CN112068424A - Discrete repetitive control method of ellipse approximation law adopting disturbance compensation - Google Patents

Abstract

A discrete repetitive control method for a motor servo system by using an ellipse approximation law of disturbance compensation, wherein a given module generates periodic reference signals, a periodic feedback link is constructed, the ellipse approximation law is provided, and equivalent disturbance compensation is introduced into the approximation law; constructing an ideal switching dynamic state based on an ellipse approach law, designing a controller according to the ideal switching dynamic state, and taking a signal obtained by current calculation as the input of a servo system; the specific controller parameter setting can be carried out according to the convergence performance index of the representation system, and a calculation formula of a monotone convergence layer boundary, an absolute convergence layer boundary and a quasi-sliding mode band boundary in the convergence process of the representation system is provided. The repetitive controller with the equivalent disturbance compensation ellipse approximation law can improve the tracking accuracy of a system and completely inhibit the existing periodic disturbance.

Description

A Discrete Repetitive Control Method Using Disturbance Compensated Ellipse Reaching Law

technical field

The invention relates to a discrete repetitive control method using an ellipse reaching law of disturbance compensation, which is used for precise position control of a motor servo system, and is also suitable for other industrial occasions with periodic running processes.

Background technique

The repetitive control uses the tracking error signal to correct the controller at the previous moment to form the input of the controller at the current moment, so the repetitive control has the characteristics of completely suppressing periodic interference and realizing precise control. At present, repetitive control methods have been widely used in various high-precision servo motor drive systems.

它可由以这个含周期时延(e^-Ts)的正反馈回路实现，不考虑输入信号的具体形式，只要给定初始段信号，内模模块就会对输入信号逐周期累加，重复输出与上周期相同的信号。The repetitive control technique is a control method based on endometrial control. The so-called endometrial principle is that when a signal is regarded as the output of an autonomous system, and the model of the signal is "embedded" in a stable closed-loop system, the controlled output can completely track the reference signal. Therefore, to design the controller according to the endometrial principle, it is necessary to construct a period T reference endometrial

It can be realized by this positive feedback loop with periodic delay (e ^-Ts ), regardless of the specific form of the input signal, as long as the initial segment signal is given, the internal model module will accumulate the input signal cycle by cycle, repeat the output and the above signals with the same period.

Using the approach law method to design the sliding mode controller of the deterministic system, the design process is clear, the controller parameter adjustment method is clear, and it is easy to implement; but for the uncertain system, the same approach law is used to design the controller, resulting in the switching dynamic Depending on the uncertainty term, the degree of influence of the uncertainty term on the switching dynamics determines the control performance of the system. In this way, it is necessary to modify the reaching law and "embed" the interference suppression measures in the switching dynamics, so as to obtain the ideal switching dynamics, so that the reaching law method can be applied to the uncertain system. When designing a discrete controller, the indicators describing the transient and steady-state behavior of the tracking error can be given by the ideal switching dynamics. Specifically, there are the following three indicators: the monotonic convergence layer boundary, the absolute convergence layer boundary and the quasi-sliding mode zone boundary. In fact, the specific values of the three indicators depend on the controller parameters. Different controller parameters have different values of the three indicators. Once the ideal switching dynamic form is given, the specific expressions of the three indicators can be given in advance, which can be used for controller parameter tuning.

SUMMARY OF THE INVENTION

In order to overcome the deficiencies of the prior art, the present invention proposes a discrete repetitive control method for a motor servo system using a disturbance-compensated ellipse reaching law. In order to make the closed-loop system have the preset expected error tracking performance, and can effectively suppress chattering, a novel reaching law-elliptic reaching law is proposed, and the motor servo is designed according to the ideal switching dynamic constructed by the reaching law. Duplicate controller. While realizing the complete suppression of periodic interference components, considering the existence of aperiodic components of the disturbance, equivalent disturbance compensation is introduced into the closed-loop system to suppress the aperiodic interference, so as to improve the control performance and enable the motor servo system to achieve high speed and high precision. track.

