equations (15) through (21), by replacing θ(k) and θ(k) [0148] One of the four algorithms described above can be selected according to the setting of the coefficients λ1′ and λ2′. In the equation (39), the coefficient λ1′ is set to a given value other than 0 or 1, and the coefficient λ2′ is set to 1, thus employing the weighted method of least squares. [0149] For the calculations of the equations (37) through (40), the values to be stored are thdefadp(k+1) and PTh(k+1) only, and no inverse matrix calculations are required. Therefore, by employing the sequential method-of-least-squares algorithm, the model parameter c1 can be statistically processed according to the method of least squares while overcoming the shortcomings of a general method of least squares. [0150] The default opening deviation thdefadp obtained as a result of the statistic process is applied to the equations (2) and (3), and the throttle valve opening deviation amount DTH(k) and the target value DTHR(k) are calculated from the following equations (41) and (42) instead of the equations (2) and (3). [0151] Using the equations (41) and (42), even when the default opening THDEF is shifted from its designed value due to characteristic variations or aging of the hardware, the shift can be compensated to perform an accurate control process. [0152] Operation processes executed by the CPU in the ECU [0153]FIG. 9 is a flowchart showing a process of the throttle valve opening control. The process is executed by the CPU in the ECU [0154] In step S [0155] In step S [0156] In step S [0157] Next, using the corrected model parameter vector θL(k) calculated in step S [0158] In step S [0159] If the stability determination flag FSMCSTAB is set to “1”, indicating that the adaptive sliding mode controller [0160] Further, when the adaptive sliding mode controller [0161] In step S [0162]FIG. 11 is a flowchart showing a process of performing calculations of the model parameter identifier [0163] In step S [0164] In step S [0165]FIG. 12 is a flowchart showing a process of calculating the identifying error idenl(k) which is carried out in step S [0166] In step S [0167] If the answer to the step S [0168] In step S [0169] The frequency characteristics of the actual controlled object having low-pass characteristics and the controlled object model thereof are represented respectively by the solid lines L [0170] By changing the frequency characteristics of the weighting of the identifying algorithm to the characteristics indicated by the broken line L [0171] 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. [0172] Since the identifying error ide(k) is calculated from the equations (17), (18), and (19), the same effect as described above can be obtained by performing the same low-pass filtering on the throttle valve opening deviation amount DTH(k) and the estimated throttle valve opening deviation amount DTHHAT(k), or by performing the same low-pass filtering on the throttle valve opening deviation amounts DTH(k−1), DTH(k−2) and the duty ratio DUT(k−d−1). [0173] Referring back to FIG. 12, the dead zone process as shown in FIG. 14 is carried out in step S [0174] In step S [0175] If the answer to step S [0176] If ide(k) is less than −EIDNRLMT, the corrected identifying error idenl(k) is calculated from the following equation (44) in step S [0177] If the identifying error ide(k) is in the range between +EIDNRLMT and −EIDNRLMT, the corrected identifying error idenl(k) is set to “0” in step S [0178]FIG. 16 is a flowchart showing a process of stabilizing the model parameter vector θ(k), which is carried out in step S [0179] In step S [0180]FIG. 17 is a flowchart showing the limit process of the model parameters a1′ and a2′, which is carried out in the step S [0181] In FIG. 18, combinations of the model parameters a1′ and a2′ which are required to be limited are indicated by “x” symbols, and the range of combinations of the model parameters a1′ and a2′ which are stable are indicated by a hatched region (hereinafter referred to as “stable region”). The limit process shown in FIG. 17 is a process of moving the combinations of the model parameters a1′ and a2′ which are in the outside of the stable region into the stable region at positions indicated by “◯” symbols. [0182] In step S [0183] If a2′ is less than XIDA2L in step S [0184] If the answer to the step S [0185] In steps S [0186] If the answers to steps S [0187] If a1′ is less than XIDA1L in step S [0188] In step S [0189] Straight lines L [0190] Therefore, in step S [0191] If the answer to step S [0192] If the corrected model parameter a1 is greater than (XA2STAB−XIDA2L) in step S [0193] In FIG. 18, the correction of the model parameter in a limit process P [0194] As described above, the limit process shown in FIG. 17 is carried out to bring the model parameters a1′ and a2′ into the stable