CN109450307B - Discrete repetitive control method for motor servo system based on similar normal distribution attraction law and adopting disturbance expansion state compensation - Google Patents

Abstract

A discrete repetitive control method for a motor servo system based on a similar normal distribution attraction law and adopting disturbance expansion state compensation is characterized in that a given link generates periodically symmetrical reference signals; constructing a periodic feedback link; according to a discrete time type normal distribution attraction law, equivalent disturbance compensation is introduced into the attraction law, the compensation amount is given by a disturbance expansion observer, an e/v signal conversion module is constructed, and an output signal of the e/v signal conversion module is used for correcting the repetitive controller; then, the output signal of the repetitive controller is calculated and used as the control signal input of the controlled object. The influence of the value of the controller parameter on the convergence process of the system tracking error is given. The specific controller parameter setting can be carried out according to the convergence performance index of the representation system, and a calculation method of a monotone decreasing area, an absolute attraction layer and a steady-state error band boundary in the process of representing the convergence of the tracking error is provided. The invention has the advantages of rapid convergence performance, accelerated interference suppression and high control precision.

Description

Discrete repetitive control method for motor servo system based on similar normal distribution attraction law and adopting disturbance expansion state compensation

Technical Field

The invention belongs to the technical field of repetitive control, in particular to a repetitive control method for precise motor servo control, which is also suitable for a periodic operation process in industrial occasions.

Background

The repetitive controller has the characteristics of 'memory' and 'learning', and can realize the periodic reference signal track tracking/periodic interference effective suppression. Which stores the previous cycle control signal to correct the previous cycle control input with the tracking error signal at that time to form the current control input. The repetitive control technology has been successfully applied to precise control of servo motors, power electronic control technology, power quality control and the like.

Repetitive control is a control method based on the internal model principle. The essence of the internal model principle is that a dynamic model (namely an internal model) of a system external signal is implanted into a control system, so that a high-precision feedback control system is formed, and the system can follow an input signal without static error. Internal model for repeating controller structure periodic signal

Where T is the period of a given signal. It is a time delay (e) with a period^-Ts) The positive feedback link of (1). Regardless of the specific form of the input signal, as long as the initial segment signal is given, the internal model output can accumulate the input signal cycle by cycle, and repeatedly output the signal with the same cycle as the previous cycle. The design of the repetitive controller using the continuous internal model is mostly a frequency domain design, and the conventional design of the discrete repetitive controller is also performed in the frequency domain. Compared with a frequency domain method, a time domain design method is visual and simple, the tracking performance of system response is easy to directly depict, the disturbance effect which can affect the controlled output can be expanded into a new state quantity by combining the existing interference observation and inhibition means, and a state which can be expanded and observed is established by using a special feedback mechanism, so that a new way is provided for designing a motor servo control system by establishing a disturbance expansion observer.

Disclosure of Invention

The invention provides a novel attraction law method, adopts a disturbance expansion state compensation technology, and is suitable for the design of a discrete repetitive controller of a motor servo system. In order to achieve the expected error tracking performance of a closed-loop system and effectively reduce the flutter, a novel attraction law, namely a normal distribution-like attraction law is provided, and an ideal error dynamic equation is constructed by taking the tracking error as the error dynamic according to the attraction law to design a motor servo repetitive controller. While the complete suppression of periodic interference components is realized, in consideration of the existence of non-periodic components in disturbance, a disturbance expansion state observation technology is introduced into a closed-loop system to compensate the non-periodic interference so as to improve the control performance of the system and enable a motor servo system to realize high-speed and high-precision track tracking.

The technical scheme adopted by the invention for solving the technical problems is as follows:

a discrete repetitive control method for a motor servo system based on a similar normal distribution attraction law and adopting disturbance expansion state compensation comprises the following steps:

1) given periodic reference signal r_kSatisfy the following requirements

r_k＝±r_k-N(1)

Where N is the period of the reference signal, r_k,r_k-NRespectively representing reference signals at k, k-N moments;

2) structural equivalent disturbance

Where N is the period of the reference signal, d_kRepresenting the equivalent disturbance signal at time k, w_k,w_k-NRespectively representing interference signals at k, k-N moments;

3) constructing discrete time class normal distribution attraction law

e_k+1＝(1-ρ)e_k-εfal_nd(e_k,σ) (3)

Wherein the content of the first and second substances,

wherein e is_k＝r_k-y_kRepresenting the tracking error at time k, y_kOutputting for the system at the moment k; rho and epsilon are adjustable parameters, sigma is a defined normal distribution function-like exponential coefficient, the value range of the coefficient satisfies epsilon > 0, rho is more than 0 and less than 1, sigma is more than 0, and

4) constructing an error dynamic equation with an interference suppression term

The interference suppression measures are embedded into the attraction law (3), and the following ideal error dynamics are constructed:

e_k+1＝(1-ρ)e_k-εfal_nd(e_k,σ)-d_k+1(4)

wherein d is_k+1Representing the equivalent disturbance at the moment k + 1;

5) constructing a repetitive controller with equivalent disturbance dilation compensation according to the ideal error dynamic (4) formula

