CN105607482A - T-S bilinear model based decentralized control method of nonlinear association large-system - Google Patents

Abstract

The invention discloses a T-S bilinear model based decentralized control method of a nonlinear association large-system. According to a Lyapunov stability analysis theory and a parallel distribution compensation algorithm, n asymptotic stability sufficient condition related with a time lag of a closed-loop association large system is obtained, a decentralized controller can be obtained through solutions of a group of linear matrix inequalities, and the design of a corresponding decentralized fuzzy controller can be converted into a convex optimization problem restricted to the linear matrix inequalities (LMIs). Finally, simulation examples verify the effectiveness of the method brought forward by the invention.

Description

A kind of decentralized control method of the Design for Large-Scale Interconnected Nonlinear Systems based on T-S bilinear model

Technical field

The invention belongs to and close the United Nations General Assembly's systems technology field, relate to a kind of nonlinear dependence the United Nations General Assembly based on T-S bilinear modelThe decentralized control method of system.

Background technology

Close the United Nations General Assembly's system ubiquity in actual life, as: power system, nuclear process, economic system, computerNetwork system etc. The existing a lot of achievements of stability analysis for the large system of linear correlation emerge, but about non-linear correlationLarge systematic research achievement is also fewer. For these large scale systems, centralized Control will make whole control system information exchangeComplex, thus cause the system integration and operating cost to improve, and the reliability of system reduces. From the real-time of system, reliableProperty and the aspect such as economy consider, there is processing the complicated decentralized control method that closes the United Nations General Assembly's system in 20 century 70s.

Fuzzy control based on T-S model is one of effective ways of research nonlinear system, is closing dividing of the United Nations General Assembly's systemThe existing a lot of achievements in loose control aspect emerge. Prior art has the stable control of having studied the large system of a class Fuzzy Correlation;Have for a class and proposed Fuzzy decentralized robust control method with the Design for Large-Scale Interconnected Nonlinear Systems of multiple time delay; Study non-lineThe dispersion H of Xing Guan the United Nations General Assembly system_∞Follow the tracks of and control; Have and proposed the shape of a class with the uncertain Design for Large-Scale Interconnected Nonlinear Systems of time lagThe decentralised control of state feedback. But the consequent part of noticing above-mentioned T-S model fuzzy rule is all a linear model.

Bilinear system is the simplest nonlinear system of one, and it can describe biology, chemical industry, social warp very naturallyMany phenomenons in the complication systems such as Ji, population. The equation of describing population change is a good example. Population change speedFor:Wherein: v is that birth rate deducts the death rate.

Consider the validity of T-S model and the feature of bilinear system, the fuzzy control based on T-S bilinear model is just drawnPlay scholar's concern. Different with common T-S model, the consequent part of Fuzzy Bilinear control system fuzzy rule is oneBilinear model. Prior art has the kinds of robust control problems of having studied a class Continuous Fuzzy bilinear system, and result is pushed awayExtensively arrive the Continuous Fuzzy bilinear system of state with time lag. There is the robust control of having studied a class discrete-time fuzzy bilinear systemProblem processed. But close the United Nations General Assembly's systematic research or blank for Fuzzy Bilinear.

Summary of the invention

The object of this invention is to provide a kind of decentralised control side of the Design for Large-Scale Interconnected Nonlinear Systems based on T-S bilinear modelMethod, the large system of fuzzy time-delay bilinear interconnection being made up of S subsystem, to each subsystem design local state feedbackController, making closed loop close the United Nations General Assembly's system is Asymptotic Stability.

The technical solution adopted in the present invention is to carry out according to following steps:

The large system of design fuzzy time-delay bilinear interconnection;

One class is by S subsystem Ω_i, i=1,2 ..., S composition band becomes Fuzzy Bilinear pass the United Nations General Assembly system of time lag sometimesΩ, i subsystem Ω_iCan be expressed as:

$\begin{array}{c}\begin{array}{ccccccccc}{R}_i^m& \mathrm{if}& {\mathrm{\ξ}}_{i1}\left(t\right)& \mathrm{is}& F_{i1}^m& \mathrm{and}...\mathrm{and}& {\mathrm{\ξ}}_{{\mathrm{iv}}_i}& \mathrm{is}& F_{{\mathrm{iv}}_i}^m\end{array}\\ \begin{array}{cc}\mathrm{then}& {\stackrel{.}{x}}_i\left(t\right)=A_{\mathrm{im}}{x}_i\left(t\right)+B_{\mathrm{im}}{u}_i\left(t\right)+N_{\mathrm{im}}{x}_i\left(t\right)u_i\left(t\right)+A_{\mathrm{idm}}{x}_i(t-d_i\left(t\right))+B_{\mathrm{idm}}{u}_i(t-d_i\left(t\right))\end{array}\\ +N_{\mathrm{idm}}{x}_i(t-d_i\left(t\right))u_i(t-d_i\left(t\right))+{\mathrm{\Σ}}_{j=1,j\≠i}^S{C}_{\mathrm{jim}}{x}_j\left(t\right)\\ \begin{array}{ccc}{x}_i\left(t\right)={\mathrm{\φ}}_i\left(t\right)& t=\left[\begin{array}{cc}-{\mathrm{\τ}}_i& 0\end{array}\right]& m=\mathrm{1,2},...,r_i\end{array}\end{array}---\left(1\right)$

Wherein:I subsystem Ω_iFuzzy rule, s is the number of subsystem; M={1,2，...，r_i}，r_iIt is the number of the fuzzy rule of i subsystem;ξ_j(t)，j＝1，2，...，v_iIt is respectively fuzzy setWith prerequisite variable;u_i(t) ∈ R is respectively state vector and control inputs; A_im，A_idm，N_im， It is known sytem matrix;That j subsystem is to i sonThe correlation matrix of system; d_i(t) being the time lag item of system, is continuously differentiable function and satisfied 0≤d_i(t)≤τ_iAnd d_i(t)≤α_i＜1；

By single-point obfuscation, the average reverse gelatinizing method in product reasoning and center, the overall model of Fuzzy control systemFor:

$\begin{array}{c}{\stackrel{\·}{x}}_i\left(t\right)={\mathrm{\Σ}}_{m=1}^{r_i}{h}_{\mathrm{im}}\left({\mathrm{\ξ}}_i\left(t\right)\right)[A_{\mathrm{im}}{x}_i\left(t\right)+B_{\mathrm{im}}{u}_i\left(t\right)+N_{\mathrm{im}}{x}_i\left(t\right)u_i\left(t\right)+A_{\mathrm{idm}}{x}_i(t-d_i\left(t\right))\\ +N_{\mathrm{idm}}{x}_i(t-d_i\left(t\right))u_i(t-d_i\left(t\right))+B_{\mathrm{idm}}{u}_i(t-d_i\left(t\right))+{\mathrm{\Σ}}_{j=1,j\≠i}^S{C}_{\mathrm{jim}}{x}_j\left(t\right)]\end{array}---\left(2\right)$

Wherein:μ_imj(ξ_i(t)) be ξ_j(t) existIn membership function; Suppose ω_im(ξ_i(t))≥0，By h_im(ξ_i(t) definition) is known:Note respectively h by abridging_im(ξ_i(t))，x_i(t-d_i(t))，u_i(t-d_i(t)) be h_im，x_id(t)，u_id(t)；

According to parallel distributed backoff algorithm, consider local feedback control device:

$\begin{array}{c}\begin{array}{cccccccc}\mathrm{if}& {\mathrm{\ξ}}_{i1}\left(t\right)& \mathrm{is}& F_{i1}^m& \mathrm{and}...\mathrm{and}& {\mathrm{\ξ}}_{{\mathrm{iv}}_i}\left(t\right)& \mathrm{is}& F_{{\mathrm{iv}}_i}^m\end{array}\\ \begin{array}{cc}\mathrm{then}& u_i\left(t\right)=\frac{{\mathrm{\ρ}}_i{K}_{\mathrm{im}}{x}_i\left(t\right)}{\sqrt{1+{x_i}^T{K}_{\mathrm{im}}^T{K}_{\mathrm{im}}{x}_i}}={\mathrm{\ρ}}_i\sin {\mathrm{\θ}}_{\mathrm{im}}={\mathrm{\ρ}}_i\cos {\mathrm{\θ}}_{\mathrm{im}}{K}_{\mathrm{im}}{x}_i\left(t\right)\end{array}\end{array}---\left(3\right)$

Here:Controller gain to be asked, ρ_i> 0 is scalar undetermined,

Can similarly be obtained by (3):

Here:

i＝1，2，...，S；m＝1，2，...，r_i

Overall decentralised control rule can be expressed as:

Under the effect of control law (5), the equation of whole closed-loop system can be expressed as:

${\stackrel{\·}{x}}_i\left(t\right)={\mathrm{\Σ}}_{m,n=1}^{r_i}{h}_{\mathrm{im}}{h}_{\mathrm{in}}[{\mathrm{\Λ}}_{i,\mathrm{mn}}{x}_i\left(t\right)+{\mathrm{\Λ}}_{i,\mathrm{dmn}}{x}_{\mathrm{id}}\left(t\right)+{\mathrm{\Σ}}_{j=1,j\≠i}^s{C}_{\mathrm{jim}}{x}_j\left(t\right)]---\left(6\right)$

Here: Λ_i，mn＝A_im+ρ_isinθ_inN_im+ρ_icosθ_inB_imK_in，

Theorem 1: for given normal number ρ_i，α_i, i=1,2 ..., S, if for given normal number ε_1i，ε_2i，i=1,2 ..., there is positive definite symmetric matrices P in S_i＞0，R_i> 0, i=1,2 ..., S and matrix K_im，i＝1，2，...，S；m＝1，2，...，r_iMeet MATRIX INEQUALITIES (7), Ze Guan the United Nations General Assembly system (5) is asymptotically stable;

Ф_i，mm＜0i＝1，2，...，S；m＝1，2，...，r_i(7a)

Ф_i，mn+Ф_i，nm＜0i＝1，2，...，S；1≤m＜n≤r_i(7b)

Wherein:

$\begin{array}{c}{\mathrm{\φ}}_{i,1\mathrm{mn}}=A_{\mathrm{im}}^T{P}_i+P_i{A}_{\mathrm{im}}+({\mathrm{\ϵ}}_{1i}+{\mathrm{\ϵ}}_{2i}){\mathrm{\ρ}}_i^2{P}_i{P}_i+{\mathrm{\ϵ}}_{1i}^{-1}{N}_{\mathrm{im}}^T{N}_{\mathrm{im}}+{\mathrm{\ϵ}}_{1i}^{-1}{\left(B_{\mathrm{im}}{K}_{\mathrm{in}}\right)}^T\left(B_{\mathrm{im}}{K}_{\mathrm{in}}\right)\\ +{\mathrm{\Σ}}_{j=1,j\≠i}^S{P}_i{C}_{\mathrm{jim}}{C}_{\mathrm{jim}}^T{P}_i+(S-1)I+R_i,\end{array}$

