A Master-slave Cross-coupled Iterative Learning Control for Repetitive Tracking of Nonlinear Contours in Multi-axis Precision Motion Systems
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摘要: 针对多轴运动系统非线性轮廓的重复跟踪，传统时域交叉耦合迭代学习控制器（Cross-coupled iterative learning control，CCILC）的设计，各轴间的耦合算子计算精度要求高，计算效率低.本文提出一种主从交叉耦合迭代学习控制方法.基于主从控制设计方法，主动轴采用时域CCILC，从动轴采用位置域交叉耦合迭代学习控制（Position domain CCILC，PDCCILC）.保证各轴间运动同步性，同时减轻对耦合算子精确性的依赖.因而可以引入轮廓误差矢量法估算耦合算子提高计算效率.采用Lifting的系统时域矩阵展开方法对所提出的算法进行了稳定性分析和性能分析.基于一个两轴毫米级运动平台，三种典型非线性轮廓跟踪（即半圆、抛物线和螺旋线）的数值仿真和实验分析验证了所提出算法的有效性.Abstract: In traditional time domain cross-coupled iterative learning control (CCILC) design, the requirements of high calculation accuracy of coupling gains between axes and low computational efficiency restrict its application to nonlinear contour tracking in repetitive tasks. This paper presents a master-slave cross-coupled iterative learning control. Based on the master-slave control design concept, the master motion axis applies time domain CCILC, while the slave motion axis adopts position domain CCILC (PDCCILC). The proposed PDCCILC control can improve synchronization between axes as well as relieve the dependence on accuracy of coupling gains, therefore, the efficient contour error vector method can be adopted to estimate the coupling gains. both stability and performance analyses are conducted using the lifted system representation method. Simulation and experimental results of the three typical nonlinear contour tracking cases (i.e., semi-circle, parabola and spiral) with a two-axis micro-motion stage have demonstrated superiority and efficacy of the proposed controller.
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1. Introduction
Since recent few decades, some researchers focus their energy on the robust stability and controller design problems about the networked-control systems (NCSs) with some uncertain parameters because some networked-control systems have been succeeded in applications in modern complicated industry processes, e.g., aircraft and space shuttle, nuclear power stations, high-performance automobiles, etc. The fuzzy-logic control based on the Takagi-Sugeno (T-S) is widely used to dealing with complex nonlinear systems because it has simple dynamic structure and highly accurate approximation to any smooth nonlinear function in any compact set. One can consult [1]$-$[8] and the other cited literature therein [9]$-$[31]. Data-packet dropout is an important issue to be addressed in the networked-control systems [6], [32]. Zhang [33] solves the problem of $H_\infty$ estimation for a class of Markov jump linear systems but he neglect possible dropout in practice. Reference [34] reports the problem of $H_\infty$ stability of discrete-time switched linear system with average dwell time and with no dropout. In [6], piecewise Lyapunov function is proposed to analyze robust of the nonlinear NCSs without time-delay issue. Random data-packet dropout and time delay are well considered but the controlled NCSs are linear systems in [32]. Reference [8] discusses the problem of robust $H_\infty$ output feedback control for a class of continuous-time Takagi-Sugeno (T-S) fuzzy affine dynamic systems with parametric uncertainties and input constraints on ignoring some nonlinearities induced by system with data-packet dropout and random time delay. Reference [5] investigates the robust $H_\infty$ stability of a class of half nonlinear NCSs with multiple probabilistic delays and multiple missing measurements regardless of the dropout in the forward path. According to above consideration, we investigate a class of new nonlinear NCSs, in which not only sensors communicate with controllers by network but also controllers do with actuator in the same manner.
The highlights of this paper, which lie primarily on the new research problems and new system models, are summarized as follows:
1) A new model is established, in which the controllers communicate with the actuator by a wireless network and the random missing control from the controller to the actuator occurs and the sensors do with the controllers in the same manner.
2) The investigation on the T-S fuzzy model is used for a class of complex systems that describe the modeling errors, disturbance rejection attenuation, probabilistic delay, missing measurements and missing control within the same framework.
The rest of this paper is organized as follows. The problem under consideration is formulated in Section 2. Development of robust $H_{\infty}$ fuzzy control performance on the exponentially stability the closed-loop fuzzy system are placed in Section 3. Section 4 gives design of robust $H_\infty$ fuzzy controller. An illustrative example is given in Section 5, and we conclude the paper in Section 6.
Notation 1: The notation used in the paper is fairly standard. %The superscript "T" stands for matrix transpose; $\mathbb{R}^n$ denotes the $n$-dimensional real vectors; $\mathbb{R}^{m\times n}$ denotes the $n$-dimensional matrix; and $I$ and 0 represent the identity matrix and zero matrix, respectively. The notation $P>0$ ($P\geq 0$) means that $P$ is real symmetric and positive definite (semi-definite), ${\rm tr}(M)$ refers to the trace of the matrix $M$, and $ \|\cdot\|_2 $ stands for the usual $l_2$ norm. In symmetric block matrices or complex matrix expressions, we use an "$\star$" to represent a term that is induced by symmetry, and ${\rm diag}\{\cdots\}$ stands for a block-diagonal matrix. In addition, ${E}\{x\}$ and ${E}\{x|y\}$ will, respectively, mean expectation of $x$ and expectation of $x $ conditional on $y$.
2. Problem Formulation
In this note, the output feedback control problem for discrete-time fuzzy systems in NCSs is taken in our consideration, where the frame-work is depicted in Fig. 1.
图 1 Framework of output feedback control systems over network environment.Fig. 1 Framework of output feedback control systems over network environment.The sensors are connected to a network, which are shared by other NCSs and susceptible to communication delays and missing measurements or pack dropouts). As Fig. 1 depicts, pack dropouts from the controller to actuator can take place stochastically. The fuzzy systems with multiple stochastic communication delays and uncertain parameters can be read as follows:
Plant Rule $i$: If $\theta_{1}(k) $ is $ M_{i1}$, and $\theta_{2}(k)$ is $M_{i2}$, and, $\ldots$, and $\theta_{p}(k)$ is $M_{ip}$, then
$ \begin{align} x(k+1)=&\ A_i(k)x(k)+A_{di}\sum\limits_{m=1}^{h}\alpha_m(k)x(k-\tau_m(k))\notag\\ & +B_{1i}u(k)+D_{1i}v(k)\notag\\ \tilde{y}(k)=&\ C_ix(k)+D_{1i}v(k)\notag\\ z(k)=&\ C_{zi}(k)+B_{2i}u(k)+D_{3i}v(k)\notag\\ x(k)=&\ \phi(k)\quad\forall\, {k}\in \mathbb{Z}^{-}, ~\, i=1, \ldots, r \end{align} $
(1) where $M_{ij}$ is the fuzzy set, $r$ stands for the number of If-then rules, and $\theta(k)=[\theta_1(k), \theta_2(k), \ldots, \theta_{p}(k)]$ is the premise variable vector, which is independent of the input variable $u(k)$. $x(k)\in \mathbb{R}^n$ is the state vector, $u(k)\in \mathbb{R}^m$, $\tilde{y}$ $\in$ $\mathbb{R}^s$ is the process output, $z(k)\in \mathbb{R}^q$ is the controlled output, $v(k)\in \mathbb{R}^p$ presents a vector of exogenous inputs, which belongs to $l_2[0, \infty)$, $\tau_m(k)$ $(m=1, 2, \ldots, h)$ are the communication delays that vary with the stochastic variables $\alpha_m(k)$, and $\phi(k)$ $(\forall\, {k}\in \mathbb{Z}^{-})$ is the initial state.
