CN1275111C - Error feedback anti-saturation static compensation controller and determining method for compensator - Google Patents

Abstract

The present invention relates to an error feedback anti-saturation static compensation controller and a determining method for a compensator, which are used for the field of control engineering. On the basis that the present invention improves the existing error feedback compensation uniform framework (referring to the picture), a structural parameter of a controller k (s) and an object P (s) is determined firstly by using a uniform framework method; then, on the basis that definite performance optimization is guaranteed, the stability of an anti-saturation system is optimized, and a stability optimization problem is converted into a convex set optimization problem, so the structure parameter of two feedback compensators lambda 1, 2 before improvement are determined; a compensator lambda 3 is changed manually finally to improve the response of the system, and the structure of the compensator lambda 3 is determined. The present invention better solves the problems of stability optimization and performance optimization by improving an original error feedback uniform framework, and the output response is improved; compared with other same arts, the design of the anti-saturation controller and the compensator is greatly simplified by the technology used in the present invention, and the present invention has better universal adaptability.

Description

Definite method of error feedback anti-saturation static compensation device

Technical field

The present invention relates to a kind of determining method for compensator, specifically is a kind of definite method of error feedback anti-saturation static compensation device, is used for the control engineering field.

Background technology

Various linear time invariant Anti-windup and bumplesstransfer (AWBT) framework before the Unified frame theory has objectively been unified.AWBT has multiple different framework, but because they do not have universality, so they itself and on their bases, develop the various technology that and also do not have universality, Kothare proposes Unified frame, with the unified difference of various LTI AWBT frameworks, develop the saturation problem comprehensive analysis method that on this basis and also can include achievement with front baffle for two parameter matrixs in this framework.Traditional method for analyzing stability mainly contains: circles theorem, and the off-line circles theorem, multivariate circles theorem etc., but they all can only represent the overachievement of research field separately without exception, and can't contain the achievement in research of other framework.

Find through open source literature retrieval prior art, Mulder E F, Kothare M V, and Morari M is at " Automatica " (2001, vol.37 (01): (the utilization LMI is handled multivariate antisaturation controller synthesis problem to the article of delivering 1407-1416) " Multivariableanti-windup controller synthesis using linear matrix inequalities ", " control automatically "), people such as Kothare have proposed comprehensive multiloop circles theorem, stability problem is converted into the LMI problem, it has easy advantage, and universality preferably arranged, but this universality neither be absolute, for example when handling common output feedback control problem, this method just can't be translated into the convex set optimization problem effectively.

Traditional performance optimization method has a lot, L ₂The gain theorem is important a kind of, and it is based on, and Unified frame proposes, and this method can turn to the performance optimization problem LMI problem, and this method has the little advantage of calculated amount equally, and its universality neither be absolute.When improving the serviceability of system, must be cost to sacrifice its stability, this is the wretched insufficiency of prior art.

Summary of the invention

The objective of the invention is to overcome the deficiencies in the prior art, a kind of definite method of error feedback anti-saturation static compensation device is provided, it is being carried out on the improved basis original AWBT Unified frame, again utilization stability and performance optimization method carry out the analysis of AWBT synthtic price index, little except having calculated amount, outside the characteristics such as universality height, can also take into account the stability and the serviceability of system simultaneously.

The present invention is achieved by the following technical solutions, by existing error feedback compensation Unified frame is improved, introduces Error Feedback compensator Λ 3, on this basis, at first uses the method for Unified frame to determine controller Structural parameters with object P (s); When guaranteeing to obtain better serviceability, the optimizing stability problem is converted into the convex set optimization problem then, thus two feedback compensator Λ before determining to improve ₁₂Structural parameters; By to another compensator Λ ₃Adjusting, improve the output response of system, determine compensator Λ simultaneously ₃Structure, comprise three parts:

(1) construction linear time invariant system

To analyze a real system, at first must carry out modeling, accurately replace real system with system chart to system.The Unified frame that the present invention adopts is based on that the control strategy of " two-step approach " forms.At first ignoring under the condition of nonlinear interaction, real system is being configured as linear time invariant (LTI) system, and then considering nonlinear interaction, determining new controller parameter with the easiest existing method

(2) stability of system and serviceability optimization

Stability and serviceability are the basic problems that any control problem all must be considered.The present invention is converted into the LMI optimization problem with the optimizing stability problem when serviceability is further optimized, obtain the optimization solution of stability problem, and this optimization solution is used to determine two compensator Λ in the former framework the most at last ₁₂Structural parameters.