The error dynamic equation with the extended state observer is:

wherein the content of the first and second substances,

an observed value representing an equivalent disturbance at time k + 1;

the extended state observer was constructed as follows:

wherein the content of the first and second substances,

to the error e_k+1Is estimated by the estimation of (a) a,

to the error e_kβ₁For observer gain coefficients with respect to error, β₂For observer gain coefficients with respect to equivalent disturbances, β₁And β₂Can be appropriately configured as long as it satisfies

All the eigenvalues of (a) are within the unit circle. r is_k+1Reference signal, y, representing the time instant k +1_k+1-N,y_k,y_k-NRespectively representing the system outputs at times k +1-N, k, k-N, u_k,u_k-NRespectively representing controller inputs at times k, k-N, where

A′(q^-1)＝a₁+a₂q^-1+…+a_nq^-n+1＝q(A(q^-1)-1)

A(q^-1)＝1+a₁q^-1+…+a_nq^-n

B(q^-1)＝b₀+b₁q^-1+…+b_mq^-m

Satisfying servo objects

A(q^-1)y_k＝q^-dB(q^-1)u_k+w_k(7)

Wherein d represents a delay, u_kAnd y_kRespectively representing input and output signals at time k, w_kInterference signals at time k; a (q)^-1) And B (q)^-1) Is q^-1Polynomial of (a), q^-1Is a one-step delay operator, n is A (q)^-1) M is B (q)^-1) The order of (a); a is₁,...,a_n,b₀,...,b_mIs a system parameter and b₀Not equal to 0, n is more than or equal to m; d is an integer and is more than or equal to 1;

the repetitive controller with equivalent disturbance dilation compensation is:

the repetitive controller (8) can also be expressed as

u_k＝±u_k-N+v_k(9)

Wherein the content of the first and second substances,

will u_kAs the controller input of the servo object, the output signal y of the servo system can be measured_kFollows the reference signal r_kAnd (4) changing.

Further, the method comprises the following steps:

6) define an extended-state compensation bound Δ for equivalent disturbances, i.e.

And Δ ∈ o (T)²) Wherein T is a discrete system sampling period; the parameters rho, epsilon and sigma of the controller are set according to indexes representing the convergence performance and the stability performance of the system, the value range meets the conditions that epsilon is more than 0 and less than 1, rho is more than 0 and less than 1, sigma is more than 0 and

to characterize the tracking error convergence performance, the monotone reduction region, absolute attraction layer and steady-state error band concepts are introduced and defined as follows:

monotonous decreasing region delta_MDR

Absolute attraction layer Δ_AAL

Steady state error band Δ_SSE

i) Monotonous decreasing area (delta)_MDR)

Δ_MDR＝max{Δ_MDR1,Δ_MDR2} (13)

In the formula,. DELTA._MDR1，Δ_MDR2Is real, and satisfies

Namely, it is

ii) absolute attraction layer (. DELTA._AAL)

Δ_AAL＝max{Δ_AAL1,Δ_AAL2} (15)

In the formula,. DELTA._AAL1，Δ_AAL2Is a real number, determined by the following equation,

namely, it is

iii) steady state error band (Δ)_SSE)

Δ_SSE＝max{Δ_SSE1,Δ_SSE2} (17)

In the formula,. DELTA._SSE1，Δ_SSE2Is a real number, determined by the following equation,

wherein

Still further, in the step 6), when the reference signal satisfies r_k＝r_k-1The discrete repetitive controller is also suitable for the constant value regulation problem, and the equivalent disturbance is d_k＝w_k-w_k-1(ii) a Wherein r is_k-1Reference signal at time k-1, w_k-1For interfering signals at time k-1

The formula (20) can also be represented as

u_k＝u_k-1+v_k(21)

Wherein the content of the first and second substances,

still further, the method further comprises the steps of:

7) the controller enables the system to converge to a smaller error band, i.e. delta, in a limited number of steps_SSEAfter entering the error band, it will not cross the error band, and the convergence step number is m^*(ii) a Defining an initial error as e₀Convergence to Δ_SSEThe number of steps is m^*。

Wherein:

the invention has the technical idea that the discrete repetitive controller for designing the motor servo system adopts the discrete time attraction law based on the class normal distribution function, is a time domain design method and is different from the currently and generally adopted frequency domain method. The given period reference signal is considered when the controller is designed, and the designed controller is more visual, simple and convenient and is easy to depict the tracking performance of the system. The time domain design of the controller is easy to combine with the existing interference suppression means, an expansion state observation method is introduced, the designed repetitive controller can completely suppress periodic interference signals, errors generated by a first period are effectively reduced, and the given reference signal is quickly and accurately tracked. The introduction of the normal distribution-like attraction law enables the system to be rapidly converged in a limited step number, and the rapidity of the system is improved.

The invention has the main effects that: the method has the advantages of quick convergence, accelerated interference suppression and high control precision.

Drawings

Fig. 1 is a schematic diagram of a repetitive controller structure.

FIG. 2 is a block diagram of a repetitive controller based on the normal-like distribution attraction law and equivalent perturbation-expansion state compensation.