The stochastic variables $\alpha_m(k)\in \mathbb{R}$ $(m=1, 2, \ldots, h)$ in (1) are assumed to satisfy mutually uncorrelated Bernoulli-distributed-white sequences described as follows:
$ \begin{align} & {\rm Prob}\{\alpha_m(k)=1\}={E}\{\alpha_m(k)\}=\bar{\alpha}_m\notag\\ & {\rm Prob}\{\alpha_m(k)=0\}=1-\bar{\alpha}_m.\notag \end{align} $
In this note, one can make the random communication-time delays satisfy the following assumption that the time-varying $\tau_m(k)$ $ (m=1, 2, \ldots, h)$ are subject to $ d_t\leq \tau_m(k)$ $\leq$ $d_T$. The matrices $A_i(k)=A_i+\Delta{A_i(k)}$, $C_{zi}(k)= C_{zi}$ $+$ $\Delta{C_{zi}}(k)$, where $ A_i, A_{di}, B_{1i}, B_{2i}, C_i, C_{zi}, D_{1i}, D_{2i}$, and $D_{3i}$ are known constant matrices with compatible dimensions. $\Delta{A_i(k)} $ and $\Delta C_{zi}(k)$ with the time-varying norm-bounded uncertainties satisfy
$ \begin{align} \left[ \begin{array}{c} \Delta A_i(k)\\ \Delta C_{zi}(k)\\ \end{array} \right]=\left[ \begin{array}{c} H_{ai}\\ H_{ci}\\ \end{array} \right]F(k)E \end{align} $
(2) with $H_{ai}$, $H_{ci}$ being constant matrices and $F^T(k)F(k)\leq I$, $\forall\, {k}$.
In this note, the packet dropout (the miss-measurement) read as
$ \begin{align} y_c(k)&= \Xi{C_i}x(k)+D_{2i}(k)\notag\\ &=\sum\limits_{l=1}^{s}\beta_lC_{il}x(k)+D_{2i}v(k)\notag\\ u(k)&=W(k)u_c(k)=W(k)C_{ki}x_c(k) \end{align} $
(3) where $\Xi=\hbox{diag}\{\beta_1, \ldots, \beta_s\}$ with $\beta_l$ $(l=1, 2, \ldots, s)$ being $s$ unrelated random variables, which are also unrelated with $\alpha_m(k)$ and $W(k)$ denoting the random packet missing from the controllers to the actuator. One can assume that $\beta_l $ has the probabilistic-density function $q_l(s)$ $(l=1, 2, \ldots, s)$ on the interval $[0, 1]$ with mathematical expectation $\mu_l$ and variance $\sigma_l^2$. $C_{il}={\rm diag}\{\underbrace{0, \ldots, 0}\limits_{l-1}, 1, \underbrace{0, \ldots, 0}\limits_{s-l}\}C_i$. We denote the stochastic pack dropouts from the controller to the actuator by $W(k)= {\rm diag}\{\omega_1(k), \ldots, \omega_m(k)\}$, where $\omega_l$ $(l=$ $1, 2, \ldots, m)$ are mutually unrelated random variables and obey Bernoulli distribution with mathematical expectation $\bar{\omega}_l$ and variance$\rho_l $and assumed to be unrelated with $\alpha_m(k)$. For a given pair of $(x(k), u(k))$, the final output of the fuzzy system is read as
$ \begin{align} x(k+1)=&\, \sum\limits_{i=1}^{r}h_i(\theta(k))[A_i(k)x(k)+B_{1, i}u(k)\notag\\ &\, +A_{di}\sum\limits_{m=1}^{h}x(k-\tau_m(k))+D_{1i}v(k)]\notag\\ y_c(k)=&\, \sum\limits_{i=1}^{r}h_i(\theta(k))[\Xi{C_i}x(k)+D_{2i}v(k)]\notag\\ z(k)=&\, \sum\limits_{i=1}^{r}h_i(\theta(k))[C_{zi}(k)x(k)+B_{2i}u(k)+D_{3i}v(k)] \end{align} $
(4) where the fuzzy-basis functions are described as
$ \begin{align} &{h_i(\theta(k))}=\frac {\vartheta_i(\theta(k))} {\sum\limits_{i=1}^{r}\vartheta_i(\theta(k))}\notag\\ &\vartheta_i(\theta(k))=\prod\limits_{j=1}^{p}M_{ij}(\theta_j(k))\notag \end{align} $
with $M_{ij}(\theta_j(k))$ being the grade of membership of $\theta_j(k)$ in $M_{ij}$. It is clear that $\vartheta_i(\theta(k))\geq 0$, $i=1, 2, \ldots, r$, $\sum_{i=1}^{r}\vartheta_i(\theta(k))>0$, $\forall\, {k}$, and $h_i(\theta(k))\geq 0$, $i=1, 2, \ldots, r$, $\sum_{i=1}^{r}h_i(\theta(k))=1$, $\forall\, {k}$. In the sequel, we denote $h_i=h_i(\theta(k))$ for brevity.
