PID control method based on DELTA operator description

Abstract

The PID control method described based on the DELTA operator solves the problem that the discretized system is unstable and the PID controller has poor control effect as the sampling frequency increases in actual engineering, and belongs to the field of embedded control. The invention comprises: determining the dynamic equation of the controlled system, time discretizing the dynamic equation, and obtaining a discrete time linearization model; obtaining a PID controller according to the discrete time linearization model, and converting the PID controller into a discrete time SOF control system; writing the discrete time SOF control system into a DELTA operator form; calculating the expected value of the SOF control system in the form of the DELTA operator, obtaining the control parameters of the SOF control system in the form of the DELTA operator, and determining the control parameters of the PID controller according to the control parameters; and controlling the controlled system using the PID controller with the determined control parameters.

Description

Technical Field

[0001] The invention relates to a PID controller and belongs to the field of embedded control. Background Art

[0002] Thanks to the rapid development of embedded controllers, the cost of implementing control systems has been greatly reduced, and various control strategies have been put into practical application. For example, more than 90% of industrial processes use PID controllers. However, in actual engineering, if you want to achieve better control effects, the system sampling period needs to be small enough. When the controlled system is discretized using the traditional shift operator, the poles of the discrete time model are concentrated near z=1, which increases the sensitivity of the discrete time model transfer function. This will lead to numerical instability problems such as sampling data quantization errors and limit oscillations, and it is easy to introduce non-minimum phase zeros, thereby reducing the control effect of the PID controller. Summary of the invention

[0003] In view of the problem that as the sampling frequency increases in actual engineering, the discretized system becomes unstable, resulting in poor control effect of the PID controller, the present invention provides a PID control method based on DELTA operator description.

[0004] A PID control method based on DELTA operator description of the present invention comprises:

[0005] S1. Determine the dynamic equation of the controlled system, discretize the dynamic equation in time, and obtain the discrete time linearization model:

[0006]

[0007] In the formula, x(t k ), is the state vector of the controlled system, including position, velocity, and acceleration, is the disturbance input, and are the controlled system output and the measured output, respectively, u(t k ) is the control quantity input of the controlled system; A z , B z1 , B z2 , C z1 , D z1 and C z2 are coefficient matrices, Represents n-dimensional real numbers;

[0008] S2, obtaining a PID controller according to a discrete time linearization model, and converting the PID controller into a discrete time SOF control system;

[0009] S3, write the discrete time SOF control system into the form of DELTA operator;

[0010] S4. Calculate the expected value of the SOF control system in the form of a DELTA operator and obtain the control parameters of the SOF control system in the form of a DELTA operator According to the control parameters Determine the control parameters of the PID controller;

[0011] S5. Control the controlled system using a PID controller with determined control parameters.

[0012] Preferably, S4 includes:

[0013] S41, set the number of iterations k = 1, select the initial value P k-1 =X k-1 =I;

[0014] S42. Find the minimum Trace(X) that satisfies the boundary conditions. k-1 P k +P k-1 X k ), get the P at this time k and X k : Boundary conditions:

[0015]

[0016] Trace(·) represents the trace of the matrix, P k and X k When X is the decision variable for k, k P k The inverse matrix of , α, σ, γ represent given values, T represents the sampling period, Controlled system Controlled system

[0017] S43、If k>N or ||Tr(X k-1 P k +P k-1 X k )-2M||<ε, M is an integer, M×M is the dimension of P, N is a predefined maximum number of iterations, ε is a specified tolerance, and the process goes to S44; otherwise, k=k+1, and the process goes to S42;

[0018] S44, when P = P k Time Solution

[0019] As a preference, α>0, ||z(t k )|| ∞ / ||w(t k )|| ∞ ≤γ,0<σ<γ 2.

[0020] As a preference, the disturbance input ||w(t k )|| ∞ ≤1, the controlled system outputs ||z(t k )|| ∞ ≤γ.

[0021] Preferably, in S2, the discrete time SOF control system is:

[0022] u(t k )=F z y a (t k )

[0023] in:

[0024]

[0025]

[0026]

[0027]

[0028]

[0029] Δy(t k )=y(t k )-y(t k-1 )

[0030] F z is the control coefficient of the discrete-time SOF control system.

