JPH0922305A - Foreseeing control function adding method - Google Patents

Abstract

PROBLEM TO BE SOLVED: To add a foreseeing control function without changing a feedback(FB) compensator by calculating a state FB gain and calculating a foreseeing feedforward(FF) gain while using this state FB gain. SOLUTION: The coefficient of a controlled system P(s) is inputted (s101) and a PID parameter found by a model matching method is inputted (s102). Next, coefficients A, B and C of a discrete state equation are calculated (s103) and coefficient matrixes (ϕ, G and GR) of an extended error system are calculated (s104). A prescribed expression is operated for changing the PID parameter into a state FB control system and by using this expression, a state FB gain F1 is operated (s105). Next, weights Q and H are set (s106: for example, Q(1, 1)=1, H=1) and Liapunov's solutions P1 , Δ1 R' and γ1 R' are calculated (s107 and s108). Afterwards, a foreseeing FF gain F1 R' is calculated (s109).

Description

Detailed Description of the Invention

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a preview control function adding method for improving control performance by effectively using future information such as target values and disturbances in the fields of servo systems, mechatronics equipment, process control and the like. .

[0002]

2. Description of the Related Art Conventionally, various methods have been proposed for preview control. Firstly, as the most basic method, optimal feedback is achieved by adding state-feedback (FB) control based on an optimal regulator and feedforward (FF) control utilizing future information of a target value or disturbance known in advance. There is a so-called servo system ("Digital predictive control" (Tsuchiya / Egami, Sangyo Tosho Co., Ltd., pp.39-4).
6, 1993)).

This seeks a preview control input that minimizes a quadratic form evaluation function relating to a state quantity including a deviation between a target value and a control quantity and an operation quantity.
And the FF gain are calculated at the same time. In this method, stability due to FB can be guaranteed, and it can be expected to follow the target value and reduce the peak value of the manipulated variable.

In obtaining the preview FF gain, the first method cannot be used except in the case of the state feedback obtained by the optimal regulator method in the FB control system.
Therefore, as a second method, the first method is applied so that it can be applied to an FB configured by a method other than the optimum regulator method.
Unlike the method described in (1), the prediction F without changing the control law of the FF is performed so as to minimize the quadratic form evaluation function regarding the state quantity including the deviation between the target value and the control quantity and the operation quantity by the FF.
There is a method of calculating the F gain (ibid., Pp. 39-4).
6).

Here, in order to design only the preview control for the predesigned output feedback control system, the second method is used.
However, in this case, it cannot be used as it is when the differential operation is calculated using the output difference like the I-PD digital control system. Therefore, there is a method that introduces up to n step past values of the output as a new state quantity, creates an expanded state equation including the integral compensator in this, and performs a preview control design for the system ("Measuring Automatic Control Society"). Collection of papers ”(Aida, VOL.24, NO.11, pp1129
~ 1136, 1988))

[0006]

The first method is the state F
It is based on B and cannot be applied as it is if all state quantities cannot be measured. In the second method, the operation amount of only FF is evaluated by the quadratic form evaluation function, and in this case, the characteristic of FB is ignored, so the movement of FB cannot be reflected in the predictive FF gain.

Further, in the third method, the form of complete feedback compensation of the I-PD control system is considered to be equivalent to state FB compensation, and the I-PD control system can be considered as a kind of state FB control system. . Therefore, this form cannot be applied to a general output FB control system.

The present invention has been made in order to solve the above problems, and an object of the present invention is to perform preview control for various output FB control systems that have already been designed without changing the FB compensator. It is to provide a preview control function adding method capable of adding a function.

[0009]

In order to achieve the above object, the invention according to claim 1 provides a transfer function of a controlled object and a PI.
In a method of performing preview feedforward compensation to a PID control system whose D parameter is known and adding a preview control function regarding a target value, the PID control system is expanded to an error system, and an expanded error system is created based on the PID parameter. The corresponding state feedback gain is calculated, and the preview feedforward gain is calculated using this state feedback gain.

Further, the invention according to claim 2 performs predictive feedforward compensation for a feedback control system in which the transfer function of the controlled object and the transfer function of the compensator in which the denominator numerator is expressed in a rational function form are known. In the method of adding a preview control function regarding a target value, a feedback control system is expanded to an error system including a control target and a compensator, a state feedback gain corresponding to this expanded error system is calculated, and this state feedback gain is used. The preview feedforward gain is calculated.