In the note, the fuzzy dynamic output-feedback controller for the fuzzy system (4) is given as
Controller Rule $i$: If $\theta_1(k)$ is $M_{il}$ and $\theta_2(k)$ is $M_{i2}$ and, $\ldots$, and $\theta_p(k)$ is $M_{ip}$ then
$ \begin{align} \begin{cases} x_c(k+1)=A_{ki}x_c(k)+B_{ki}y_c(k)\\ u(k)= W(k)C_{ki}x_c(k) \end{cases} \end{align} $
(5) with $x_c(k)\in \mathbb{R}^n$ being the controller state along with the controller parameters $A_{ki}$, $B_{ki}$ and $C_{ki}$ to be determined. Naturally, the overall fuzzy output-feedback controller is read as
$ \begin{align} \begin{cases} x_c(k+1)=\sum\limits_{i=1}^{r}h_i[A_{ki}x_c(k)+B_{ki}y(k)]\\ u(k)=\sum\limits_{i=1}^{r}h_iW(k)C_{ki}x_c(k), \ \ i=1, 2, \ldots, r. \end{cases} \end{align} $
(6) Combining (6) with (4), we can obtain the closed-loop system described as
$ \begin{align} \begin{cases} \bar{x}(k+1)=\sum\limits_{i-1}^{r}\sum\limits_{j=1}^{r}h_ih_j[(A_{ij}+B_{ij})\bar{x}(k)+D_{ij}v(k) \\ \qquad \qquad \quad\, +\sum\limits_{m=1}^{h}(\bar{A}_{dmi}+\tilde{A}_{dmi})\bar{x}(k-\tau_m(k)]\\ z(k)=\sum\limits_{i=1}^{r}\sum\limits_{j=1}^{r}h_ih_j[\bar{C}_{ij}(k)+\bar{\bar{C}}_{ij}]\bar{x}(k) +D_{3i}v(k) \end{cases} \end{align} $
(7) where
$ \begin{align*} &\bar{x}(k)=\left[ \begin{array}{c} x(k) \\ x_c(k) \\ \end{array} \right], \quad A_{ij}=\left[ \begin{array}{cc} A_i(k)&B_{1i}\bar{W}C_{kj} \\ B_{ki}\bar{\Xi}C_j&A_{ki} \\ \end{array} \right]\\[1mm] &B_{ij}=\left[ \begin{array}{cc} 0& B_{1i}\tilde{W}(k)C_{kj}\\ B_{ki}\tilde{\Xi}C_j& 0\\ \end{array} \right]\\[1mm] &\bar{A}_{dmi}=\left[ \begin{array}{cc} \bar{\alpha}_mA_{di}&0 \\ 0&0 \\ \end{array} \right], \quad \tilde{A}_{dmi}=\left[ \begin{array}{cc} \tilde{\alpha}_mA_{di}&0 \\ 0&0 \\ \end{array} \right]\\[1mm] &D_{ij}=\left[ \begin{array}{c} D_{1i} \\ B_{ki}D_{2j} \\ \end{array} \right], \quad \bar{C}_{ij}(k)=\bigg[ \begin{array}{cc} C_{zi}(k)&B_{2i}\bar{W}C_{kj} \\ \end{array} \bigg]\\[1mm] &\bar{\bar{C}}_{ij}(k)=\bigg[ \begin{array}{cc} 0&B_{2i}\tilde{W}(k)C_{kj} \\ \end{array} \bigg] \end{align*} $
with $\tilde{\alpha}_m(k)=\alpha_m(k)-\bar{\alpha}_m(k)$ and $\tilde{\omega}_j(k)={\omega}_j(k)-\bar{\omega}_j(k)$. It is evident that $E\{\tilde{\alpha}_m(k)\}=0$ and that $E\{\tilde{\omega}_j(k)\}=0$ and that $E\{\tilde{\alpha}_m^2(k)\}=\bar{\alpha}_m(1-\bar{\alpha}_m)=\sigma_m^2$ and that $E\{\tilde{\omega}_j^2(k)\}$ $=$ $\bar{\omega}_j(1-\bar{\omega}_j)=\rho_j^2$.
Denote
$ \begin{align*} &\bar{x}(k-\tau)\\ &=\left[ \!\!\begin{array}{cccc} \ \ \bar{x}^T(k-\tau_1(k)) &\!\bar{x}^T(k-\tau_2(k))&\! \cdots &\!\bar{x}^T(k-\tau_h(k))\ \ \\ \end{array} \!\!\right]^T\\ &\xi(k)=\left[ \begin{array}{ccc} \bar{x}^T(k)&\bar{x}^T(k-\tau) &v^T(k) \\ \end{array} \right]^T\end{align*} $
then (7) can also be rewritten as
$ \begin{align} \begin{cases} \bar{x}(k+1) =\sum\limits_{i=1}^{r}\sum\limits_{j=1}^{r}h_ih_j\left[A_{ij}\!+B_{ij}, \hat{Z}_{mi}\!+\Delta\hat{Z}_{mi}, D_{ij}\right]\xi(k) \\ z(k)=\sum\limits_{i=1}^{r}\sum\limits_{j=1}^{r}h_ih_j\left[\bar{C}_{ij}+ \bar{\bar{C}}_{ij}, 0, D_{3i}\right]\xi(k) \end{cases} \end{align} $
(8) where $\hat{Z}_{mi}=[\bar{A}_{d1i}, \ldots, \bar{A}_{dhi}]$ and $\Delta\hat{Z}_{mi}=[\tilde{A}_{d1i}, \ldots, \tilde{A}_{dhi}]$. In order to smoothly formulate the problem in the note, we introduce the following definition.
Definition 1: For the system (7) and every initial conditions $\phi$, the trivial solution is said to be exponentially mean square stable if, in the case of $v(k)=0$, there exist constants $\delta>0$ and $0<\kappa<1$ such that $E\{\|\bar{x}(k)\|^2\}$ $\leq$ $\delta\kappa^k \sup_{-d_M\leq i\leq 0}E\{\|{\phi(i)}\|^2\}$, $\forall\, {k}\geq 0$.
We will develop techniques to settle the robust $H_{\infty}$ dynamic output feedback problem for the discrete-time fuzzy system (7) subject to the following conditions:
1) The fuzzy system (7) is exponentially stable in the mean square.
2) Under zero-initial condition, the controlled output $z(k)$ satisfies
$ \begin{align} \sum\limits_{k=0}^{\infty}E\left\{\|{z(k)}\|^2\right\}\leq \gamma^2\sum\limits_{k=0}^{\infty}E\left\{\|{v(k)}\|^2\right\} \end{align} $
(9) for all nonzero $v(k)$, where $\gamma>0$ is a prescribed scalar.
Remark 1: The proposed new model has the function that not only the controllers communicate with the actuator by wireless but also the sensors do with the controllers by the same manner.
3. Development of Robust ${\pmb H}_{\pmb \infty}$ Fuzzy Control Performance
At first, we give the following lemma, which will be adopted in obtaining our main results.