[0031] Preferably, in S3, the SOF control system in the form of a DELTA operator is:

[0032]

[0033]

[0034]

[0035] Preferably, in S2, the PID controller is:

[0036]

[0037] K p ,K i ,K d are the coefficients of the proportional term, integral term, and differential term respectively.

[0038] Preferably, the controlled system is a piezoelectric ceramic actuator, an electromagnetic motor, a piezoelectric fast-tilt mirror for beam control, or a satellite laser communication precision tracking system.

[0039] The beneficial effects of the present invention are as follows: when sampling at high speed, the present invention uses the DELTA operator to discretize the model to approach the original continuous model, and the result of the model analysis is close to the result in the continuous system. The PID controller designed by the present invention has stable performance and improves the control effect. BRIEF DESCRIPTION OF THE DRAWINGS

[0040] Figure 1 is the block diagram of the generalized object G(s);

[0041] Figure 2 This is the block diagram of the l1 standard problem;

[0042] Figure 3 (a) is x1 of Example 1, (b) is x2 of Example 1, and (c) is u of Example 1 under the initial condition x0 = [π / 4 0] T The following response, Figure 3 The horizontal axis in represents time;

[0043] Figure 4 is the response of z in the time interval 8-16, and the abscissa represents time;

[0044] Figure 5 Where (a) is the response of u in Example 1, (b) is the response of z in Example 1, Figure 5 The horizontal axis represents time. DETAILED DESCRIPTION

[0045] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0046] It should be noted that, in the absence of conflict, the embodiments of the present invention and the features in the embodiments may be combined with each other.

[0047] The present invention will be further described below in conjunction with the accompanying drawings and specific embodiments, but they are not intended to limit the present invention.

[0048] The PID control method based on DELTA operator description of this embodiment includes:

[0049] Step 1: Determine the dynamic equation of the controlled system, discretize the dynamic equation in time, and obtain the discrete time linearization model:

[0050]

[0051] In the formula, x(t k ), is the state vector of the controlled system, including position, velocity, and acceleration, is the disturbance input, and They are the controlled system output and the measurement output respectively. The measurement output can be position output or angle output. u(t k ) is the control quantity input of the controlled system; A z , B z1 , B z2 , C z1 , D z1 and C z2 are coefficient matrices, Represents n-dimensional real numbers;

[0052] The block diagram of the generalized object G(s) is as follows Figure 1 shown.

[0053] in

[0054] The block diagram of the l1 standard problem is as follows Figure 2 As shown. l1 performance is from w(t k ) to z(t k ) satisfies the performance ||z(t k )|| ∞ / ||w(t k )|| ∞ ≤γ, where γ is a given value. The PID controller function is given by

[0055]

[0056] Where K p ,K i , are the coefficients of the proportional term, integral term, and differential term respectively.

[0057] Step 2, obtaining a PID controller according to the discrete time linearization model, and converting the PID controller into a discrete time SOF control system;

[0058] Step 3: Write the discrete-time SOF control system in the form of a DELTA operator;

[0059] Step 4: Calculate the expected value of the SOF control system in the form of a DELTA operator and obtain the control parameters of the SOF control system in the form of a DELTA operator. According to the control parameters Determine the control parameters of the PID controller;

[0060] Step 5: Use the PID controller with determined control parameters to control the controlled system.

[0061] After discretizing the controlled system, this embodiment proposes a PID controller with L1 performance based on the DELTA operator for discrete time systems. First, the PID controller is converted into a static output feedback (SOF) control system using system enhancement technology, and then the SOF control system is represented in the DELTA domain. Finally, the expected value of the SOF controller is calculated to obtain the PID controller. The controlled system of this embodiment can be an electromagnetic motor, piezoelectric ceramics, and satellite laser communication precision tracking system in engineering.

