CN109188908B - Digital controller design method based on exponential type non-switching attraction law - Google Patents

Abstract

A digital controller non-switching attraction law design method adopting an interference differential suppression strategy is characterized in that a given module generates a reference signal; constructing a corresponding interference difference compensation feedback link according to the specific form of a given reference signal, wherein the output signal of the interference difference compensation feedback link is used for correcting a digital controller; and constructing an ideal error dynamic state based on a non-switching attraction law, designing a digital controller according to the ideal error dynamic state, and taking a signal obtained by calculation of the current controller as the input of a servo object. The specific controller parameter setting can be carried out according to the indexes of the convergence performance of the representation system, and a steady-state error band, an absolute attraction layer and a monotone decreasing area of the representation tracking error convergence process are provided. And then the anti-interference capability and the tracking precision of the system are improved, so that the servo system can follow the change of the reference signal. The invention provides a method for designing a non-switching suction law of a digital controller, which can eliminate the phenomenon of system buffeting and has good control precision.

Description

Digital controller design method based on exponential type non-switching attraction law

Technical Field

The invention relates to an exponential type non-switching attraction law design method adopting an interference difference suppression strategy digital controller, which is suitable for a position servo system and other industrial application occasions.

Background

The approach law method is an effective tool for designing a sliding mode controller of a servo system, and due to the adoption of the approach law, the dynamic process of a closed-loop system is represented by an approach process and a sliding mode, and the stability and the convergence of the closed-loop system are determined by the specific approach law and a switching function form. The actual controller design needs to consider the influence of various disturbances, the disturbance suppression measures are 'embedded' in the original approach law, and the modified approach law forms ideal switching dynamics. In this way, a controller dynamically designed according to ideal switching can effectively suppress disturbances.

The attraction law method directly adopts the tracking error signal without defining a switching function, and the design of the controller is more direct and simpler. The attraction law reflects the expected dynamic characteristics of the system error when disturbance is not considered; in the presence of disturbances, a controller directly based on the attraction law cannot be implemented. The interference suppression measures can be 'embedded' into the attraction law, and ideal error dynamics with the disturbance suppression effect are constructed. The digital controller is designed according to the constructed ideal error dynamic equation, and the dynamic process of the closed-loop system is determined by the ideal error dynamic equation and has the expected tracking performance represented by the ideal error dynamic equation.

The attraction law method is different from an approach law method of discrete sliding mode control. The main differences between the two are as follows: replacing a switching function by the tracking error and replacing a switching surface by the original point by the attraction law method; the approach law method requires a finite time to reach the switching surface, while the attraction law method requires a finite time to reach the origin; the closed-loop system designed by the attraction law method still has robust performance related to parameter drift and external interference, only the sliding mode control focuses on invariance of sliding mode motion, and the attraction law method pursues invariance of system steady state.

When the discrete controller is designed by an attraction law method, indexes describing transient and steady-state behaviors of the tracking error can be dynamically given by an ideal error, and the indexes specifically comprise the following four indexes: a steady state error band, an absolute attraction layer, a monotonically decreasing area, and a maximum number of steps required for the tracking error to first enter the steady state error band. In fact, the specific values of the four indexes depend on the controller parameters, the controller parameters are different, and the values of the four indexes are also different. Once the ideal error dynamic form is given, specific expressions of four indexes can be given in advance for parameter setting of the controller. In the currently published attraction law method, four indexes all depend on the boundary of the equivalent interference signal. The boundary that effectively inhibits interference and reduces equivalent interference signals is a difficult problem to be solved urgently by an attraction law method.