(3) performance of improvement system

The present invention is by introducing Error Feedback compensator Λ ₃And manual adjustments can be improved the performance of system.

Below the invention will be further described, particular content is as follows:

1, construction linear time invariant system

At first, determine the actuator of linear system and the structural parameters of controller when not considering saturation, these parameters generally are to determine by the method for mathematical modeling.According to the controller architecture parameter K (s) that obtains, determine then at the new controller that has under the saturation conditions

Structural parameters, wherein,

For the needs of further analyzing, will Be converted into state space form:

$\stackrel{\·}{x}=\mathrm{Ax}+(B_v-B_{\mathrm{\ξ}}\mathrm{\Λ})v+B_{w}w$

u＝C _ux+(D _uv-D _uξΛ)v+D _uww

z＝C _zx+(D _zv-D _zξΛ)v+D _zww

A wherein, B _v, B _ξ, B _wDeng the function that is the initial parameter of feedback/feed-forward interconnection system.W represents all input signals of the system that enters from surrounding environment, comprises interference, sensor noise etc.; Z represents the controlled output signal that system is all, tracking error for example, and the purpose of CONTROLLER DESIGN is exactly as much as possible z to be remained less value.

2, the stability of system and serviceability optimization

At Error Feedback compensator Λ ₃(Λ when not having an effect ₃=0), utilization method for optimizing stability and L ₂The gain performance optimization, stability and the serviceability to system is optimized simultaneously, and determines two compensator Λ ₁And Λ ₂Structural parameters.Suppose L from input w to controlled output z ₂Norm is:

$\underset{{\left|\right|w\left|\right|}_2\≠0}{\sup }\frac{{\left|\right|{\mathrm{\γ}}^{1/2}z\left|\right|}_2}{{\left|\right|w\left|\right|}_2}\≤\sqrt{\mathrm{\Γ}}---\left(1\right)$

By minimizing this L ₂Norm (is promptly determined

), determine the performance optimization index of system, on the basis of this performance optimization index, again system is carried out optimizing stability.The optimizing stability problem is converted into LMI the most at last, but separating of obtaining is that local optimum is separated.

Whether the question resolves itself into exists matrix $M=\mathrm{diag}(M_1,M_2,K,M_{n_u})$ Following LMI is set up, and satisfied M＞0, δ ₁=δ ^-1＞0, Q=P ^-1In the time of＞0 and Γ＞0, it is minimum making Γ

$\left[\begin{array}{ccccc}{\mathrm{QA}}^T+\mathrm{AQ}& B_w& B_{v}M-B_{\mathrm{\&xi;}}X+{\mathrm{QC}}_u^T& {\mathrm{QC}}_z^T& 0\\ B_w^T& -\mathrm{\&Gamma;}& D_{\mathrm{uw}}^T& D_{\mathrm{zw}}^T& 0\\ {\mathrm{MB}}_v^T-X^T{B}_{\mathrm{\&xi;}}^T+C_{u}Q& D_{\mathrm{uw}}& l& {\mathrm{MD}}_{\mathrm{zv}}^T-X^T{D}_{\mathrm{z\&xi;}}^T& M\\ C_{z}Q& D_{\mathrm{zw}}& D_{\mathrm{zv}}M-D_{\mathrm{z\&xi;}}X& {-\mathrm{\&gamma;}}^{-1}& 0\\ 0& 0& M& 0& -{\mathrm{\&delta;}}_{1}I\end{array}\right]<0---\left(2\right)$

Wherein: $l=-2M+D_{\mathrm{uv}}M+{\mathrm{MD}}_{\mathrm{uv}}^T-D_{\mathrm{u\&xi;}}X-X^T{D}_{\mathrm{u\&xi;}}^T\&CenterDot;$

Feasible solution M and X by formula (2) determine compensator Λ ₁And Λ ₂Gain matrix Λ=XM ^-1

The present invention again can be by (γ is the diagonal angle real matrix, represents the importance of every road output to weight factor γ.For example, if the output of each road is of equal importance, then the selection of γ=I) is carried out manual adjustments to the performance index of system and (is promptly determined by formula (1)

Minimum value), like this, the performance of the output signal that some are important will obtain emphasis optimization.