[0011]

First, the operation of the present invention will be described.
In the invention described in claim 1, the preview control function is added to the PID control system as shown in FIG. In FIG. 3, r is a target value, e is an error signal (control deviation), u is an operation amount, y is a control amount, and the controlled object P is modeled by a transfer function P (s) as shown in Formula 1. Can be converted into. In Equation _{1, a 1 ~a n, b} 1 ~b n is a coefficient, s
Is the Laplace operator.

[0012]

[Equation 1]

This controlled object P is converted into a continuous state equation shown in equation (2). However, the coefficient matrix A ', B', C '
Is as shown in Equation 3, and in Equation 3, I _r is r
It is an identity matrix with rows and r columns.

[0014]

[Equation 2]

[0015]

(Equation 3)

Equation 2 is discretized with a sampling period Δt and converted into Equation 4 which is a discrete state equation. Also,
The error signal e (k) is represented by Equation 5. The coefficients A, B, and C in Expression 4 are as shown in Expression 6.

[0017]

(Equation 4)

[0018]

(Equation 5)

[0019]

(Equation 6)

In these equations, x (k): state variable (n × 1), y (k): output variable (m × 1), u (k): input variable (r × 1), R (k) ): Target value signal (m × 1), A: n × n, B: n × r, C: n × m (n, r, and m are the number of state variables and the number of input variables, respectively)
It is the number of output variables and r ≧ m). Further, the system represented by Equation 2 is assumed to be controllable and observable.

Here, the deviation and the deviation one step before are defined as e (k) and e ₁ (k), respectively, and when expanded to an error system in which the state quantities are added, Equation 7 is obtained. 8 is obtained. However, the coefficient matrix Φ, G, G _R in the equation 8 is as in the equation 9.

[0022]

(Equation 7)

[0023]

(Equation 8)

[0024]

[Equation 9]

Further, when the control law of the discretized PID compensator is developed, the following equation 10 is obtained. In Equation 10, K _P is a proportional gain, K _I is an integral gain, K _D1 is a differential gain, T _I is an integration time, and T _D1 is a differential time. Also,
The condition is Equation 11.

[0026]

(Equation 10)

[0027]

[Equation 11]

A block diagram of the PID preview control system at this time is shown in FIG. In FIG. 4, z ⁻¹ is a sampling delay operator.

Now, let us consider the case of preview control for a change in the target value. That is, the design of the preview feedforward term is defined as a quadratic form evaluation function J ′ including an error term and a forecast input term assuming that the target value from the current time to the MR step future is known as shown in Expression 12. Consider minimizing this evaluation function J ′. However, control input Δu
(T) is as shown in Expression 13. Equation 1
2 and 13, Q and H are weight matrices, and F ₁ is state FB.
The gain, F _1R, is the preview FF gain.

[0030]

(Equation 12)

[0031]

(Equation 13)

The second item of Expression 12 becomes Expression 14, and from this, Expression 12 is converted into Expression 15.

[0033]

[Equation 14]

[0034]

(Equation 15)

Therefore, the evaluation function to be obtained is as shown in Expression 16. It should be noted that F _1R ', Δ _1R ' and Γ _1R 'in the equation 16 are as shown in the equation 17.

[0036]

(Equation 16)

[0037]

[Equation 17]

However, in these equations, ξ ₁ = Φ +
GF ₁ , Lyapunov equation P ₁ = η + ξ ₁ P ₁ ξ ₁ ^T solution,
η = Q + F ₁ HF ₁ ^T. Further, R ₀ is the value of the target value change of ΔR (1) = R ₀ at k = 0. The preview gain F _1R 'which minimizes the evaluation function of Expression 16 is obtained by Expression 16 by partial differentiation.