${\mathrm{\φ}}_{i,2\mathrm{mn}}={\mathrm{\ϵ}}_{2i}^{-1}{N}_{\mathrm{idm}}^T{N}_{\mathrm{idm}}+{\mathrm{\ϵ}}_{2i}^{-1}{\left(B_{\mathrm{idm}}{K}_{\mathrm{in}}\right)}^T\left(B_{\mathrm{idm}}{K}_{\mathrm{in}}\right)-(1-{\mathrm{\α}}_i)R_i.$

Prove: choose following Lyapunov function:

$V\left(t\right)={\mathrm{\Σ}}_{i=1}^S{V}_i\left(t\right)={\mathrm{\Σ}}_{i=1}^S[x_i^T\left(t\right)P_i{x}_i\left(t\right)+{\∫}_{i-d_i\left(i\right)}^i{x}_i^T\left(s\right)R_i{x}_i\left(s\right)\mathrm{ds}]---\left(8\right)$

Along the path of system (6), to V (t) differentiate, can obtain:

$\begin{array}{c}\stackrel{.}{V}\left(t\right)={\mathrm{\Σ}}_{i=1}^S{\mathrm{\Σ}}_{i=1}^{r_i}{h}_{\mathrm{im}}{h}_{\mathrm{in}}[{\stackrel{.}{x}}_i^T\left(t\right)P_i{x}_i\left(t\right)+x_i^T\left(t\right)P_i{\stackrel{.}{x}}_i\left(t\right)+x_i^T\left(t\right)R_i{x}_i\left(t\right)-(1-{\stackrel{.}{d}}_i)x_{\mathrm{id}}^T\left(t\right)R_i{x}_{\mathrm{id}}\left(t\right)]\\ \≤{\mathrm{\Σ}}_{i=1}^S{\mathrm{\Σ}}_{m,n=1}^{r_i}{h}_{\mathrm{im}}{h}_{\mathrm{in}}[x_i^T\left(t\right)({\mathrm{\Λ}}_{i,\mathrm{mn}}^T{P}_i+P_i{\mathrm{\Λ}}_{i,\mathrm{mn}})x_i\left(t\right)+x_{\mathrm{id}}^T\left(t\right){\mathrm{\Λ}}_{i,\mathrm{dmn}}^T{P}_i{x}_i\left(t\right)+x_i^T\left(t\right)P_i{\mathrm{\Λ}}_{i,\mathrm{dmn}}{x}_{\mathrm{id}}\left(t\right)\\ +{\mathrm{\Σ}}_{j=1,j\≠i}^S{x}_j^T\left(t\right)C_{\mathrm{jim}}^T{P}_i{x}_i\left(t\right)+x_i^T\left(t\right)P_i{\mathrm{\Σ}}_{j=1,j\≠i}^s{C}_{\mathrm{jim}}{x}_j\left(t\right)+x_i^T\left(t\right)R_i{x}_i\left(t\right)-(1-{\mathrm{\α}}_i)x_{\mathrm{id}}^T\left(t\right)R_i{x}_{\mathrm{id}}\left(t\right)\end{array}---\left(9\right)$

Consider following formula, and can be obtained by lemma 1:

$\begin{array}{c}{\mathrm{\Λ}}_{i,\mathrm{mn}}^T{P}_i+P_i{\mathrm{\Λ}}_{i,\mathrm{mn}}=A_{\mathrm{im}}^T{P}_i+P_i{A}_{\mathrm{im}}+{\left({\mathrm{\ρ}}_i\sin {\mathrm{\θ}}_{\mathrm{in}}{N}_{\mathrm{im}}\right)}^T{P}_i+P_i\left({\mathrm{\ρ}}_i\sin {\mathrm{\θ}}_{\mathrm{in}}{N}_{\mathrm{im}}\right)\\ +{\left({\mathrm{\ρ}}_i\cos {\mathrm{\θ}}_{\mathrm{in}}{B}_{\mathrm{im}}{K}_{\mathrm{in}}\right)}^T{P}_i+P_i\left({\mathrm{\ρ}}_i\cos {\mathrm{\θ}}_{\mathrm{in}}{B}_{\mathrm{im}}{K}_{\mathrm{in}}\right)\\ \≤A_{\mathrm{im}}^T{P}_i+P_i{A}_{\mathrm{im}}+{\mathrm{\ϵ}}_{1i}{\mathrm{\ρ}}_i^2{P}_i{P}_i+{\mathrm{\ϵ}}_{1i}^{-1}{N_{\mathrm{im}}^{T}N}_{\mathrm{im}}+{\mathrm{\ϵ}}_{1i}^{-1}{\left(B_{\mathrm{im}}{K}_{\mathrm{in}}\right)}^T\left(B_{\mathrm{im}}{K}_{\mathrm{in}}\right),\\ x_{\mathrm{id}}^T\left(T\right){\mathrm{\Λ}}_{i,\mathrm{dmn}}^T{P}_i{x}_i\left(t\right)+x_i^T\left(t\right)P_i{\mathrm{\Λ}}_{i,\mathrm{dmn}}{x}_{\mathrm{id}}\left(t\right)\\ \≤x_{\mathrm{id}}^T\left(t\right)A_{\mathrm{idm}}^T{P}_i{x}_i\left(t\right)+x_i^T\left(t\right)P_i{A}_{\mathrm{idm}}{x}_{\mathrm{id}}\left(t\right)+x_i^T\left(t\right){\mathrm{\ϵ}}_{2i}{\mathrm{\ρ}}_i^2{P}_i{P}_i{x}_i\left(t\right)\\ +x_{\mathrm{id}}^T\left(t\right){\mathrm{\ϵ}}_{2i}^{-1}{N}_{\mathrm{idm}}^T{N}_{\mathrm{idm}}{x}_{\mathrm{id}}\left(t\right)+x_{\mathrm{id}}^T\left(t\right){\mathrm{\ϵ}}_{2i}^{-1}{\left(B_{\mathrm{idm}}{K}_{\mathrm{in}}\right)}^T\left(B_{\mathrm{idm}}{K}_{\mathrm{in}}\right)x_{\mathrm{id}}\left(t\right),\end{array}---\left(10\right)$

In like manner can obtain:

$\begin{array}{c}{\mathrm{\Σ}}_{i=1}^S{\mathrm{\Σ}}_{m=1}^{r_i}{h}_{\mathrm{im}}[{\mathrm{\Σ}}_{j=1,j\≠i}^S{x}_j^T\left(t\right)C_{\mathrm{jim}}^T{P}_i{x}_i\left(t\right)+x_i^T\left(t\right)P_i{\mathrm{\Σ}}_{j=1,j\≠i}^s{C}_{\mathrm{jim}}{x}_j\left(t\right)]\\ ={\mathrm{\Σ}}_{i=1}^S{\mathrm{\Σ}}_{m=1}^{r_i}{h}_{\mathrm{im}}[x_1^T\left(t\right)C_{1\mathrm{im}}^t{P}_i{x}_i\left(t\right)+...+x_{i-1}^T\left(t\right)C_{i-1\mathrm{im}}^T{P}_i{x}_i\left(t\right)+x_{i+1}^T\left(t\right)C_{i+1\mathrm{im}}^T{P}_i{x}_i\left(t\right)+...\\ +x_S^T\left(t\right)C_{\mathrm{Sim}}^T{P}_i{x}_i\left(t\right)+x_i^T\left(t\right)P_i{C}_{1\mathrm{im}}{x}_1\left(t\right)+...+x_i^T\left(t\right)P_i{C}_{i-1\mathrm{im}}{x}_{i-1}\left(t\right)+x_i^T\left(t\right)P_i{C}_{i+1\mathrm{im}}{x}_{i+1}\left(t\right)\\ +...x_i^T\left(t\right)P_i{C}_{\mathrm{Sim}}{x}_S\left(t\right)]\\ \≤{\mathrm{\Σ}}_{i=1}^S{\mathrm{\Σ}}_{m=1}^{r_i}{h}_{\mathrm{im}}[x_i^T\left(t\right)P_i{\mathrm{\Σ}}_{j=1,j\≠i}^S{C}_{\mathrm{jim}}{C}_{\mathrm{jim}}^T{P}_i{x}_i\left(t\right)+{\mathrm{\Σ}}_{j=1,j\≠i}^S{x}_j^T\left(t\right)x_j\left(t\right)]\\ ={\mathrm{\Σ}}_{i=1}^S{\mathrm{\Σ}}_{m=1}^{r_i}{h}_{\mathrm{im}}[x_i^T\left(t\right)P_i{\mathrm{\Σ}}_{j=1,j\≠i}^S{C}_{\mathrm{jim}}{C}_{\mathrm{jim}}^T{P}_i{x}_i\left(t\right)+(S-1)x_i^T\left(t\right)x_i\left(t\right)]\end{array}---\left(11\right)$

(8) are brought into in (9), (10), (11), and noteCan obtain:

$\begin{array}{c}\stackrel{.}{V}\left(t\right)\&le;{\mathrm{\&Sigma;}}_{i=1}^S{\mathrm{\&Sigma;}}_{m,n=1}^{r_i}{h}_{\mathrm{im}}{h}_{\mathrm{in}}{\mathrm{\&eta;}}_i^T\left(t\right){\mathrm{\&Phi;}}_{i,\mathrm{mn}}{x}_i\left(t\right){\mathrm{\&eta;}}_i\left(t\right)\\ ={\mathrm{\&Sigma;}}_{i=1}^S[{\mathrm{\&Sigma;}}_{m=1}^{r_i}{h}_{\mathrm{im}}^2{\mathrm{\&eta;}}_i^T\left(t\right){\mathrm{\&Phi;}}_{i,\mathrm{mm}}{\mathrm{\&eta;}}_i\left(t\right)+{\mathrm{\&Sigma;}}_{1=m<n}^{r_i}{h}_{\mathrm{im}}{h}_{\mathrm{in}}{\mathrm{\&eta;}}_i^T\left(t\right)({\mathrm{\&Phi;}}_{i,\mathrm{mn}}+{\mathrm{\&Phi;}}_{i,\mathrm{nm}}){\mathrm{\&eta;}}_i\left(t\right)]\end{array}---\left(12\right)$

Known according to (7) in theorem 1So known pass the United Nations General Assembly's system (6) is asymptotically stable;