FIG. 3 is a block diagram of a discrete-time extended state observer design based on tracking error.

Fig. 4-7 are numerical simulations under the action of a repetitive controller based on the normal-like distribution attraction law and equivalent perturbed dilation state compensation when ρ is 0.8, e is 0.6, σ is 1, and Δ is 0.2:

fig. 4 shows the position signals when ρ is 0.8, ∈ 0.6, σ is 1, and Δ is 0.2.

Fig. 5 shows the controller signal u when ρ is 0.8, ∈ 0.6, σ is 1, and Δ is 0.2.

Fig. 6 shows the equivalent perturbation expansion state compensation signal when ρ is 0.8, ∈ is 0.6, σ is 1, and Δ is 0.2

Fig. 7 shows the tracking error signal e when ρ is 0.8, ∈ is 0.6, σ is 1, and Δ is 0.2.

Fig. 8-11 are numerical simulations under the influence of a repetitive controller based on the normal distribution-like attraction law when ρ is 0.8, ∈ is 0.6, σ is 1, and Δ is 0.2:

fig. 8 shows the position signals when ρ is 0.8, ∈ is 0.6, σ is 1, and Δ is 0.2.

Fig. 9 shows the controller signal u when ρ is 0.8, ∈ 0.6, σ is 1, and Δ is 0.2.

Fig. 10 is an equivalent perturbed dilated state compensation signal when ρ is 0.8, ∈ is 0.6, σ is 1, and Δ is 0.2

Fig. 11 shows a tracking error signal e when ρ is 0.8, ∈ is 0.6, σ is 1, and Δ is 0.2.

Fig. 12-15 are numerical simulations under the action of a repetitive controller based on the normal-like distribution attraction law and equivalent perturbed dilation state compensation when ρ is 0.8, e is 0.3, σ is 1, and Δ is 0.2:

fig. 12 shows position signals when ρ is 0.8, ∈ is 0.3, σ is 1, and Δ is 0.2.

Fig. 13 shows the controller signal u when ρ is 0.8, ∈ 0.3, σ is 1, and Δ is 0.2.

Fig. 14 shows an equivalent disturbance expansion state compensation signal when ρ is 0.8, ∈ is 0.3, σ is 1, and Δ is 0.2

Fig. 15 shows a tracking error signal e when ρ is 0.8, ∈ is 0.3, σ is 1, and Δ is 0.2.

Fig. 16 to 19 are numerical simulations under the action of a repetitive controller based on the normal distribution-like attraction law when ρ is 0.8, ∈ is 0.3, σ is 1, and Δ is 0.2:

fig. 16 shows position signals when ρ is 0.8, ∈ is 0.3, σ is 1, and Δ is 0.2.

Fig. 17 shows the controller signal u when ρ is 0.8, ∈ is 0.3, σ is 1, and Δ is 0.2.

Fig. 18 shows an equivalent disturbance expansion state compensation signal when ρ is 0.8, ∈ is 0.3, σ is 1, and Δ is 0.2

Fig. 19 shows a tracking error signal e when ρ is 0.8, ∈ is 0.3, σ is 1, and Δ is 0.2.

Fig. 20 to 22 show the results of experiments of the permanent magnet synchronous motor control system (cycle 0.8s) under the action of the repetitive controller when ρ is 0.7, ∈ is 0.1, and σ is 5:

fig. 20 shows position signals when ρ is 0.7, ∈ is 0.1, and σ is 5.

Fig. 21 shows the controller signal u when ρ is 0.7, ∈ is 0.1, and σ is 5.

Fig. 22 shows a tracking error signal e when ρ is 0.7, ∈ is 0.1, and σ is 5.

Fig. 23 to 25 show the results of experiments performed by the permanent magnet synchronous motor control system (cycle 4s) under the action of the repetitive controller when ρ is 0.7, ∈ is 0.1, and σ is 5:

fig. 23 shows position signals when ρ is 0.7, ∈ is 0.1, and σ is 5.

Fig. 24 shows the controller signal u when ρ is 0.7, ∈ is 0.1, and σ is 5.

Fig. 25 shows a tracking error signal e when ρ is 0.7, ∈ is 0.1, and σ is 5.

Fig. 26 to 28 show the results of experiments of the permanent magnet synchronous motor control system (cycle 0.8s) under the action of the repetitive controller when ρ is 0.3, ∈ is 0.1, and σ is 5:

fig. 26 shows position signals when ρ is 0.3, ∈ is 0.1, and σ is 5.

Fig. 27 shows the controller signal u when ρ is 0.3, ∈ is 0.1, and σ is 5.

Fig. 28 shows a tracking error signal e when ρ is 0.3, ∈ is 0.1, and σ is 5.

Fig. 29 to 31 show the results of experiments of the permanent magnet synchronous motor control system (cycle 4s) under the action of the repetitive controller when ρ is 0.3, ∈ is 0.1, and σ is 5:

fig. 29 shows position signals when ρ is 0.3, ∈ is 0.1, and σ is 5.

Fig. 30 shows the controller signal u when ρ is 0.3, ∈ is 0.1, and σ is 5.