[0039]

(Equation 18)

Next, the operation of the invention described in claim 2 will be described. According to the second aspect of the invention, the preview control function is added to the FB control system having a general compensator in which the denominator numerator is represented by a transfer function in the rational function form. FIG. 5 is a block diagram of the FB control system represented by the general compensator C (s) and the controlled object P (s). The controlled object P (s) is modeled by Expression 1 as described above, and this discrete state equation expression is expressed as Expression 19. In Expression 19, A _P , B _P , and C _P are coefficient matrices. The discretization procedure is the same as the conversion procedure from Equation 2 to Equation 4.

[0041]

[Equation 19]

Next, the general compensator C (s) is defined by the transfer function shown in Expression 20. In this formula _{20, Ca 1 ~Ca k, Cb} 1 ~Cb k is a coefficient.

[0043]

(Equation 20)

In order to form a servo system for the preview control design, the general compensator C (s) is separated into [integrator + general compensator C '] as shown in FIG. Bowl C '
The discrete state equation expression of is expressed as Expression 21. This formula 2
In 1, A _C ′, B _C ′ and C _C ′ are coefficients. The discretization procedure is the same as the conversion procedure from Expression 2 to Expression 4.

[0045]

(Equation 21)

Here, if Equation 19 and Equation 21 are put together, Equation 22 is obtained. Then, if an enlarged error system is constructed for this system, the following formula 23 is obtained. However, the condition is Equation 24.

[0047]

(Equation 22)

[0048]

(Equation 23)

[0049]

(Equation 24)

The speed type control law of the FB control portion at this time is as shown in Expression 25. Where Δt is the sampling interval, KI = TI · Δt, TI is the gain of the controller integrator in the transfer function expression, and F ₁ = [F _e F _x ].
Corresponds to the optimum servo system gain in this control system.

[0051]

(Equation 25)

FIG. 7 shows a block diagram of this general compensator preview control system. This FIG. 7 is based on the FB control system shown in FIG. 5 and has gains F _e and F _{x of} new plants A and B and compensators.
It is the block diagram decomposed into. Using the enlarged error system shown in Expression 23 and F ₁ in Expression 25, the preview control design is performed according to the procedure from Expression 12 to Expression 18.

[0053]

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS FIG. 1 shows an embodiment of the invention described in claim 1 and is a flow chart of a design calculation procedure for adding a preview control function to a PID control system. Hereinafter, a design example of the preview control system will be described with reference to this flowchart. First, the control target P (s) is, for example, Equation 2
Consider PID preview control as represented by 6.

[0054]

(Equation 26)

First, the coefficient of the controlled object P (s) is input (S101 in FIG. 1), and the PID parameter of Equation 27 obtained by the model matching method is input (S10).
2).

[0056]

[Equation 27]

Next, the coefficient A of the discrete state equations by Equation 6, B, calculates the C (S103), the coefficient matrix of the enlargement error system using Equation _{9 (Φ, G, G R} ) is calculated (S104 ). Formula 28 is calculated in order to convert the PID parameter to the state FB control system (Δt = 0.64 in formula 28), and the state FB gain F ₁ is calculated using this formula as shown in formula 29 (S105).

[0058]

[Equation 28]

[0059]

(Equation 29)

Next, the weights Q and H are set (S106:
For example, Q (1,1) = 1, H = 1), Lyapunov solution P ₁ and Δ _1R ′, Γ _1R ′ are calculated (S107, S108).
After that, the preview FF gain F _1R 'is calculated by the formula 18 (S109).

Here, when the preview FF gain F _1R is obtained by the second method explained in the prior art, the following formula 30 is obtained. Further, according to the embodiment of the invention of claim 1, the prediction F
The F gain F _1R 'is obtained as shown in Expression 31.

[0062]

[Equation 30]

[0063]

(Equation 31)

The step responses of the respective control amounts using the preview FF gains F _1R and F _1R 'only for the FB which does not have the preview control function are shown in Fig. 8 and the step responses of the manipulated variables at this time. It is shown in FIG. However,
The number of preview steps is 4, and the step change of the target value is given 9.6 seconds after the start. From these figures, it is understood that the response performance is improved in the present invention.

Next, FIG. 2 shows an embodiment of the invention described in claim 2, and is a flow chart of a design calculation procedure for adding a preview control function to an FB control system having a general compensator. Hereinafter, a design example of the preview control system will be described with reference to this flowchart.