Consider that (7) in theorem 1 are bilinearity MATRIX INEQUALITIES, can not directly be solved by LMI tool box, two-wireProperty MATRIX INEQUALITIES converts LMI to, provides the method for designing of controller:

Theorem 2: for given normal number α_i，ρ_i, i=1,2 ..., s, if for given normal number ε_1i，ε_2i，i=1,2 ..., s exists matrixWith matrix G_im，i＝1，2，...，S；m＝1，2，...，r_iMeet MATRIX INEQUALITIES (13), Ze Guan the United Nations General Assembly system (5) is Asymptotic Stability, and controller gain is: K_im＝G_imZ^-1，i＝1，2，...，S；m＝1，2，...，r_i.；

$\begin{array}{c}\left[\begin{array}{ccccccc}{T}_{i,m}& *& *& *& *& *& *\\ Z_i{A}_{\mathrm{idm}}^T& -(1-{\mathrm{\&alpha;}}_i){\stackrel{\&OverBar;}{R}}_i& *& *& *& *& *\\ Z_i& 0& -\frac{I}{s-1}& *& *& *& *\\ N_{\mathrm{im}}{Z}_i& 0& 0& -{\mathrm{\&epsiv;}}_{1i}I& *& *& *\\ B_{\mathrm{im}}{G}_{\mathrm{im}}& 0& 0& 0& -{\mathrm{\&epsiv;}}_{1i}I& *& *\\ 0& N_{\mathrm{idm}}{Z}_i& 0& 0& 0& -{\mathrm{\&epsiv;}}_{2i}I& *\\ 0& B_{\mathrm{idm}}{G}_{\mathrm{im}}& 0& 0& 0& 0& -{\mathrm{\&epsiv;}}_{2i}I\end{array}\right]<0,\\ i=\mathrm{1,2},...,S;m=\mathrm{1,2},...,r_i\end{array}---\left(13a\right)$

$\begin{array}{c}\left[\begin{array}{ccccccc}{T}_{i,m}+T_{i,n}& *& *& *& *& *& *\\ Z_i{A}_{\mathrm{idm}}^T+Z_i{A}_{\mathrm{idm}}^T& -2(1-{\mathrm{\&alpha;}}_i){\stackrel{\&OverBar;}{R}}_i& *& *& *& *& *\\ 2Z_i& 0& -\frac{2I}{s-1}& *& *& *& *\\ \stackrel{\&OverBar;}{{\left(\mathrm{NZ}\right)}_{i,\mathrm{mn}}}& 0& 0& -t_{55}& *& *& *\\ \stackrel{\&OverBar;}{{\left(\mathrm{BG}\right)}_{i,\mathrm{mn}}}& 0& 0& 0& -t_{66}& *& *\\ 0& \stackrel{\&OverBar;}{{\left(N_{d}Z\right)}_{i,\mathrm{mn}}}& 0& 0& 0& -t_{77}& *\\ 0& \stackrel{\&OverBar;}{{\left(B_{d}G\right)}_{i,\mathrm{mn}}}& 0& 0& 0& 0& -t_{88}\end{array}\right]<0,\\ i=\mathrm{1,2},...,S;1\&le;m<n\&le;r_i\end{array}---\left(13b\right)$

Here:

$\stackrel{\&OverBar;}{{\left(\mathrm{NZ}\right)}_{i,\mathrm{mn}}}=\left[\begin{array}{c}{N}_{\mathrm{im}}{Z}_i\\ N_{\mathrm{in}}{Z}_i\end{array}\right],\stackrel{\&OverBar;}{{\left(\mathrm{BG}\right)}_{i,\mathrm{mn}}}=\left[\begin{array}{c}{B}_{\mathrm{im}}{G}_{\mathrm{in}}\\ B_{\mathrm{in}}{G}_{\mathrm{im}}\end{array}\right],\stackrel{\&OverBar;}{{\left(N_{d}Z\right)}_{i,\mathrm{mn}}}=\left[\begin{array}{c}{N}_{\mathrm{idm}}{Z}_i\\ N_{\mathrm{idn}}{Z}_i\end{array}\right],\stackrel{\&OverBar;}{{\left(B_{d}G\right)}_{i,\mathrm{mn}}}=\left[\begin{array}{c}{B}_{\mathrm{idm}}{G}_{\mathrm{in}}\\ B_{\mathrm{idn}}{G}_{\mathrm{im}}\end{array}\right],$

t₄₄＝t₅₅＝diag{ε_1iI，ε_1iI}，t₆₆＝t₇₇＝diag{ε_2iI，ε_2iI}.

Prove: chooseAnd noteBy K_im＝G_imZ^-1Known, M_im＝K_imZ；

(13a) taken advantage of to diag{P in left and right simultaneously_i，P_i, I, I, I, I, I} can obtain:

$\begin{array}{c}\left[\begin{array}{ccccccc}{P_{i}T}_{i,m}{P}_i& *& *& *& *& *& *\\ A_{\mathrm{idm}}^T{P}_i& -(1-{\mathrm{\&alpha;}}_i)R_i& *& *& *& *& *\\ I& 0& -\frac{I}{s-1}& *& *& *& *\\ N_{\mathrm{im}}& 0& 0& -{\mathrm{\&epsiv;}}_{1i}I& *& *& *\\ B_{\mathrm{im}}{K}_{\mathrm{im}}& 0& 0& 0& -{\mathrm{\&epsiv;}}_{1i}I& *& *\\ 0& N_{\mathrm{idm}}& 0& 0& 0& -{\mathrm{\&epsiv;}}_{2i}I& *\\ 0& B_{\mathrm{idm}}{K}_{\mathrm{im}}& 0& 0& 0& 0& -{\mathrm{\&epsiv;}}_{2i}I\end{array}\right]<0,\\ i=\mathrm{1,2},...,S;m=\mathrm{1,2},...,r_i\end{array}---\left(14a\right)$

Set up and can be equivalent to (7a) establishment by Schur complement fixed reason known (14a); Same reason (13b) can be derived (7b)Vertical; Known under designed controller by theorem 1 like this, Fuzzy Bilinear is closed the United Nations General Assembly's system Asymptotic Stability.

The invention has the beneficial effects as follows that the band that a class is described by T-S bilinear model becomes the non-linear correlation of time lag sometimesLarge system, has studied its dispersity feedback control problem. Theoretical and the parallel distributed compensation according to Lyapunov stability analysisAlgorithm, has obtained closed loop and has closed the adequate condition of the United Nations General Assembly's system delay dependent asymptotic stability. Decentralized controller can be by one group of linear momentThe solution of battle array inequality obtains. The design of adopting corresponding Decentralized Fuzzy controller can change into one and be subject to LMI(LMI) the protruding optimization problem of constraint. Finally, emulation numerical example has been verified the validity of institute's extracting method.

Brief description of the drawings

Fig. 1 is the condition responsive curve map of subsystem 1.

Fig. 2 is the condition responsive curve map of subsystem 2.

Fig. 3 controls curve map.

Detailed description of the invention

Below in conjunction with the drawings and specific embodiments, the present invention is described in detail.

Study a class input and state all close the United Nations General Assembly's system dispersion control with the Fuzzy Bilinear that sometimes becomes time lag hereinProblem processed. Based on Lyapunov Theory of Stability, obtain closed loop and closed the asymptotically stable abundant bar that the United Nations General Assembly's system time lag is relevantPart. The design of Decentralized Fuzzy controller can change into a protruding optimization problem that is subject to LMI (LMI) constraint.

In this article, RⁿRepresent n dimension Euclidean space, P > 0 (P >=0) represents that P is that a positive definite (positive semidefinite) is real rightClaim matrix. In matrix expression: A^TRepresent the transposed matrix of A, A^-1Represent the inverse matrix of A, * represents symmetrical, diag... .} represents diagonal matrix, and I represents the unit matrix of suitable dimension.RepresentIf do not specified,Matrix all represents the matrix of suitable dimension.

1. system is described

One class is by S subsystem Ω_i, i=1,2 ..., S composition band becomes Fuzzy Bilinear pass the United Nations General Assembly system of time lag sometimesΩ, i subsystem Ω_iCan be expressed as:

$\begin{array}{c}\begin{array}{ccccccccc}{R}_i^m& f& {\mathrm{\&xi;}}_{i1}\left(t\right)& \mathrm{is}& F_{i1}^m& \mathrm{and}...\mathrm{and}& {\mathrm{\&xi;}}_{{\mathrm{iv}}_i}\left(t\right)& \mathrm{is}& F_{{\mathrm{iv}}_i}^m\end{array}\\ \begin{array}{cc}\mathrm{then}& {\stackrel{\&CenterDot;}{x}}_i\left(t\right)=A_{\mathrm{im}}{x}_i\left(t\right)+B_{\mathrm{im}}{u}_i\left(t\right)+N_{\mathrm{im}}{x}_i\left(t\right)u_i\left(t\right)+A_{\mathrm{idm}}{x}_i(t-d_i\left(t\right))+B_{\mathrm{idm}}{u}_i(t-d_i\left(t\right))\end{array}\\ +N_{\mathrm{idm}}{x}_i(t-d_i\left(t\right))u_i(t-d_i\left(t\right))+{\mathrm{\&Sigma;}}_{j=1,j\&NotEqual;i}^S{C}_{\mathrm{jim}}{x}_j\left(t\right)\\ \begin{array}{ccc}{x}_i(t={\mathrm{\&phi;}}_i\left(t\right)& t=\left[\begin{array}{cc}-{\mathrm{\&tau;}}_i& 0\end{array}\right]& m=\mathrm{1,2},...,r_i\end{array}\end{array}---\left(1\right)$

Wherein:I subsystem Ω_iFuzzy rule, s is the number of subsystem. M={1,2，...，r_i}，r_iIt is the number of the fuzzy rule of i subsystem.It is respectively fuzzy set and frontCarry variable.Respectively state vector and control inputs. It is known sytem matrix.The correlation matrix of j subsystem to i subsystem.d_i(t) being the time lag item of system, is continuously differentiable function and satisfied 0≤d_i(t)≤τ_iWith

By single-point obfuscation, the average reverse gelatinizing method in product reasoning and center, the overall model of Fuzzy control systemFor:

$\begin{array}{c}{\stackrel{\&CenterDot;}{x}}_i\left(t\right)={\mathrm{\&Sigma;}}_{m=1}^{r_i}{h}_{\mathrm{im}}\left({\mathrm{\&xi;}}_i\left(t\right)\right)[A_{\mathrm{im}}{x}_i\left(t\right)+B_{\mathrm{im}}{u}_i\left(t\right)+N_{\mathrm{im}}{x}_i\left(t\right)u_i\left(t\right)+A_{\mathrm{idm}}{x}_i(t-d_i\left(t\right))\\ +N_{\mathrm{idm}}{x}_i\left({t-d}_i\left(t\right)\right)u_i\left({t-d}_i\left(t\right)\right)+B_{\mathrm{idm}}{u}_i\left({t-d}_i\left(t\right)\right)+{\mathrm{\&Sigma;}}_{j=1,j\&NotEqual;i}^S{C}_{\mathrm{jim}}{x}_j\left(t\right)]\end{array}---\left(2\right)$

Wherein:μ_inj(ξ_i(t)) be ξ_j(t) existIn membership function. In literary composition, supposeBy h_im(ξ_i(t) definition) is known:Note respectively h obscure in the situation that by abridging not causing below_im(ξ_i(t))，x_i(t-d_i(t))，u_i(t-d_i(t)) be h_im，x_id(t)，u_id(t)。

According to parallel distributed backoff algorithm, consider local feedback control device:

$\begin{array}{c}\begin{array}{cccccccccc}\mathrm{if}& {\mathrm{\&xi;}}_{i1}\left(t\right)& \mathrm{is}& F_{i1}^m& \mathrm{and}& ...& \mathrm{and}& {\mathrm{\&xi;}}_{{\mathrm{iv}}_i}\left(t\right)& \mathrm{is}& F_{i{v}_i}^m\end{array}\\ \begin{array}{cc}\mathrm{then}& u_i\left(t\right)=\frac{{\mathrm{\&rho;}}_i{K}_{\mathrm{im}}{x}_i\left(t\right)}{\sqrt{1+{x_i}^T{K}_{\mathrm{im}}^T{K}_{\mathrm{im}}{x}_i}}{\mathrm{\&rho;}}_i\sin {\mathrm{\&theta;}}_{\mathrm{im}}={\mathrm{\&rho;}}_i\cos {\mathrm{\&theta;}}_{\mathrm{im}}{K}_{\mathrm{im}}{x}_i\left(t\right)\end{array}\end{array}---\left(3\right)$

Here:Controller gain to be asked, ρ_i> 0 is scalar undetermined,

Can similarly be obtained by (3):

Here:

i＝1，2，...，S；m＝1，2，...，r_i

Overall decentralised control rule can be expressed as:

Under the effect of control law (5), the equation of whole closed-loop system can be expressed as:

$\stackrel{\&CenterDot;}{x_i}\left(t\right)={\mathrm{\&Sigma;}}_{m,n=1}^{\mathrm{\&eta;}}{h}_{\mathrm{im}}{h}_{\mathrm{in}}[{\mathrm{\&Lambda;}}_{i,\mathrm{mn}}{x}_i\left(t\right)+{\mathrm{\&Lambda;}}_{i,\mathrm{dmn}}{x}_{\mathrm{id}}\left(t\right)+{\mathrm{\&Sigma;}}_{j=1,j\&NotEqual;i}^S{C}_{\mathrm{jim}}{x}_j\left(t\right)]---\left(6\right)$

Here: Λ_i，mn＝A_im+ρ_isinθ_inN_im+ρ_icosθ_inB_imK_in，

Target herein: for the large system of fuzzy time-delay bilinear interconnection (6) being formed by S subsystem, to each heightSystem local state feedback controller, it is asymptotically stable making closed loop close the United Nations General Assembly's system (5).

Below be given in the lemma that will use in proof:

Lemma 1: establish M, N is the real matrix that is applicable to dimension,, for scalar ε > 0, has M^TN+N^TM≤εM^TM+ε^-1N^TN becomesVertical.

2. Main Conclusions

Theorem 1: for given normal number ρ_i，α_i, i=1,2 ..., S, if for given normal number ε_1i，ε_2i，i=1,2 ..., there is positive definite symmetric matrices P in S_i＞0，R_i> 0, i=1,2 ..., S and matrix K_im，i＝1，2，...，S；m＝1，2，...，r_iMeet MATRIX INEQUALITIES (7), Ze Guan the United Nations General Assembly system (5) is asymptotically stable.

Φ_i，mm＜0i＝1，2，...，S；m＝1，2，...，r_i(7a)

Φ_i，mn+Φ_i，nm＜0i＝1，2，...，S；1≤m＜n≤r_i(7b)

Wherein: ${\mathrm{\&Phi;}}_{i,\mathrm{mn}}=\left[\begin{array}{cc}{\mathrm{\&phi;}}_{i,1\mathrm{mn}}& P_i{A}_{\mathrm{idm}}\\ *& {\mathrm{\&phi;}}_{i,2\mathrm{mn}}\end{array}\right],$

$\begin{array}{c}{\mathrm{\&phi;}}_{i,1\mathrm{mn}}=A_{\mathrm{im}}^T{P}_i+P_i{A}_{\mathrm{im}}+({\mathrm{\&epsiv;}}_{1i}+{\mathrm{\&epsiv;}}_{2i}){\mathrm{\&rho;}}_i^2{P}_i{P}_i+{\mathrm{\&epsiv;}}_{1i}^{-1}{N}_{\mathrm{im}}^T{N}_{\mathrm{im}}+{\mathrm{\&epsiv;}}_{1i}^{-1}{\left(B_{\mathrm{im}}{K}_{\mathrm{in}}\right)}^T\left(B_{\mathrm{im}}{K}_{\mathrm{in}}\right)\\ +{\mathrm{\&Sigma;}}_{j=1,j\&NotEqual;i}^S{P}_i{C}_{\mathrm{jim}}{C}_{\mathrm{jim}}^T{P}_i+(S-1)I+R_i,\end{array}$

${\mathrm{\&phi;}}_{i,2\mathrm{mn}}={\mathrm{\&epsiv;}}_{2i}^{-1}{N}_{\mathrm{idm}}^T{N}_{\mathrm{idm}}+{\mathrm{\&epsiv;}}_{2i}^{-1}{\left(D_{\mathrm{idm}}{K}_{\mathrm{in}}\right)}^T\left(B_{\mathrm{idm}}{K}_{\mathrm{in}}\right)-(1-{\mathrm{\&alpha;}}_i)R_i.$

Prove: choose following Lyapunov function:

$V\left(t\right)={\mathrm{\&Sigma;}}_{i=1}^S{V}_i\left(t\right)={\mathrm{\&Sigma;}}_{i=1}^S[x_i^T\left(t\right)P_i{x}_i\left(t\right)+{\&Integral;}_{t-d_i\left(t\right)}^t{x}_i^T\left(s\right)R_i{x}_i\left(s\right)\mathrm{ds}]---\left(8\right)$

Along the path of system (6), to V (t) differentiate, can obtain:

$\begin{array}{c}\stackrel{\&CenterDot;}{V}\left(t\right)={\mathrm{\&Sigma;}}_{i=1}^S{\mathrm{\&Sigma;}}_{m,n=1}^{r_i}{h}_{\mathrm{im}}{h}_{\mathrm{in}}[\stackrel{\&CenterDot;}{x_i^T}\left(t\right)P_i{x}_i\left(t\right)+x_i^T\left(t\right)P_i\stackrel{\&CenterDot;}{x_i}\left(t\right)+x_i^T\left(t\right)R_i{x}_i\left(t\right)-(1-\stackrel{\&CenterDot;}{d_i})x_{\mathrm{id}}^T\left(t\right)R_i{x}_{\mathrm{id}}\left(t\right)]\\ \&le;{\mathrm{\&Sigma;}}_{i=1}^S{\mathrm{\&Sigma;}}_{m,n=1}^{r_i}{h}_{\mathrm{im}}{h}_{\mathrm{in}}[x_i^T\left(t\right)({\mathrm{\&Lambda;}}_{i,\mathrm{mn}}^T{P}_i+P_i{\mathrm{\&Lambda;}}_{i,\mathrm{mn}})x_i\left(t\right)+x_{\mathrm{id}}^T\left(t\right){\mathrm{\&Lambda;}}_{i,\mathrm{dmn}}^T{P}_i{x}_i\left(t\right)+x_i^T\left(t\right)P_i{\mathrm{\&Lambda;}}_{i,\mathrm{dmn}}{x}_{\mathrm{id}}\left(t\right)\\ +{\mathrm{\&Sigma;}}_{j=1j\&NotEqual;i}^S{x}_j^T\left(t\right)C_{\mathrm{jim}}^T{P}_i{x}_i\left(t\right)+x_i^T\left(t\right)P_i{\mathrm{\&Sigma;}}_{j=1,j\&NotEqual;i}^s{C}_{\mathrm{jim}}{x}_j\left(t\right)+x_i^T\left(t\right)R_i{x}_i\left(t\right)-(1-{\mathrm{\&alpha;}}_i)x_{\mathrm{id}}^T\left(t\right)R_i{x}_{\mathrm{id}}\left(t\right)]\end{array}---\left(9\right)$

Consider following formula, and can be obtained by lemma 1:

$\begin{array}{c}{\mathrm{\&Lambda;}}_{i,\mathrm{mn}}^T{P}_i+P_i{\mathrm{\&Lambda;}}_{i,\mathrm{mn}}=A_{\mathrm{im}}^T{P}_i+P_i{A}_{\mathrm{im}}+{\left({\mathrm{\&rho;}}_i\sin {\mathrm{\&theta;}}_{\mathrm{in}}{N}_{\mathrm{im}}\right)}^T{P}_i+P_i\left({\mathrm{\&rho;}}_i\sin {\mathrm{\&theta;}}_{\mathrm{in}}{N}_{\mathrm{im}}\right)\\ +{\left({\mathrm{\&rho;}}_i\cos {\mathrm{\&theta;}}_{\mathrm{in}}{B}_{\mathrm{im}}{K}_{\mathrm{in}}\right)}^T{P}_i+P_i\left({\mathrm{\&rho;}}_i\cos {\mathrm{\&theta;}}_{\mathrm{in}}{B}_{\mathrm{im}}{K}_{\mathrm{in}}\right)\\ \&le;A_{\mathrm{im}}^T{P}_i+P_i{A}_{\mathrm{im}}+{\mathrm{\&epsiv;}}_{1i}{\mathrm{\&rho;}}_i^2{P}_i{P}_i+{\mathrm{\&epsiv;}}_{1i}^{-1}{N}_{\mathrm{im}}^T{N}_{\mathrm{im}}+{\mathrm{\&epsiv;}}_{1i}^{-1}{\left(B_{\mathrm{im}}{K}_{\mathrm{im}}\right)}^T\left(B_{\mathrm{im}}{K}_{\mathrm{in}}\right),\\ x_{\mathrm{id}}^T\left(t\right){\mathrm{\&Lambda;}}_{i,\mathrm{dmn}}^T{P}_i{x}_i\left(t\right)+x_i^T\left(t\right)P_i{\mathrm{\&Lambda;}}_{i,\mathrm{dmn}}{x}_{\mathrm{id}}\left(t\right)\\ \&le;x_{\mathrm{id}}^T\left(t\right)A_{\mathrm{idm}}^T{P}_i{x}_i\left(t\right)+x_i^T\left(t\right)P_i{A}_{\mathrm{idm}}{x}_{\mathrm{id}}\left(t\right)+x_i^T\left(t\right){\mathrm{\&epsiv;}}_{2i}{\mathrm{\&rho;}}_i^2{P}_i{P}_i{x}_i\left(t\right)\\ +x_{\mathrm{id}}^T\left(t\right){\mathrm{\&epsiv;}}_{2i}^{-1}{N}_{\mathrm{idm}}^T{N}_{\mathrm{idm}}{x}_{\mathrm{id}}\left(t\right)+x_{\mathrm{id}}^T\left(t\right){\mathrm{\&epsiv;}}_{2i}^{-1}{\left(B_{\mathrm{idm}}{K}_{\mathrm{in}}\right)}^T\left(B_{\mathrm{idm}}{K}_{\mathrm{in}}\right)x_{\mathrm{id}}\left(t\right),\end{array}---\left(10\right)$