region shown in FIG. 18, thus calculating the corrected model parameters a1 and a2. [0195]FIG. 19 is a flowchart showing a limit process of the model parameter b1′, which is carried out in step S [0196] In steps S [0197] If the answers to steps S [0198] If b1′ is less than XIDB1L in step S [0199]FIG. 20 is a flowchart showing a limit process of the model parameter c1′, which is carried out in step S [0200] In steps S [0201] If the answers to steps S [0202] If c1′ is less than XIDC1L in step S [0203]FIG. 21 is a flowchart showing a process of calculations of the state predictor, which is carried out in step S [0204] In step S [0205] In step S [0206]FIG. 22 is a flowchart showing a process of calculation of the control input Usl (=DUT) applied to the throttle valve actuating device [0207] In step S [0208] In step S [0209] If FSMCSTAB is “0” in step S [0210] If FSMCSTAB is “1” in step S [0211] In steps S [0212]FIG. 23 is a flowchart showing a process of calculating the predicted switching function value σpre, which is carried out in step S [0213] In step S [0214] In steps S [0215]FIG. 24 is a flowchart showing a process of calculating the switching function setting parameter VPOLE, which is carried out in step S [0216] In step S [0217] If FSMCSTAB is “0”, indicating that the adaptive sliding mode controller [0218] In step S [0219] Specifically, when the target value DTHR for the throttle valve opening changes greatly in the decreasing direction, the switching function setting parameter VPOLE is set to a relatively small value. This makes it possible to prevent the throttle valve [0220] As shown in FIG. 25C, the VPOLE map may be set so that the switching function setting parameter VPOLE decreases when the throttle valve opening TH is in the vicinity of the fully closed opening or the fully open opening. Therefore, when the throttle valve opening TH is in the vicinity of the fully closed opening or the fully open opening, the speed for the throttle valve opening TH to follow up the target opening THR is reduced. As a result, collision of the throttle valve [0221] In steps S [0222]FIG. 26 is a flowchart showing a process of calculating an integrated value of σpre, SUMSIGMA, of the predicted switching function value σpre. This process is carried out in step S [0223] In step S [0224] In steps S [0225]FIG. 27 is a flowchart showing a process of calculating the reaching law input Urch, which is carried out in step S [0226] In step S [0227] If the stability determination flag FSMCSTAB is “1”, indicating that the adaptive sliding mode controller [0228] In steps S [0229] As described above, when the adaptive sliding mode controller [0230]FIG. 28 is a flowchart showing a process of calculating the adaptive law input Uadp, which is carried out in step S [0231] In step S [0232] If the stability determination flag FSMCSTAB is “1”, indicating that the adaptive sliding mode controller [0233] As described above, when the adaptive sliding mode controller [0234]FIG. 29 is a flowchart showing a process of determining the stability of the sliding mode controller, which is carried out in step S [0235] In step S [0236] In step S [0237] In step S [0238] In step S [0239] If the count of the stability determining period counter CNTJUDST subsequently becomes “0”, the process goes from step S [0240] In step S [0241] In step S [0242]FIG. 30 is a flowchart showing a process of calculating the default opening deviation thdefadp, which is carried out in step S [0243] In step S [0244] where PTH(k−1) represents a gain parameter calculated in step S [0245] In step S [0246] In step S [0247] The equation (56) is obtained by setting λ1′ and λ2′ in the equation (39) respectively to a predetermined value XDEFADP and “1”. [0248] According to the process shown in FIG. 30, the model parameter c1′ is statistically processed by the sequential method-of-weighted-least-squares algorithm to calculate the default opening deviation thdefadp. [0249] In the present embodiment, the throttle valve actuating device [0250] Second Embodiment [0251] In the first embodiment described above, the controlled object model is defined by the equation (1) including the dead time d, and the predicted deviation amount PREDTH after the elapse of the dead time d is calculated with the state predictor [0252] In order to further reduce the calculation load on the CPU, the fixed gain algorithm is employed as the algorithm for identifying the model parameters. [0253] For further stabilizing the control, another process instead of the dead zone process is employed as the process for preventing the drift of the model parameters. [0254] The second embodiment will be described below primarily with respect to details which are different from those of the first embodiment. Other details except for what will be described below are identical to those of the first