CN106773695A - Non-linear switching two-time scale system synovial membrane control method - Google Patents

Abstract

A kind of non-linear switching two-time scale system synovial membrane control method, belongs to control of complex systems field, it is adaptable to there are non-linear, switching and three kinds of high-precision controls of specialty systemizations of double donors.The method describes non-linear switching two-time scale system using discrete-time fuzzy singular perturbation switching model, fusion fuzzy logic, singular perturbation technology and sliding mode control theory, design synovial membrane controller, the adequate condition for solving synovial membrane controller gain is provided, for non-linear switching two-time scale system provides high-precision control scheme.Advantage is to solve the problems, such as that existing control method cannot take into account the steady-state error that the free switching between treatment each subsystem of controlled system triggers with fast mode, so as to greatly improve the control performance of non-linear switching two-time scale system.

Description

Non-linear switching two-time scale system synovial membrane control method

Technical field

The invention belongs to control of complex systems technical field, in particular, provide a kind of non-linear switching two-time scale system and slide Film control method, suitable for the modeling of the complication systems such as board rolling, flexible arm of robot, medical mechanism arm and complicated circuit With high-precision control.

Background technology

Non-linear switching two-time scale system control problem is that a class is related to switching and markers to convert two kinds of complicated systems of problem System control problem, is widely present in the fields such as board rolling, intelligent robot and power electronics.It is general at present because of such problem Growing control accuracy demand cannot have been met all over the classical control theory for using and method, in the urgent need to proposing new theory With method.

Research for nonlinear system, switching system and two-time scale system in recent years obtains greater advance, but considers Non-linear, switching and double donors simultaneously deposit the complex system modeling and control problem of situation still in the elementary step, because on Stating three kinds of presence of characteristic will significantly increase system controller design difficulty, need multi-disciplinary fusion and crossing research.It is unusual Perturbation technique is the effective tool for processing double donors problem, but when also there is switching characteristic in system, solves small perturbation parameter The numerical value ill-conditioning problem for causing is the difficult point for switching two-time scale system controller design problem.

Synovial membrane control method is a kind of by designing switching law, makes free conversion operation between subsystem, so as to realize The control method of whole closed-loop control system asymptotically stability, after being proposed by Soviet Union scholar Emelyanov from the 1950's, obtains Obtain widely studied, but be applied to the research of two-time scale system and not yet find.In sum, research switches two-time scale system Modeling and synovial membrane control problem, have larger contribution, with important theory significance and reality to the high-precision control of such system Border application value.

It is double that the present invention proposes non-linear switching under the subsidy of state natural sciences fund general project (51374082) The modeling of discrete-time fuzzy singular perturbation and the synovial membrane control method of timing system.

The content of the invention

It is a kind of non-linear it is an object of the invention to provide the non-linear switching two-time scale system synovial membrane control method of one kind Switch the modeling of discrete-time fuzzy singular perturbation and the synovial membrane control method of two-time scale system, solve existing control method controlled in treatment Accessible switching between object subsystem and greatly reduce the problem in the steady-state error that fast time scale causes.

The technical scheme is that：A kind of discrete-time fuzzy singular perturbation switching model builds and synovial membrane control method, should Method sets up discrete singular perturbation switching model, and then the dynamics of the controlled switching two-time scale system of description designs high-precision Degree synovial membrane controller.The above method is applied to the overall hardware configuration used during real system and classical control system method phase Together, mainly include：Controlled device, sensor, controller, communication component and actuator.

Step 1, the kinetics equation according to controlled device, set up its discrete-time fuzzy singular perturbation switching model.

By the small parameter of controlled system is related or change state variable sees fast variable as faster, change it is relatively slow or Measured state variable sees slow variable as, sets up the discrete-time fuzzy singular perturbation switching model with multiple subsystems.