In like manner can obtain:

$\begin{array}{c}{\mathrm{\&Sigma;}}_{i=1}^S{\mathrm{\&Sigma;}}_{m=1}^{r_i}{h}_{\mathrm{im}}[{\mathrm{\&Sigma;}}_{j=1,j\&NotEqual;i}^S{x}_j^T\left(t\right)C_{\mathrm{jim}}^T{P}_i{x}_i\left(t\right)+x_i^T\left(t\right)P_i{\mathrm{\&Sigma;}}_{j=1,j\&NotEqual;i}^s{C}_{\mathrm{jim}}{x}_j\left(t\right)]\\ ={\mathrm{\&Sigma;}}_{i=1}^S{\mathrm{\&Sigma;}}_{m=1}^{r_i}{h}_{\mathrm{im}}[x_1^T\left(t\right)C_{1\mathrm{im}}^T{P}_i{x}_i\left(t\right)+...+x_{i-1}^T\left(t\right)C_{i-1\mathrm{im}}^T{P}_i{x}_i\left(t\right)+x_{i+1}^T\left(t\right)C_{i+1\mathrm{im}}^T{P}_i{x}_i\left(t\right)+...\\ +x_S^T\left(t\right)C_{\mathrm{Sim}}^T{P}_i{x}_i\left(t\right)+x_i^T\left(t\right)P_i{C}_{1\mathrm{im}}{x}_1\left(t\right)+...+x_i^T\left(t\right)P_i{C}_{i-1\mathrm{im}}{x}_{i-1}\left(t\right)+x_i^T\left(t\right)P_i{C}_{i+1\mathrm{im}}{x}_{i+1}\left(t\right)\\ +...x_i^T\left(t\right)P_i{C}_{\mathrm{Sim}}{x}_S\left(t\right)]\\ \&le;{\mathrm{\&Sigma;}}_{i=1}^S{\mathrm{\&Sigma;}}_{m=1}^{r_i}{h}_{\mathrm{im}}\left[x_i^T\left(t\right)P_i{\mathrm{\&Sigma;}}_{j=1,j\&NotEqual;i}^S{C}_{\mathrm{jim}}{C}_{\mathrm{jim}}^T{P}_i{x}_i\left(t\right){+\mathrm{\&Sigma;}}_{j=1,j\&NotEqual;i}^S{x}_j^T\left(t\right)x_j\left(t\right)\right]\\ ={\mathrm{\&Sigma;}}_{i=1}^S{\mathrm{\&Sigma;}}_{m=1}^{r_i}{h}_{\mathrm{im}}[x_i^T\left(t\right)P_i{\mathrm{\&Sigma;}}_{j=1,j\&NotEqual;i}^S{C}_{\mathrm{jim}}{C}_{\mathrm{jim}}^T{P}_i{x}_i\left(t\right)+(S-1)x_i^T\left(t\right)x_i\left(t\right)]\end{array}---\left(11\right)$

(8) are brought into in (9), (10), (11), and noteCan obtain:

$\begin{array}{c}\stackrel{\&CenterDot;}{V}{\left(t\right)\&le;{\mathrm{\&Sigma;}}_{i=1}^S{\mathrm{\&Sigma;}}_{m,n=1}^{r_i}h}_{\mathrm{im}}{h}_{\mathrm{in}}{\mathrm{\&eta;}}_i^T\left(t\right){\mathrm{\&Phi;}}_{i,\mathrm{mn}}{x}_i\left(t\right){\mathrm{\&eta;}}_i\left(t\right)\\ ={\mathrm{\&Sigma;}}_{i=1}^S[{\mathrm{\&Sigma;}}_{m=1}^{r_i}{h}_{\mathrm{im}}^2{\mathrm{\&eta;}}_i^T\left(t\right){\mathrm{\&Phi;}}_{i,\mathrm{mn}}{\mathrm{\&eta;}}_i\left(t\right)+{\mathrm{\&Sigma;}}_{1=m<n}^{r_i}{h}_{\mathrm{im}}{h}_{\mathrm{in}}{\mathrm{\&eta;}}_i^T\left(t\right)({\mathrm{\&Phi;}}_{i,\mathrm{mn}}+{\mathrm{\&Phi;}}_{i,\mathrm{nm}}){\mathrm{\&eta;}}_i\left(t\right)]\end{array}---\left(12\right)$

Known according to (7) in theorem 1So known pass the United Nations General Assembly's system (6) is asymptotically stable.

Consider that (7) in theorem 1 are bilinearity MATRIX INEQUALITIES, can not directly be solved by LMI tool box, belowBilinearity MATRIX INEQUALITIES converts LMI to, provides the method for designing of controller:

Theorem 2: for given normal number α_i，ρ_i, i=1,2 ..., s, if for given normal number ε_1i，ε_2i，i=1,2 ..., s exists matrixS and matrix G_im，i＝1，2，...，S；m＝1，2，...，r_iMeet MATRIX INEQUALITIES (13), Ze Guan the United Nations General Assembly system (5) is Asymptotic Stability, and controller gain is: K_im＝G_imZ^-1，i＝1，2，...，S；m＝1，2，...，r_i.。

$\begin{array}{c}\left[\begin{array}{ccccccc}{T}_{i,m}& *& *& *& *& *& *\\ Z_i{A}_{\mathrm{idm}}^T& -(1-{\mathrm{\&alpha;}}_i){\stackrel{\&OverBar;}{R}}_i& *& *& *& *& *\\ Z_i& 0& -\frac{I}{s-1}& *& *& *& *\\ N_{\mathrm{im}}{Z}_i& 0& 0& -{\mathrm{\&epsiv;}}_{1i}I& *& *& *\\ B_{\mathrm{im}}{G}_{\mathrm{im}}& 0& 0& 0& -{\mathrm{\&epsiv;}}_{1i}I& *& *\\ 0& N_{\mathrm{idm}}{Z}_i& 0& 0& 0& -{\mathrm{\&epsiv;}}_{2i}I& *\\ 0& B_{\mathrm{idm}}{G}_{\mathrm{im}}& 0& 0& 0& 0& -{\mathrm{\&epsiv;}}_{2i}I\end{array}\right]<0,\\ i=\mathrm{1,2},...,S;m=\mathrm{1,2},...,r_i\end{array}---\left(13a\right)$

$\begin{array}{c}\left[\begin{array}{ccccccc}{T}_{i,m}+T_{i,n}& *& *& *& *& *& *\\ Z_i{A}_{\mathrm{idm}}^T+Z_i{A}_{\mathrm{idn}}^T& -2(1-{\mathrm{\&alpha;}}_i){\stackrel{\&OverBar;}{R}}_i& *& *& *& *& *\\ {2Z}_i& 0& -\frac{2I}{s-1}& *& *& *& *\\ \stackrel{\&OverBar;}{{\left(\mathrm{NZ}\right)}_{i,\mathrm{mn}}}& 0& 0& -t_{55}& *& *& *\\ \stackrel{\&OverBar;}{{\left(\mathrm{BG}\right)}_{i,\mathrm{mn}}}& 0& 0& 0& -t_{66}& *& *\\ 0& \stackrel{\&OverBar;}{{\left(N_{d}Z\right)}_{i,\mathrm{mn}}}& 0& 0& 0& -t_{77}& *\\ 0& \stackrel{\&OverBar;}{{\left(B_{d}G\right)}_{i,\mathrm{mn}}}& 0& 0& 0& 0& -t_{88}\end{array}\right]<0,\\ i=\mathrm{1,2},...,S;1\&le;m<n\&le;r_i\end{array}---\left(13b\right)$

Here:

$\stackrel{\&OverBar;}{{\left(\mathrm{NZ}\right)}_{i,\mathrm{mn}}}=\left[\begin{array}{c}{N}_{\mathrm{im}}{Z}_i\\ N_{\mathrm{in}}{Z}_i\end{array}\right],\stackrel{\&OverBar;}{{\left(\mathrm{BG}\right)}_{i,\mathrm{mn}}}=\left[\begin{array}{c}{B}_{\mathrm{im}}{G}_{\mathrm{in}}\\ B_{\mathrm{in}}{G}_{\mathrm{im}}\end{array}\right],\stackrel{\&OverBar;}{{\left(N_{d}Z\right)}_{i,\mathrm{mn}}}=\left[\begin{array}{c}{N}_{\mathrm{idm}}{Z}_i\\ N_{\mathrm{idn}}{Z}_i\end{array}\right],\stackrel{\&OverBar;}{{\left(B_{d}G\right)}_{i,\mathrm{mn}}}=\left[\begin{array}{c}{B}_{\mathrm{idm}}{G}_{\mathrm{in}}\\ B_{\mathrm{idn}}{G}_{\mathrm{im}}\end{array}\right],$

t₄₄＝t₅₅＝diag{ε_1iI，ε_1iI}，t₆₆＝t₇₇＝diag{ε_2iI，ε_2iI}.