embodiment. [0255]FIG. 31 is a functional block diagram of a throttle valve control system which is realized by the ECU [0256] The adaptive sliding mode controller [0257] Using the adaptive sliding mode controller [0258] The model parameter identifier [0259] The model parameter scheduler [0260] In the present embodiment, since the controlled object model is defined by the above equation (1a), the adaptive sliding mode controller [0261] The equations (9b), (10b), and (11b) are obtained by setting the dead time d to “0” in the equations (9), (10), and (11). [0262] The model parameter identifier θ( [0263] The identifying error ide(k) in the equation (15) is defined by the following equations (17), (18), and (19a). The gain coefficient vector KP(k) is defined by the following equation (20), and the square matrix P(k) in the equation (20) is calculated from the following equation (21). ζ( [0264] In the present embodiment, the following requirements B4 and B5 are required to be satisfied, in addition to the requirements B1 through B3 which should be satisfied like the first embodiment should be satisfied. [0265] B1) Adaptation to Quasi-Static Dynamic Characteristic Changes and Hardware Characteristics Variations [0266] “Quasi-static dynamic characteristics changes” mean slow-rate characteristics changes such as power supply voltage fluctuations or hardware degradations due to aging. [0267] B2) Adaptation to High-Rate Dynamic Characteristics Changes [0268] Specifically, this means adaptation to dynamic characteristics changes depending on changes in the throttle valve opening TH. [0269] B3) Prevention of a Drift of Model Parameters [0270] The drift, which is an excessive increase of the absolute values of model parameters, should be prevented. The drift of model parameters is caused by the effect of the identifying error, which should not be reflected to the model parameters, due to nonlinear characteristics of the controlled object. [0271] B4) Matching with the Calculating Capability of the ECU [0272] Specifically, the amount of calculations is required to be further reduced. [0273] B5) The Stabilization of Model Parameters (Control Performance) [0274] Specifically, variations of identified model parameters should be minimized. [0275] In order to satisfy the requirement B4, the coefficients λ1 and λ2 are set respectively to “1” and “0”, to thereby employ the fixed gain algorithm. Accordingly, the square matrix P(k) is made constant, and the calculation of the equation (21) can be omitted. As a result, the amount of calculations can greatly be reduced. [0276] Specifically, when the fixed gain algorithm is employed, the equation (20) is simplified into the following equation (20a) where P represents a square matrix having constants as diagonal elements.
[0277] According to the algorithm thus simplified, the amount of calculations can be reduced. However, the identifying ability is also slightly reduced. Further, the equation (15) for calculating the model parameter vector θ(k) can be rewritten to the following equation (15b) and has an integral structure of the identifying error ide(k). Therefore, the identifying error ide(k) is likely to be integrated to the model parameters to cause the drift of the model parameters. θ( [0278] where θ(0) represents an initial vector having elements of initial values of the model parameters. [0279] In the present embodiment, in order to prevent the drift of the model parameters, the model parameter vector θ(k) is calculated from the following equation (15c) instead of the above equation (15b). θ( [0280] where DELTA represents a forgetting coefficient vector having forgetting coefficients DELTAi (i=1 through 4) as elements, as indicated by the following equation. [0281] The forgetting coefficients DELTAi are set to a value between 0 and 1 (0<DELTAi<1) and have a function to gradually reduce the influence of past identifying errors. One of the coefficient DELTA3 which is relevant to the calculation of the model parameter b1, and the coefficient DELTA4 which is relevant to the calculation of the model parameter c1 is set to “1”. In other words, one of the forgetting coefficients DELTA3 and DELTA4 is made noneffective. By setting one or more of the elements of the forgetting coefficient vector DELTA to “1”, it is possible to prevent a steady-state deviation between the target value DTHR and the throttle valve opening deviation amount DTH from occurring. If both of the coefficients DELTA3 and DELTA4 are set to “1”, the effect of preventing the model parameters from drifting becomes insufficient. Accordingly, it is preferable to set only one of the coefficients DELTA3 or DELTA4 to “1”. [0282] If the equation (15) is rewritten into a recursive form, the following equations (15d) and (15e) are obtained. A process of calculating the model parameter vector θ(k) from