[0062] In a preferred embodiment, step 4 includes:

[0063] Step 41, set the number of iterations k, the initial value of k is 1, and select the initial value P k-1 =X k-1 =I;

[0064] Step 42: Find the minimum Trace(X) that satisfies the boundary conditions. k-1 P k +P k-1 X k ), get the P at this time k and X k : Boundary conditions:

[0065]

[0066] Trace(·) represents the trace of the matrix, P k and X k When X is the decision variable for k, k P k The inverse matrix of , α, σ, γ represent given values, T represents the sampling period, Controlled system Controlled system

[0067] Step 43: If k>N or ||Tr(X k-1 P k +P k-1 X k )-2M||<ε, M is an integer, M×M is the dimension of P, N is a predefined maximum number of iterations, ε is a specified tolerance, and the process goes to step 44; otherwise, k=k+1, and the process goes to step 42;

[0068] Step 44: When P=P k Time Solution

[0069] In this embodiment, let s k =Trace(X k-1 P k +P k-1 X k ) and assume that s k+1 is the optimal value, then s k+1 ≤Trace(X k P k-1 +P k X k-1 )=s k On the other hand, there are the following facts:

[0070]

[0071] Obviously, the sequence S is monotonically decreasing, and its lower bound is 2M. The above analysis means that the minimum Trace(X) under the boundary conditions is satisfied. k-1 P k +P k-1 X k ) converges and is a local optimization method, which means that even if no solution can be found, there are feasible solutions.

[0072] The purpose of this implementation is to discretize the controlled system in time and design a PID controller for the discrete time system based on the DELTA operator so that the closed-loop system is stable and the channel is from w(t k ) to z(t k ) satisfies the l1 performance || z(t k )|| ∞ / ||w(t k )|| ∞ ≤γ, where ||w(t k )|| ∞ ≠0, under zero initial conditions. Since the controlled system of this embodiment is a linear system, the l1 performance can be expressed in another way, that is, for any disturbance input ||w(t k )|| ∞ ≤1 satisfies the controlled output ||z(t k )|| ∞ ≤γ.

[0073] Principle derivation:

[0074] In step 2 of this embodiment, the discrete time SOF control system is:

[0075] u(t k )=F z y a (t k ) (1)

[0076] in:

[0077]

[0078]

[0079]

[0080]

[0081]

[0082] Δy(t k )=y(t k )-y(t k-1 )

[0083] F z is the control coefficient of the discrete-time SOF control system.

[0084] Although the PID controller has been transformed into a SOF control system, the dynamic characteristics of the two systems are exactly the same. In addition, the two systems are k ) to z(t k ) have the same transfer function. Therefore, they have the same performance indicators. In view of this, it can be said that the two systems are equivalent.

[0085] In step 3 of this embodiment, the DELTA operator form SOF control system is:

[0086]

[0087]

[0088] It can be observed that if the SOF matrix is ​​determined, the coefficients of the PID controller can be derived.

[0089]

[0090] in Select the quadratic function As a Lyapunov function, and for V(x a (t k )) Perform DELTA operator operation

[0091]

[0092] in If ||w(t k )|| ∞ ≤1 and satisfies

[0093] δV(x a (t k))-α(w T (t k )w(t k )-V(x a (t k )))<0 (7)

[0094] Then the collection

[0095] Where α>0 is a predefined scalar. Substituting (6) into (7) yields

[0096]

[0097] Applying Schur's complement theorem to (8), we can obtain

[0098]

[0099] If (9) is satisfied, for zero initial conditions, ||w(t k )|| ∞ ≤1 for any disturbance input, the state vector x a (t k ) will never exceed In addition, if

[0100]

[0101] For a fixed scalar σ(0<σ<γ 2 ) is established, from w(t k ) to z(t k ) satisfies the l1 performance γ>0. To prove this, we have

[0102]

[0103] According to the above analysis, we can conclude that:

[0104] Theorem 1. The SOF control system given by is asymptotically stable under the specified l1 performance γ>0 if there exists a predefined scalar α>0, 0<σ<γ 2 , matrices P>0 and X>0, so that (10) and

[0105] PX=I (12)

[0106]

[0107]

[0108] Established, of which

[0109] Proof The result of Theorem 1 can be derived from (9) and (10). Next, we will prove that (9) can be derived from (12)-(14). In fact, inequality (9) can be rewritten as:

[0110]

[0111] where D = diag{P,I,P} and

[0112]

[0113] From the projection theorem, we can see that inequality (15) holds if and only if

[0114]

[0115]

[0116] Note that (16) is the same as (13). Since (ΛD) ⊥ =D -1 Λ ⊥ and X = P -1 (see (12)), inequality (18) is equal to

[0117]

[0118] in

[0119]

[0120] Note that (18) is the same as (14).

