CN110161857B - Design method of active disturbance rejection controller suitable for non-minimum phase system - Google Patents

Abstract

The invention discloses a design method of an active disturbance rejection controller suitable for a non-minimum phase system, which comprises the following steps: 1) providing a stable non-minimum phase system model G(s); 2) setting a closed-loop reference model, the transfer function H of which_R(s); 3) with H_R(s) constructing an ESO for the target; 4) total disturbance d of the system_K+ f frequency domain estimation; 5) constructing a disturbance compensation control rate; 6) giving a stability judgment condition; 7) designing parameters of K(s); 8) calculating time constantτAnd in the stable regionτSelecting tau value to realize ESO and adjusting tau to realize expected performance in tau < infinity. The invention provides a design method of an active disturbance rejection controller suitable for a non-minimum phase system, and provides a new disturbance rejection control rate, which has a simple structure and is easy to realize.

Description

Design method of active disturbance rejection controller suitable for non-minimum phase system

Technical Field

The invention relates to an industrial process control system and a motion control system, belongs to the technical field of anti-interference control, and discloses a design method of an active-anti-interference controller suitable for a non-minimum phase system.

Background

In actual industrial production, an industrial system is often affected by multiple disturbances, one is internal disturbance caused by mismatch of a physical model and the actual system, and the other is external disturbance, mainly disturbance generated by the outside to a control system. Modern industrial production and manufacturing processes are continually demanding increased performance and accuracy from control systems. By designing the anti-interference controller, the non-static-error regulation and tracking control are realized, and the method has important functions and significance.

The non-minimum phase system is widely used in industrial objects, such as robot flexible structure control, concentration control in chemical process, ship course control, aircraft attitude control and the like. The non-minimum phase system is mainly characterized in that the zero point of the right half plane is provided, the one-to-one correspondence relation between amplitude frequency and phase frequency is not satisfied, and the design and analysis method of the minimum phase system is not applicable any more. The pole-zero of the right half-plane adversely affects the stability, robustness and dynamic performance of the system. The control of non-minimum phase systems is much more difficult than minimum phase systems.

Active Disturbance Rejection Control (ADRC) is an effective control strategy for external disturbances and internal uncertainties of the system. The basic idea of the method is to take the uncertainty, the external disturbance and even the nonlinearity of the model as a total disturbance, and actively estimate and compensate by an extended state observer, so that the dependence on the model is eliminated. The active disturbance rejection controller is originally provided for a minimum phase system, realizes linear regulation of a control system by regulating two bandwidth parameters and an input gain parameter, has the characteristics of strong robustness, simple structure, high response speed, small overshoot and the like, and greatly promotes the development and application of disturbance rejection control theory. For a non-minimum phase system, a conventional active disturbance rejection controller is easy to make the system unstable, the relation between the controller parameter and the control performance is not clear enough, and the system is difficult to popularize and apply in an industrial control system.

Therefore, it is important to provide an active disturbance rejection controller suitable for non-minimum phase system in the present invention.

Disclosure of Invention

The present invention provides a design method of an active disturbance rejection controller suitable for non-minimum phase systems, which overcomes the deficiencies of the prior art described in the background.

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

an active disturbance rejection controller design method suitable for a non-minimum phase system comprises the following steps:

1) providing a stable non-minimum phase system model G(s);

2) setting a closed-loop reference model, the transfer function H of which_R(s) is:

with predistorters K(s) representing augmented objects G(s) K(s) as perturbed closed-loop reference modelsForm, there are:

y^(r)＝-k₀y⁽⁰⁾-k₁y⁽¹⁾-…-k_r-1y^(r-1)+k₀(u_K+d_K+f)；

wherein the relative order r.deg [ G ] is determined according to a non-minimum phase system model G(s)]Selecting a closed-loop reference model H_R(s) relative order to the predistorter K(s) to satisfy r.deg [ K ]]+r.deg[G]＝r.deg[H_R]；d_Kd/K, d is the external disturbance of the system, K is the pre-compensator, f is the internal disturbance of the system, d_K+ f is the total disturbance of the system, defining the expansion state x_(r+1)＝d_K+f，u＝Ku_K(ii) a The state space model is:

wherein h is the total disturbance d of the system_KA differential of + f;