Fig. 31 shows a tracking error signal e when ρ is 0.3, ∈ is 0.1, and σ is 5.

Fig. 32 to 34 show the results of experiments of the permanent magnet synchronous motor control system (cycle 0.8s) under the action of the repetitive controller when ρ is 0.7, ∈ is 0.5, and σ is 5:

fig. 32 shows position signals when ρ is 0.7, ∈ is 0.5, and σ is 5.

Fig. 33 shows the controller signal u when ρ is 0.7, ∈ is 0.5, and σ is 5.

Fig. 34 shows a tracking error signal e when ρ is 0.7, ∈ is 0.5, and σ is 5.

Fig. 35 to 37 are experimental results of the permanent magnet synchronous motor control system (cycle 4s) under the action of the repetitive controller when ρ is 0.7, ∈ is 0.5, and σ is 5:

fig. 35 shows position signals when ρ is 0.7, ∈ is 0.5, and σ is 5.

Fig. 36 shows the controller signal u when ρ is 0.7, ∈ is 0.5, and σ is 5.

Fig. 37 shows a tracking error signal e when ρ is 0.7, ∈ is 0.5, and σ is 5.

Detailed Description

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

Referring to fig. 1-3, a discrete repetitive control method for a motor servo system based on the attraction law of a normal-like distribution function and adopting disturbance expansion state compensation, wherein fig. 1 is a schematic structural diagram of a repetitive controller; FIG. 2 is a block diagram of a repetitive controller based on the attraction law and equivalent perturbed dilation state compensation of a normal-like distribution function; FIG. 3 is a block diagram of a discrete-time extended state observer design based on tracking error.

A discrete repetitive control method for a motor servo system based on a normal distribution function-like attraction law and equivalent disturbance expansion state compensation comprises the following steps:

1) given periodic reference signal r_kSatisfy the following requirements

r_k＝±r_k-N(1)

Where N is the period of the reference signal, r_k,r_k-NRespectively representing reference signals at k, k-N moments;

2) structural equivalent disturbance

Where N is the period of the reference signal, d_kRepresenting the equivalent disturbance signal at time k, w_k,w_k-NRespectively representing interference signals at k, k-N moments;

3) constructing discrete time class normal distribution attraction law

e_k+1＝(1-ρ)e_k-εfal_nd(e_k,σ) (3)

Wherein the content of the first and second substances,

wherein e is_k＝r_k-y_kRepresenting the tracking error at time k, y_kOutputting for the system at the moment k; rho and epsilon are adjustable parameters, sigma is a defined normal distribution function-like exponential coefficient, the value range of the coefficient satisfies epsilon > 0, rho is more than 0 and less than 1, sigma is more than 0, and

4) constructing an error dynamic equation with an interference suppression term

The interference suppression measures are embedded into the attraction law (3), and the following ideal error dynamics are constructed:

e_k+1＝(1-ρ)e_k-εfal_nd(e_k,σ)-d_k+1(4)

wherein d is_k+1Representing the equivalent disturbance at the moment k + 1;

5) constructing a repetitive controller with equivalent disturbance dilation compensation according to the ideal error dynamic (4) formula

The error dynamic equation with the extended state observer is:

wherein d is_k+1Is the equivalent perturbation at the time k +1,

at time d of k +1_k+1The observed value of (a);

using the observation error, a state observer of the form:

wherein the content of the first and second substances,

to the error e_k+1Is estimated by the estimation of (a) a,

to the error e_kβ₁For observer gain coefficients with respect to error, β₂For observer gain coefficients with respect to equivalent disturbances, β₁And β₂Can be appropriately configured as long as it satisfies

Is within the unit circle, observer gain coefficients β for the error₁Observer gain coefficient β for equivalent disturbance, designed as 0.25₂Designed to be 0.5 and sigma set to be 0.6. r is_k+1Reference signal, y, representing the time instant k +1_k+1-N,y_k,y_k-NRespectively representing the system outputs at times k +1-N, k, k-N, u_k,u_k-NRespectively, representing the controller inputs at times k, k-N, where,

A′(q^-1)＝a₁+a₂q^-1+…+a_nq^-n+1＝q(A(q^-1)-1)

A(q^-1)＝1+a₁q^-1+…+a_nq^-n

B(q^-1)＝b₀+b₁q^-1+…+b_mq^-m

satisfying servo objects

A(q^-1)y_k＝q^-dB(q^-1)u_k+w_k(7)

Wherein d represents a delay, u_kAnd y_kRespectively representing input and output signals at time k, w_kInterference signals at time k; a (q)^-1) And B (q)^-1) Is q^-1Polynomial of (a), q^-1Is a one-step delayLate operator, n being A (q)^-1) M is B (q)^-1) The order of (a); a is₁,...,a_n,b₀,...,b_mIs a system parameter and b₀Not equal to 0, n is more than or equal to m; d is an integer and is more than or equal to 1;

second order difference equation model of motor servo object of the embodiment

y_k+1+a₁y_k+a₂y_k-1＝b₁u_k+b₂u_k-1+w_k+1

Wherein, y_k+1y_kOutput position signal u representing the time of servo system k +1, k_k,u_k-1Input control signal at time k, k-1, w_k+1Interference signal (satisfying matching condition) at the moment of k +1 of servo system₁,a₂,b₁,b₂The values of the servo system model parameters are obtained through parameter estimation.