Prove: chooseAnd noteBy K_im＝G_imZ^-1Known, M_im＝K_imZ。

(13a) taken advantage of to diag{P in left and right simultaneously_i，P_i, I, I, I, I, I} can obtain:

$\begin{array}{c}\left[\begin{array}{ccccccc}{P_{i}T}_{i,m}{P}_i& *& *& *& *& *& *\\ A_{\mathrm{idm}}^T{P}_i& -(1-{\mathrm{\&alpha;}}_i)R_i& *& *& *& *& *\\ I& 0& -\frac{I}{s-1}& *& *& *& *\\ N_{\mathrm{im}}& 0& 0& -{\mathrm{\&epsiv;}}_{1i}I& *& *& *\\ B_{\mathrm{im}}{K}_{\mathrm{im}}& 0& 0& 0& -{\mathrm{\&epsiv;}}_{1i}I& *& *\\ 0& N_{\mathrm{idm}}& 0& 0& 0& -{\mathrm{\&epsiv;}}_{2i}I& *\\ 0& B_{\mathrm{idm}}{K}_{\mathrm{im}}& 0& 0& 0& 0& -{\mathrm{\&epsiv;}}_{2i}I\end{array}\right]<0,\\ i=\mathrm{1,2},...,S;m=\mathrm{1,2},...,r_i\end{array}---\left(14a\right)$

Set up and can be equivalent to (7a) establishment by Schur complement fixed reason known (14a). Same reason (13b) can be derived (7b)Vertical. Known under designed controller by theorem 1 like this, it is asymptotically stable that Fuzzy Bilinear is closed the United Nations General Assembly's system.

3. sample calculation analysis

In order to set forth method above, consider that 2 sub-system relationships form the Fuzzy Bilinear pass system of the United Nations General Assembly with time lagSystem:

Subsystem1：

$\begin{array}{c}\begin{array}{ccccc}{R}_1^1:& \mathrm{if}& x_{11}& \mathrm{is}& F_{11}^1\end{array}\\ \begin{array}{cc}\mathrm{then}& {\stackrel{\&CenterDot;}{x}}_1\end{array},\left(t\right)=A_{11}{x}_1\left(t\right)+A_{1d1}{x}_{1d}\left(t\right)+N_{11}{x}_1\left(t\right)u_1\left(t\right)+N_{1d1}{x}_{1d}\left(t\right)u_{1d}\left(t\right)+B_{11}{u}_{1d}\left(t\right)\\ +B_{1d1}{u}_{1d}\left(t\right)+C_{211}{x}_2\left(t\right);\\ \begin{array}{ccccc}{R}_1^2:& \mathrm{if}& x_{11}& \mathrm{is}& F_{11}^2\end{array}\\ \begin{array}{cc}\mathrm{then}& {\stackrel{\&CenterDot;}{x}}_1\left(t\right)=A_{12}{x}_1\left(t\right)+A_{1d2}{x}_{1d}\left(t\right)+N_{12}{x}_1\left(t\right)u_1\left(t\right)+N_{1d2}{x}_{1d}\left(t\right)u_{1d}\left(t\right)+B_{12}{u}_{1d}\left(t\right)\end{array}\\ +B_{1d2}{u}_{1d}\left(t\right)+C_{212}{x}_2\left(t\right);\end{array}$

Subsystem2：

$\begin{array}{c}\begin{array}{ccccc}{R}_2^{1:}& \mathrm{if}& x_{21}& \mathrm{is}& F_{21}^1\end{array}\\ \mathrm{then}\stackrel{\&CenterDot;}{x_2}\left(t\right)=A_{21}{x}_2\left(t\right)+A_{2d1}{x}_{2d}\left(t\right)+N_{21}{x}_2\left(t\right)u_2\left(t\right)+N_{2d1}{x}_{2d}\left(t\right)u_{2d}\left(t\right)+B_{21}{u}_{2d}\left(t\right)\\ +B_{2d1}{u}_{2d}\left(t\right)+C_{121}{x}_1\left(t\right);\\ \begin{array}{ccccc}{R}_2^{2:}& \mathrm{if}& x_{21}& \mathrm{is}& F_{21}^2\end{array}\\ \mathrm{then}\stackrel{\&CenterDot;}{x_2}\left(t\right)=A_{22}{x}_2\left(t\right)+A_{2d2}{x}_{2d}\left(t\right)+N_{22}{x}_2\left(t\right)u_2\left(t\right)+N_{2d2}{x}_{2d}\left(t\right)u_{2d}\left(t\right)+B_{22}{u}_{2d}\left(t\right)\\ +B_{2d2}{u}_{2d}\left(t\right)+C_{122}{x}_1\left(t\right);\end{array}$

Wherein:

$A_{11}=\left[\begin{array}{cc}-95& 7\\ -35& -97\end{array}\right],A_{12}=\left[\begin{array}{cc}-82& 9\\ -30& -90\end{array}\right],A_{21}=\left[\begin{array}{cc}-110& 15\\ -10& -100\end{array}\right],A_{22}=\left[\begin{array}{cc}-102& 9\\ -5& -90\end{array}\right];$

$N_{11}=\left[\begin{array}{cc}-3& 0\\ 0& 0\end{array}\right],N_{12}=\left[\begin{array}{cc}-3& -3\\ 1& -3\end{array}\right],N_{21}=\left[\begin{array}{cc}0& 0\\ 0& -4\end{array}\right],N_{22}=\left[\begin{array}{cc}0& 0\\ 4& -1\end{array}\right];B_{11}=\left[\begin{array}{c}-1\\ 0\end{array}\right],B_{12}=\left[\begin{array}{c}1\\ 1\end{array}\right],$

$B_{21}=B_{22}=\left[\begin{array}{c}1\\ 1\end{array}\right];A_{1d1}=\left[\begin{array}{cc}-10& 0\\ 5& 2\end{array}\right],A_{1d2}=\left[\begin{array}{cc}-15& 0\\ 0& 7\end{array}\right],A_{2d1}=\left[\begin{array}{cc}-12& 0\\ 10& 24\end{array}\right],A_{2d2}=\left[\begin{array}{cc}10& 0\\ 14& 7\end{array}\right];$

$N_{1d1}=\left[\begin{array}{cc}0.5& 0\\ 0& 0.2\end{array}\right],N_{1d2}=\left[\begin{array}{cc}0.3& 0\\ 0& 0.5\end{array}\right],N_{2d1}=\left[\begin{array}{cc}0.3& 0\\ 0& 0.5\end{array}\right],N_{2d2}=\left[\begin{array}{cc}0.3& 0\\ 0& 0.5\end{array}\right];$

$B_{1d1}=B_{1d2}=\left[\begin{array}{c}1\\ 0\end{array}\right],B_{2d1}=B_{2d2}=\left[\begin{array}{c}-1\\ 0\end{array}\right];C_{211}=C_{212}=\left[\begin{array}{cc}1& -1\\ 0& 1\end{array}\right],C_{121}=C_{122}=\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right],$

Selecting All Parameters ρ₁＝0.75，ρ₂＝0.63，a₁＝a₂＝0，ε₁₁＝ε₁₂＝ε₂₁＝ε₂₂=1. Choose membership functionWith RootAccording to theorem 2, solve corresponding LMIs and can obtain 2

K₁₁＝[-1.1233-1.0443]；K₁₂＝[-1.8542-1.0211]；

K₂₁＝[-1.4735-1.2671]；K₂₂＝[-1.8013-1.2024].

It is x that two subsystems are chosen respectively to initial value₁₀＝[-2.81.7]^T，x₂₀＝[3.0-1.6]^T, utilizeMATLAB emulation, Fig. 1 is the state variable response curve of subsystem 1, and Fig. 2 is the condition responsive curve of subsystem 2, and Fig. 3 is twoThe control curve of individual subsystem. Can be found out by simulation result, under designed controller, it is asymptotic that closed loop is closed the United Nations General Assembly's systemStable.

Claims (1)

1. a decentralized control method for the Design for Large-Scale Interconnected Nonlinear Systems based on T-S bilinear model, is characterized in that, according toLower step is carried out:

The large system of design fuzzy time-delay bilinear interconnection;

One class is by S subsystem Ω_i, i=1,2 ..., S composition band becomes the Fuzzy Bilinear pass system Ω of the United Nations General Assembly of time lag sometimes, theI subsystem Ω_iCan be expressed as:

$\begin{array}{c}\begin{array}{ccccccccc}{R}_i^m& f& {\mathrm{\&xi;}}_{i1}\left(t\right)& \mathrm{is}& F_{i1}^m& \mathrm{and}...\mathrm{and}& {\mathrm{\&xi;}}_{{\mathrm{iv}}_i}\left(t\right)& \mathrm{is}& F_{{\mathrm{iv}}_i}^m\end{array}\\ \begin{array}{cc}\mathrm{then}& {\stackrel{.}{x}}_i\left(t\right)=A_{\mathrm{im}}{x}_i\left(t\right)+B_{\mathrm{im}}{u}_i\left(t\right)+N_{\mathrm{im}}{x}_i\left(t\right)u_i\left(t\right)+A_{\mathrm{idm}}{x}_i(t-d_i\left(t\right))+B_{\mathrm{idm}}{u}_i(t-d_i\left(t\right))\end{array}\\ +N_{\mathrm{idm}}{x}_i(t-d_i\left(t\right))u_i(t-d_i\left(t\right))+{\mathrm{\&Sigma;}}_{j=1,j\&NotEqual;i}^S{C}_{\mathrm{jim}}{x}_j\left(t\right)\\ \begin{array}{ccc}{x}_i\left(t\right)={\mathrm{\&phi;}}_i\left(t\right)& t=\begin{array}{}\end{array}\left[\begin{array}{cc}-{\mathrm{\&tau;}}_i& 0\end{array}\right]& m=\mathrm{1,2},...,\end{array}{r}_i\end{array}---\left(1\right)$

Wherein:I subsystem Ω_iFuzzy rule, s is the number of subsystem; M={1,2 ..., r_i}，r_iIt is the number of the fuzzy rule of i subsystem;Respectively fuzzy set and prerequisite variable;Respectively state vector and control inputs; It is known system squareBattle array;The correlation matrix of j subsystem to i subsystem; d_i(t) being the time lag item of system, is continuousDifferentiable function and satisfied 0≤d_i(t)≤τ_iWith

By single-point obfuscation, the average reverse gelatinizing method in product reasoning and center, the overall model of Fuzzy control system is:

$\begin{array}{c}{\stackrel{.}{x}}_i\left(t\right)={\mathrm{\&Sigma;}}_{m=1}^{r_i}{h}_{\mathrm{im}}\left({\mathrm{\&xi;}}_i\left(t\right)\right)[A_{\mathrm{im}}{x}_i\left(t\right)+B_{\mathrm{im}}{u}_i\left(t\right)+N_{\mathrm{im}}{x}_i\left(t\right)u_i\left(t\right)+A_{\mathrm{idm}}{x}_i(t-d_i\left(t\right))\\ +N_{\mathrm{idm}}{x}_i(t-d_i\left(t\right))u_i(t-d_i\left(t\right))+B_{\mathrm{idm}}{u}_i(t-d_i\left(t\right))+{\mathrm{\&Sigma;}}_{j=1,j\&NotEqual;i}^S{C}_{\mathrm{jim}}{x}_j\left(t\right)]\end{array}---\left(2\right)$