3) with H_R(s) targeting, constructing an ESO, having

Wherein x is_o＝[x_o1,x_o2,…,x_o(r+1)]^TFor ESO states, L is observer gain, y_pFor the actual output of the system, the pole allocation condition is met:

wherein, ω is_oFor the ESO bandwidth, τ is 1/ω_oIs the ESO time constant, I is the identity matrix;

4) total disturbance d of the system_KFrequency of + fDomain estimation; in active disturbance rejection control, expanded state x_o(r+1)Can realize the system disturbance estimation, and has x in the frequency domain_o(r+1)(s)＝-F₁(s)u_K(s)+F₂(s)y_p(s) wherein:

5) constructing a disturbance compensation control rate u ═ K(s) u_k＝K(s)·(y*-x_o(r+1)) Where y is a given input signal;

6) giving stability judgment conditions, writing the involved parts into a form of a relatively prime polynomial, having

The characteristic equation of the closed loop system is

p_c(s,τ)＝a_F(τs)a_k(s)a_g(s)b_h(s)(a_F(τs)-b_F(τs))+a_F(τs)b_k(s)b_g(s)a_h(s)b_F(τs)

＝a_F(τs)a_k(s)a_g(s)b_h(s)φ(s,τ)；

Wherein the content of the first and second substances,

construction of

If phi (S) is Hurwitz polynomial, the system is stable at tau ∞ and there is a stable regionτ< tau < ∞, the system remains stable and has

Wherein the content of the first and second substances,

7) design parameters of K(s) taking into account

And

two cases, in which z, p, α, β are design parameters of the predistorter K, can be designed as the following two cases, respectively:

the first condition is as follows:

let λ be such that φ (S) is a Hurwitz polynomial, then z ═ a₀λ and p ═ b₀；

Case two:

let λ be such that φ (S) is a Hurwitz polynomial, then z ═ a₀λ and p ═ b₀Alpha and beta are undetermined parameters;

introducing a low-pass filter W(s), optimally calculating alpha and beta through low-frequency model matching,

8) calculating time constantτAnd in the stable regionτSelecting tau value to realize ESO and adjusting tau to realize expected performance in tau < infinity.

Compared with the background technology, the technical scheme has the following advantages:

1. by means of a pre-compensator K(s), 1/b of the conventional active disturbance rejection control is replaced₀The dynamic characteristics of the system can be compensated better, so that G(s) K(s) and the closed-loop reference model H_R(s) matching at low frequency band, canThe stability of the system is ensured, and a theoretical basis is provided for the parameter design of the predistorter K(s).

2. The time constant τ of the ESO is given (i.e. the observer bandwidth inverse, τ ═ 1/ω)_o) The method of calculating a stability interval of (1).

3. A new anti-interference control rate is provided, the structure is simple, and the implementation is easy.

4. A design method of an active disturbance rejection controller suitable for a non-minimum phase system is provided.

Drawings

The invention is further illustrated by the following figures and examples.

Fig. 1 shows a block diagram of a proposed active disturbance rejection controller system;

FIG. 2 is a diagram of the distance phi (τ) of a chemical process control system;

FIG. 3 is a Nyquist plot of L (s, τ) at different bandwidths in a chemical process control system;

FIG. 4 shows a variation c of the chemical process control system_A0Set value and output concentration c_BAnd (5) a relational graph.