The repetitive controller with equivalent disturbance dilation compensation is:

note the book

Input signal

Equation (8) can be written as

In the formula, v_kRepresenting an input signal

The amount of correction of (a) is,

will u_kAs the controller input of the servo object, the output signal y of the servo system can be measured_kFollows the reference signal r_kAnd (4) changing.

6) Monotone decreasing area delta according to system tracking error_MDRAbsolute attraction layer Δ_AALAnd steady state error band Δ_SSESetting the controller parameters to achieve the optimal control effect, wherein the controller parameters mainly comprise: normal distribution function index coefficient sigma, adjustable parameters rho, epsilon and expansion state compensation boundary delta of equivalent disturbance, wherein the indexes are monotonous decreasing area delta_MDRAbsolute attraction layer Δ_AALAnd steady state error band Δ_SSEThe definition is as follows:

monotonous decreasing region delta_MDR

Absolute attraction layer Δ_AAL

Steady state error band Δ_SSE

According to the above-mentioned Delta_MDR、Δ_AALAnd delta_SSEThe determined boundary values are as follows:

i) monotonous decreasing area (delta)_MDR)

Δ_MDR＝max{Δ_MDR1,Δ_MDR2} (13)

In the formula,. DELTA._MDR1，Δ_MDR2Is real, and satisfies

Namely, it is

ii) absolute attraction layer (. DELTA._AAL)

Δ_AAL＝max{Δ_AAL1,Δ_AAL2} (15)

In the formula,. DELTA._AAL1，Δ_AAL2Is a real number, can be determined by the following formula,

iii) steady state error band (Δ)_SSE)

Δ_SSE＝max{Δ_SSE1,Δ_SSE2} (17)

In the formula,. DELTA._SSE1，Δ_SSE2Is a real number, can be determined by the following formula,

wherein

For the above repetitive controller design, the following description is made:

introduction of d into class normal distribution attraction law_k+1Reflecting a suppression measure for the disturbing signal of a given periodic pattern, introducing

Reflecting the error compensation after adding the extended state observer.

(6) In the formula (8), e_k,y_k,y_k-1,y_k-1-NAll can be obtained by measurement, u_k-1,u_k-1-NThe stored value of the control signal may be read from memory.

When the reference signal satisfies r_k＝r_k-1The discrete repetitive controller is also suitable for the constant value regulation problem, and the equivalent disturbance is d_k＝w_k-w_k-1(ii) a Wherein r is_k+1Reference signal at time k +1, w_k,w_k-1Interfering signals are at time k, k-1.

The formula (20) can also be represented as

u_k＝u_k-1+v_k(21)

Wherein the content of the first and second substances,

the above repetitive controllers are given for a second order system and in the same way also give the design results for a higher order system.

7) The controller designed by the invention can make the system converge to an error band in a limited step number, wherein the error band is delta_SSEAfter entering the error band, it will not cross the error band, and the convergence step number is m^*(ii) a Defining an initial error as e₀Error converges from delta to delta_SSEThe number of steps is m^*；

Wherein:

example (b): in the embodiment, a permanent magnet synchronous motor servo system is taken as an example to execute a repeated tracking task on a fixed interval, a position reference signal of the servo system has a periodically symmetrical characteristic, the servo motor adopts three-loop control, and a current loop and a speed loop controller are provided by an ELMO driver; the position loop controller is provided by a DSP development board TMS320F 2812.

Designing a position loop controller requires establishing a mathematical model of a servo object except for a position loop, including a current loop, a speed loop, a power driver, an alternating current permanent magnet synchronous servo motor body and a detection device (see fig. 2). Obtaining a mathematical model of the servo object by parameter estimation

y_k+1-1.9376y_k+0.9376y_k-1＝2.5504u_k-1.7307u_k-1+w_k+1(24)

Wherein, y_k,u_kPosition output and velocity set signal (control input), w, respectively, for a position servo system_kIs an interference signal.

Since the present embodiment uses a sinusoidal signal as the reference signal of the system, the repetitive controller may take the form of a controller given by equation (8), and a specific expression thereof may be written as

The effectiveness of the repetitive controller given by the present invention will be illustrated in this example by numerical simulation and experimental results.

Numerical simulation: given a position reference signal of r_k＝20sin(2kπfT_s) Unit rad, frequency f 0.5Hz, sampling period T_sThe number of cycles N used is 800, 0.002 s. During simulation, the selected disturbance amount w (k) is composed of periodic interference and aperiodic random interference, and the specific form is

w(k)＝-5*sin(2*pi*(k)/N)+0.06*rand() (26)

Under the action of a repetitive controller (25), different controller parameters rho, epsilon and sigma are selected, and three boundary layers of a servo system are different. For purposes of illustrating the invention patent with respect to the monotonically decreasing region Δ_MDRAbsolute attraction layer Delta_AALAnd steady state error band Δ_SSEThe theoretical correctness of (1) is given in FIGS. 7 and 15 as Δ_MDR，Δ_AALAnd Δ_SSEThe specific value of (a).