Wherein:μ_inj(ξ_i(t)) be ξ_j(t) existIn membership function; FalseIfBy h_im(ξ_i(t) definition) is known:Note respectively h by abridging_im(ξ_i(t))，x_i(t-d_i(t))，u_i(t-d_i(t)) be h_im，x_id(t)，u_id(t)；

According to parallel distributed backoff algorithm, consider local feedback control device:

$\begin{array}{c}\begin{array}{cccccccc}f& {\mathrm{\&xi;}}_{i1}\left(t\right)& \mathrm{is}& F_{i1}^m& \mathrm{and}...\mathrm{and}& {\mathrm{\&xi;}}_{{\mathrm{iv}}_i}\left(t\right)& \mathrm{is}& F_{{\mathrm{iv}}_i}^m\end{array}\\ \begin{array}{cc}\mathrm{then}& u_i\left(t\right)=\frac{{\mathrm{\&rho;}}_i{K}_{\mathrm{im}}{x}_i\left(t\right)}{\sqrt{1+{x_i}^T{K}_{\mathrm{im}}^T{K}_{\mathrm{im}}{x}_i}}={\mathrm{\&rho;}}_i\sin {\mathrm{\&theta;}}_{\mathrm{im}}={\mathrm{\&rho;}}_i\cos {\mathrm{\&theta;}}_{\mathrm{im}}{K}_{\mathrm{im}}{x}_i\left(t\right)\end{array}\end{array}---\left(3\right)$

Here:Controller gain to be asked, ρ_i> 0 is scalar undetermined,

Can similarly be obtained by (3):

Here:

Overall decentralised control rule can be expressed as:

Under the effect of control law (5), the equation of whole closed-loop system can be expressed as:

${\stackrel{.}{x}}_i\left(t\right)={\mathrm{\&Sigma;}}_{m,n=1}^{r_i}{h}_{\mathrm{im}}{h}_{\mathrm{in}}[{\mathrm{\&Lambda;}}_{i,\mathrm{mn}}{x}_i\left(t\right)+{\mathrm{\&Lambda;}}_{i,\mathrm{dmn}}{x}_{\mathrm{id}}\left(t\right){+\mathrm{\&Sigma;}}_{j=1,j\&NotEqual;i}^s{C}_{\mathrm{jim}}{x}_j\left(t\right)]---\left(6\right)$

Here: Λ_i，mn＝A_im+ρ_isinθ_inN_im+ρ_icosθ_inB_imK_in，

Theorem 1: for given normal number ρ_i，α_i, i=1,2 ..., S, if for given normal number ε_1i，ε_2i，i＝1，2 ..., there is positive definite symmetric matrices P in S_i＞0，R_i> 0, i=1,2 ..., S and matrix K_im，i＝1，2，...，S；m＝1，2，...，r_iMeet MATRIX INEQUALITIES (7), Ze Guan the United Nations General Assembly system (5) is asymptotically stable;

Φ_i，mm＜0i＝1，2，...，S；m＝1，2，...，r_i(7a)

Φ_i，mn+Φ_i，nm＜0i＝1，2，...，S；1≤m＜n≤r_i(7b)

Wherein:

$\begin{array}{c}{\mathrm{\&phi;}}_{i,1\mathrm{mn}}=A_{\mathrm{im}}^T{P}_i+P_i{A}_{\mathrm{im}}+({\mathrm{\&epsiv;}}_{1i}+{\mathrm{\&epsiv;}}_{2i}){\mathrm{\&rho;}}_i^2{P}_i{P}_i+{\mathrm{\&epsiv;}}_{1i}^{-1}{N}_{\mathrm{im}}^T{N}_{\mathrm{im}}+{\mathrm{\&epsiv;}}_{1i}^{-1}{\left(B_{\mathrm{im}}{K}_{\mathrm{in}}\right)}^T\left(B_{\mathrm{im}}{K}_{\mathrm{in}}\right)\\ +{\mathrm{\&Sigma;}}_{j=1,j\&NotEqual;i}^S{P}_i{C}_{\mathrm{jim}}{C}_{\mathrm{jim}}^T{P}_i+(S-1)I+R_i,\end{array}$

${\mathrm{\&phi;}}_{i,2\mathrm{mn}}={\mathrm{\&epsiv;}}_{2i}^{-1}{N}_{\mathrm{idm}}^T{N}_{\mathrm{idm}}+{\mathrm{\&epsiv;}}_{2i}^{-1}{\left(B_{\mathrm{idm}}{K}_{\mathrm{in}}\right)}^T\left(B_{\mathrm{idm}}{K}_{\mathrm{in}}\right)-(1-{\mathrm{\&alpha;}}_i)R_i.$

Prove: choose following Lyapunov function:

$V\left(t\right)={\mathrm{\&Sigma;}}_{i=1}^S{V}_i\left(t\right)={\mathrm{\&Sigma;}}_{i=1}^S[x_i^T\left(t\right)P_i{x}_i\left(t\right)+{\&Integral;}_{t-d_i\left(t\right)}^t{x}_i^T\left(s\right)R_i{x}_i\left(s\right)\mathrm{ds}]---\left(8\right)$

Along the path of system (6), to V (t) differentiate, can obtain:

$\begin{array}{c}\stackrel{.}{V}\left(t\right)={\mathrm{\&Sigma;}}_{i=1}^S{\mathrm{\&Sigma;}}_{m,n=1}^{r_i}{h}_{\mathrm{im}}{h}_{\mathrm{in}}[{\stackrel{.}{x}}_i^T\left(t\right)P_i{x}_i\left(t\right)+x_i^T\left(t\right)P_i{\stackrel{.}{x}}_i\left(t\right)+x_i^T\left(t\right)R_i{x}_i\left(t\right)-(1-{\stackrel{.}{d}}_i)x_{\mathrm{id}}^T\left(t\right)R_i{x}_{\mathrm{id}}\left(t\right)]\\ \&le;{\mathrm{\&Sigma;}}_{i=1}^S{\mathrm{\&Sigma;}}_{m,n=1}^{r_i}{h}_{\mathrm{im}}{h}_{\mathrm{in}}[x_i^T\left(t\right)({\mathrm{\&Lambda;}}_{i,\mathrm{mn}}^T{P}_i+P_i{\mathrm{\&Lambda;}}_{i,\mathrm{mn}})x_i\left(t\right)+x_{\mathrm{id}}^T\left(t\right){\mathrm{\&Lambda;}}_{i,\mathrm{dmn}}^T{P}_i{x}_i\left(t\right)+x_i^T\left(t\right)P_i{\mathrm{\&Lambda;}}_{i,\mathrm{dmn}}{x}_{\mathrm{id}}\left(t\right)\\ +{\mathrm{\&Sigma;}}_{j=1j\&NotEqual;i}^S{x}_j^T\left(t\right)C_{\mathrm{jim}}^T{P}_i{x}_i\left(t\right)+x_i^T\left(t\right)P_i{\mathrm{\&Sigma;}}_{j=1,j\&NotEqual;i}^s{C_{\mathrm{jim}}}^{}{x}_j\left(t\right)+x_i^T\left(t\right)R_i{x}_i\left(t\right)-(1-{\mathrm{\&alpha;}}_i)x_{\mathrm{id}}^T\left(t\right)R_i{x}_{\mathrm{id}}\left(t\right)]\end{array}---\left(9\right)$

Consider following formula, and can be obtained by lemma 1:

$\begin{array}{c}{\mathrm{\&Lambda;}}_{i,\mathrm{mn}}^T{P}_i+P_i{\mathrm{\&Lambda;}}_{i,\mathrm{mn}}=A_{\mathrm{im}}^T{P}_i+P_i{A}_{\mathrm{im}}+{\left({\mathrm{\&rho;}}_i\sin {\mathrm{\&theta;}}_{\mathrm{in}}{N}_{\mathrm{im}}\right)}^T{P}_i+P_i\left({\mathrm{\&rho;}}_i\sin {\mathrm{\&theta;}}_{\mathrm{in}}{N}_{\mathrm{im}}\right)\\ +{\left({\mathrm{\&rho;}}_i\cos {\mathrm{\&theta;}}_{\mathrm{in}}{B}_{\mathrm{im}}{K}_{\mathrm{in}}\right)}^T{P}_i+P_i\left({\mathrm{\&rho;}}_i\cos {\mathrm{\&theta;}}_{\mathrm{in}}{B}_{\mathrm{im}}{K}_{\mathrm{in}}\right)\\ \&le;A_{\mathrm{im}}^T{P}_i+P_i{A}_{\mathrm{im}}+{\mathrm{\&epsiv;}}_{1i}{\mathrm{\&rho;}}_i^2{P}_i{P}_i+{\mathrm{\&epsiv;}}_{1i}^{-1}{N}_{\mathrm{im}}^T{N}_{\mathrm{im}}+{\mathrm{\&epsiv;}}_{1i}^{-1}{\left(B_{\mathrm{im}}{K}_{\mathrm{in}}\right)}^T\left(B_{\mathrm{im}}{K}_{\mathrm{in}}\right),\\ x_{\mathrm{id}}^T\left(t\right){\mathrm{\&Lambda;}}_{i,\mathrm{dmn}}^T{P}_i{x}_i\left(t\right)+x_i^T\left(t\right)P_i{\mathrm{\&Lambda;}}_{i,\mathrm{dmn}}{x}_{\mathrm{id}}\left(t\right)\\ \&le;x_{\mathrm{id}}^T\left(t\right)A_{\mathrm{idm}}^T{P}_i{x}_i\left(t\right)+x_i^T\left(t\right)P_i{A}_{\mathrm{idm}}{x}_{\mathrm{id}}\left(t\right)+x_i^T\left(t\right){\mathrm{\&epsiv;}}_{2i}{\mathrm{\&rho;}}_i^2{P}_i{P}_i{x}_i\left(t\right)\\ +x_{\mathrm{id}}^T\left(t\right){\mathrm{\&epsiv;}}_{2i}^{-1}{N}_{\mathrm{idm}}^T{N}_{\mathrm{idm}}{x}_{\mathrm{id}}\left(t\right)+x_{\mathrm{id}}^T\left(t\right){\mathrm{\&epsiv;}}_{2i}^{-1}{\left(B_{\mathrm{idm}}{K}_{\mathrm{in}}\right)}^T\left(B_{\mathrm{idm}}{K}_{\mathrm{in}}\right)x_{\mathrm{id}}\left(t\right),\end{array}---\left(10\right)$