3, improve the performance of system

The Error Feedback compensator that the present invention introduces will act on tracking error signal z, influence output signal the most at last.The present invention is by manual adjustments Λ ₃The method of gain matrix can improve the performance of system.

Generally speaking, the present invention can be resolved the antisaturation problem of system owing to adopted the two-step approach of technical maturation in a comparatively simple framework.Not only solved the stability and the performance optimization problem of system reliably,, further improved the output response of system also by introducing the Error Feedback compensator.Empirical tests uses the present invention can handle a series of saturation problem, for the user constructs the system of a stable and function admirable, and prolongs the serviceable life of control system topworks to a certain extent.

Description of drawings

The improved Error Feedback Unified frame of Fig. 1 synoptic diagram

Fig. 2 embodiment: four water pot devices equivalence block diagram

Embodiment

As shown in Figure 1, the content in conjunction with definite method of error feedback anti-saturation static compensation device provides following examples:

The present invention is used for the comprehensive Design (comprising optimizing stability and performance optimization) of following framework, and determines the structural parameters of three static compensators simultaneously.

1, construction linear time invariant system

As shown in Figure 2, four water pot experimental provisions are by four interconnected water pots, and two water pumps and a plurality of valve are formed.The input signal of device is input voltage amplitude (v), the V of water pump ₁And V ₂The output signal of device is current potential (v), the Y of height of water level of two water pots of respective base ₁And Y ₂Wherein, the discharge that flows into each water pot is controlled by related valve.

If obtain by certain system is carried out mathematical modeling

$P\left(s\right)=\left[\begin{array}{cc}\frac{0.794}{(29.37s+1)}& \frac{0.454}{(\mathrm{33,91}s+1)(29.37s+1)}\\ \frac{0.305}{(33.91s+1)(29.37s+1)}& \frac{0.610}{(33.91s+1)}\end{array}\right]$

The mathematical model of controller is selected (decoupling zero) for use:

$\left[\begin{array}{c}{u}_1\left(s\right)\\ u_2\left(s\right)\end{array}\right]=\left[\begin{array}{cc}20+\frac{4}{s}+s& 0\\ 0& 20+\frac{4}{3}+s\end{array}\right]\left[\begin{array}{c}{y}_1\left(s\right)-y_1^{*}\\ y_2\left(s\right)-y_2^{*}\end{array}\right]$

Because valve can not unrestrictedly be opened greatly, so limited this practical problems that makes of valve opening degree has become saturation problem.The initial input set point of supposing this system is $\left[\begin{array}{cc}{y}_1^{*}& y_2^{*}\end{array}\right]=\left[\begin{array}{cc}12& 10\end{array}\right],$ The output potential of the present actuator of the limited issue table of valve is limited, makes $\hat{u}\&le;50\&CenterDot;$ Here, as long as just can obtain by the present invention

With

.Suppose z=e and w=r, the transport function of said system

With

Be converted into state space form, can obtain the structural parameters of each matrix.

2, the stability of system and serviceability optimization

Suppose Λ ₃=0, suppose γ=I, substitution formula (1) obtains performance index

With every loop parameter substitution formula (2) that the first step obtains, obtain the feasible solution X and the M of LMI problem, thereby determine Λ _1,2Structural parameters.