Lemma 1 (Schur complement): Given constant matrices $S_1$, $S_2$, $S_3$, where $S_1=S_1^T$ and $0<S_2=S_2^T$, then $ S_1$ $+$ $S_3^TS_2^{-1}S_3$ $<$ $0$ if and only if
$ \begin{align*} \left[ \begin{array}{cc} S_1&S_3^T \\ S_3 &-S_2 \\ \end{array} \right]<0~~ \hbox{or}~~ \left[ \begin{array}{cc} -S_2&S_3 \\ S_3^T&S_1 \\ \end{array} \right]<0. \end{align*} $
Lemma 2 (S-procedure) [5]: Letting $L=L^T$ and $H$ and $E$ be real matrices of appropriate dimensions with $F$ satisfying $FF^T\leq I$, then $ L+HFE+E^TF^TH^T<0$ if and only if there exists a positive scalar $\varepsilon>0$ such that $L$ $+$ $\varepsilon^{-1}HH^T+\varepsilon E^TE<0$, or equivalently
$ \begin{align*} \left[ \begin{array}{ccc} L&H&\varepsilon{E^T} \\ H^T &-\varepsilon{I}&0 \\ \varepsilon{E}&0 &-\varepsilon{I} \\ \end{array} \right]<0. \end{align*} $
Lemma 3: For any real matrices $X_{ij}$ for $i$, $j=1, 2, \ldots, $ $r$ and $n>0$ with appropriate dimensions, we have [35]
$ \sum\limits_{i=1}^r\sum\limits_{j=1}^r\sum\limits_{l=1}^r\sum\limits_{l=1}^rh_ih_jh_kh_lX_{ij}^T\Lambda{X_{kl}}\leq\sum\limits_{i=1}^r\sum\limits_{j=1}^rh_ih_jX_{ij}^T\Lambda X_{ij}. $
Theorem 1: For given controller parameters and a prescribed $H_{\infty}$ performance $\gamma>0$, the nominal fuzzy system (7) is exponentially stable if there exist matrices $P>0$ and $Q_k$ $>$ $0$, $k=1, 2, \ldots, h$, satisfying
$ \left[ \begin{array}{cc} \Pi_i&\star \\ 0.5\Sigma_{ii}&\bigwedge \\ \end{array} \right]<0 $
(10) $ \left[ \begin{array}{cc} 4\Pi_i&\star \\ \Sigma_{ij}&\bigwedge \\ \end{array} \right]<0, \quad 1\leq i<j\leq r $
(11) where
$ \Pi_i =\ {\rm diag}\bigg\{-P+\sum\limits_{k=1}^h(d_T-d_t+1)Q_k, \hat{\alpha}\breve{A}_{di}^T\breve{P} \breve{A}_{di}\notag\\ \ \ \ \ \ \ -{\rm diag}\{Q_1, Q_2, \ldots, Q_h\}, -\gamma^2I\bigg\} $
(12) $\begin{align*} \hat{\alpha}=&\ {\rm diag}\left\{\bar{\alpha}_1(1-\bar{\alpha}_1), \ldots, \bar{\alpha}_h(1-\bar{\alpha}_h)\right\}\notag\\ \breve{A}_{di}=&\ {\rm diag}\{\underbrace{\hat{A}_{di}, \ldots, \hat{A}_{di}}\limits_h\}\notag\\ \check{C}_{ij}=&\ \left[\sigma_1\hat{C}_{11ij}^TP, \ldots\!, \sigma_s\hat{C}_{1sij}^TP, \rho_1\hat{C}_{k1ij}^TP, \ldots\!, \rho_m\hat{C}_{kmij}^TP\right]^T\notag\\ &\check{P}=\hbox{diag}\{\underbrace{P, \ldots, P}\limits_{s+m}\}\\ &{\small\bigwedge}=\hbox{diag}\{-\check{P}, -P, -I, \hbox{diag}\{\underbrace{-I, \ldots, -I}\limits_m\}\}\\ &\breve{P}=\hbox{diag}\{\underbrace{P, \ldots, P}\limits_h\}\\ &\hat{A}_{di}=\left[ \begin{array}{cc} A_{di}&0\\ 0&0\\ \end{array} \right] \\ &\Sigma_{ij}=\\ &\!\!\!\left[\!\!{\small \begin{array}{ccccc} \check{C}_{ij}\!+\!\check{C}_{ji}\! &\! 0\!&\!0 \\[2mm] PA_{ij}\!+\!PA_{ji} \! &\! P\hat{Z}_{mi}\!+\!P\hat{Z}_{mj} \! &\!PD_{ij}\!+\!PD_{ji}\\[2mm] \bar{C}_{ij}\!+\!\bar{C}_{ji}\! &\!0\! &\!D_{3i}\!+\!D_{3j}\\[2mm] \, [0 ~~ \rho_1B_{2i}C_{kj1}\!+\!\rho_1B_{2j}C_{ki1}] \! &\!0\! &\!0\\[2mm] \vdots\! &\!\vdots\! &\!\vdots\\[2mm] \, [0 ~~ \rho_mB_{2i}C_{kjm}\!+\!\rho_mB_{2j}C_{kim}]\! &\!0\! &\!0\\ \end{array}}\!\!\!\! \right]. \end{align*} $
Proof:
Let
$ \begin{align*} &\Theta_j(k)=\{x(k-\tau_j(k), x(k-\tau_j(k)+1, \ldots, x(k)\}\\ &\chi(k)=\{\Theta_1(k)\bigcup\Theta_2(k)\bigcup\ldots\bigcup\Theta_h(k)\}=\bigcup\limits_{j=1}^{h}\Theta_j(k) \end{align*} $
where $j=1, 2, \ldots, h$. We consider the following Lyapunov functional for the system of (7): $V(\chi(k))=\sum_{i=1}^3V_i(k)$, where
$ \begin{align*} &V_1(k)=\bar{x}^T(k)P\bar{x}\\ &V_2(k)=\sum\limits_{j=1}^{h}\sum\limits_{i=k-\tau_j(k)}^{k-1}\bar{x}^T(i)Q_j\bar{x}(i)\\ &V_3(k)=\sum\limits_{j=1}^h\sum\limits_{m=-d_M+1}^{-d_m}\sum\limits_{i=k+m}^{k-1}\bar{x}^T(i)Q_j\bar{x}(i) \end{align*} $
with $P>0$, $Q_j>0$ $(j=1, 2, \ldots, h)$ being matrices to be determined.