1) When the controller parameter ρ is 0.8, e is 0.6, σ is 1, and Δ is 0.2 (see fig. 7)

Δ_MDR＝Δ_AAL＝0.2174

Δ_SSE＝0.2117

2) When the controller parameter ρ is 0.8, e is 0.3, σ is 1, and Δ is 0.2 (see fig. 15)

Δ_MDR＝Δ_AAL＝Δ_SSE＝0.2008

The simulation results are shown in fig. 7 and fig. 15. The numerical results verify the monotonous reduction area delta of the tracking error of the system under the action of the repetitive controller given by the patent under the condition of a given system model, a reference signal and an interference signal_MDRAbsolute attraction layer Delta_AALAnd steady state error band Δ_SSE。

3) Comparing the tracking error signal e of fig. 7 with fig. 11 in the case of controller parameters where ρ is 0.8, ε is 0.6, σ is 1, and Δ is 0.2, fig. 7 is the result of a simulation with the controller embedded in the extended state observer, clearly the tracking error of the first cycle in fig. 7 is much smaller than in fig. 11 compared to the case without the extended state observer, which is also illustrated by comparing the position signals of fig. 4 and 8. The introduction of the extended state observer takes the tracking error as an extended state, and under the action of the repetitive controller provided by the patent, the tracking error of the first period is greatly reduced, so that the rapid tracking is realized.

4) Comparing the tracking error signal e of fig. 15 with fig. 19 in the case of controller parameters where ρ is 0.8, ε is 0.3, σ is 1, and Δ is 0.2, fig. 15 is the result of a simulation with the controller embedded in the extended state observer, and it is clear that the tracking error of the first cycle in fig. 15 is much smaller than in fig. 19 compared to the case without the extended state observer, which is also illustrated by comparing the position signals of fig. 12 and 16. The introduction of the extended state observer takes the tracking error as an extended state, and under the action of the repetitive controller provided by the patent, the tracking error of the first period is greatly reduced, so that the rapid tracking is realized.

5) Monotonous reduction region delta of tracking error under different controller parameters_MDRAbsolute attraction layer Delta_AALAnd steady state error band Δ_SSEAll the parameters are different, and each parameter is properly adjusted according to the mutual relation of the control parameters, so that a better tracking effect can be achieved.

The experimental results are as follows: the block diagram of the permanent magnet synchronous motor control system used in the experiment is shown in figure 1. And verifying the tracking performance of discrete repetitive control based on the similar normal distribution function attraction law by setting different controller parameters. Giving the position signal a sinusoidal signal r_k＝Asin(2πfT_sk) And (7) rad. Wherein the amplitude is

The experiment was carried out in two groups, one group having a frequency f of 0.25Hz and a sampling time T_sThe sampling number N in one sampling period is 800 s; one set of frequency f is 1.25Hz and sampling time T_sThe number N of samples in one sampling period is 800.

1) The controller parameters are taken as ρ 0.7, ε 0.1, and σ 5.

(i) Frequency f is 1.25Hz, and sampling time is T_sThe sampling number N of one sampling period is 800 s;

using servo motors in repetitive controllersThe system tracking error, position output signal and controller signal are shown in fig. 20-22 as a function of equation (25). It can be seen from fig. 22 that the system tracking error is greatly reduced after one reference signal period (T ═ 0.8s), converging on | e_kL is less than or equal to 0.3deg, after two reference signal periods (2T is 1.6s), the system enters a steady state, and the tracking error is less than or equal to e and less than or equal to-0.2 deg_kFluctuation is less than or equal to 0.2 deg. As can be seen from fig. 20, the system can achieve better tracking at a frequency of f-1.25 Hz.

(ii) Frequency f is 0.25Hz and sampling time is T_sThe sampling number N in one sampling period is 800 s;

with the servo motor under the action of a repetitive controller, as shown in equation (25), the system tracking error, position output signal and controller signal are shown in fig. 23-25. As can be seen from fig. 25, the system tracking error is greatly reduced after one reference signal period (T ═ 4s), and converges to | e_kL is less than or equal to 0.2deg, after two reference signal periods (2T is 8s), the steady state is entered, and the tracking error is less than or equal to e at-0.1 deg_kFluctuation is less than or equal to 0.1 deg. As can be seen from fig. 23, the system can achieve better tracking even at a frequency of 0.25 Hz. Comparing fig. 22 and fig. 25, under the condition that the controller parameters are the same, the tracking error of the first reference signal period of fig. 25 is obviously smaller than that of fig. 22, and after the second reference signal period, the fluctuation of the tracking error of fig. 25 is obviously smaller than that of fig. 22, the fluctuation range is relatively smaller, the tracking effect is relatively better, and the control accuracy is relatively higher. And by comparing fig. 21 and fig. 24, in the case where the expected tracking trajectory is consistent, the controller output at the frequency f of 0.25Hz is significantly smaller than the controller output at the frequency f of 1.25 Hz. By combining the data analysis, the system can realize better tracking at the frequency f of 0.25 Hz.