Obtain ${\mathrm{\&Lambda;}}_1=\left[\begin{array}{cc}59.4804& -59.4275\\ -9.9421& 10.9903\\ -67.0548& 86.0632\\ 12.6603& -15.5621\end{array}\right],$ ${\mathrm{\&Lambda;}}_2=\left[\begin{array}{cc}93.4235& -84.16\\ -96.1038& 124.91112\end{array}\right]$

3, improve the performance of system

For the further performance of improvement system, the error input of system controller is compensated, i.e. manual adjustments Λ ₃Value.Actual proof: work as setting ${\mathrm{\&Lambda;}}_3=\left[\begin{array}{cc}-0.3& 0\\ 0& -0.3\end{array}\right]$ The time, the output response of system has improvement.Add that promptly the overshoot of system further reduces behind the anti-saturation static compensation device that is proposed, Xiang Ying rapidity increases simultaneously, and reduce stabilization time.

Claims (3)

1, a kind of definite method of error feedback anti-saturation static compensation device is characterized in that, existing error feedback compensation Unified frame is improved, and introduces Error Feedback compensator Λ ₃, Λ ₃With sending into feedback channel after the processing of the error signal between the actuator input and output signal that collects, on this basis, at first use the method for Unified frame to determine controller

Structural parameters with object P (s); When guaranteeing to obtain better serviceability, the optimizing stability problem is converted into the convex set optimization problem then, thus two feedback compensator Λ before determining to improve _1,2Structural parameters; By to another compensator Λ ₃Adjusting, improve the output response of system, determine compensator Λ simultaneously ₃Structure, comprise three parts:

(1) construction linear time invariant system

To analyze a real system, at first must carry out modeling to system, accurately replace real system with system chart, the Unified frame that adopts is based on that the control strategy of " two-step approach " forms: at first ignoring under the condition of nonlinear interaction, real system is configured as linear time invariant (LTI) system, and then the consideration nonlinear interaction, determine new controller parameter with the easiest existing method

(2) stability of system and serviceability optimization

Stability and serviceability are the basic problems that any control problem all must be considered, when serviceability is further optimized, the optimizing stability problem is converted into the LMI optimization problem, obtain the optimization solution of stability problem, this optimization solution is used to determine two compensator Λ in the former framework the most at last _1,2Structural parameters;

(3) performance of improvement system

By introducing Error Feedback compensator Λ ₃And manual adjustments can be improved the performance of system.

2, definite method of error feedback anti-saturation static compensation device according to claim 1 is characterized in that,

Determine by K (s) is unique, promptly

Then

3, definite method of error feedback anti-saturation static compensation device according to claim 1 is characterized in that, is guaranteeing certain system works performance index The basis on, stability problem is converted into the convex set optimization problem, specifically be whether the question resolves itself into is existed matrix $M=\mathrm{diag}(M_1,M_2,K,M_{n_u})$ Following LMI is set up, and satisfied M＞0, δ ₁=δ ^-1＞0, Q=P ^-1In the time of＞0 and Г＞0, it is minimum making Г

$\left[\begin{array}{ccccc}{\mathrm{QA}}^T+\mathrm{AQ}& B_w& B_{v}M-B_{\mathrm{\&xi;}}X+{\mathrm{QC}}_u^T& {\mathrm{QC}}_z^T& 0\\ B_w^T& -\mathrm{\&Gamma;}& D_{\mathrm{uw}}^T& D_{\mathrm{zw}}^T& 0\\ {\mathrm{MB}}_v^T-X^T{B}_{\mathrm{\&xi;}}^T+C_{u}Q& D_{\mathrm{uw}}& l& {\mathrm{MD}}_{\mathrm{zv}}^T-X^T{D}_{\mathrm{z\&xi;}}^T& M\\ C_{z}Q& D_{\mathrm{zw}}& D_{\mathrm{zv}}M-D_{\mathrm{z\&xi;}}X& -{\mathrm{\&gamma;}}^{-1}& 0\\ 0& 0& M& 0& -{\mathrm{\&delta;}}_{1}I\end{array}\right]<0$

Wherein: $l=-2M+D_{\mathrm{uv}}M+{\mathrm{MD}}_{\mathrm{uv}}^T-D_{\mathrm{u\&xi;}}X-X^T{D}_{\mathrm{u\&xi;}}^T.$ From the feasible solution X and the M of following formula, determine two compensator Λ in original framework _1,2Structural parameters, compensator determines that formula is: Λ=XM ^-1

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