The technical scheme adopted by the present invention to solve the above-mentioned technical problems is:

A discrete repetitive control method using a disturbance-compensated ellipse reaching law, the control method comprising the following steps:

1) Differential equation model of servo system

e(k+1)=Ae(k)+b(u(k)+w(k)) (1)

Among them, e(k), e(k+1) represent the tracking error of the servo system at time k, k+1, u(k) represents the input control of the servo system at time k, w(k) represents the servo system at time k The interference signal, A represents the servo system matrix, b represents the control coefficient of the system;

2) Given a periodic reference signal, satisfy

r(k)=r(k-N) (2)

Among them, N represents the period of the reference signal, r(k) represents the reference signal of the servo system at time k, and r(k-N) represents the reference signal of the servo system at the kth time of the previous cycle;

3) Define the tracking error

e(k)=y(k)-r(k)

(3)

Among them, y(k) represents the system output of the servo system at time k;

4) The linear switching function is selected as s(k)=c ^T e(k), c ^T is the gain parameter, and its selection determines the convergence and convergence speed of the system on the sliding mode surface;

5) Construct equivalent disturbance

d(k)=c ^T b(w(k)-w(kN)) (4)

Among them, d(k) represents the equivalent disturbance signal of the servo system at time k, and w(k-N) represents the disturbance signal of the servo system at the kth time of the previous cycle;

6) Constructing discrete-time ellipse reaching law

s(k+1)=(1-ρ)s(k)-εfal(s(k),δ) (5)

ρ、ε和δ均为可调参数，且0＜ρ＜1，δ＞0，ε＞0；Among them, s(k+1) represents the switching function at time k+1; the elliptic function constructed in the reaching law is

ρ, ε and δ are all adjustable parameters, and 0<ρ<1, δ>0, ε>0;

7) According to the above-mentioned ellipse reaching law, construct the ideal switching dynamics

s(k+1)=(1-ρ)s(k)-εfal(s(k),δ)+d(k)-d ^* (k) (6)

得|d^*(k)-d(k)|≤Δ，其中Δ为理想切换动态公式(6)中干扰的界；Among them, d ^* (k) represents the compensation amount of d(k) at time k; denote d _u and d _l as the upper and lower limits of d(k) respectively, and d(k) satisfies d _u ≥d(k)≥d _l , Pick

get |d ^* (k)-d(k)|≤Δ, where Δ is the boundary of interference in the ideal switching dynamic formula (6);

8) Construct the model of discrete sliding mode repetitive controller according to the described ideal switching dynamics:

u(k)=u(kN)+(c ^T b) ^-1 [(1-ρ)s(k)-εfal(s(k),δ)-s(k+1-N)-c ^T A (e(k)-e(kN))-d ^* (k)] (7)

Among them, u(kN) represents the control variable of the servo system at the kth time of the previous cycle, (c ^T b) ^-1 represents the matrix coefficient, and s(k+1-N) represents the kth time of the servo system in the previous cycle. The switching function at time +1, c ^T A represents the gain coefficient, and e(kN) represents the tracking error variable of the servo system at the k-th time of the previous cycle;

Taking u(k) as the input signal of the controller of the servo object, the output signal y(k) of the servo system is obtained by measurement, which changes with the reference signal r(k).

Further, after the repetitive controller design is completed, the controller parameters are tuned according to the indicators that characterize the convergence performance of the system. In order to characterize the convergence performance of the system, the concepts of monotonic convergence layer boundary, absolute convergence layer boundary and pseudo-sliding mode zone boundary are introduced. as follows:

Monotonic convergence layer boundary Δ _MDR : outside the monotone convergence layer boundary, s(k) monotonically decreases with the same sign, that is

Absolute convergence layer boundary Δ _AL : outside the absolute convergence layer, |s(k)| decreases monotonically, namely

The quasi-sliding mode band boundary Δ _QSM : the system moves in a field where the switching plane s(k)=c ^T e(k)=0, once it enters the field, it will stabilize in it, that is,

① Monotonic convergence layer boundary Δ _MDR

in

②Absolute convergence layer boundary Δ _AL

in

③ Pseudo sliding mode zone boundary Δ _QSM

(i)δ≤Z ₁

(ii) Z ₁ <δ<Z ₂

(iii) Z ₁ <Z ₂ <δ

in

Still further, the adjustable parameters of the controller include ρ, ε and δ; the parameter setting is based on the index characterizing the convergence process.