Detailed Description

An active disturbance rejection controller design method suitable for a non-minimum phase system comprises the following steps:

1) providing a stable non-minimum phase system model G(s);

i.e. the pole of the object is located in the left half-plane (object stable) but there is a right half-plane zero, according to the system relative order r.deg G]Selecting a closed-loop reference model H_R(s) relative order to the predistorter K(s) to satisfy r.deg [ K ]]+r.deg[G]＝r.deg[H_R]Such that G(s) K(s) is at zero frequency with H_R(s) are approximately equal, with G (j0) K (j0) ≈ H_R(j0) In that respect In the active-disturbance-rejection control,

the parameter is a gain compensation located in front of the controlled object, having

In the prior art, the parameter is not given

The relationship between this parameter and stability is also not clear in a reasonable selection method in non-minimum phase systems. In the present invention, the parameter is generally designed, i.e. a dynamic compensator K(s) is introduced to replace

Forming an augmented object G(s) K(s), and designing a predistorter K(s).

2) Setting a closed-loop reference model, the transfer function H of which_R(s) is:

with a predistorter K(s) representing the augmented object G(s) K(s) as a perturbed version of a closed-loop reference model, comprising:

y^(r)＝-k₀y⁽⁰⁾-k₁y⁽¹⁾-…-k_r-1y^(r-1)+k₀(u_K+d_K+f)；

wherein d is_Kd/K, d is the external disturbance of the system, K is the pre-compensator, f is the internal disturbance of the system, d_K+ f is the total disturbance of the system, defining the expansion state x_(r+1)＝d_K+f，u＝Ku_K(ii) a The state space model is:

wherein h is the total disturbance d of the system_KA differential of + f;

3) with H_R(s) targeting, constructing an ESO, having

Wherein x is_o＝[x_o1,x_o2,…,x_o(r+1)]^TFor ESO states, L is observer gain, y_pFor the actual output of the system, the pole allocation condition is met:

wherein ω is_oFor the ESO bandwidth, τ is 1/ω_oFor the ESO time constant, I is the identity matrix.

4) Total disturbance d of the system_K+ f frequency domain estimation; in active disturbance rejection control, expanded state x_o(r+1)Can realize the system disturbance estimation, and has x in the frequency domain_o(r+1)(s)＝-F₁(s)u_K(s)+F₂(s)y_p(s) wherein:

5) constructing a disturbance compensation control rate u ═ K(s) u_k＝K(s)·(y^*-x_o(r+1)) Wherein y is^*For a given input signal; compared to conventional active disturbance rejection control, this control rate only contains feedback compensation for the dilated state, because: ESO was designed for H_R(s) rather than for the controlled object G(s), the total disturbance d of the system_K+ f from expanded state x_o(r+1)After estimation, the system is compensated, so that the input-output relation of the system is close to H_R(s), the control structure is very simple.

6) Give out a stability judgmentConditional, writing the respective parts involved in a co-prime polynomial form, of

The characteristic equation of the closed loop system is

p_c(s,τ)＝a_F(τs)a_k(s)a_g(s)b_h(s)(a_F(τs)-b_F(τs))+a_F(τs)b_k(s)b_g(s)a_h(s)b_F(τs)

＝a_F(τs)a_k(s)a_g(s)b_h(s)φ(s,τ)；

Wherein the content of the first and second substances,

construction of

S ═ τ S, τ → ∞ is equivalent to S → 0. If phi (S) is Hurwitz polynomial, the system is stable at tau ∞, there is a stable region tau < tau ∞, the system remains stable, and there is

Wherein the content of the first and second substances,

7) design parameters of K(s) taking into account

And

two cases, in which z, p, α, β are design parameters of the predistorter K, can be designed as the following two cases, respectively:

the first condition is as follows:

design parameter z ═ a₀λ and p ═ b₀Such that phi (S) is a Hurwitz polynomial;

case two:

design parameter z ═ a₀λ and p ═ b₀Alpha and beta are undetermined parameters, and corresponding values can make phi (S) be Hurwitz polynomial;

introducing a low-pass filter W(s), optimally calculating alpha and beta through low-frequency model matching,

8) calculating the time constant tau and in the stable intervalτSelecting tau value to realize ESO and adjusting tau to realize expected performance in tau < infinity.