Regular i:If ξ₁K () is φ_i1..., ξ_gK () is φ_ig, then

X (k+1)=E_εA_iσ(k)x(k)+E_εB_iσ(k)u(k) (1)

Wherein,

x_s(k)∈RⁿIt is slow variable, x_f(k)∈R^mIt is fast variable, u (k) ∈ R^qIt is control input, φ_i1..., φ_ig(i= 1,2 ..., r) it is fuzzy set, ξ₁(k) ..., ξ_gK () is measurable system variable, A_iσ(k),B_iσ(k)It is appropriate dimension Matrix, switching signal σ (k):[0 ,+∞) → { 1,2 ..., N }, σ (k)=j represents j-th subsystem in k moment switching systems It is activated, N is subsystem number, ε is singular perturbation parameter, I_n×n,I_m×mRespectively n ranks unit matrix and m rank unit matrix.

Given [x (k)；U (k)], can obtain Global fuzzy model using standard fuzzy reasoning is

X (k+1)=E_ε[A_σ(k)(μ)x(k)+B_σ(k)(μ)u(k)] (2)

Wherein,

Membership functionφ_ij(ξ_j(k)) it is ξ_jK () exists φ_ijIn degree of membership, if w_i(ξ (k)) >=0, i=1,2 ..., r, r are regular number, μ_i(ξ (k)) >=0,For It is easy to record us and makes μ_i=μ_i(ξ(k))。

Step 2, design synovial membrane controller

Assuming that system mode can be surveyed completely, following sliding formwork function is constructed：

Wherein,G_σ(k)∈R^(n+m)×qFor controller parameter matrix and Make

A_σ(k)(μ)-B_σ(k)(μ)[H_σ(k)(μ)B_σ(k)(μ)]^-1[H_σ(k)(μ)A_σ(k)(μ)+G_σ(k)] (5)

It is Hurwitz's.

Consider following sliding formwork difference of function：

It can be seen from sliding mode control theory, when system mode reaches sliding-mode surface, there is S (k+1)-S (k)=0, therefore, can Obtain equivalence control rule

Bring control law (7) into formula (2), obtain following sliding mode equation：

Step 3, solution controller gain.

Fusion switching control theory, Lyapunov stability theorems, LMI approach, derive such as theorem 1 Shown synovial membrane controller existence condition.

Theorem 1：For fully small perturbation parameter ε>0, control rate (7) causes switching two-time scale system (2) asymptotically stability, And if only if has matrix G_σ(k)∈R^(n+m)×qFormula (9) is set to be Hurwitz, positive definite symmetric matrices P_σ(k)Make linear matrix inequality technique Formula (10) is set up.

A_σ(k)(μ)-B_σ(k)(μ)[H_σ(k)(μ)B_σ(k)(μ)]^-1[H_σ(k)(μ)A_σ(k)(μ)+G_σ(k)] (9)

Wherein,I=1,2 ..., r, q are the dimension of control input, switching signal

Zone switched β_jFor

β_j=x (k) | x^T(k)P_j X (k) >=0 }, j=1,2 ..., N (12)

N is the switching subsystem number of controlled system.

Step 4, above-mentioned switching model and control law are described as C language code, implant controller realizes that controlled system is high Precision controlling.

Advantages of the present invention：

1), fusion fuzzy logic, switching system and singular perturbation theory, propose fuzzy singular perturbation switching model structure side Method, solve existing Modeling Theory cannot accurate description it is non-linear switching two-time scale system dynamics asked with kinematics characteristic Topic, for the modeling of non-linear switching two-time scale system provides new approaches.

2), with reference to synovial membrane theory and fuzzy control method, the synovial membrane control based on fuzzy singular perturbation switching model is proposed Method, solves existing control technology and is difficult to eliminate the steady-state error problem that controlled system switching characteristic and fast variable trigger, and is The high-precision control of non-linear switching two-time scale system provides new method.

Brief description of the drawings

The flow chart of Fig. 1 the inventive method.

Fig. 2 closed-loop system condition responsive curve maps.

Fig. 3 sliding formwork function response curves.

Fig. 4 switching signal figures.

Specific implementation method

Apply the inventive method to that there are two switching systems of mode as follows below, illustrate its implementation.

Step 1, there are two switching systems of mode for a kind of, set up following discrete-time fuzzy singular perturbation switching mould Type.