the equations (15d) and (15e) instead of the equation (15) is hereinafter referred to as a δ correcting method, and dθ(k) defined by the equation (15e) is referred to as “updating vector”. θ( [0283] According to the algorithm using the δ correcting method, the drift preventing effect which satisfies the requirement B3 and the model parameter stabilizing effect which satisfies the requirement B5 are obtained. Specifically, the initial vector θ(0) is maintained at all times, and the values which can be taken by the elements of the updating vector dθ(k) are limited by the effect of the coefficient vector DELTA. Therefore, the model parameters can be stabilized in the vicinity of their initial values. [0284] Furthermore, since the model parameters are calculated while adjusting the updating vector dθ(k) according to the identifying process based on the input and output data of the actual controlled object, the model parameters matching the actual controlled object can be obtained, and hence the above requirement B1 is satisfied. [0285] In order to satisfy the requirement B2, the model parameter vector θ(k) is calculated from the following equation (15f) which uses the reference model parameter vector θbase, instead of the initial vector θ(0) in the equation (15d). θ( [0286] The reference model parameter vector θbase is set according to the throttle valve opening deviation amount DTH (TH−THDEF) by the model parameter scheduler [0287] As described above, according to the present embodiment, the amount of calculations by the ECU is reduced by employing the fixed gain algorithm (the requirement B4). The adaptation to quasi-static dynamic characteristics changes and hardware characteristics variations (the requirement B1), the stabilization of the model parameters (control performance) (the requirement B5), and the prevention of the drift of the model parameter (the requirement B3) are achieved by employing the algorithm using the δ correcting method. Further, the adaptation to dynamic characteristics changes corresponding to changes in the throttle opening TH is achieved by employing the model parameter scheduler [0288] The elements a1′, a2′, b1′, and c1′ of the model parameter vector θ(k) calculated from the equation (15f) are subjected to the limit process so that the corrected model parameter vector θL(k) (θL(k) [0289] In addition, the model parameter c1′ is statistically processed, and the central value of its variations is calculated as the default opening deviation thdefadp and used to calculate the throttle valve opening deviation amount DTH and the target value DTHR from the equations (41) and (42) like the first embodiment. [0290] Operations of the CPU of the ECU [0291]FIG. 32 is a flowchart showing a throttle valve opening control process. The throttle valve opening control process differs from the throttle valve opening control process shown in FIG. 9 in that step S [0292] In step S [0293] In step S [0294] In step S [0295]FIG. 33 is a flowchart showing a process of calculations of the model parameter identifier [0296] In step S [0297] In step S [0298] In step S [0299]FIG. 35 is a flowchart showing a process of calculating the identifying error ide(k) which is carried out in step S [0300] In step S [0301] In the present embodiment, the predetermined value XCNTIDST in step S [0302]FIG. 36 is a flowchart showing a process of calculating the control input Usl (DUT) applied to the throttle valve actuating device [0303] In step S [0304]FIG. 37 is a flowchart showing a process of calculating the switching function value σ which is carried out in step S [0305] In step S [0306]FIG. 38 is a flowchart showing a process of calculating an integrated value SUMSIGMAa of the switching function value σ which is carried out in step S [0307] In step S [0308] In steps S [0309]FIG. 39 is a flowchart showing a process of calculating the reaching law input Urch, which is carried out in step S [0310] In the present embodiment, using the switching function value σ instead of the predicted switching function value σpre, the reaching law input Urch is calculated in step S [0311]FIG. 40 is a flowchart showing a process of calculating the adaptive law input Uadp which is carried out in step S [0312] In the present embodiment, using the integrated value SUMSIGMAa of the switching function value σ, the adaptive law input Uadp is calculated in step S [0313]FIG. 41 is a flowchart showing a process of determining the stability of the sliding mode controller which is carried out in step S [0314] In step S [0315] In the present embodiment, the throttle valve actuating device [0316] Third Embodiment [0317]FIG. 42 is a block diagram showing a control system according to the third embodiment of the present invention. [0318] As shown in FIG. 42, the control system comprises a plant [0319] The subtractor [0320] The