First, the controlled object (second-order stable system) P (s) is shown in Equation 32, and the general compensator C (s) is shown in Equation 33 as H∞.
Use a controller based on control design.

[0067]

(Equation 32)

[0068]

[Equation 33]

In FIG. 2, first, the controlled object P (s)
Are input (S201), and the coefficients A _P , B _P , and C _P of the discrete state equation in Equation 19 are calculated (S20).
2). Next, the coefficient of the general compensator C (s) is input (S2
03), separate the integrator and replace the compensator C (s) with the compensator C (s) '+ to form a servo system for preview control design.
It is an integrator (S204). Then, the coefficient A _C ′ of the discrete state equation of the compensator C (s) ′ shown in Expression 21
B _C 'and C _C ' are calculated (S205).

Next, A _P , B _P and A _C ′, B _C ′ are combined to form new control targets A and B (S206).
After that, the coefficient matrix (Φ, G, G _R ) of the expansion error system in Expression 24 is calculated (S207), and the state FB gain F ₁ (= [F _e F _x ]) is calculated (S208).

F in the controller of Expression 33
The control rule of B control is as shown in Formula 34. However,
KI = TI · Δt, where Δt = 0.1, TI
= 0.1088 to KI = 0.01088.

[0072]

(Equation 34)

Using the control rule based on this state FB gain F ₁ , weights Q and H are set (S209: For example, Q (1,
1) = 0.01, H = 1), and the predictive FF gain F _1R 'is calculated as in FIG. 1 (S210 to S212).

Here, when the preview FF gain F _1R is obtained by the second method described in the prior art, the following formula 35 is obtained. Further, according to the embodiment of the invention of claim 2, the prediction F
When the F gain F _1R 'is obtained, the formula 36 is obtained.

[0075]

(Equation 35)

[0076]

[Equation 36]

The step responses of the respective control amounts using the preview FF gains F _1R and F _1R 'only for the FB without the preview control function are shown in Fig. 10 and the step responses of the manipulated variables at this time are shown. It is shown in FIG. However, the step change of the target value is given 5 seconds after the start. The sampling interval Δt is 0.1 second as described above. From these figures, the superiority of the present invention over the prior art was confirmed.

[0078]

As described above, according to the present invention, if the transfer function of the controlled object and the PID parameter or the transfer function of the compensator are known, the preview FF gain is calculated without changing the PID parameter or the compensator. Then, a preview control function can be added to various FB control systems, and an already operating FB control system can be used as a preview servo system.

[Brief description of drawings]

FIG. 1 is a flowchart showing a PID preview control system design calculation procedure according to an embodiment of the present invention.

FIG. 2 is a flowchart showing a general compensator preview control system design calculation procedure according to the embodiment of the invention of claim 2;

FIG. 3 is a block diagram of a PID control system to which the invention of claim 1 is applied.

FIG. 4 is a block diagram of a PID preview control system.

FIG. 5 is a block diagram of a control system to which the invention of claim 2 is applied.

FIG. 6 is a block diagram of a control system in which the general compensator in FIG. 5 is separated.

FIG. 7 is a block diagram of a general compensator preview control system.

FIG. 8 is a diagram showing a step response of a control amount by PID preview control.

FIG. 9 is a diagram showing a step response of an operation amount by PID preview control.

FIG. 10 is a diagram showing a step response of a control amount by H∞ control + preview control.

FIG. 11 is a diagram showing a step response of an operation amount by H∞ control + preview control.

Claims (2)

[Claims] 1. A method of adding a preview control function for a target value by performing preview feedforward compensation to a PID control system whose transfer function and PID parameter to be controlled are known, wherein the PID control system is expanded to an error system. And the PI
A preview control function adding method characterized by calculating a state feedback gain corresponding to an enlarged error system based on the D parameter and calculating a preview feedforward gain using this state feedback gain. 2. A feedback control system in which a transfer function of an object to be controlled and a transfer function of a compensator in which a denominator and a numerator are expressed in a rational function form is known is subjected to preview feedforward compensation to add a preview control function for a target value. In the method, the feedback control system is expanded to an error system including a control target and a compensator, a state feedback gain corresponding to the expanded error system is calculated, and a preview feedforward gain is calculated using the state feedback gain. A method for adding a preview control function characterized by the above.