Regular i:If ξ₁T () is φ_i1, then

X (k+1)=E_εA_iσ(k)x(k)+E_εB_iσ(k)u(k) (13)

Wherein,

x_sK () ∈ R are slow variable, x_fK () ∈ R are fast variable, u (k) ∈ R^qIt is control input, ξ₁(k)=x_s(t), φ_i1 It is fuzzy set, i=1,2, ε=0.01,

When σ (k)=1, the systematic parameter of correspondence subsystem 1 is：

When σ (k)=2, the systematic parameter of correspondence subsystem 2 is：

Given [x (k)；U (k)], can obtain Global fuzzy model using standard fuzzy reasoning is

X (k+1)=E_ε[A_σ(k)(μ)x(k)+B_σ(k)(μ)u(k)] (14)

Wherein,

μ₂(x_s(k))=1- μ₁(x_s(k)) (17)

Assuming that the original state of system is x (0)=[- 0.1 0.2]^T,

Wherein,

Sliding mode controller is

Wherein, σ (k)=1,2,

Simulation result (such as Fig. 1-Fig. 4) shows that the inventive method can not only make controlled non-linear switching two-time scale system each Free switching between subsystem and the steady-state error that fast mode causes can also be reduced, so as to reach its high-precision control mesh Mark.

Claims (1)

1. one kind is non-linear switches two-time scale system synovial membrane control method, it is characterised in that following steps：

Step 1, the kinetics equation according to controlled device, set up its discrete-time fuzzy singular perturbation switching model

By the small parameter correlation of controlled system or change, state variable sees fast variable as faster, and change is relatively slow or can survey State variable sees slow variable as, sets up the discrete-time fuzzy singular perturbation switching model with multiple subsystems；

Regular i:If ξ₁K () is φ_i1..., ξ_gK () is φ_ig, then

X (k+1)=E_εA_iσ(k)x(k)+E_εB_iσ(k)u(k) (1)

Wherein,

$E_{\mathrm{\ϵ}}=\left[\begin{array}{cc}{I}_{n\×n}& 0\\ 0& {\mathrm{\ϵI}}_{m\×m}\end{array}\right],x\left(k\right)=\left[\begin{array}{c}{x}_s\left(k\right)\\ x_f\left(k\right)\end{array}\right],$

x_s(k)∈RⁿIt is slow variable, x_f(k)∈R^mIt is fast variable, u (k) ∈ R^qIt is control input, φ_i1..., φ_ig(i=1, 2 ..., r) it is fuzzy set, ξ₁(k) ..., ξ_gK () is measurable system variable, A_iσ(k),B_iσ(k)It is appropriate dimension square Battle array, switching signal σ (k):[0 ,+∞) → { 1,2 ..., N }, σ (k)=j represents j-th subsystem quilt in k moment switching systems Activation, N is subsystem number, and ε is singular perturbation parameter, I_n×n,I_m×mRespectively n ranks unit matrix and m rank unit matrix；

Given [x (k)；U (k)], can obtain Global fuzzy model using standard fuzzy reasoning is

X (k+1)=E_ε[A_σ(k)(μ)x(k)+B_σ(k)(μ)u(k)] (2)

Wherein,

$A_{\mathrm{\σ}\left(k\right)}\left(\mathrm{\μ}\right)=\underset{i=1}{\overset{r}{\Σ}}{\mathrm{\μ}}_i{A}_{i\mathrm{\σ}\left(k\right)},B_{\mathrm{\σ}\left(k\right)}\left(\mathrm{\μ}\right)=\underset{i=1}{\overset{r}{\Σ}}{\mathrm{\μ}}_i{B}_{i\mathrm{\σ}\left(k\right)},---\left(3\right)$

Membership functionφ_ij(ξ_j(k)) it is ξ_jK () is in φ_ijIn Degree of membership, if ω_i(ξ (k)) >=0, i=1,2 ..., r, r are regular number, μ_i(ξ (k)) >=0,In order to just μ is made in us are recorded_i=μ_i(ξ(k))；

Step 2, design synovial membrane controller

Assuming that system mode can be surveyed completely, following sliding formwork function is constructed：