The concentration control problem of a chemical process control system is considered, and simulation verification proves that the anti-interference controller provided by the invention has a good effect. The chemical system mainly controls the concentration of a product B by adjusting a dilution ratio F, wherein the inflow contains a class of reactants A with the concentration c_A0Has certain influence on the reaction process and can be regarded as the interference existing in the system. The active disturbance rejection controller is now designed for the specific embodiment:

(1) when the temperature T of the reaction kettle is assumed to be constant, the system is in an equilibrium state, and the dilution F and the product concentration y are c_BThe transfer function of the relationship between:

and has the following input and output requirements:

c_A0c is not less than 4.5mol/l_A0≤5.7mol/l。

It can be seen that the chemical engineering control system has a zero point of the right half plane, and is a non-minimum phase system. The relative order of the system is denoted r.deg.G]Selecting and determining a closed-loop referential model H as 2_RAnd predistorter K(s) are each r.deg [ H ]_R]＝r.deg[G]2 and r.deg [ K ]]0, such that: r.deg [ K ]]+r.deg[G]＝r.deg[H_R]。

(2) Designing the desired closed loop transfer function H_R(s) setting the parameter to ω_c50. So that the closed loop system has certain response speed. The transfer function of the closed-loop reference model is:

the augmented object G(s) K(s) is written in a form of a closed-loop reference model containing disturbance input, and a differential equation is expressed as:

y^(r)＝-100y⁽¹⁾-2500y⁽⁰⁾+2500(u_K+d_K+f)

wherein d is_Kd/K, f is the internal disturbance of the system, d_K+ f is the total disturbance of the system, defined as the expansion state x_(r+1)＝d_K+f，u＝Ku_K. As shown in fig. 1, the ESO performs an estimated compensation of the dilated state, thereby performing an elimination of the disturbance.

The state space model is:

(3) for the closed-loop reference model, model H_R(s) to target, construct the ESO and feedback active disturbance rejection control structure, having:

wherein x_o＝[x_o1,x_o2,x_o3]^TIs a system state variable, y₀For the actual output of the system, L is the gain of the extended state observer, and the condition of satisfying the pole allocation is as follows:

wherein ω is_oDenotes the ESO bandwidth, τ ═ 1/ω_oIs the time constant of the ESO.

(4) In active disturbance rejection control, expanded state x_o3Can realize the system disturbance estimation, and has x in the frequency domain_o3＝-F₁u_K+F₂y_pWherein:

(5) constructing a disturbance compensation control law u ═ K(s) u_k＝K(s)·(y^*-x_o3)。

(6) And (3) stability judgment conditions: there is a time constantτ> 0, for allτ< τ < ∞, the closed loop system has robust internal stability if the following conditions are met:

(C1)r.deg[K]+r.deg[G]＝r.deg[H_R]；

(C2) g(s) has no pole of the right half-plane, or a_g(s) is a Hurwitz polynomial;

(C3) k(s) has no pole on the right half-plane, or a_k(s) is a Hurwitz polynomial;

(C4) φ (S) is a Hurwitz polynomial.

Writing the various parts involved in a co-prime polynomial of the form:

the characteristic polynomial of the closed loop system is:

p_c(s,τ)＝a_F(τs)a_k(s)a_g(s)b_h(s)φ(s,τ)

wherein: phi (S) ═ S³+3S²+3S + λ, S ═ τ S. Due to a_F(ts) automatically satisfies the Hurwitz condition, a_k(s) satisfies the stability condition (C3), a_g(s) satisfies the stability condition (C2), b_h(s) is a constant value so that the Hurwitz condition is satisfied for the closed loop system characteristic polynomial. If K is known, the time constant τ is calculated as the lower bound:

wherein

FIG. 3

The point where the distance between the nyquist curve L (s, τ) and (-1,0) is the smallest is represented, and when Φ is 0, the nyquist curve passes through (-1,0), and the corresponding minimum value of τ is τ.

(7) Designing parameters of K(s) according to the stability judgment condition,

consider that

In this example, φ (S) ═ S³+3S²+3S + λ, λ is chosen such that φ (S) is a Hurwitz polynomial, where λ is 0 < λ < 9, and z is a₀λ 115370 λ and p b₀＝5142.8。

In this example, λ is 1, and K is 22.43. Fig. 2 shows the values of k and τ in the predistorter for different λ.