Disclosure of Invention

In order to overcome the defects that the influence of interference signals on the performance of a servo system cannot be inhibited and the tracking control precision is low in the existing design method suitable for the digital controller of the position servo system, the invention provides an attraction law method suitable for designing the digital controller of the position servo system. In order to inhibit the influence of interference signals on the performance of a servo system and improve the tracking control precision, an interference differential compensation technology is adopted and embedded into an exponential attraction law so as to construct an ideal error dynamic state with disturbance inhibition capability. The digital controller is dynamically designed according to the ideal error, so that the closed-loop system has the characteristic of being dynamically drawn by the ideal error, and the anti-interference capability and the tracking performance of the position servo system are improved. The controller is designed by adopting a non-switching suction law, so that the buffeting phenomenon can be eliminated. The invention specifically provides a specific expression of four indexes, namely a steady-state error band, an absolute attraction layer, a monotone decreasing area, the maximum step number required for a tracking error to enter the steady-state error band for the first time, and the like, and can be used for guiding the parameter setting of the controller.

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

an exponential attraction law design for a digital controller with interference differential compensation, comprising the following steps:

1) given a reference signal r_kFor a polynomial of the time variable k, M representing the highest power of the polynomial, three specific reference signals are as follows:

square wave signal, M ═ 0:

triangular wave signal, M ═ 1:

s-curve, M ═ 3:

wherein, A is amplitude, and N is sampling times of the reference signal in one period;

2) structure ideal error dynamics

For exponential type no-switching attraction law

e_k+1＝(1-ρ(e_k))e_k (4)

Wherein e is_k＝r_k-y_kSystematic tracking error at time k, y_kThe actual output signal of the system at the moment k;

embedding interference suppression measures into the attraction law to construct an ideal error dynamic

Wherein, delta is more than 0 and less than 1, alpha is more than 0, beta is more than 1, d_k+1Is the equivalent interference at the time k +1,

compensating the signal for equivalent interference; due to the parameter ρ (e)_k) At e_kAway from the origin, approaches δ at e_kApproaches to the origin and approaches to 1, so that the system error starts from the initial position under the action of an ideal error dynamic state (6), the convergence speed is gradually increased, and delta, alpha and beta are parameters for adjusting the suction speed;

3) interference differential compensation strategy

Take the equivalent interference compensation effect as

Defining an equivalent interference for a particular reference signal in the form of

Equivalent interference signal when M is 0

d_k+1＝w_k+1 (7)

Equivalent interference signal when M is 1

d_k+1＝w_k+1-w_k (8)

Equivalent interference signal when M is 3

d_k+1＝[(w_k+1-w_k)-(w_k-w_k-1)]-[(w_k-w_k-1)-(w_k-1-w_k-2)] (9)

Wherein, w_k+1Interference at time k + 1;

defining the step number of the interference difference as L, wherein L represents that the interference compensation error comprises L interferences at successive moments, and the two-step interference difference d is shown as formula (7)_k+1-d_k＝w_k+1-w_kComprising w_k+1And w_kTwo-time interference, in order to effectively suppress interference, when selecting equivalent interference, the following condition should be satisfied

Wherein the content of the first and second substances,

is the smallest integer not less than;

4) controller design

According to the ideal error dynamics (6) and the equivalent interference signal d_k+1The following controller expression is given:

for a square wave reference signal equation (1),

wherein, F (q)^-1)＝B(q^-1)-b₀；

For the triangular wave reference signal equation (2),

wherein the content of the first and second substances,

for the S-curve reference signal equation (3),

in formulae (11) to (13), A (q)^-1)、B(q^-1) For serving systems

A(q^-1)y_k＝q^-1B(q^-1)u_k+w_k (14)

With respect to q^-1The parameter polynomial of (2):

A(q^-1)＝1+a₁q^-1+a₂q^-2+……+a_nq^-n

B(q^-1)＝b₀+b₁q^-1+b₂q^-2+……+b_mq^-m

wherein u is_kAnd y_kInput and output signals, q, respectively, at time k of the servo system^-1For one-step delay operator, m and n are respectively A (q)^-1)、B(q^-1) Order of (b)₀≠0，1≤m≤n。

Further, the method comprises the following steps:

5) analysis of properties

And giving specific expressions of a steady-state error band, an absolute attraction layer, a monotone subtraction area and at most four indexes of steps required by the tracking error entering the steady-state error band for the first time, and being used for describing the tracking performance of the system and guiding the parameter setting of the controller, wherein the steady-state error band, the absolute attraction layer and the monotone subtraction area are defined as follows:

steady state error band Δ_SSE

Absolute attraction layer Δ_AAL

Monotonous decreasing region delta_MDR

Equivalent interference compensation error satisfaction

The expression of each index is as follows:

steady state error band (Δ)_SSE)

Δ_SSE＝Δ (15)

Absolute attraction layer (. DELTA.)_AAL)

Δ_AAL＝max{Δ_AAL1,Δ_AAL2} (16)

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

Monotonous decreasing area (delta)_MDR)

Δ_MDR＝max{Δ_MDR1,Δ_MDR2} (18)

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

Number of convergence steps

Wherein e is₀Is the initial value of the tracking error.

The technical conception of the invention is as follows: an exponential type no-switching attraction law method for the design of a tracking controller of a position servo system is provided. Equivalent interference is defined according to a given reference signal, and interference suppression measures are embedded into the non-switching attraction law to form ideal error dynamics with interference suppression effect. The digital controller is dynamically designed according to the ideal error to realize accurate tracking of a given reference signal.

The control effect of the invention is mainly shown in that: according to the given reference signal, a corresponding interference difference compensation measure is adopted, and the tracking accuracy is improved by suppressing interference. Meanwhile, a non-switching discrete time suction law is adopted to eliminate system buffeting.

Drawings

FIG. 1 is a block diagram of a controller system.

In FIG. 2 to FIG. 3, r is_k＝10sin(2πfkT_s)deg，f＝2Hz，T_sWhen δ is 0.01, δ is 0.1, α is 5, β is 10, and Δ is 0.6, the numerical simulation result of controller equation (22) is used:

FIG. 2 shows a tracking error signal e_k；

Fig. 3 shows interference compensation errors

FIGS. 4-7 show a reference signal r_kAs in formula (1), A ═ 5, T_sWhen δ is 0.1, δ is 0.5, α is 10, and β is 10, the numerical simulation result of controller equation (22) is used:

FIG. 4 shows a reference signal r_kAnd the output signal y_k；

FIG. 5 shows a tracking error signal e_k；

FIG. 6 shows interference compensation errors

FIG. 7 shows a control signal u_k。

FIGS. 8-11 show a reference signal r_kWhen a is 20, δ is 0.5, α is 10, and β is 10, the numerical simulation result of the controller equation (22) is used as equation (2):

FIG. 8 shows a reference signal r_kAnd the output signal y_k；

FIG. 9 shows a tracking error signal e_k；

FIG. 10 shows interference compensation errors

FIG. 11 shows a control signal u_k。

FIGS. 12-15 show a reference signal r_kAs in formula (2), A ═ 20, T_sWhen δ is 0.1, δ is 0.5, α is 10, and β is 10, the numerical simulation result of controller equation (23) is used:

FIG. 12 shows a reference signal r_kAnd the output signal y_k；

FIG. 13 shows a tracking error signal e_k；

FIG. 14 shows the interference compensation error d_k-d_k-1；

FIG. 15 shows a control signal u_k。

FIGS. 16-19 show a reference signal r_kAs in formula (3), A ═ 5, T_sWhen δ is 0.1, δ is 0.5, α is 10, and β is 10, the numerical simulation result of controller equation (22) is used:

FIG. 16 shows a reference signal r_kAnd the output signal y_k；

FIG. 17 shows a tracking error signal e_k；

FIG. 18 shows interference compensation errors

FIG. 19 shows a control signal u_k。

FIGS. 20-23 show a reference signal r_kWhen a is 5, δ is 0.5, α is 10, β is 10, and δ is 0.5, the numerical simulation result of the controller formula (23) is adopted as formula (3):

FIG. 20 shows a reference signal r_kAnd the output signal y_k；

FIG. 21 shows a tracking error signal e_k；

FIG. 22 shows interference compensation errors

FIG. 23 shows a control signal u_k。

FIGS. 24-27 show a reference signal r_kAs in formula (3), A ═ 5, T_sWhen α is 0.1, β is 10, and δ is 0.5, the numerical simulation result of controller equation (24) is used:

FIG. 24 shows a reference signal r_kAnd the output signal y_k；

FIG. 25 shows a tracking error signal e_k；

FIG. 26 shows interference compensation errors

FIG. 27 shows a control signal u_k。

FIGS. 28-29 show reference signals r_kAs in formula (1), A ═ 15deg, T_sWhen α is 0.2ms, β is 10, and δ is 0.2, the experimental result of the controller equation (22) is used:

FIG. 28 shows the reference signal r in turn_kAnd the output signal y_kControl signal u_kTracking error signal e_kAnd interference compensation error

FIG. 29 shows a tracking error signal e_kA histogram of (a).

FIGS. 30-31 show a reference signal r_kSuch as formula (2), A ═ 180deg, T_sWhen α is 0.2ms, β is 10, and δ is 0.2, the experimental result of the controller equation (22) is used:

FIG. 30 shows the reference signal r in turn_kAnd the output signal y_kControl signal u_kTracking error signal e_kAnd interference compensation error

FIG. 31 shows a tracking error signal e_kA histogram of (a).

FIGS. 32-33 show a reference signal r_kSuch as formula (2), A ═ 180deg, T_sWhen α is 0.2ms, β is 10, and δ is 0.2, the experimental result of the controller equation (23) is used:

FIG. 32 shows the reference signal r in turn_kAnd the output signal y_kControl signal u_kTracking error signal e_kAnd interference compensation error

FIG. 33 shows a tracking error signal e_kA histogram of (a).

FIGS. 34-35 show a reference signal r_kSuch as formula (3), A ═ 180deg, T_sWhen α is 0.2ms, β is 10, and δ is 0.2, the experimental result of the controller equation (22) is used:

FIG. 34 shows the reference signal r in turn_kAnd the output signal y_kControl signal u_kTracking error signal e_kAnd interference compensation error

FIG. 35 shows a tracking error signal e_kA histogram of (a).

FIGS. 36-37 show a reference signal r_kSuch as formula (3), A ═ 180deg, T_sWhen α is 0.2ms, β is 10, and δ is 0.2, the experimental result of the controller equation (23) is used:

FIG. 36 shows the reference signal r in turn_kAnd the output signal y_kControl signal u_kTracking error signal e_kAnd interference compensation error

FIG. 37 shows a tracking error signal e_kA histogram of (a).

FIGS. 38-39 show the reference signal r_kSuch as formula (3), A ═ 180deg, T_sWhen α is 0.2ms, β is 10, and δ is 0.2, the experimental result of the controller formula (24) is used:

FIG. 38 shows the reference signal r in turn_kAnd the output signal y_kControl signal u_kTracking error signal e_kAnd interference compensation error

FIG. 39 shows a tracking error signal e_kA histogram of (a).

Detailed Description

The following further describes embodiments of the present invention with reference to the accompanying drawings.

FIG. 1 is a block diagram of a servo system. Referring to fig. 2 to 39, an exponential type non-switching attraction law method for a discrete servo system with interference differential compensation and effective interference suppression is designed by a controller, and comprises the following steps:

1) given reference signal r_k

Reference signal r_kIs a polynomial of a time variable k, M representing the highest power of the polynomial; three specific reference signals are as follows:

square wave signal, M ═ 0:

triangular wave signal, M ═ 1:

s-curve, M ═ 3:

wherein, A is amplitude, and N is sampling times of the reference signal in one period;

2) structure ideal error dynamics

For exponential type no-switching attraction law

e_k+1＝(1-ρ(e_k))e_k (4)

Wherein e is_k＝r_k-y_kSystematic tracking error at time k, y_kThe actual output signal of the system at the moment k;

embedding interference suppression measures into the attraction law to construct an ideal error dynamic