Further, in the step 5), assuming that d _u ≥ d(k) ≥ d _l , then

The technical idea of the present invention is that the discrete repetitive controller of the motor servo system is designed based on the discrete time ellipse reaching law, which is a time domain design method, which is different from the frequency domain method commonly used at present. Considering the given reference signal when designing the controller, the designed controller is more intuitive and simple, and it is easy to describe the tracking performance of the system. The time domain design of the controller is also easy to combine with the existing interference suppression methods. The designed repetitive controller can completely suppress the periodic interference signal and reduce the error caused by the aperiodic interference signal, so as to realize the fast response to the given reference signal. High precision tracking.

The effects of the invention are mainly manifested in: fast convergence performance, accelerated interference suppression and high control precision.

Description of drawings

Figure 1 is a block diagram of an AC permanent magnet synchronous motor servo system.

Fig. 2 is the structure block diagram of the repetitive control controller of the ellipse reaching law.

3 is a schematic diagram of _ΔMDR , _ΔAL and _ΔQSM when c=−0.5, ε=0.1, ρ=0.6, δ=0.8, Δ=0.3.

FIG. 4 is a schematic diagram of _ΔMDR , _ΔAL and _ΔQSM when c=−0.5, ε=0.1, ρ=0.5, δ=0.9, Δ=0.3.

Figure 5-12 shows the experimental results of the PMSM control system when the feedback controller parameters are c=-0.5, ρ=0.5, ε=0.5, and δ=0.9, where:

Figure 5 shows the reference position signal and the actual position signal under the action of the feedback controller based on the ellipse reaching law.

Figure 6 is the controller voltage signal under the action of the feedback controller based on the elliptic reaching law.

FIG. 7 is a graph of the switching function under the action of the feedback controller based on the ellipse reaching law.

Fig. 8 is a histogram of switching function distribution under the action of a feedback controller based on elliptic reaching law.

Figure 9 shows the reference position signal and the actual position signal under the action of the feedback controller based on the ellipse reaching law and equivalent disturbance compensation.

Figure 10 is the controller voltage signal under the action of the feedback controller based on elliptic reaching law and equivalent disturbance compensation.

Figure 11 is a graph of the switching function under the action of a feedback controller based on elliptic reaching law and equivalent disturbance compensation.

Figure 12 is a histogram of switching function distribution under the action of a feedback controller based on elliptic reaching law and equivalent disturbance compensation.

Figure 13-20 shows the experimental results of the PMSM control system when the repeated controller parameters are c=-0.5, ρ=0.5, ε=0.5, and δ=0.9, where:

Fig. 13 is the reference position signal and the actual position signal under the action of the repetitive controller based on the ellipse reaching law.

Figure 14 is the controller voltage signal under the action of the repetitive controller based on the elliptic reaching law.

Figure 15 is a graph of the switching function under the action of the repetitive controller based on the ellipse reaching law.

Figure 16 is a histogram of the distribution of switching functions under the action of a repetitive controller based on the ellipse reaching law.

Fig. 17 is the reference position signal and the actual position signal under the action of the repetitive controller based on the ellipse reaching law and equivalent disturbance compensation.

Figure 18 is the controller voltage signal under the action of the repetitive controller based on elliptic reaching law and equivalent disturbance compensation.

Figure 19 is a graph of the switching function under the action of a repetitive controller based on elliptic reaching law and equivalent disturbance compensation.

Figure 20 is a histogram of the switching function distribution under the action of a repetitive controller based on elliptic reaching law and equivalent disturbance compensation.

Figures 21-28 are the experimental results of the PMSM control system when the feedback controller parameters are c=-0.5, ρ=0.7, ε=0.3, and δ=0.8, where:

Figure 21 shows the reference position signal and the actual position signal under the action of the feedback controller based on the ellipse reaching law.

Figure 22 is the controller voltage signal under the action of the feedback controller based on the elliptic reaching law.

FIG. 23 is a graph of the switching function under the action of the feedback controller based on the ellipse reaching law.

Figure 24 is a histogram of switching function distribution under the action of a feedback controller based on elliptic reaching law.

Figure 25 shows the reference position signal and the actual position signal under the action of the feedback controller based on the ellipse reaching law and equivalent disturbance compensation.

Figure 26 is the controller voltage signal under the action of the feedback controller based on elliptic reaching law and equivalent disturbance compensation.

Figure 27 is a graph of the switching function under the action of a feedback controller based on elliptic reaching law and equivalent disturbance compensation.

Figure 28 is a histogram of the switching function distribution under the action of a feedback controller based on the elliptic attraction law and equivalent disturbance compensation.