2) The controller parameters are taken as ρ 0.3, ε 0.1, and σ 5.

(i) Frequency f is 1.25Hz, and sampling time is T_sThe sampling number N of one sampling period is 800 s;

the system tracking error, position output signal and controller signal using a servo motor under repetitive controller, as shown in equation (25), are shown in fig. 26-28. FIG. 28 shows that the system tracks errorsThe difference decreases after one reference signal period (T ═ 0.8s), converging to | e_kL is less than or equal to 1.5deg, after two reference signal periods (2T is 1.6s), the system enters a steady state, and the tracking error is less than or equal to e and less than or equal to-0.35 deg_kFluctuation is less than or equal to 0.35 deg. As can be seen from fig. 26, the system can perform tracking at a frequency f of 1.25 Hz. Comparing fig. 22 and fig. 28, when the frequency and the desired trajectory match, different controller parameters have a large influence on the tracking effect, and the control effect is better when the controller parameter ρ is 0.7, and ∈ is 0.1, and σ is 5, than when the controller parameter ρ is 0.3, and ∈ is 0.1, and σ is 5.

(ii) Frequency f is 0.25Hz and sampling time is T_sThe sampling number N in one sampling period is 800 s;

with the servo motor under the action of a repetitive controller, as shown in equation (25), the system tracking error, position output signal and controller signal are shown in fig. 29-31. It can be seen from fig. 31 that the tracking error of the system is greatly reduced after one reference signal period (T ═ 4s), and enters a steady state, and the tracking error is-0.5 deg ≦ e_kFluctuation is less than or equal to 0.5 deg. As can be seen from fig. 29, the system can achieve better tracking even at a frequency of 0.25 Hz. Comparing fig. 31 and fig. 28, under the condition that the controller parameters are the same, the tracking error of the first reference signal period of fig. 31 is significantly smaller than that of fig. 28, and after the second reference signal period, the fluctuation of the tracking error of fig. 31 is significantly smaller than that of fig. 28, the fluctuation range is relatively smaller, the tracking effect is relatively better, and the control accuracy is relatively higher. And by comparing fig. 30 and fig. 27, in the case where the expected tracking trajectory is consistent, the controller output at the frequency f of 0.25Hz is significantly smaller than the controller output at the frequency f of 1.25 Hz. By combining the data analysis, the system can realize better tracking at the frequency f of 0.25 Hz.

3) The controller parameters are taken as ρ 0.7, ε 0.5, and σ 5.

(i) Frequency f is 1.25Hz, and sampling time is T_sThe sampling number N of one sampling period is 800 s;

with the servo motor under the action of a repetitive controller, as shown in equation (25), the system tracking error, position output signal and controller signal are shown in fig. 32-34. From the figure34 it can be seen that the system tracking error is greatly reduced after one reference signal period (T ═ 0.8s), converging to | e_kL is less than or equal to 1.5deg, after two reference signal periods (2T is 1.6s), the system enters a steady state, and the tracking error is less than or equal to e at-0.5 deg_kFluctuation is less than or equal to 0.5 deg. As can be seen from fig. 34, the system can achieve better tracking at a frequency of f-1.25 Hz.

(ii) Frequency f is 0.25Hz and sampling time is T_sThe sampling number N in one sampling period is 800 s;

with the servo motor under the action of a repetitive controller, as shown in equation (25), the system tracking error, position output signal and controller signal are shown in fig. 35-37. It can be seen from fig. 37 that the tracking error of the system is greatly reduced after one reference signal period (T ═ 4s), and enters a steady state, and the tracking error is-0.1 deg ≦ e_kFluctuation is less than or equal to 0.1 deg. Comparing fig. 37 and fig. 34, in the case of the same controller parameter, the tracking error of the first reference signal period of fig. 37 is significantly smaller than that of fig. 34, and after the second reference signal period, the tracking error of fig. 37 fluctuates significantly smaller than that of fig. 34, the fluctuation range is relatively small, the tracking effect is relatively good, and the control accuracy is relatively high. By combining the data analysis, the system can realize better tracking at the frequency f of 0.25 Hz.

Claims (4)

1. A discrete repetitive control method for a motor servo system based on a similar normal distribution attraction law and adopting disturbance expansion state compensation is characterized in that: the method comprises the following steps:

1) given periodic reference signal r_kSatisfy the following requirements

r_k＝±r_k-N(1)

Where N is the period of the reference signal, r_k,r_k-NRespectively representing reference signals at k, k-N moments;

2) structural equivalent disturbance

Wherein N is a referencePeriod of signal, d_kRepresenting the equivalent disturbance signal at time k, w_k,w_k-NRespectively representing interference signals at k, k-N moments;

3) constructing discrete time class normal distribution attraction law

e_k+1＝(1-ρ)e_k-εfal_nd(e_k,σ) (3)

Wherein the content of the first and second substances,

wherein e is_k＝r_k-y_kRepresenting the tracking error at time k, y_kOutputting for the system at the moment k; rho and epsilon are adjustable parameters, sigma is a defined normal distribution function-like exponential coefficient, the value range of the coefficient satisfies epsilon > 0, rho is more than 0 and less than 1, sigma is more than 0, and