plant [0321] The control quantity determining unit [0322] The relations between the components and parameters of the third embodiment and the components and parameters of the first embodiment will be described below. [0323] The pH sensor [0324] The second reference value V2BASE is added to bias the central value of the first control quantity U [0325] The flow rate control valve [0326] Because of the above relations, the plant [0327] In the present embodiment, the identifier [0328] Modification of Third Embodiment [0329]FIG. 43 shows a modification of the control system shown in FIG. 42. In the modification, a plant [0330] The modeling and the control process which are the same as those of the third embodiment are also applicable to the plant including the local feedback loop as shown in FIG. 43. [0331] Since the circuit for energizing the motor in the first embodiment is already known, this circuit has not been described in detail. The circuit for energizing the motor may include a current sensor for detecting an output current of the switching element that is turned on and off, and a feedback control process may be carried out to make a detected current value ID coincide with a current value IR corresponding to the control quantity Usl. The present modification corresponds to a structure where the above circuit arrangement is applied to the first embodiment. [0332] Fourth Embodiment [0333]FIG. 44 is a block diagram of a control system according to the fourth embodiment of the present invention. The control system shown in FIG. 44 corresponds to the control system according to the second embodiment, and is similar to the control system shown in FIG. 42 except that the control quantity determining unit [0334] The control quantity determining unit [0335] The identifier [0336] The parameter scheduler [0337] In the present embodiment, the identifier [0338] Modification of Fourth Embodiment [0339]FIG. 45 shows a modification of the control system shown in FIG. 44. In the modification, a plant [0340] The modeling and the control process which are the same as those of the fourth embodiment are also applicable to the plant including the local feedback loop as shown in FIG. 45. [0341] Other Embodiments [0342] The δ correcting method may be replaced with a ε correcting method described below, as a method of calculating the identifying error ide(k) of the model parameters. Specifically, the following equation (15g) may be used instead of the equation (15c) to calculate the model parameter vector θ(k). θ( [0343] where EPS represents a forgetting coefficient vector having forgetting coefficients EPSi (i=1 through 4) as its elements, as indicated by the following equation. [0344] Like the forgetting coefficients DELTAi, the forgetting coefficients EPS1, EPS2, and EPS4 are set to a value between “0” and “1” (0<EPSi<1) and have a function to gradually reduce the influence of past identifying errors. [0345] In the ε correcting method, the coefficient EPS3 which is relevant to the calculation of the model parameter b1 must be set to “1” for the following reasons. In the ε correcting method, the all values of the model parameters becomes closer to zero, as the identifying error ide(k) becomes less. Since the model parameter b1 is applied to the denominator of the equations (9b), (10b), and (11b), the input Usl applied to the controlled object diverges as the model parameter b1 becomes closer to “0”. [0346] The equation (15g) is different from the equation (15c) in that the initial vector θ(0) is also multiplied by the forgetting coefficient vector EPS. [0347] If the equation (15g) is rewritten into a recursive form, the following equation (15h) is obtained. The method of calculating the model parameter vector θ(k) from the equation (15h) instead of the equation (15) is referred to as the ε correcting method. θ( [0348] The ε correcting method also has a function to gradually reduce the influence of past identifying errors eid. Accordingly, the drift of the model parameters are prevented by the ε correcting method. [0349] In the second embodiment, the drift of the model parameters is prevented by the δ correcting method. However, like the first embodiment, the corrected identifying error idenl(k) may be calculated according to the dead zone process (FIG. 14), and the model parameter vector θ(k) may be calculated using the corrected identifying error idenl(k). [0350] In the first embodiment, the dead zone process may be replaced with the δ correcting method or the ε correcting method. [0351] Although certain preferred embodiments of the present invention have been shown and described in detail, it should be understood that various changes and modifications may be made therein without departing from the scope of the appended claims. Referenced by
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