Figure 29-36 shows the experimental results of the permanent magnet synchronous motor control system when the repeated controller parameters are c=-0.5, ρ=0.7, ε=0.3, and δ=0.8, where:

Figure 29 is the reference position signal and the actual position signal under the action of the repetitive controller based on the ellipse reaching law.

Figure 30 is the controller voltage signal under the action of the repetitive controller based on the elliptic reaching law.

Figure 31 is a graph of the switching function under the action of the repetitive controller based on the ellipse reaching law.

Figure 32 is a histogram of the distribution of switching functions under the action of a repetitive controller based on the ellipse reaching law.

Figure 33 shows the reference position signal and the actual position signal under the action of the repetitive controller based on the ellipse reaching law and equivalent disturbance compensation.

Figure 34 is the controller voltage signal under the action of the repetitive controller based on elliptic reaching law and equivalent disturbance compensation.

Figure 35 is a graph of the switching function under the action of a repetitive controller based on elliptic reaching law and equivalent disturbance compensation.

Figure 36 is a histogram of the switching function distribution under the action of a repetitive controller based on elliptic reaching law and equivalent disturbance compensation.

Detailed ways

The specific embodiments of the present invention will be further described with reference to the accompanying drawings.

With reference to Fig. 1 and Fig. 2, a kind of discrete repetitive control method adopting the ellipse reaching law of equivalent disturbance compensation, wherein, Fig. 1 is a block diagram of a motor servo system; Fig. 2 is the repetitive control controller structure schematic diagram of ellipse reaching law; The control method includes the following steps:

1) Differential equation model of servo system

e(k+1)=Ae(k)+b(u(k)+w(k)) (1)

Among them, e(k), e(k+1) represent the tracking error of the servo system at time k, k+1, u(k) represents the input control of the servo system at time k, w(k) represents the servo system at time k The interference signal, A represents the servo system matrix, b represents the control coefficient of the system;

2) Given a periodic reference signal, satisfy

r(k)=r(k-N) (2)

Among them, N represents the period of the reference signal, r(k) represents the reference signal of the servo system at time k, and r(k-N) represents the reference signal of the servo system at the kth time of the previous cycle;

3) Define the tracking error

e(k)=y(k)-r(k) (3)

Among them, y(k) represents the system output of the servo system at time k;

4) The linear switching function is selected as s(k)=c ^T e(k), c ^T is the gain parameter, and its selection determines the convergence and convergence speed of the system on the sliding mode surface;

5) Construct equivalent disturbance

d(k)=c ^T b(w(k)-w(kN)) (4)

Among them, d(k) represents the equivalent disturbance signal of the servo system at time k, and w(k-N) represents the disturbance signal of the servo system at the kth time of the previous cycle;

6) Constructing discrete-time ellipse reaching law

s(k+1)=(1-ρ)s(k)-εfal(s(k),δ) (5)

ρ、ε和δ均为可调参数，且0＜ρ＜1，δ＞0，ε＞0；Among them, s(k+1) represents the switching function at time k+1; the elliptic function constructed in the reaching law is

ρ, ε and δ are all adjustable parameters, and 0<ρ<1, δ>0, ε>0;

7) According to the above-mentioned ellipse reaching law, construct the ideal switching dynamics

s(k+1)=(1-ρ)s(k)-εfal(s(k),δ)+d(k)-d ^* (k) (6)

得|d^*(k)-d(k)|≤Δ，其中Δ为理想切换动态公式(6)中干扰的界；Among them, d ^* (k) represents the compensation amount of d(k) at time k; denote d _u and d _l as the upper and lower limits of d(k) respectively, and d(k) satisfies d _u ≥d(k)≥d _l , Pick

get |d ^* (k)-d(k)|≤Δ, where Δ is the boundary of interference in the ideal switching dynamic formula (6);

8) Construct the model of discrete sliding mode repetitive controller according to the described ideal switching dynamics:

u(k)=u(kN)+(c ^T b) ^-1 [(1-ρ)s(k)-εfal(s(k),δ)-s(k+1-N)-c ^T A (e(k)-e(kN))-d ^* (k)] (7)

Among them, u(kN) represents the control variable of the servo system at the kth time of the previous cycle, (c ^T b) ^-1 represents the matrix coefficient, and s(k+1-N) represents the kth time of the servo system in the previous cycle. The switching function at time +1, c ^T A represents the gain coefficient, and e(kN) represents the tracking error variable of the servo system at the k-th time of the previous cycle;

Taking u(k) as the input signal of the controller of the servo object, the output signal y(k) of the servo system is obtained by measurement, which changes with the reference signal r(k).

