一种基于T-S双线性模型的非线性关联大系统的分散控制方法

摘要

本发明公开了一种基于T-S双线性模型的非线性关联大系统的分散控制方法，根据Lyapunov稳定性分析理论和并行分布补偿算法，得到了闭环关联大系统时滞相关渐近稳定的充分条件，分散控制器可由一组线性矩阵不等式的解得到，采相应的分散模糊控制器的设计可转化成一个受线性矩阵不等式(LMI)约束的凸优化问题。最后，仿真数例验证了所提方法的有效性。

主权项

一种基于T‑S双线性模型的非线性关联大系统的分散控制方法，其特征在于，按照以下步骤进行：设计时滞模糊双线性关联大系统；一类由S个子系统Ω_i，i＝1，2，...，S组成带有时变时滞的模糊双线性关联大系统Ω，第i个子系统Ω_i可表示为：$\begin{array}{c}\begin{array}{ccccccccc}{R}_i^m& f& {\xi }_{i1}\left(t\right)& \mathrm{is}& F_{i1}^m& \mathrm{and}...\mathrm{and}& {\xi }_{{\mathrm{iv}}_i}\left(t\right)& \mathrm{is}& F_{{\mathrm{iv}}_i}^m\end{array}\\ \begin{array}{cc}\mathrm{then}& {\stackrel{.}{x}}_i\left(t\right)=A_{\mathrm{im}}{x}_i\left(t\right)+B_{\mathrm{im}}{u}_i\left(t\right)+N_{\mathrm{im}}{x}_i\left(t\right)u_i\left(t\right)+A_{\mathrm{idm}}{x}_i(t-d_i\left(t\right))+B_{\mathrm{idm}}{u}_i(t-d_i\left(t\right))\end{array}\\ +N_{\mathrm{idm}}{x}_i(t-d_i\left(t\right))u_i(t-d_i\left(t\right))+{\Sigma }_{j=1,j\ne i}^S{C}_{\mathrm{jim}}{x}_j\left(t\right)\\ \begin{array}{ccc}{x}_i\left(t\right)={\phi }_i\left(t\right)& t=\begin{array}{}\end{array}\left[\begin{array}{cc}-{\tau }_i& 0\end{array}\right]& m=\mathrm{1,2},...,\end{array}{r}_i\end{array}---\left(1\right)$其中：是第i个子系统Ω_i的模糊规则，s是子系统的数目；m＝{1，2，...，r_i}，r_i是第i个子系统的模糊规则的数目；分别是模糊集合和前提变量；分别是状态向量和控制输入；是已知的系统矩阵；是第j个子系统对第i个子系统的关联作用矩阵；d_i(t)是系统的时滞项，是连续可微函数且满足0≤d_i(t)≤τ_i和通过单点模糊化，乘积推理和中心平均反模糊化方法，模糊控制系统的总体模型为：$\begin{array}{c}{\stackrel{.}{x}}_i\left(t\right)={\Sigma }_{m=1}^{r_i}{h}_{\mathrm{im}}\left({\xi }_i\left(t\right)\right)[A_{\mathrm{im}}{x}_i\left(t\right)+B_{\mathrm{im}}{u}_i\left(t\right)+N_{\mathrm{im}}{x}_i\left(t\right)u_i\left(t\right)+A_{\mathrm{idm}}{x}_i(t-d_i\left(t\right))\\ +N_{\mathrm{idm}}{x}_i(t-d_i\left(t\right))u_i(t-d_i\left(t\right))+B_{\mathrm{idm}}{u}_i(t-d_i\left(t\right))+{\Sigma }_{j=1,j\ne i}^S{C}_{\mathrm{jim}}{x}_j\left(t\right)]\end{array}---\left(2\right)$其中：μ_inj(ξ_i(t))是ξ_j(t)在中的隶属度函数；假设由h_im(ξ_i(t))的定义可知：分别简记h_im(ξ_i(t))，x_i(t‑d_i(t))，u_i(t‑d_i(t))为h_im，x_id(t)，u_id(t)；根据并行分布补偿算法，考虑局部反馈控制器：$\begin{array}{c}\begin{array}{cccccccc}f& {\xi }_{i1}\left(t\right)& \mathrm{is}& F_{i1}^m& \mathrm{and}...