Wherein, delta is more than 0 and less than 1, alpha is more than 0, beta is more than 1, d_k+1Is the equivalent interference at the time k +1,

the signal is compensated for equivalent interference. Due to the parameter ρ (e)_k) At e_kAway from the origin, approaches δ at e_kApproaches to the origin and approaches to 1, so under the action of an ideal error dynamic state (6), the system error starts from an initial position, the convergence speed gradually increases, and delta, alpha and beta are parameters for adjusting the suction speed;

3) differential compensation measures for interference

The equivalent interference compensation measures are provided as

Defining an equivalent interference for a given reference signal in the form of

Equivalent interference signal when M is 0

d_k+1＝w_k+1 (7)

Equivalent interference signal when M is 1

d_k+1＝w_k+1-w_k (8)

Equivalent interference signal when M is 3

d_k+1＝[(w_k+1-w_k)-(w_k-w_k-1)]-[(w_k-w_k-1)-(w_k-1-w_k-2)] (9)

Wherein, w_k+1Interference at time k + 1;

and defining the interference difference step number as L, wherein L represents that the interference compensation error comprises L interferences at successive time instants. As in equation (7), a two-step interference differential d_k+1-d_k＝w_k+1-w_kComprising w_k+1And w_kTwo moments interfere; in order to effectively suppress interference, the following condition should be satisfied when selecting the equivalent interference

Wherein the content of the first and second substances,

is the smallest integer not less than.

4) Controller design

According to the ideal error dynamics (6) and the equivalent interference signal d_k+1The following controller is easily given:

for the square wave reference signal formula (1), the controller is

For the triangular wave reference signal formula (2), the controller is

For the S-curve reference signal formula (3), the controller is

Formula (11) and formula(12) And in formula (13), a₁，a₂，b₀，b₁For serving systems

y_k+1+a₁y_k+a₂y_k-1＝b₀u_k+b₁u_k-1+w_k+1 (14)

The parameters of (1);

5) analysis of properties

Giving specific expressions of four indexes such as a steady state error band, an absolute attraction layer, a monotone decreasing area and the maximum step number required for the tracking error to enter the steady state error band for the first time, and the like, and being used for describing the tracking performance of the system and guiding the parameter setting of the controller, wherein the steady state error band, the absolute attraction layer and the monotone decreasing area are defined as follows

Steady state error band Δ_SSE

Absolute attraction Δ_AAL

Monotonous decreasing region delta_MDR

Equivalent interference compensation error satisfaction

Specific expressions of the indexes are as follows:

steady state error band (Δ)_SSE)

Δ_SSE＝Δ (15)

Absolute attraction layer (. DELTA.)_AAL)

Δ_AAL＝max{Δ_AAL1,Δ_AAL2} (16)

Wherein, Delta_AAL1，Δ_AAL2Is a real number, determined by

Monotonous decreasing area (delta)_MDR)

Δ_MDR＝max{Δ_MDR1,Δ_MDR2} (18)

Wherein, Delta_MDR1，Δ_MDR2Is real, and satisfies

Number of convergence steps

Wherein e is₀Is the initial value of the tracking error.

As is clear from equation (15), since the steady-state error band of the tracking error decreases with a decrease in Δ, the control accuracy can be improved by adopting the disturbance difference compensation measure.

In the embodiment, the permanent magnet synchronous motor device executes a position accurate tracking task, and a digital controller is designed for position loop control, wherein 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.

The mathematical model of the servo system is as follows

y_k+1-1.5001y_k+0.4989y_k-1＝2.1589u_k-0.5113u_k-1+w_k+1 (21)

When M is 0, the compound is represented by formula (10)

When M is 1, the compound is represented by formula (11)

When M is 3, the compound is represented by formula (12)

The effectiveness of interference difference compensation measures in a discrete servo system is verified through numerical simulation and experimental results.

The simulation is divided into two parts, the first part verifies the specific expression of the performance index given by the formula (15), and the second part verifies the interference suppression effect of the interference difference compensation measure.