Further, after the repetitive controller design is completed, the controller parameters are tuned according to the indicators that characterize the convergence performance of the system. In order to characterize the convergence performance of the system, the concepts of monotonic convergence layer boundary, absolute convergence layer boundary and pseudo-sliding mode zone boundary are introduced. as follows:

Monotonic convergence layer boundary Δ _MDR : outside the monotone convergence layer boundary, s(k) monotonically decreases with the same sign, that is

Absolute convergence layer boundary Δ _AL : outside the absolute convergence layer, |s(k)| decreases monotonically, namely

The quasi-sliding mode band boundary Δ _QSM : the system moves in a field where the switching plane s(k)=c ^T e(k)=0, once it enters the field, it will stabilize in it, that is,

① Monotonic convergence layer boundary Δ _MDR

in

②Absolute convergence layer boundary Δ _AL

in

③ Pseudo sliding mode zone boundary Δ _QSM

(i)δ≤Z ₁

(ii) Z ₁ <δ<Z ₂

(iii) Z ₁ <Z ₂ <δ

in

Still further, the adjustable parameters of the controller include ρ, ε and δ; the parameter setting is based on the index characterizing the convergence process.

Further, in the step 5), assuming that d _u ≥ d(k) ≥ d _l , then

Embodiment: a discrete repetitive control method for a motor servo system using an ellipse reaching law of equivalent disturbance compensation, comprising the following steps:

Step 1. Second-order differential equation model of the motor servo object

y(k+1)+a ₁ y(k)+a ₂ y(k-1)=b ₁ u(k)+b ₂ u(k-1)+w(k)

Among them, y(k+1), y(k) and y(k-1) represent the output signals of the servo system at times k+1, k and k-1, respectively, u(k) and u(k-1 ) represent the input signal of the servo system at time k and k-1, respectively, w(k) represents the disturbance signal of the system at time k, a ₁ , a ₂ , b ₁ , and b ₂ represent the system parameters of the servo system, which are obtained by parameter estimation is obtained;

Step 2. Given a periodic reference signal, satisfy

r(k)=r(k-N) (2)

Among them, N represents the period of the reference signal, r(k) represents the reference signal of the servo system at time k, and r(k-N) represents the reference signal of the servo system at the kth time of the previous cycle;

Step 3. Construct discrete-time ellipse reaching law

s(k+1)=(1-ρ)s(k)-εfal(s(k),δ) (5)

ρ、ε和δ均为可调参数，且0＜ρ＜1，δ＞0，ε＞0；Among them, the elliptic function in the reaching law is

ρ, ε and δ are all adjustable parameters, and 0<ρ<1, δ>0, ε>0;

Step 4. Introduce equivalent disturbance to construct an ellipse reaching law with disturbance suppression

s(k+1)=(1-ρ)s(k)-εfal(s(k),δ)+d(k)-d ^* (k) (6)

Among them, d(k) represents the equivalent disturbance signal of the permanent magnet synchronous motor at time k, and d ^* (k) represents the compensation amount of d(k) at time k;

Step 5. Given the reference signal r(k), write the system equation from the second-order differential equation model of the motor servo object

e(k+1)=Ae(k)+b(v(k)+w(k)) (1)

为系统跟踪误差方程中定义的控制输入，且系统矩阵和控制系数分别为

Where, e(k)=[e ₁ (k) e ₂ (k)] ^T , e ₁ (k)=y(k-1)-e(k-1), e ₂ (k)=y(k )-e(k), e ₁ (k) is the tracking error at time k-1, e ₂ (k) is the tracking error at time k, input

is the control input defined in the system tracking error equation, and the system matrix and control coefficient are respectively

Step 6. Select the linear switching function as s(k)=c ^T e(k), where c is selected as c=[-0.5 1] ^T ;

Step 7. For the second-order difference equation model of the motor servo object, in order to achieve the ideal switching dynamics, the discrete sliding mode repetitive controller should be designed as:

For the repeating controller design above, do the following:

1) e(k), y(k), y(k-1) and y(k-1-N) can be obtained by measurement, u(k-1) and u(k-1-N) represent control The stored value of the signal, which can be read from memory.