$S\left(k\right)=H_{\mathrm{\σ}\left(k\right)}\left(\mathrm{\μ}\right)E_{\mathrm{\ϵ}}^{-1}x\left(k\right)+G_{\mathrm{\σ}\left(k\right)}{E}_{\mathrm{\ϵ}}^{-1}\underset{h=0}{\overset{k-1}{\Σ}}x\left(h\right)---\left(4\right)$

Wherein,G_σ(k)∈R^(n+m)×qFor controller parameter matrix and make

A_σ(k)(μ)-B_σ(k)(μ)[H_σ(k)(μ)B_σ(k)(μ)]^-1[H_σ(k)(μ)A_σ(k)(μ)+G_σ(k)] (5)

It is Hurwitz's；

Consider following sliding formwork difference of function：

$\begin{array}{c}S\left(k+1\right)-S\left(k\right)\\ =\[H_{\mathrm{\σ}\left(k\right)}\left(\mathrm{\μ}\right)A_{\mathrm{\σ}\left(k\right)}\left(\mathrm{\μ}\right)-H_{\mathrm{\σ}\left(k\right)}\left(\mathrm{\μ}\right)E_{\mathrm{\ϵ}}^{-1}+G_{\mathrm{\σ}\left(k\right)}{E}_{\mathrm{\ϵ}}^{-1}\]x\left(k\right)+H_{\mathrm{\σ}\left(k\right)}\left(\mathrm{\μ}\right)B_{\mathrm{\σ}\left(k\right)}\left(\mathrm{\μ}\right)u\left(k\right)\end{array}---\left(6\right)$

It can be seen from sliding mode control theory, when system mode reaches sliding-mode surface, there is S (k+1)-S (k)=0, therefore, can wait Valency control law

$u\left(k\right)=-\[H_{\mathrm{\σ}\left(k\right)}\left(\mathrm{\μ}\right)A_{\mathrm{\σ}\left(k\right)}\left(\mathrm{\μ}\right)-H_{\mathrm{\σ}\left(k\right)}\left(\mathrm{\μ}\right)E_{\mathrm{\ϵ}}^{-1}+G_{\mathrm{\σ}\left(k\right)}{E}_{\mathrm{\ϵ}}^{-1}\]x\left(k\right)---\left(7\right)$

Bring control law (7) into formula (2), obtain following sliding mode equation：

$x(k+1)=\[I-E_{\mathrm{\ϵ}}{B}_{\mathrm{\σ}\left(k\right)}\left(\mathrm{\μ}\right)G_{\mathrm{\σ}\left(k\right)}{E}_{\mathrm{\ϵ}}^{-1}\]x\left(k\right)---\left(8\right)$

Step 3, solution controller gain

Fusion switching control theory, Lyapunov stability theorems, LMI approach, derive as shown in theorem 1 Synovial membrane controller existence condition；

Theorem 1：For fully small perturbation parameter ε>0, control rate (7) cause switching two-time scale system (2) asymptotically stability, when and Only when there is matrix G_σ(k)∈R^(n+m)×qFormula (9) is set to be Hurwitz, positive definite symmetric matrices P_σ(k)Make LMI (10) set up,

A_σ(k)(μ)-B_σ(k)(μ)[H_σ(k)(μ)B_σ(k)(μ)]^-1[H_σ(k)(μ)A_σ(k)(μ)+G_σ(k)] (9)

$\left[\begin{array}{cc}-P_{\mathrm{\&sigma;}\left(k\right)}& *\\ {\mathrm{\&Lambda;P}}_{\mathrm{\&sigma;}\left(k\right)}& -P_{\mathrm{\&sigma;}\left(k\right)}\end{array}\right]<0---\left(10\right)$

Wherein,I=1,2 ..., r, q are the dimension of control input, switching signal

Zone switched β_jFor

β_j=x (k) | x^T(k)P_jX (k) >=0 }, j=1,2 ..., N (12)

N is the switching subsystem number of controlled system；

Step 4, above-mentioned switching model and control law are described as C language code, implant controller realizes controlled system in high precision Control.