[0121] Since there is a product term of decision variables P and X in (13), the condition of Theorem 1 is BMIs and cannot be directly solved by existing convex optimization methods. Considering that X is the inverse matrix of P, equation constraint (17) is equivalent to

[0122] Trace(PX)=2M

[0123]

[0124] Where M is an integer and M×M is the dimension of P. Then find the required feedback matrix

[0125] Embodiment 1:

[0126] The well-known inverted pendulum example is used in this example. The dynamics of the pendulum are as follows

[0127]

[0128] Where g = 9.8 m / s 2, a=1 / (M+m), M=8kg, m=2kg, l=0.5m, x1 is the angle of the pendulum, x2 is the angular velocity, C1=[1 1], D1=0.1, C2=[1 0].

[0129] Given a sampling period T = 0.001s, a discrete time linear model can be obtained, where

[0130]

[0131]

[0132] C z1 =[1 1],D z1 =0.1,C z2 =[1 0]

[0133] The PID controller is used to control the inverted pendulum. Assuming the l1 performance γ = 0.4, we will show how to use Algorithm 1 to design the PID controller.

[0134] Step (1): Convert the PID controller composed of discrete time linearization model into SOF control system, where

[0135]

[0136]

[0137]

[0138]

[0139] F z =[K p K i K d ] is the SOF matrix that needs to be determined. In addition, rewrite the above discrete-time SOF in the delta domain

[0140] Control system. Delta operator SOF control system can be written as DELTA operator form SOF control system, where

[0141]

[0142]

[0143]

[0144]

[0145] Step (2): Set α = 0.8, σ = 0.05 and ε = 10 -5From Algorithm 1, the following SOF controller can be derived after 279 iterations.

[0146]

[0147] It is easy to reproduce the PID controller from this, where K p =113.6616, K i =4.9653×10 -3 , K d =14817.2985.

[0148] In order to prove the effectiveness of the designed PID controller, a simulation is carried out. The closed-loop system is under the initial condition x0 = [π / 4 0] T The following response is Figure 3 As shown in Figure 2, a uniform random perturbation w is added at t ≥ 8s. Figure 3 a and Figure 3 b It can be seen that the PID controller can stabilize the inverted pendulum near the origin. Figure 3 c, when w is imposed on the system, the controller adjusts its output to reduce the effect of the disturbance. In this figure, the detailed response of u over the time interval [10 10.1] is also depicted. The response of the controlled output Z is shown in Figure 4 As shown, where ||z|| ∞ =0.1040<γ=0.4.

[0149] This method is also applicable to multivariable PID controllers. In this embodiment, for example, a four-cylinder process system. Assume that the discrete time linearization model of the nonlinear system at the equilibrium point of T = 0.01s can be described by the discrete time linearization model of this embodiment, where

[0150]

[0151]

[0152]

[0153] Among them γ = 0.6, α = 0.001, σ = 0.3, ε = 10 -5 , the coefficients of the PID controller are as follows:

[0154]

[0155]

[0156]

[0157] The simulation results under zero initial conditions are as follows Figure 5 As shown, where ||z||∞ =0.2001<γ=0.6. Obviously, the designed controller achieves satisfactory performance.

[0158] Embodiment 2:

[0159] In this embodiment, a piezoelectric ceramic actuator is used as an example. Given a sampling period T = 0.0002s, its discrete time linear system can be described by a discrete time linearization model, where

[0160]

[0161] C z1 =[0.01 0.01],C z2 =[95 6.908],D z1 =0

[0162] The PID controller is obtained according to the discrete time linearization model. The PID controller is used to control the position of the piezoelectric ceramic. Assuming that the l1 performance γ=130.484, the following will show how to design the PID controller.

[0163] Convert the PID controller to a discrete-time SOF control system where

[0164]

[0165]

[0166]

[0167]

[0168] F z =[K p K i K d ] is the SOF matrix that needs to be determined. In addition, rewrite the above discrete-time SOF in the delta domain

[0169] Control system. Delta operator SOF control system can be expressed by DELTA operator form SOF control system, where

[0170]

[0171]

[0172]

[0173]

[0174] Set α = 1.6826 × 10 4, σ = 200 and ε = 10 -5 , the following SOF controller can be deduced through iteration

[0175]

[0176] It is easy to reproduce the PID controller from it, where K p = 0.9, K i = 750, K d = 0.