2) When the reference signal satisfies r(k)=r(k-1), the discrete repetitive controller is also suitable for the constant regulation problem, and the equivalent disturbance at this time is d(k)=w(k)-w( k-1); wherein, r(k-1) represents the reference signal of the servo system at time k-1, and w(k-1) represents the interference signal of the servo system at time k-1;

3) The above-mentioned repetitive controller is aimed at the second-order system, and the design result of the higher-order system can also be given according to the same method.

Step 8. Set the controller parameters according to the monotonic convergence layer boundary Δ _MDR , the absolute convergence layer boundary Δ _AL and the quasi-sliding mode zone boundary Δ _QSM that characterize the system convergence speed to achieve the best control effect. The controller parameters mainly include: ellipse parameter δ, adjustable parameters ρ, ε and equivalent disturbance bound Δ;

According to the above definitions of Δ _MDR , Δ _AL and Δ _QSM , the determined boundary values are as follows:

Monotonic Convergence Layer Boundary Δ _MDR

Absolutely convergent layer boundary Δ _AL

Pseudo-Sliding Mode Band Boundary Δ _QSM

① Monotonic convergence layer boundary Δ _MDR

in

②Absolute convergence layer boundary Δ _AL

in

③ Pseudo sliding mode zone boundary Δ _QSM

(i)δ≤Z ₁

(ii) Z ₁ <δ<Z ₂

(iii) Z ₁ <Z ₂ <δ

in

According to equations (11)-(16), the value of each boundary is calculated to determine the tracking performance of the closed-loop system.

In this embodiment, the permanent magnet synchronous motor servo system performs the repetitive tracking task in a fixed interval as an example, and the position reference signal thereof has the characteristic of periodic symmetry. With TMS320F2812DSP as the controller, AC servo motor APM-SB01AGN as the control object, and ELMO AC servo driver and host computer to form a permanent magnet synchronous motor servo system to control the motor position. The servo system adopts three-loop control, the current loop and speed loop controllers are provided by the ELMO driver, and the position loop is provided by the DSP development board.

To design a position loop controller, it is necessary to establish mathematical models of servo objects other than the position loop, including current loop, speed loop, power driver, AC permanent magnet synchronous servo motor body and detection device. The mathematical model of the servo object obtained by parameter estimation is

y(k+1)-1.8949y(k)+0.8949y(k-1)=1.7908u(k)-0.5704u(k-1)+w(k) (17)

Among them, y(k), u(k) respectively represent the position output and speed given signal (control input) of the position servo system, and w(k) represent the interference signal.

Since the sinusoidal signal is used as the reference signal of the system in this embodiment, the repetitive controller can take the form of the controller given by equation (6), and its specific expression can be written as

In this example, numerical simulation and experimental results will be used to illustrate the effectiveness of the repetitive controller given by the present invention.

The given position reference signal is r(k)=10sin( _2πfkTs ), the unit is degree (deg), the frequency f=1Hz, the sampling period _Ts =0.001s, and the sampling period N=1000. During the simulation, the selected interference signal w(k) is composed of periodic disturbance and aperiodic random disturbance, and its specific form is

w(k)=2sin(2πfkT _s )+0.15sgn(sin(2kπ/150)) (19)

Under the action of the repetitive controller (18), different controller parameters ρ, ε and δ are selected, and the three boundary layers of the servo system are also different. In order to illustrate the theoretical correctness of the patent of the present invention about the monotonic convergence layer boundary Δ _MDR , the absolute convergence layer boundary Δ _AL and the quasi-sliding mode zone boundary Δ _QSM , numerical simulations are carried out.

1) When the controller parameters c=-0.5, ε=0.1, ρ=0.6, δ=0.8, Δ=0.3, according to the calculation formula of the three boundary values, we can get

Δ _MDR = 0.9926, Δ _AL = 0.6642, Δ _QSM = 0.3145

2) When the controller parameters c=-0.5, ε=0.1, ρ=0.5, δ=0.9, Δ=0.3, according to the calculation formula of the three boundary values, we can get

Δ _MDR = Δ _AL = 0.7987, Δ _QSM = 0.3109

The simulation results are shown in Figure 3-4. Given the system model, reference signal and interference signal, the above numerical results verify the monotonic convergence layer boundary Δ _MDR , the absolute convergence layer boundary Δ _AL and the pseudo-convergence layer boundary Δ MDR , which characterize the system convergence rate under the action of the repetitive controller given in this patent. Accuracy of the sliding mode band boundary Δ _QSM .