(1) Given reference letter r_k＝10sin(2πfkT_s) deg, frequency f 2Hz, sampling period T_s0.01 interference w_k0.28| mod (k,20) -10| +0.32| mod (k +7,20) -10 |. Under the action of the controller formula (22), when the controller parameters Δ is 0.6, δ is 0.1, α is 5, and β is 10 (see fig. 2 and 3), the performance index is

Δ_SSE＝0.6

Through simulation, the result shows that the steady-state error band delta_SSEThe formula (15) is satisfied.

(2) The reference signals are respectively a square wave signal formula (1), a triangular wave signal formula (2) and an S curve formula (3), and the amplitude A is respectively 5, 10 and 5. To verify that the interference difference step number L is in the full equation (10), the controller is designed to accurately track the corresponding reference signal, α is 10, β is 10, and δ is 0.5, and the disturbance signal is selected as w_k＝0.1r_kSampling period T_s＝0.1。

1) Reference signal r_kFor equation (1), controller equation (22) is used, and the simulation results are shown in FIGS. 4-7, where Δ_SSE＝0deg。

2) Reference signal r_kFor equation (2), the controller is adopted as equation (22), and the simulation results are shown in FIGS. 8-11, in which Δ_SSE＝0.1deg。

3) Reference signal r_kFor equation (2), the controller is adopted as equation (23), and the simulation results are shown in FIGS. 12-15, in which Δ_SSE＝0deg。

4) Reference signal r_kFor equation (3), the controller is adopted as equation (22), and the simulation results are shown in FIGS. 16-19, in which Δ_SSE＝0.2deg。

5) Reference signal r_kFor equation (3), the controller is adopted as equation (23), and the simulation results are shown in fig. 20-23, wherein Δ is_SSE＝0.07deg。

6) Reference signal r_kFor equation (3), the controller is adopted as equation (24), and the simulation results are shown in FIGS. 24-27, in which Δ_SSE＝0deg。

Through simulation (2), it is shown that a controller adopting the interference difference compensation design can realize accurate tracking of a given reference signal when M and L satisfy the formula (10), and the closer L is to each other when M and L do not satisfy the formula (10)

The better the tracking effect and the control process has no buffeting.

The control method provided by the invention is verified on a position servo device, and FIG. 1 is a block diagram of a position servo system. The reference signals are respectively a square wave signal formula (1), a triangular wave signal formula (2) and an S curve formula (3), and the maximum powers of the 3 reference signals with respect to the variable k are respectively 0, 1 and 3. The effects of the interference difference compensation technique were verified using controller equations (20), (21), and (22), respectively. Sampling period T in the experiment_sThe controller parameter α is 10, β is 10, δ is 0.2ms, and the amplitudes a of the reference signals equation (1), equation (2), and equation (3) are 15deg, 180deg, and 180deg, respectively. The experimental results are as follows:

1) reference signal r_kAs shown in formula (1), the controller (22) is adopted to test the knotThe results are shown in fig. 28-29. In the figure, Δ_SSE＝0.05deg。

2) Reference signal r_kThe experimental results are shown in fig. 30-fig. 31 using the controller (22) as in formula (2). In the figure, Δ_SSE＝0.2deg。

3) Reference signal r_kThe experimental results are shown in fig. 32-33 using the controller (23) as in formula (2). In the figure, Δ_SSE＝0.05deg。

4) Reference signal r_kThe experimental results are shown in fig. 34-35 using the controller (22) as in equation (3). In the figure, Δ_SSE＝0.1deg。

5) Reference signal r_kThe results of the experiment are shown in fig. 36-37 using the controller (23) as in equation (3). In the figure, Δ_SSE＝0.07deg。

6) Reference signal r_kThe experimental results are shown in fig. 38-39 using the controller (24) as in equation (3). In the figure, Δ_SSE＝0.05deg。

Experimental results show that when M and L satisfy the formula (10), the controller adopting the interference difference compensation design can realize accurate tracking of a given reference signal, and when M and L do not satisfy the formula (10), the closer L is

The better the system tracking performance and the control process is buffeting free.