$ \begin{align} {E}[\Delta{V}|x(k)]&={E}[V(\chi(k+1))|\chi(k)]-V(\chi(k))\notag\\ & ={E}[(V(\chi(k+1))-V(\chi(k)))|\chi(k)]\notag\\ & =\sum\limits_{i=1}^{3}{E}[\Delta{V_i}|\chi(k)]. \end{align} $
(13) According to (7), we have
$ \begin{align*} &{E}\{\Delta{V_1}|\chi(k)\}\\ &\qquad={E} \left[(\bar{x}^T(k+1)P\bar{x}(k+1)-\bar{x}^T(k)P\bar{x}(k))|\chi(k)\right]\\ &\qquad\leq\xi^T(k)\sum\limits_{i=1}^{r}\sum\limits_{j=1}^{r}\Omega_{ij}\xi(k) \end{align*} $
where
$ \begin{align} & {{\Omega }_{ij}}=E\left\{ \left[\begin{matrix} A_{ij}^{T}P{{A}_{ij}}+B_{ij}^{T}P{{B}_{ij}}-P & {} \\ \star & {} \\ \star & {} \\ \end{matrix} \right. \right. \\ & \left. \left. \begin{matrix} {} & A_{ij}^{T}P{{{\hat{Z}}}_{mi}} & A_{ij}^{T}P{{D}_{ij}} \\ {} & \hat{Z}_{mi}^{T}P{{{\hat{Z}}}_{mi}}+\Delta \hat{Z}_{mi}^{T}P\Delta {{{\hat{Z}}}_{mi}} & \hat{Z}_{mi}^{T}P{{D}_{ij}} \\ {} & \star & D_{ij}^{T}P{{D}_{ij}} \\ \end{matrix} \right] \right\} \\ \end{align} $
$ {{B}_{ij}}=\left[\begin{matrix} 0 & 0 \\ {{B}_{ki}}\tilde{\Xi }{{C}_{j}} & 0 \\ \end{matrix} \right]+\left[\begin{matrix} 0 & {{B}_{1i}}\tilde{\omega }(k){{C}_{kj}} \\ 0 & 0 \\ \end{matrix} \right] $
$ \begin{align} & E\{B_{ij}^{T}P{{B}_{ij}}\} \\ & \ \ \ \ \ =\sum\limits_{l=1}^{s}{\sigma _{l}^{2}}{{\left[\begin{matrix} 0 & 0 \\ {{B}_{ki}}{{C}_{jl}} & 0 \\ \end{matrix} \right]}^{T}}P\left[\begin{matrix} 0 & 0 \\ {{B}_{ki}}{{C}_{jl}} & 0 \\ \end{matrix} \right] \\ & \ \ \ \ \ +\sum\limits_{l=1}^{m}{\rho _{l}^{2}}{{\left[\begin{matrix} 0 & {{B}_{1i}}{{C}_{kjl}} \\ 0 & 0 \\ \end{matrix} \right]}^{T}}P\left[\begin{matrix} 0 & {{B}_{1i}}{{C}_{kjl}} \\ 0 & 0 \\ \end{matrix} \right] \\ & \ \ \ ={{({{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{P}}}^{-1}}{{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{C}}}_{lij}})}^{T}}\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{P}({{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{P}}}^{-1}}{{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{C}}}_{lij}}) \\ \end{align} $
$ \begin{align} & \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{P}=\rm{diag}\{\underbrace{\mathit{P}, \ldots, \mathit{P}}_{\mathit{s}+\mathit{m}}\} \\ & {{{\hat{C}}}_{1lij}}=\left[\begin{matrix} 0 & 0 \\ {{B}_{ki}}{{C}_{jl}} & 0 \\ \end{matrix} \right] \\ & {{{\hat{C}}}_{klij}}=\left[\begin{matrix} 0 & {{B}_{1i}}{{C}_{kjl}} \\ 0 & 0 \\ \end{matrix} \right] \\ & {{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{C}}}_{ij}}={{\left[{{\sigma }_{1}}\hat{C}_{11ij}^{T}P, \ldots, {{\sigma }_{s}}\hat{C}_{1sij}^{T}P, {{\rho }_{1}}\hat{C}_{k1ij}^{T}P, \ldots, {{\rho }_{m}}\hat{C}_{kmij}^{T}P \right]}^{T}} \\ \end{align} $
$ \begin{align} & E\left\{ \Delta \hat{Z}_{mi}^{T}P\Delta {{{\hat{Z}}}_{mi}} \right\} \\ & \ \ \ \ \ =\sum\limits_{m=1}^{h}{{{{\bar{\alpha }}}_{m}}}(1-{{{\bar{\alpha }}}_{m}}){{\left[ \begin{matrix} {{A}_{di}} & 0 \\ 0 & 0 \\ \end{matrix} \right]}^{T}}P\left[ \begin{matrix} {{A}_{di}} & 0 \\ 0 & 0 \\ \end{matrix} \right] \\ & \ \ \ \ \ \ =\sum\limits_{m=1}^{h}{\hat{A}_{di}^{T}}P{{{\hat{A}}}_{di}}=\hat{\alpha }\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{A}_{di}^{T}\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{P}{{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{A}}}_{di}} \\ \end{align} $
$ \begin{align} & \hat{\alpha }=\rm{diag}\{{{{\bar{\alpha }}}_{1}}(1-{{{\bar{\alpha }}}_{1}}), \ldots, {{{\bar{\alpha }}}_\mathit{h}}(1-{{{\bar{\alpha }}}_\mathit{h}})\} \\ & {{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{A}}}_{di}}=\rm{diag}\{\underbrace{\mathit{{{\hat{A}}}_{di}}, \ldots, \mathit{{{\hat{A}}}_{di}}}_\mathit{h}\} \\ & E\{\Delta {{V}_{2}}|\chi (k)\}\le E\{\sum\limits_{j=1}^{h}{({{{\bar{x}}}^{T}}(}k){{Q}_{j}}\bar{x}(k) \\ & \ \ \ \ \ -{{{\bar{x}}}^{T}}(k-{{\tau }_{j}}(k)){{Q}_{j}}\bar{x}(k-{{\tau }_{j}}(k)) \\ & \ \ \ \ \ +\sum\limits_{i=k-{{d}_{M}}+1}^{k-{{d}_{m}}}{{{{\bar{x}}}^{T}}}(i){{Q}_{j}}\bar{x}(i))|\chi (k)\} \\ & E\{\Delta {{V}_{3}}|\chi (k)\}=E\{\sum\limits_{j=1}^{h}{((}{{d}_{T}}-{{d}_{t}}){{{\bar{x}}}^{T}}(k){{Q}_{j}}\bar{x}(k) \\ & \ \ \ \ \ -\sum\limits_{i=k-{{d}_{m}}+1}^{k-{{d}_{m}}}{{{{\bar{x}}}^{T}}}(i){{Q}_{j}}\bar{x}(i))|\chi (k)\}. \\ \end{align} $