(8) Calculating the minimum value of the time constant tau as tau, and calculating the minimum value in the embodimentτDetermining the stable interval to be 0.0033 & lttau & lt infinity, and selecting proper ESO bandwidth omega_oE [0, 303.03)), and fig. 3 shows the nyquist plot of L (s, τ) at different bandwidths when K is 22.43, and the ESO parameter ω is selected_o150. FIG. 4 shows the difference c_A0Under the condition, a system response diagram of the concentration adjusting process realizes better control performance.

The above description is only a preferred embodiment of the present invention, and therefore should not be taken as limiting the scope of the invention, which is defined by the appended claims and their equivalents.

Claims (1)

1. An active disturbance rejection controller design method suitable for a non-minimum phase system is characterized by comprising the following steps: the method comprises the following steps:

1) providing a stable non-minimum phase system model G(s);

2) setting a closed-loop reference model, the transfer function H of which_R(s) is:

with a predistorter K(s) representing the augmented object G(s) K(s) as a perturbed version of a closed-loop reference model, comprising:

y^(r)＝-k₀y⁽⁰⁾-k₁y⁽¹⁾-…-k_r-1y^(r-1)+k₀(u_K+d_K+f)；

wherein the relative order r.deg [ G ] is determined according to a non-minimum phase system model G(s)]Selecting a closed-loop reference model H_R(s) relative order to the predistorter K(s) to satisfy r.deg [ K ]]+r.deg[G]＝r.deg[H_R]；d_Kd/K, d is the external disturbance of the system, K is the pre-compensator, f is the internal disturbance of the system, d_K+ f is the total disturbance of the system, defining the expansion state x_(r+1)＝d_K+f，u＝Ku_K(ii) a The state space model is:

wherein h is the total disturbance d of the system_KA differential of + f;

3) with H_R(s) targeting, constructing an ESO, having

Wherein x is_o＝[x_o1,x_o2,…,x_o(r+1)]^TFor ESO states, L is observer gain, y_pFor the actual output of the system, the pole allocation condition is met:

wherein, ω is_oFor the ESO bandwidth, τ is 1/ω_oIs the ESO time constant, I is the identity matrix;

4) total disturbance d of the system_K+ f frequency domain estimation; in active disturbance rejection control, expanded state x_o(r+1)Can realize the system disturbance estimation, and has x in the frequency domain_o(r+1)(s)＝-F₁(s)u_K(s)+F₂(s)y_p(s) wherein:

5) constructing a disturbance compensation control rate u ═ K(s) u_k＝K(s)·(y^*-x_o(r+1)) Wherein y is^*For a given input signal;

6) giving stability judgment conditions, writing the involved parts into a form of a relatively prime polynomial, having

The characteristic equation of the closed loop system is

p_c(s,τ)＝a_F(τs)a_k(s)a_g(s)b_h(s)(a_F(τs)-b_F(τs))+a_F(τs)b_k(s)b_g(s)a_h(s)b_F(τs)

＝a_F(τs)a_k(s)a_g(s)b_h(s)φ(s,τ)；

Wherein the content of the first and second substances,

construction of

If phi (S) is Hurwitz polynomial, the system is stable at tau ∞ and there is a stable regionτ< tau < ∞, the system remains stable and has

Wherein the content of the first and second substances,

7) design parameters of K(s) taking into account

And

two cases, in which z, p, α, β are design parameters of the predistorter K, can be designed as the following two cases, respectively:

the first condition is as follows:

let λ be such that φ (S) is a Hurwitz polynomial, then z ═ a₀λ and p ═ b₀；

Case two:

let λ be such that φ (S) is a Hurwitz polynomial, then z ═ a₀λ and p ═ b₀Alpha and beta are undetermined parameters;

introducing a low-pass filter W(s), optimally calculating alpha and beta through low-frequency model matching,

8) calculating time constantτAnd in the stable regionτSelecting tau value to realize ESO and regulating tauAnd (4) realizing the expected performance.

Images