\mathrm{and}& {\xi }_{{\mathrm{iv}}_i}\left(t\right)& \mathrm{is}& F_{{\mathrm{iv}}_i}^m\end{array}\\ \begin{array}{cc}\mathrm{then}& u_i\left(t\right)=\frac{{\rho }_i{K}_{\mathrm{im}}{x}_i\left(t\right)}{\sqrt{1+{x_i}^T{K}_{\mathrm{im}}^T{K}_{\mathrm{im}}{x}_i}}={\rho }_i\sin {\theta }_{\mathrm{im}}={\rho }_i\cos {\theta }_{\mathrm{im}}{K}_{\mathrm{im}}{x}_i\left(t\right)\end{array}\end{array}---\left(3\right)$这里：是待求的控制器增益，ρ_i＞0是待定的标量，由(3)可类似的得到：这里：则全局分散控制律可表示为：在控制律(5)的作用下，整个闭环系统的方程可表示为：${\stackrel{.}{x}}_i\left(t\right)={\Sigma }_{m,n=1}^{r_i}{h}_{\mathrm{im}}{h}_{\mathrm{in}}[{\Lambda }_{i,\mathrm{mn}}{x}_i\left(t\right)+{\Lambda }_{i,\mathrm{dmn}}{x}_{\mathrm{id}}\left(t\right){+\Sigma }_{j=1,j\ne i}^s{C}_{\mathrm{jim}}{x}_j\left(t\right)]---\left(6\right)$这里：Λ_i，mn＝A_im+ρ_isinθ_inN_im+ρ_icosθ_inB_imK_in，定理1：对于给定的正常数ρ_i，α_i，i＝1，2，...，S，如果对于给定的正常数ε_1i，ε_2i，i＝1，2，...，S存在正定对称矩阵P_i＞0，R_i＞0，i＝1，2，...，S和矩阵K_im，i＝1，2，...，S；m＝1，2，...，r_i满足矩阵不等式(7)，则关联大系统(5)是渐近稳定的；Φ_i，mm＜0 i＝1，2，...，S；m＝1，2，...，r_i   (7a)Φ_i，mn+Φ_i，nm＜0 i＝1，2，...，S；1≤m＜n≤r_i   (7b)其中：$\begin{array}{c}{\phi }_{i,1\mathrm{mn}}=A_{\mathrm{im}}^T{P}_i+P_i{A}_{\mathrm{im}}+({ϵ}_{1i}+{ϵ}_{2i}){\rho }_i^2{P}_i{P}_i+{ϵ}_{1i}^{-1}{N}_{\mathrm{im}}^T{N}_{\mathrm{im}}+{ϵ}_{1i}^{-1}{\left(B_{\mathrm{im}}{K}_{\mathrm{in}}\right)}^T\left(B_{\mathrm{im}}{K}_{\mathrm{in}}\right)\\ +{\Sigma }_{j=1,j\ne i}^S{P}_i{C}_{\mathrm{jim}}{C}_{\mathrm{jim}}^T{P}_i+(S-1)I+R_i,\end{array}$${\phi }_{i,2\mathrm{mn}}={ϵ}_{2i}^{-1}{N}_{\mathrm{idm}}^T{N}_{\mathrm{idm}}+{ϵ}_{2i}^{-1}{\left(B_{\mathrm{idm}}{K}_{\mathrm{in}}\right)}^T\left(B_{\mathrm{idm}}{K}_{\mathrm{in}}\right)-(1-{\alpha }_i)R_i.$证明：选取如下Lyapunov函数：$V\left(t\right)={\Sigma }_{i=1}^S{V}_i\left(t\right)={\Sigma }_{i=1}^S[x_i^T\left(t\right)P_i{x}_i\left(t\right)+{\int }_{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)$沿着系统(6)的轨线，对V(t)求导，可得到：$\begin{array}{c}\stackrel{.}{V}\left(t\right)={\Sigma }_{i=1}^S{\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 {\Sigma }_{i=1}^S{\Sigma }_{m,n=1}^{r_i}{h}_{\mathrm{im}}{h}_{\mathrm{in}}[x_i^T\left(t\right)({\Lambda }_{i,\mathrm{mn}}^T{P}_i+P_i{\Lambda }_{i,\mathrm{mn}})x_i\left(t\right)+x_{\mathrm{id}}^T\left(t\right){\Lambda }_{i,\mathrm{dmn}}^T{P}_i{x}_i\left(t\right)+x_i^T\left(t\right)P_i{\Lambda }_{i,\mathrm{dmn}}{x}_{\mathrm{id}}\left(t\right)\\ +{\Sigma }_{j=1j\ne 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{\Sigma }_{j=1,j\ne i}^s{C_{\mathrm{jim}}}^{}{x}_j\left(t\right)+x_i^T\left(t\right)R_i{x}_i\left(t\right)-(1-{\alpha }_i)x_{\mathrm{id}}^T\left(t\right)R_i{x}_{\mathrm{id}}\left(t\right)]\end{array}---\left(9\right)$考虑下式，并由引理1可得到：$\begin{array}{c}{\Lambda }_{i,\mathrm{mn}}^T{P}_i+P_i{\Lambda }_{i,\mathrm{mn}}=A_{\mathrm{im}}^T{P}_i+P_i{A}_{\mathrm{im}}+{\left({\rho }_i\sin {\theta }_{\mathrm{in}}{N}_{\mathrm{im}}\right)}^T{P}_i+P_i\left({\rho }_i\sin {\theta }_{\mathrm{in}}{N}_{\mathrm{im}}\right)\\ +{\left({\rho }_i\cos {\theta }_{\mathrm{in}}{B}_{\mathrm{im}}{K}_{\mathrm{in}}\right)}^T{P}_i+P_i\left({\rho }_i\cos {\theta }_{\mathrm{in}}{B}_{\mathrm{im}}{K}_{\mathrm{in}}\right)\\ \le A_{\mathrm{im}}^T{P}_i+P_i{A}_{\mathrm{im}}+{ϵ}_{1i}{\rho }_i^2{P}_i{P}_i+{ϵ}_{1i}^{-1}{N}_{\mathrm{im}}^T{N}_{\mathrm{im}}+{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){\Lambda }_{i,\mathrm{dmn}}^T{P}_i{x}_i\left(t\right)+x_i^T\left(t\right)P_i{\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){ϵ}_{2i}{\rho }_i^2{P}_i{P}_i{x}_i\left(t\right)\\ +x_{\mathrm{id}}^T\left(t\right){ϵ}_{2i}^{-1}{N}_{\mathrm{idm}}^T{N}_{\mathrm{idm}}{x}_{\mathrm{id}}\left(t\right)+x_{\mathrm{id}}^T\left(t\right){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)$同理可得到：$\begin{array}{c}{\Sigma }_{i=1}^S{\Sigma }_{m=1}^{r_i}{h}_{\mathrm{im}}[{\Sigma }_{j=1,j\ne 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{\Sigma }_{j=1,j\ne i}^s{C}_{\mathrm{jim}}{x}_j\left(t\right)]\\ ={\Sigma }_{i=1}^S{\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 {\Sigma }_{i=1}^S{\Sigma }_{m=1}^{r_i}{h}_{\mathrm{im}}[x_i^T\left(t\right)P_i{\Sigma }_{j=1,j\ne i}^S{C}_{\mathrm{jim}}{C}_{\mathrm{jim}}^T{P}_i{x}_i\left(t\right)+{\Sigma }_{j=1,j\ne i}^S{x}_j^T\left(t\right)x_j\left(t\right)]\\ ={\Sigma }_{i=1}^S{\Sigma }_{m=1}^{r_i}{h}_{\mathrm{im}}[x_i^T\left(t\right)P_i{\Sigma }_{j=1,j\ne 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)$把(9)、(10)、(11)带入(8)，并记可得到：$\begin{array}{c}\stackrel{.