It is clear that
$ {E}\{\Delta{V_2}|\chi(k)\}+{E}\{\Delta{V_3}|\chi(k)\}\leq\xi^T(k)T_{ij}\xi(k) $
with
$ \begin{align*} T_{ij}=&\ \hbox{diag}\Bigg\{\sum\limits_{k=1}^h(d_T-d_t+1)Q_k, \\ &-\hbox{diag}\{Q_1, Q_2, \ldots, Q_h\}, 0\Bigg\}.\end{align*} $
Therefore, we have ${E}\{\Delta{V}|\chi(k)\}\leq\xi^T(k)\Gamma_{ij}\xi(k)$, where $\Gamma_{ij}$ $=$ $\Omega_{ij}+T_{ij}$. Due to
$ \begin{align*} &{E}\left\{z^T(k)z(k)-\gamma^2v^T(k)v(k)\right\}\\ &\qquad\leq\xi(k)\sum\limits_{i=1}^r\sum\limits_{j=1}^rh_ih_j {E}\left\{[\bar{C}_{ij}+\bar{\bar{C}}_{ij}, 0, D_{3i}]^T\right.\\ &\qquad\quad \left.\times[\bar{C}_{ij}+\bar{\bar{C}}_{ij}, 0, D_{3i}] - \hbox{diag}\{0, 0, \gamma^2I\}\right\}\xi(k) \end{align*} $
we can obtain
$ \begin{align*} &{E}\left\{z^T(k)z(k)-\gamma^2v^T(k)v(k)+\Delta{V(k)}\right\}\\ &\qquad \leq\xi^T(k)({\Omega}_{ij}^T\hbox{diag} \{P, I\}{\Omega}_{ij}\\ &\qquad\quad +\mathcal{Z}_{ij}^T\hbox{diag}\{\check{P}, I\}\mathcal{Z}_{ij}+\bar{P})\xi(k) \end{align*} $
where
$ \begin{align*} &{\Omega}_{ij}=\left[ \begin{array}{ccc} A_{ij}&\hat{Z}_{mi}&D_{ij}\\ \bar{C}_{ij}&0&D_{3i}\\ \end{array} \right]\\ & \Game _{kijt}= \bigg[ \begin{array}{ccc} \left[ \begin{array}{cc} 0&\rho_tB_{2i}C_{kjt} \end{array} \right]&0&0 \end{array} \bigg]^T \\ &\mathfrak{D}_{ij}=\bigg[ \begin{array}{ccc} \Game_{kij1}&\ldots&\Game_{kijm} \end{array} \bigg]^T \\ &\mathcal{Z}_{ij}=\left[ \begin{array}{c} [\check{P}^{-1}\check{C}_{ij}, 0, 0]\\ \mathfrak{D}_{ij} \end{array} \right]\\ &\bar{P}=\hbox{diag}\bigg\{-P+\sum\limits_{k=1}^h(d_T-d_t+1)Q_k, \hat{\alpha}\breve{A}_{di}^T\breve{P} \breve{A}_{di}\\ &\qquad -\hbox{diag}\{Q_1, Q_2, \ldots, Q_h\}, -\gamma^2I\bigg\}. \end{align*} $
Define $J(n)={E}\sum\nolimits_{k=0}^n[z^T(k)z(k)-\gamma^2v^T(k)v(k)]$, we have
$ \begin{align*} J(n)=&\ {E}\sum\limits_{k=0}^n\left[z^T(k)z(k)-\gamma^2v^T(k)v(k)+\Delta{V(\chi(k))}\right] \\ &-{E}V(\chi(n+1))\\ \leq&\ {E}\sum\limits_{k=0}^n\left[z^T(k)z(k)-\gamma^2v^T(k)v(k)+\Delta{V(\chi(k))}\right]\\ \leq&\ \sum\limits_{k=0}^n\sum\limits_{i=1}^r\sum\limits_{j=1}^rh_ih_j\xi^T(k)({\Omega}_{ij}^T \hbox{diag} \{P, I\}{\Omega}_{ij}\\ &\ +\mathcal{Z}_{ij}^T\hbox{diag}\{\check{P}, I\}\mathcal{Z}_{ij}+\bar{P})\xi(k)\\ =&\ \sum\limits_{k=0}^n\sum\limits_{i=1}^rh_i^2\xi^T(k)({\Omega}_{ii}^T \hbox{diag} \{P, I\}{\Omega}_{ii}\\ &\ +\mathcal{Z}_{ii}^T\hbox{diag}\{\check{P}, I\}\mathcal{Z}_{ii}+\bar{P})\xi(k)\\ &\ +\frac{1}{2}\sum\limits_{k=0}^n\sum\limits_{j=1, i<j}^rh_ih_j\xi^T(k)\\ &\ \times\left[({\Omega}_{ij} +{\Omega}_{ji})^T\hbox{diag}\{P, I\}({\Omega}_{ij}+{\Omega}_{ji})\right.\\ &\ +\left. (\mathcal{Z}_{ij}+\mathcal{Z}_{ji})^T\hbox{diag}\{\check{P}, I\} (\mathcal{Z}_{ij}+\mathcal{Z}_{ji})+4\bar{P}\right]\xi(k). \end{align*} $
According to Schur complement, we can conclude from (10) and (11) that $J(n)<0$. Letting $n\rightarrow\infty$, we have
$ \begin{align*} \sum\limits_n^\infty{E}\left\{\|z(k)\|^2\right\}\leq\gamma^2\sum\limits_n^\infty{E}\left\{\|v(k)\|^2\right\}. \end{align*} $
According to Schur complement again, we know that ${E}\{\Delta{V}|x(k)\}$ $<$ $0$ if and only if (10) and (11) hold true. Furthermore, one can easily verify the fact that the discrete-time nominal (7) with $v(k)=0$ is exponentially stable.
4. Design of Robust ${\pmb H}_{\pmb\infty}$ Fuzzy Controller
In this section, we are devoted to how to determine the controller parameters in (6) such that the closed-loop system (7) is exponentially stable with $H_\infty$ performace.