In like manner can obtain:

$\begin{array}{c}{\mathrm{\&Sigma;}}_{i=1}^S{\mathrm{\&Sigma;}}_{m=1}^{r_i}{h}_{\mathrm{im}}[{\mathrm{\&Sigma;}}_{j=1,j\&NotEqual;i}^S{x}_j^T\left(t\right)C_{\mathrm{jim}}^T{P}_i{x}_i\left(t\right)+x_i^T\left(t\right)p_i{\mathrm{\&Sigma;}}_{j=1,j\&NotEqual;i}^s{C}_{\mathrm{jim}}{x}_j\left(t\right)]\\ ={\mathrm{\&Sigma;}}_{i=1}^S{\mathrm{\&Sigma;}}_{m=1}^{r_i}{h}_{\mathrm{im}}[x_1^T\left(t\right)C_{1\mathrm{im}}^T{P}_i{x}_i\left(t\right)+...+x_{i-1}^T\left(t\right)C_{i-1\mathrm{im}}^T{P}_i{x}_i\left(t\right)+x_{i+1}^T\left(t\right)C_{i+1\mathrm{im}}^T{P}_i{x}_i\left(t\right)+...\\ +x_S^T\left(t\right)C_{\mathrm{Sim}}^T{P}_i{x}_i\left(t\right)+x_i^T\left(t\right)P_i{C}_{1\mathrm{im}}{x}_1\left(t\right)+...+x_i^T\left(t\right)P_i{C}_{i-1\mathrm{im}}{x}_{i+1}\left(t\right)+x_i^T\left(t\right)P_i{C}_{i+1\mathrm{im}}{x}_{i+1}\left(t\right)\\ +...x_i^T\left(t\right)P_i{C}_{\mathrm{Sim}}{x}_S\left(t\right)]\\ \&le;{\mathrm{\&Sigma;}}_{i=1}^S{\mathrm{\&Sigma;}}_{m=1}^{r_i}{h}_{\mathrm{im}}[x_i^T\left(t\right)P_i{\mathrm{\&Sigma;}}_{j=1,j\&NotEqual;i}^S{C}_{\mathrm{jim}}{C}_{\mathrm{jim}}^T{P}_i{x}_i\left(t\right)+{\mathrm{\&Sigma;}}_{j=1,j\&NotEqual;i}^S{x}_j^T\left(t\right)x_j\left(t\right)]\\ ={\mathrm{\&Sigma;}}_{i=1}^S{\mathrm{\&Sigma;}}_{m=1}^{r_i}{h}_{\mathrm{im}}[x_i^T\left(t\right)P_i{\mathrm{\&Sigma;}}_{j=1,j\&NotEqual;i}^S{C}_{\mathrm{jim}}{C}_{\mathrm{jim}}^T{P}_i{x}_i\left(t\right)+(S-1)x_i^T\left(t\right)x_i\left(t\right)]\end{array}---\left(11\right)$

(8) are brought into in (9), (10), (11), and noteCan obtain:

$\begin{array}{c}\stackrel{.}{V}\left(t\right)\&le;{\mathrm{\&Sigma;}}_{i=1}^S{\mathrm{\&Sigma;}}_{m,n=1}^{r_i}{h}_{\mathrm{im}}{h}_{\mathrm{in}}{\mathrm{\&eta;}}_i^T\left(t\right){\mathrm{\&Phi;}}_{i,\mathrm{mn}}{x}_i\left(t\right){\mathrm{\&eta;}}_i\left(t\right)\\ ={\mathrm{\&Sigma;}}_{i=1}^S[{\mathrm{\&Sigma;}}_{m=1}^{r_i}{h}_{\mathrm{im}}^2{\mathrm{\&eta;}}_i^T\left(t\right){\mathrm{\&Phi;}}_{i,\mathrm{mm}}{\mathrm{\&eta;}}_i\left(t\right)+{\mathrm{\&Sigma;}}_{i=m<n}^{r_i}{h}_{\mathrm{im}}{h}_{\mathrm{in}}{\mathrm{\&eta;}}_i^T\left(t\right)({\mathrm{\&Phi;}}_{i,\mathrm{mn}}+{\mathrm{\&Phi;}}_{i,\mathrm{nm}}){\mathrm{\&eta;}}_i\left(t\right)]\end{array}---\left(12\right)$

Known according to (7) in theorem 1So known pass the United Nations General Assembly's system (6) is asymptotically stable;

Consider that (7) in theorem 1 are bilinearity MATRIX INEQUALITIES, can not directly be solved by LMI tool box, bilinearity squareBattle array inequality converts LMI to, provides the method for designing of controller:

Theorem 2: for given normal number α_i，ρ_i, i=1,2 ..., s, if for given normal number ε_1i，ε_2i，i＝1，2 ..., s exists matrixAnd matrixMeet matrix notEquation (13), Ze Guan the United Nations General Assembly system (5) is Asymptotic Stability, and controller gain is: K_im＝G_imZ^-1，i＝1，2，...，S；m＝1，2，...，r_i.；

$\begin{array}{c}\left[\begin{array}{ccccccc}{T}_{i,m}& *& *& *& *& *& *\\ Z_i{A}_{\mathrm{idm}}^T& -(1-{\mathrm{\&alpha;}}_i){\stackrel{\&OverBar;}{R}}_i& *& *& *& *& *\\ Z_i& 0& -\frac{I}{s-1}& *& *& *& *\\ N_{\mathrm{im}}{Z}_i& 0& 0& -{\mathrm{\&epsiv;}}_{1i}I& *& *& *\\ B_{\mathrm{im}}{G}_{\mathrm{im}}& 0& 0& 0& -{\mathrm{\&epsiv;}}_{1i}I& *& *\\ 0& N_{\mathrm{idm}}{Z}_i& 0& 0& 0& -{\mathrm{\&epsiv;}}_{2i}I& *\\ 0& B_{\mathrm{idm}}{G}_{\mathrm{im}}& 0& 0& 0& 0& -{\mathrm{\&epsiv;}}_{2i}I\end{array}\right]<0,\\ i=\mathrm{1,2},...,S;m=\mathrm{1,2},..,r_i\end{array}---\left(13a\right)$

$\begin{array}{c}\left[\begin{array}{ccccccc}{T}_{i,m}+T_{i,n}& *& *& *& *& *& *\\ Z_i{A}_{\mathrm{idm}}^T+Z_i{A}_{\mathrm{idn}}^T& -2(1-{\mathrm{\&alpha;}}_i){\stackrel{\&OverBar;}{R}}_i& *& *& *& *& *\\ {2Z}_i& 0& -\frac{2I}{s-1}& *& *& *& *\\ \stackrel{\&OverBar;}{{\left(\mathrm{NZ}\right)}_{i,\mathrm{mn}}}& 0& 0& -t_{55}& *& *& *\\ \stackrel{\&OverBar;}{{\left(\mathrm{BG}\right)}_{i,\mathrm{mn}}}& 0& 0& 0& -t_{66}& *& *\\ 0& \stackrel{\&OverBar;}{{\left(N_{d}Z\right)}_{i,\mathrm{mn}}}& 0& 0& 0& -t_{77}& *\\ 0& \stackrel{\&OverBar;}{{\left(B_{d}G\right)}_{i,\mathrm{mn}}}& 0& 0& 0& 0& -t_{88}\end{array}\right]<0,\\ i=\mathrm{1,2},...,S;1\&le;m<n\&le;r_i\end{array}---\left(13b\right)$

Here:

$\stackrel{\&OverBar;}{{\left(\mathrm{NZ}\right)}_{i,\mathrm{mn}}}=\left[\begin{array}{c}{N}_{\mathrm{im}}{Z}_i\\ N_{\mathrm{in}}{Z}_i\end{array}\right],\stackrel{\&OverBar;}{{\left(\mathrm{BG}\right)}_{i,\mathrm{mn}}}=\left[\begin{array}{c}{B}_{\mathrm{im}}{G}_{\mathrm{in}}\\ B_{\mathrm{in}}{G}_{\mathrm{im}}\end{array}\right],\stackrel{\&OverBar;}{{\left(N_{d}Z\right)}_{i,\mathrm{mn}}}=\left[\begin{array}{c}{N}_{\mathrm{idm}}{Z}_i\\ N_{\mathrm{idn}}{Z}_i\end{array}\right],\stackrel{\&OverBar;}{{\left(B_{d}G\right)}_{i,\mathrm{mn}}}=\left[\begin{array}{c}{B}_{\mathrm{idm}}{G}_{\mathrm{in}}\\ B_{\mathrm{idn}}{G}_{\mathrm{im}}\end{array}\right],$

t₄₄＝t₅₅＝diag{ε_1iI，ε_1iI}，t₆₆＝t₇₇＝diag{ε_2iI，ε_2iI}.

Prove: chooseAnd noteBy K_im＝G_imZ^-1Known, M_im＝K_imZ；

(13a) taken advantage of to diag{P in left and right simultaneously_i，P_i, I, I, I, I, I} can obtain:

$\begin{array}{c}\left[\begin{array}{ccccccc}{P_{i}T}_{i,m}{P}_i& *& *& *& *& *& *\\ A_{\mathrm{idm}}^T{P}_i& -(1-{\mathrm{\&alpha;}}_i)R_i& *& *& *& *& *\\ I& 0& -\frac{I}{s-1}& *& *& *& *\\ N_{\mathrm{im}}& 0& 0& -{\mathrm{\&epsiv;}}_{1i}I& *& *& *\\ B_{\mathrm{im}}{K}_{\mathrm{im}}& 0& 0& 0& -{\mathrm{\&epsiv;}}_{1i}I& *& *\\ 0& N_{\mathrm{idm}}& 0& 0& 0& -{\mathrm{\&epsiv;}}_{2i}I& *\\ 0& B_{\mathrm{idm}}{K}_{\mathrm{im}}& 0& 0& 0& 0& -{\mathrm{\&epsiv;}}_{2i}I\end{array}\right]<0,\\ i=\mathrm{1,2},...,S;m=\mathrm{1,2},..,r_i\end{array}---\left(14a\right)$

Set up and can be equivalent to (7a) establishment by Schur complement fixed reason known (14a); Same reason (13b) can be derived (7b) and set up; ThisSample is known under designed controller by theorem 1, and Fuzzy Bilinear is closed the United Nations General Assembly's system Asymptotic Stability.