[0177] Similarly, the method provided in this application is also applicable to the angle control of a fast steering mirror driven by a piezoelectric ceramic.

[0178] Although the present invention has been described herein with reference to specific embodiments, it should be understood that these embodiments are merely examples of the principles and applications of the present invention. Therefore, it should be understood that many modifications can be made to the exemplary embodiments, and other arrangements can be designed, as long as they do not depart from the spirit and scope of the present invention as defined by the appended claims. It should be understood that the different dependent claims and the features herein can be combined in a manner different from that described in the original claims. It should also be understood that the features described in connection with a single embodiment can be used in other embodiments.

Claims

1. A PID control method based on DELTA operator description, characterized in that: The method comprises: S1. Determine the dynamic equation of the controlled system, discretize the dynamic equation in time, and obtain the discrete time linearization model: In the formula, x(t k ), is the state vector of the controlled system, including position, velocity, and acceleration, is the disturbance input, and are the controlled system output and the measured output, respectively, u(t k ) is the control quantity input of the controlled system; A z , B z1 , B z2 , C z1 , D z1 and C z2 are coefficient matrices, Represents n-dimensional real numbers; S2, obtaining a PID controller according to a discrete time linearization model, and converting the PID controller into a discrete time SOF control system; S3, write the discrete time SOF control system into the form of DELTA operator; S4. Calculate the expected value of the SOF control system in the form of a DELTA operator and obtain the control parameters of the SOF control system in the form of a DELTA operator According to the control parameters Determine the control parameters of the PID controller; S5, controlling the controlled system using a PID controller that determines control parameters; The S4 includes: S41, set the number of iterations k = 1, select the initial value P k-1 =X k-1 =I; S42. Find the minimum Trace(X) that satisfies the boundary conditions. k-1 P k +P k-1 X k ), obtain Pk and Xk at this time: boundary conditions: Trace(·) represents the trace of the matrix, P k and X k When X is the decision variable for k, k P k The inverse matrix of , α, σ, γ represent given values, T represents the sampling period, S43、If k>N or ||Tr(X k-1 P k +P k-1 X k )-2M||<ε, M is an integer, M×M is the dimension of P, N is a predefined maximum number of iterations, ε is a specified tolerance, and the process goes to S44; otherwise, k=k+1, and the process goes to S42; S44, when P = P k Time Solution 2. The PID control method based on DELTA operator description according to claim 1, characterized in that: α>0,||z(t k )|| ∞ / ||w(t k )|| ∞ ≤γ,0<σ<γ 2 。 3. The PID control method based on DELTA operator description according to claim 2 is characterized in that: Perturbation input ||w(t k )|| ∞ ≤1, the controlled system outputs ||z(t k )|| ∞ ≤γ.

4. The PID control method based on DELTA operator description according to claim 1, characterized in that: In S2, the discrete time SOF control system is: u(t k )=F z y a (t k ) in: Δy(t k )=y(t k )-y(t k-1 ) F z is the control coefficient of the discrete-time SOF control system.

5. The PID control method based on DELTA operator description according to claim 4 is characterized in that: In S3, the SOF control system in the form of a DELTA operator is:

6. The PID control method based on DELTA operator description according to claim 5, characterized in that: In S2, the PID controller is: K p ,K i ,K d are the coefficients of the proportional term, integral term, and differential term respectively.

7. The PID control method based on DELTA operator description according to claim 1, characterized in that: The controlled system is a piezoelectric ceramic actuator, an electromagnetic motor, a piezoelectric fast tilt mirror for beam control, or a satellite laser communication precision tracking system.

8. A computer-readable storage device storing a computer program, characterized in that: When the computer program is executed, the PID control method described based on the DELTA operator as described in any one of claims 1 to 7 is implemented.

9. A PID control device based on DELTA operator description, comprising a storage device, a processor, and a computer program stored in the storage device and executable on the processor, characterized in that: The processor executes the computer program to implement the PID control method described based on the DELTA operator as described in any one of claims 1 to 7.

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