4) constructing an error dynamic equation with an interference suppression term

The interference suppression measures are embedded into the attraction law (3), and the following ideal error dynamics are constructed:

e_k+1＝(1-ρ)e_k-εfal_nd(e_k,σ)-d_k+1(4)

wherein d is_k+1Representing the equivalent disturbance at the moment k + 1;

5) constructing a repetitive controller with equivalent disturbance expansion compensation according to the ideal error dynamic formula (4)

The error dynamic equation with the extended state observer is:

wherein the content of the first and second substances,

an observed value representing an equivalent disturbance at time k + 1;

the extended state observer was constructed as follows:

wherein the content of the first and second substances,

to the error e_k+1Is estimated by the estimation of (a) a,

to the error e_kβ₁For observer gain coefficients with respect to error, β₂For observer gain coefficients with respect to equivalent disturbances, β₁And β₂Can be appropriately configured as long as it satisfies

All the characteristic values of the data are within the unit circle; r is_k+1Reference signal, y, representing the time instant k +1_k+1-N,y_k,y_k-NRespectively representing the system outputs at times k +1-N, k, k-N, u_k,u_k-NController inputs representing times k, k-N, respectively;

in the formula (I), the compound is shown in the specification,

A′(q^-1)＝a₁+a₂q^-1+…+a_nq^-n+1＝q(A(q^-1)-1)

A(q^-1)＝1+a₁q^-1+…+a_nq^-n

B(q^-1)＝b₀+b₁q^-1+…+b_mq^-m

satisfying servo objects

A(q^-1)y_k＝q^-dB(q^-1)u_k+w_k(7)

Wherein d represents a delay, u_kAnd y_kRespectively representing input and output signals at time k, w_kInterference signals at time k; a (q)^-1) And B (q)^-1) Is q^-1Polynomial of (a), q^-1Is a one-step delay operator, n is A (q)^-1) M is B (q)^-1) The order of (a); a is₁,…,a_n,b₀,…,b_mIs a system parameter and b₀Not equal to 0, n is more than or equal to m; d is an integer and is more than or equal to 1;

the repetitive controller with equivalent disturbance dilation compensation is:

the repetitive controller (8) can also be expressed as

u_k＝±u_k-N+v_k(9)

Wherein the content of the first and second substances,

will u_kAs the controller input of the servo object, the output signal y of the servo system can be measured_kFollows the reference signal r_kAnd (4) changing.

2. The discrete repetitive control method for the motor servo system based on the normal distribution-like attraction law and using the disturbance expansion state compensation as claimed in claim 1, wherein: the method further comprises the steps of: 6) define an extended-state compensation bound Δ for equivalent disturbances, i.e.

And Δ ∈ o (T)²) Wherein T is a discrete system sampling period; the parameters rho, epsilon and sigma of the controller are set according to indexes representing the convergence performance and the stability performance of the system, the value range meets the conditions that epsilon is more than 0 and less than 1, rho is more than 0 and less than 1, sigma is more than 0 and

to characterize the tracking error convergence performance, the monotone reduction region, absolute attraction layer and steady-state error band concepts are introduced and defined as follows:

monotonous decreasing region delta_MDR

Absolute attraction layer Δ_AAL

Steady state error band Δ_SSE

i) Monotonous decreasing region delta_MDR

Δ_MDR＝max{Δ_MDR1,Δ_MDR2} (13)

In the formula,. DELTA._MDR1，Δ_MDR2Is real, and satisfies

ii) Absolute attraction layer Δ_AAL

Δ_AAL＝max{Δ_AAL1,Δ_AAL2} (15)

In the formula,. DELTA._AAL1，Δ_AAL2Is a real number, can be determined by the following formula,

iii) steady state error band Δ_SSE

Δ_SSE＝max{Δ_SSE1,Δ_SSE2} (17)

In the formula,. DELTA._SSE1，Δ_SSE2Is a real number, can be determined by the following formula,

wherein

3. The discrete repetitive control method for the motor servo system based on the normal distribution-like attraction law and using the disturbance expansion state compensation as claimed in claim 2, wherein: in the step 6), when the reference signal satisfies r_k＝r_k-1The discrete repetitive controller is also suitable for the constant value regulation problem, and the equivalent disturbance is d_k＝w_k-w_k-1(ii) a Wherein r is_k-1Reference signal at time k-1, w_k-1Interference signals at the k-1 moment;

the formula (20) can also be represented as

u_k＝u_k-1+v_k(21)

Wherein the content of the first and second substances,

4. the discrete repetitive control method for the motor servo system based on the normal distribution-like attraction law and using the disturbance expansion state compensation as claimed in claim 3, wherein: the method further comprises the steps of: 7) the controller enables the system to converge to a smaller error band, defined as delta, in a finite number of steps_SSEAfter entering the error band, it will not cross the error band, and the convergence step number is m^*(ii) a Defining an initial error as e₀The number of steps from the initial error convergence to the boundary δ is m^*；

Wherein:

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