By Theorem 1, one can easily draw the conclusion as follow:
Theorem 2: For a prescribed constant $\gamma>0$, the nominal fuzzy system (7) is exponentially stable if there exist positive definite matrices $P>0$, $L>0$, $Q_k>0$ $(k=1, 2, $ $\ldots, $ $h)$, and $K_i$ and $\bar{C}_{ki}$ such that
$ \Gamma_1=\left[ \begin{array}{cc} \Pi_i&\star \\ 0.5\bar{\Sigma}_{ii}& \bar{\Lambda} \\ \end{array} \right]<0, \ \ i=1, 2, \ldots, r $
(14) $ \Gamma_2=\left[ \begin{array}{cc} 4\Pi_i&\star \\ \bar{\Sigma}_{ij}&\bar{\Lambda} \\ \end{array} \right]<0, \ \ 1\leq i<j\leq r $
(15) $ PL=I $
(16) hold, then the nominal system (7) is exponentially stable with disturbance attenuation $\gamma$, where $\overline{\bigwedge}=\hbox{diag}\{-\bar{L}, -L, $ $-I, $ $\hbox{diag}\{\underbrace{-I, \ldots, -I}\limits_m\}\}$
$ \bar{\Sigma}_{ij}=\left[ \begin{array}{ccc} \Phi_{11ij}+\Phi_{11ji}&0&0 \\ \Phi_{21ij}+\Phi_{21ji}&\Phi_{22ij}+\Phi_{22ji}& \Phi_{23ij}+\Phi_{23ji} \\ \Phi_{31ij}+\Phi_{31ji}&0&\Phi_{33ij}+\Phi_{33ji} \\ \Phi_{41ij}+\Phi_{41ji}&0&0 \\ \end{array} \right] $
(17) $\begin{align} &I_l=\hbox{diag}\{\underbrace{0, \ldots, 0}\limits_{l-1}, 1, \underbrace{0, \ldots, 0}\limits_{m-l}\}, \quad K_i=\bigg[ \begin{array}{cc} A_{ki}&B_{ki}\\ \end{array}\bigg] \notag\\[1mm] &\bar{C}_{ki}=\bigg[ \begin{array}{cc} 0&C_{ki}\\ \end{array} \bigg], \quad \bar{E}=\left[ \begin{array}{c} 0 \\ I \\ \end{array} \right], \quad \bar{\bar{E}}=\left[ \begin{array}{l} I \\ 0 \\ \end{array} \right] \notag\\[1mm] &\bar{A}_i=\left[ \begin{array}{cc} A_i&0 \\ 0&0 \\ \end{array} \right], \quad \bar{B}_{1i}=\left[ \begin{array}{c} B_{1i} \\ 0 \\ \end{array} \right], \quad R_{il}=\left[ \begin{array}{cc} 0&0 \\ C_{il}&0 \\ \end{array} \right] \notag\\[1mm] &\bar{D}_{1i}=\left[ \begin{array}{c} D_{1i} \\ 0 \\ \end{array} \right], \quad \bar{D}_{2i}=\left[ \begin{array}{c} 0 \\ D_{2i} \\ \end{array} \right]\notag\\[1mm] & \Phi_{11ij}=\left[ \begin{array}{c} \sigma_1\bar{E}K_iR_{j1} \\ \vdots \\ \sigma_s\bar{E}K_iR_{js} \\ \rho_1\bar{E}\beta_{1i}I_1\bar{C}_{kj} \\ \vdots \\ \rho_m\bar{E}\beta_{1i}I_m\bar{C}_{kj} \\ \end{array} \right], \ \ \Phi_{41ij}=\left[ \begin{array}{c} \rho_1B_{2i}I_1\bar{C}_{kj} \\ \vdots \\ \rho_mB_{2i}I_m\bar{C}_{kj} \\ \end{array} \right]\notag\\[1mm] & \Phi_{21ij}=\bar{A}_i+\bar{E}K_i\bar{R}_j+\bar{B}_{1i}\hbox{diag}\{w_1, \ldots, w_m\}\bar{C} _{kj} \notag\\[1mm] &\Phi_{31ij}=\bar{C}_{zi}+B_{2i}\hbox{diag}\{w_1, \ldots, w_m\}\bar{C}_{kj}\notag \\[1mm] & \bar{C}_{zi}=\left[ \begin{array}{cc} C_{zi}&0 \\ \end{array} \right], \quad \bar{L}=\hbox{diag}\{\underbrace{L, \ldots, L} \limits_{s+m}\}\notag \\[1mm] & \Phi_{22ij}=\hat{Z}_{mi}, \quad \Phi_{23ij}=D_{ij}, \quad \Phi_{33ij}=D_{3i}.\notag \end{align} $
Proof: We rewrite the parameters in Theorem 1 in the following form:
$ \begin{align*} & A_{ij}=\bar{A}_i+\bar{E}K_i\bar{R}_j+\bar{B}_{1i}\hbox{diag}\{w_1, \ldots, w_m\}\bar{C}_{kj} \\ &\hat{C}_{lij}=\bar{E}K_i{R}_{jl} \\ & \bar{C}_{ij}=\bar{C}_{zi}+B_{2i}\hbox{diag}\{w_1, \ldots, w_m\}\bar{C}_{kj} \\ & D_{ij}=\bar{D}_{1i}+\bar{D}_{1i}K_i\bar{D}_{2j}. \end{align*} $
Pre-and post-multiplying the (10) and (11) by $ \hbox{diag}\{I, $ $I, $ $I, $ $\check{P}^{-1}, $ $P^{-1}, $ $\underbrace{I, \ldots, I}\limits_m\}$ and Letting $L=P^{-1}$, we have (14)$-$(16) and complete the proof easily. Now we will point out that the robust $H_\infty$ controller parameters can be determined in light of Theorem 2.
Theorem 3: For given scalar $\gamma>0$, if there exist positive define matrices $P>0$, $L>0$, $Q_k>0$ $(k=1, 2, \ldots, h)$, and matrices $K_i$, $\bar{C}_{ki}$ of proper dimensions and a constant $\varepsilon>0$ such that
$ \left[ \begin{array}{cc} \Gamma_1&\star \\ \Xi_{ii}&\hbox{diag}\{-\varepsilon{I}, -\varepsilon{I}\} \\ \end{array} \right]<0, \notag\\ \qquad\qquad\qquad\qquad\qquad i=1, 2, \ldots, r $
(18) $ \left[ \begin{array}{cc} \Gamma_2& \star \\ \Xi_{ij}&\hbox{diag}\{-\varepsilon{I}, -\varepsilon{I}\} \\ \end{array} \right]<0, \notag\\ \qquad\qquad\qquad\qquad\qquad 1\leq i<j\leq r $
(19) $ PL=I $
(20) hold, where
$ \begin{align*}&\Xi_{ii}=\left[ \begin{array}{ccccccc} 0&0&0&0&[H_{ai}^T ~~ 0] &H_{ci}^T&0 \\ \varepsilon[ E ~~ 0] &0&0&0&0&0&0 \\ \end{array} \right]\\ &\Xi_{ij}=\left[ \begin{array}{ccccccc} 0&0&0&0&[H_{ai}^T+H_{aj}^T ~~ 0] &H_{ci}^T+H_{cj}^T&0 \\ \varepsilon[E ~~ 0] &0&0&0&0&0&0 \\ \end{array} \right] \end{align*} $
then the uncertain fuzzy system (7) is exponentially stable and the controller parameters $K_i$ and $\bar{C}_{ki} $ can be obtained naturally.