The block diagram of the permanent magnet synchronous motor control system used in the experiment is shown in Figure 1. By setting different controller parameters, the tracking performance of discrete repetitive control based on elliptic reaching law is verified. The given position signal is r _k =A(sin(2πfkT _s )+1), wherein, the amplitude A=135deg, the sampling period T _s =5ms, and the frequency f=1Hz.

Using a feedback controller takes the form

The feedback controller based on disturbance compensation is adopted in the following form

Use a repeating controller of the form

The repetitive controller based on disturbance compensation is of the following form

A. The controller parameters take c=-0.5, ρ=0.5, ε=0.5, δ=0.9, the reference signal period T=4s, and the sampling period Ts=0.005s.

Using the feedback controller (20), the position signal, the controller voltage, the switching function curve and the switching function histogram are shown in Figures 5-8, where _ΔQSM = 0.12deg.

B. The controller parameters take c=-0.5, ρ=0.5, ε=0.5, δ=0.9, the reference signal period T=4s, and the sampling period Ts=0.005s.

Using the feedback controller (21) based on disturbance compensation, the position signal, controller voltage, switching function curve and switching function histogram are shown in Figures 9-12, where _ΔQSM = 0.1deg.

C. The controller parameters take c=-0.5, ρ=0.5, ε=0.5, δ=0.9, the reference signal period T=4s, and the sampling period Ts=0.005s.

Using the repetitive controller (22), the position signal, controller voltage, switching function curve and switching function histogram are shown in Figures 13-16, where _ΔQSM = 0.07deg.

D. The controller parameters take c=-0.5, ρ=0.5, ε=0.5, δ=0.9, the reference signal period T=4s, and the sampling period Ts=0.005s.

Using the repetitive controller (23) based on disturbance compensation, the position signal, controller voltage, switching function curve and switching function histogram are shown in Figures 17-20, where _ΔQSM = 0.05deg.

E. The controller parameters take c=-0.5, ρ=0.7, ε=0.3, δ=0.8, the reference signal period T=4s, and the sampling period Ts=0.005s.

Using the feedback controller (20), the position signal, the controller voltage, the switching function curve and the switching function histogram are shown in Figures 21-24, where Δ _QSM =0.11deg.

F. The controller parameters are taken as c=-0.5, ρ=0.7, ε=0.3, δ=0.8, reference signal period T=4s, sampling period Ts=0.005s.

Using the feedback controller (21) based on disturbance compensation, the position signal, controller voltage, switching function curve and switching function histogram are shown in Figures 25-28, where _ΔQSM = 0.09deg.

G. The controller parameters take c=-0.5, ρ=0.7, ε=0.3, δ=0.8, the reference signal period T=4s, and the sampling period Ts=0.005s.

Using the repetitive controller (22), the position signal, controller voltage, switching function curve and switching function histogram are shown in Figures 29-32, where _ΔQSM = 0.07deg.

H. The controller parameters take c=-0.5, ρ=0.7, ε=0.3, δ=0.8, the reference signal period T=4s, and the sampling period Ts=0.005s.

Using the repetitive controller (23) based on disturbance compensation, the position signal, controller voltage, switching function curve and switching function histogram are shown in Figures 33-36, where _ΔQSM =0.05deg.

I. The controller parameters take c=-0.5, ρ=0.8, ε=0.1, δ=0.7, the reference signal period T=4s, and the sampling period Ts=0.005s.

The above experimental results show that an ellipse reaching law proposed in the present invention can effectively suppress the chattering phenomenon during system trajectory tracking; introduce equivalent disturbances and use equivalent disturbance compensation as the compensation for modeling characteristics and external unknown disturbances, effectively The unknown disturbance is suppressed; combined with the repetitive control, the periodic disturbance is completely suppressed, which further improves the control performance of the system.