}{V}\left(t\right)\le {\Sigma }_{i=1}^S{\Sigma }_{m,n=1}^{r_i}{h}_{\mathrm{im}}{h}_{\mathrm{in}}{\eta }_i^T\left(t\right){\Phi }_{i,\mathrm{mn}}{x}_i\left(t\right){\eta }_i\left(t\right)\\ ={\Sigma }_{i=1}^S[{\Sigma }_{m=1}^{r_i}{h}_{\mathrm{im}}^2{\eta }_i^T\left(t\right){\Phi }_{i,\mathrm{mm}}{\eta }_i\left(t\right)+{\Sigma }_{i=m<n}^{r_i}{h}_{\mathrm{im}}{h}_{\mathrm{in}}{\eta }_i^T\left(t\right)({\Phi }_{i,\mathrm{mn}}+{\Phi }_{i,\mathrm{nm}}){\eta }_i\left(t\right)]\end{array}---\left(12\right)$根据定理1中的(7)可知所以可知关联大系统(6)是渐近稳定的；考虑定理1中的(7)是一个双线性矩阵不等式，不能直接由LMI工具箱求解，把双线性矩阵不等式转换成LMI，给出控制器的设计方法：定理2：对于给定的正常数α_i，ρ_i，i＝1，2，...，s，如果对于给定的正常数ε_1i，ε_2i，i＝1，2，...，s存在着矩阵和矩阵满足矩阵不等式(13)，则关联大系统(5)是渐近稳定，且控制器增益为：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-{\alpha }_i){\bar{R}}_i& *& *& *& *& *\\ Z_i& 0& -\frac{I}{s-1}& *& *& *& *\\ N_{\mathrm{im}}{Z}_i& 0& 0& -{ϵ}_{1i}I& *& *& *\\ B_{\mathrm{im}}{G}_{\mathrm{im}}& 0& 0& 0& -{ϵ}_{1i}I& *& *\\ 0& N_{\mathrm{idm}}{Z}_i& 0& 0& 0& -{ϵ}_{2i}I& *\\ 0& B_{\mathrm{idm}}{G}_{\mathrm{im}}& 0& 0& 0& 0& -{ϵ}_{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-{\alpha }_i){\bar{R}}_i& *& *& *& *& *\\ {2Z}_i& 0& -\frac{2I}{s-1}& *& *& *& *\\ \overline{{\left(\mathrm{NZ}\right)}_{i,\mathrm{mn}}}& 0& 0& -t_{55}& *& *& *\\ \overline{{\left(\mathrm{BG}\right)}_{i,\mathrm{mn}}}& 0& 0& 0& -t_{66}& *& *\\ 0& \overline{{\left(N_{d}Z\right)}_{i,\mathrm{mn}}}& 0& 0& 0& -t_{77}& *\\ 0& \overline{{\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)$这里：$\overline{{\left(\mathrm{NZ}\right)}_{i,\mathrm{mn}}}=\left[\begin{array}{c}{N}_{\mathrm{im}}{Z}_i\\ N_{\mathrm{in}}{Z}_i\end{array}\right],\overline{{\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],\overline{{\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],\overline{{\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],$t₄₄＝t₅₅＝diag{ε_1iI，ε_1iI}，t₆₆＝t₇₇＝diag{ε_2iI，ε_2iI}.证明：选取并记由K_im＝G_imZ^‑1可知，M_im＝K_imZ；对(13a)同时左右乘diag{P_i，P_i，I，I，I，I，I}可得到：$\begin{array}{c}\left[\begin{array}{ccccccc}{P_{i}T}_{i,m}{P}_i& *& *& *& *& *& *\\ A_{\mathrm{idm}}^T{P}_i& -(1-{\alpha }_i)R_i& *& *& *& *& *\\ I& 0& -\frac{I}{s-1}& *& *& *& *\\ N_{\mathrm{im}}& 0& 0& -{ϵ}_{1i}I& *& *& *\\ B_{\mathrm{im}}{K}_{\mathrm{im}}& 0& 0& 0& -{ϵ}_{1i}I& *& *\\ 0& N_{\mathrm{idm}}& 0& 0& 0& -{ϵ}_{2i}I& *\\ 0& B_{\mathrm{idm}}{K}_{\mathrm{im}}& 0& 0& 0& 0& -{ϵ}_{2i}I\end{array}\right]<0,\\ i=\mathrm{1,2},...,S;m=\mathrm{1,2},..,r_i\end{array}---\left(14a\right)$由Schur补定理可知(14a)成立可等价于(7a)成立；同理由(13b)可推导出(7b)成立；这样由定理1可知在所设计的控制器下，模糊双线性关联大系统渐近稳定。