Claims (2)

1. An exponential type non-switching attraction law design method of a discrete time controller adopting an interference difference suppression strategy is characterized by comprising the following steps:

1) given a reference signal r_kFor a polynomial of the time variable k, M representing the highest power of the polynomial, three specific reference signals are as follows:

square wave signal, M ═ 0:

triangular wave signal, M ═ 1:

s-curve, M ═ 3:

wherein, A is amplitude, and N is sampling times of the reference signal in one period;

2) structure ideal error dynamics

For exponential type no-switching attraction law

e_k+1＝(1-ρ(e_k))e_k (4)

Wherein e is_k＝r_k-y_kSystematic tracking error at time k, y_kThe actual output signal of the system at the moment k;

embedding interference suppression measures into the attraction law to construct an ideal error dynamic

Wherein, delta is more than 0 and less than 1, alpha is more than 0, beta is more than 1, d_k+1Is the equivalent interference at the time k +1,

compensating the signal for equivalent interference; due to the parameter ρ (e)_k) At e_kAway from the origin, approaches δ at e_kApproaches to the origin, and thus the system error starts from the initial position under the action of the ideal error dynamics (6), the convergence rate gradually increases, and delta, alpha and beta are used for adjusting the suction speedA parameter of degree;

3) interference differential compensation strategy

Take the equivalent interference compensation effect as

Defining an equivalent interference for a particular reference signal in the form of

Equivalent interference signal when M is 0

d_k+1＝w_k+1 (7)

Equivalent interference signal when M is 1

d_k+1＝w_k+1-w_k (8)

Equivalent interference signal when M is 3

d_k+1＝[(w_k+1-w_k)-(w_k-w_k-1)]-[(w_k-w_k-1)-(w_k-1-w_k-2)] (9)

Wherein, w_k+1Interference at time k + 1;

defining the step number of the interference difference as L, wherein L represents that the interference compensation error comprises L interferences at successive moments, and the two-step interference difference d is shown as formula (7)_k+1-d_k＝w_k+1-w_kComprising w_k+1And w_kTwo-time interference, in order to effectively suppress interference, when selecting equivalent interference, the following condition should be satisfied

Wherein the content of the first and second substances,

is the smallest integer not less than;

4) controller design

According to the ideal error dynamics (6) and the equivalent interference signal d_k+1The following controller expression is given:

for a square wave reference signal equation (1),

wherein, F (q)^-1)＝B(q^-1)-b₀；

For the triangular wave reference signal equation (2),

wherein the content of the first and second substances,

for the S-curve reference signal equation (3),

in formulae (11) to (13), A (q)^-1)、B(q^-1) For serving systems

A(q^-1)y_k＝q^-1B(q^-1)u_k+w_k (14)

With respect to q^-1The parameter polynomial of (2):

A(q^-1)＝1+a₁q^-1+a₂q^-2+……+a_nq^-n

B(q^-1)＝b₀+b₁q^-1+b₂q^-2+……+b_mq^-m

wherein u is_kAnd y_kInput and output signals, q, respectively, at time k of the servo system^-1For one-step delay operator, m and n are respectively A (q)^-1)、B(q^-1) Order of (b)₀≠0，1≤m≤n。

2. The method of claim 1, wherein the method further comprises the following steps:

5) analysis of properties

And giving specific expressions of a steady-state error band, an absolute attraction layer, a monotone subtraction area and at most four indexes of steps required by the tracking error entering the steady-state error band for the first time, and being used for describing the tracking performance of the system and guiding the parameter setting of the controller, wherein the steady-state error band, the absolute attraction layer and the monotone subtraction area are defined as follows:

steady state error band Δ_SSE

Absolute attraction layer Δ_AAL

Monotonous decreasing region delta_MDR

Equivalent interference compensation error satisfaction

The expression of each index is as follows:

steady state error band Δ_SSE

Δ_SSE＝Δ (15)

Absolute attraction layer Δ_AAL

Δ_AAL＝max{Δ_AAL1,Δ_AAL2} (16)

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

Monotonous decreasing region delta_MDR

Δ_MDR＝max{Δ_MDR1,Δ_MDR2} (18)

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

Number of convergence steps

Wherein e is₀Is the initial value of the tracking error.

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