Claims (4)

1. a discrete repetitive control method employing the ellipse reaching law of disturbance compensation, is characterized in that: described control method comprises the following steps: 1) Differential equation model of servo system e(k+1)=Ae(k)+b(u(k)+w(k)) (1) Among them, e(k), e(k+1) represent the tracking error of the servo system at time k, k+1, u(k) represents the input control of the servo system at time k, w(k) represents the servo system at time k The interference signal, A represents the servo system matrix, b represents the control coefficient of the system; 2) Given a periodic reference signal, satisfy r(k)=r(k-N) (2) Among them, N represents the period of the reference signal, r(k) represents the reference signal of the servo system at time k, and r(k-N) represents the reference signal of the servo system at the kth time of the previous cycle; 3) Define the tracking error e(k)=y(k)-r(k) (3) Among them, y(k) represents the system output of the servo system at time k; 4) The linear switching function is selected as s(k)=c ^T e(k), c ^T is the gain parameter, and its selection determines the convergence and convergence speed of the system on the sliding mode surface; 5) Construct equivalent disturbance d(k)=c ^T b(w(k)-w(kN)) (4) Among them, d(k) represents the equivalent disturbance signal of the servo system at time k, and w(k-N) represents the disturbance signal of the servo system at the kth time of the previous cycle; 6) Constructing discrete-time ellipse reaching law s(k+1)=(1-ρ)s(k)-εfal(s(k),δ) (5) Among them, s(k+1) represents the switching function at time k+1; the elliptic function constructed in the reaching law is

ρ, ε and δ are all adjustable parameters, and 0<ρ<1, δ>0, ε>0; 7) According to the above-mentioned ellipse reaching law, construct the ideal switching dynamics s(k+1)=(1-ρ)s(k)-εfal(s(k),δ)+d(k)-d ^* (k) (6) Among them, d ^* (k) represents the compensation amount of d(k) at time k; denote d _u and d _l as the upper and lower limits of d(k) respectively, and d(k) satisfies d _u ≥d(k)≥d _l , Pick

get |d ^* (k)-d(k)|≤Δ, where Δ is the boundary of interference in the ideal switching dynamic formula (6); 8) Construct a discrete sliding mode repetitive controller according to the ideal switching dynamics u(k)=u(kN)+(c ^T b) ^-1 [(1-ρ)s(k)-εfal(s(k),δ)-s(k+1-N)-c ^T A (e(k)-e(kN))-d ^* (k)] (7) Among them, u(kN) represents the control variable of the servo system at the kth time of the previous cycle, (c ^T b) ^-1 represents the matrix coefficient, and s(k+1-N) represents the kth time of the servo system in the previous cycle. The switching function at time +1, c ^T A represents the gain coefficient, and e(kN) represents the tracking error variable of the servo system at the k-th time of the previous cycle; Taking u(k) as the input signal of the controller of the servo object, the output signal y(k) of the servo system is obtained by measurement, which changes with the reference signal r(k). 2. a kind of discrete repetitive control method adopting the ellipse reaching law of disturbance compensation as claimed in claim 1 is characterized in that: after the repetitive controller design is completed, the controller parameter setting is carried out according to the index characterizing the system convergence performance; In order to characterize the convergence performance of the system, the concepts of monotonic convergence layer boundary, absolute convergence layer boundary and quasi-sliding mode zone boundary are introduced. The specific definitions are as follows: Monotonic convergence layer boundary Δ _MDR : outside the monotone convergence layer boundary, s(k) monotonically decreases with the same sign, that is

Absolute convergence layer boundary Δ _AL : outside the absolute convergence layer, |s(k)| decreases monotonically, namely

The quasi-sliding mode belt boundary Δ _QSM : the system moves in a field where the switching plane s(k)=c ^T e(k)=0, once it enters the field, it will stabilize in it, that is

① Monotonic convergence layer boundary Δ _MDR

in

②Absolute convergence layer boundary Δ _AL

in

③ Pseudo-sliding mode belt boundary Δ _QSM

(i)δ≤Z ₁

(ii) Z ₁ <δ<Z ₂

(iii) Z ₁ <Z ₂ <δ

in

3. A discrete repetitive control method using an ellipse reaching law of disturbance compensation as claimed in claim 1 or 2, wherein the adjustable parameters of the controller include ρ, ε and δ; Process indicators. 4. A kind of discrete repetitive control method adopting the ellipse reaching law of disturbance compensation as claimed in claim 1 or 2, it is characterized in that: in described step 5), suppose d _u ≥ d(k) ≥ d _l , but

Images