Proof: Replace $\bar{A}_i$, $\bar{A}_j$, $\bar{C}_{zi}, $ and $ \bar{C}_{zj}$ in Theorem 2 by $\bar{A}_i+\triangle\bar{A}_i(k)$, $\bar{A}_j\triangle\bar{A}_j(k)$, $\bar{C}_{zi}+\triangle\bar{C}_{zi}(k), $ and $ \bar{C}_{zj}\, +\, \triangle\bar{C}_{zj}(k)$, respectively, where
$ \begin{align} & \triangle\bar{A}_i(k)=\left[ \begin{array}{cc} \triangle{A}_i(k)&0 \\ 0&0 \\ \end{array} \right], \quad \triangle\bar{C}_{zi}(k)=[ \triangle{C}_{zi}(k) ~~ 0].\!\notag \end{align} $
According to Lemma 1, (18) and (19) can be rewritten as follows:
$ \begin{align} &\Gamma_1+{H}_1F(k){E}+{E}^TF(k)^T{H}_1^T<0\notag\\ &\Gamma_2+{H}_2F(k){E}+{E}^TF(k)^T{H}_2^T<0\notag \end{align} $
where
$ \begin{align*} &{E}=[E ~~ 0]\\ &{H}_1=\left[ \begin{array}{ccccccc} 0& 0&0&0&[H_{ai}^T ~~ 0] &H_{ci}^T&0 \\ \end{array} \right]\\ & {H}_2=\left[ \begin{array}{ccccccc} 0& 0&0&0 &[H_{ai}^T+H_{aj}^T ~~ 0] &H_{ci}^T+H_{cj}^T&0 \\ \end{array} \right]. \end{align*} $
According to Lemma 1 along with Schur complement, we can easily obtain (18) and (19).
In order to solve (18), (19) and (20), the cone-complementarity linearization (CCL) algorithm proposed in [36] and [37] is used in this note.
The nonlinear minimization problem: $\min\hbox{tr}(PL) $ subject to (18) and (19) and
$ \left[ \begin{matrix} P & I \\ I & L \\ \end{matrix} \right]\ge 0. $
(21) The following algorithm [5] is borrowed to solve the above problem.
Algorithm 1:
Step 1: Find a feasible set $(P_0, L_0, Q_{k(0)}, K_{i(0)}, \bar{C}_{ki(0)})$ satisfying (18), (19) and (21). Set $q=0$.
Step 2: Solving the linear matrix inequality (LMI) problem, $\min\hbox{tr}(PL_{(0)}+P_{(0)}L) $ subject to (18), (19) and (21).
Step 3: Substitute the obtained matrix variables $(P$, $L$, $Q_{k}, K_{i(0)}, \bar{C}_{ki})$ into (14) and (15). If conditions(14) and (15) are satisfied with $|\hbox{tr}(PL)-n|<\delta$ for some sufficiently small scalar $\delta >0$, then output the feasible solutions. Exit.
Step 4: If $q>N$, where $N$ is the maximum number of iterations allowed, then output the feasible solutions $(P$, $L$, $Q_{k}, K_{i}$, $\bar{C}_{ki})$, and exit. Else, set $q=q+1$, and goto Step 2.
5. An Illustrative Example
we give an illustrative examples to explain the proposed model is effective and feasible in this section.
Example 1: Consider a T-S fuzzy model (1). The rules are given as follows:
Plant Rule 1: If $x_1(k)$ is $h_1(x_1(k))$ then
$ \begin{align} \begin{cases} x(k+1) = A_1(k)x(k)+A_{d1}\sum\limits_{m=1}^h\alpha_m(k)x(k-\tau_m(k))\\ \qquad\qquad\quad +~B_{11}u(k)+D_{11}v(k) \\[2mm] y(k) = \Xi C_1x(k) +D_{21}v(k) \\[2mm] z(k) = C_{z1}(k)x(k)+B_{21}u(k)+D_{31}v(k) \end{cases} \end{align} $
(21) Plant Rule 2: If $x_1(k)$ is $h_2(x_1(k))$ then
$ \begin{align} \begin{cases} x(k+1) = A_2(k)x(k)+A_{d2}\sum\limits_{m=1}^h\alpha_m(k)x(k-\tau_m(k))\\ \qquad\qquad\quad +~B_{12}u(k)+D_{12}v(k) \\[2mm] y(k) =\Xi C_2x(k) +D_{22}v(k) \\[2mm] z(k) =C_{z2}(k)x(k)+B_{22}u(k)+D_{32}v(k) \end{cases} \end{align} $
(22) The given model parameters are written as follows:
$ \begin{align} & {{A}_{1}}=\left[ \begin{matrix} 1 & 0.2 & 0 \\ 0.1 & 0.1 & 0.1 \\ 0.1 & 0.2 & 0.2 \\ \end{matrix} \right],\quad {{D}_{11}}=\left[ \begin{matrix} 0.1 \\ 0 \\ 0 \\ \end{matrix} \right] \\ & {{A}_{d1}}=\left[ \begin{matrix} 0.03 & 0 & -0.01 \\ 0.02 & 0.03 & 0 \\ 0.04 & 0.05 & -0.1 \\ \end{matrix} \right], \quad {{B}_{11}}=\left[ \begin{matrix} 1 & 1 \\ 0.4 & 1 \\ 0 & 1 \\ \end{matrix} \right] \\ & {{D}_{31}}=\left[ \begin{matrix} -0.1 \\ 0 \\ 0.1 \\ \end{matrix} \right], \quad \ {{C}_{1}}=\left[ \begin{matrix} 1 & 0.8 & 0.7 \\ -0.6 & 0.9 & 0.6 \\ \end{matrix} \right] \\ & {{C}_{2}}=\left[ \begin{matrix} 0.1 & 0.8 & 0.7 \\ -0.6 & 0.9 & 0.6 \\ \end{matrix} \right],\quad {{D}_{21}}=\left[ \begin{matrix} 0.15 \\ 0 \\ \end{matrix} \right] \\ & {{D}_{22}}=\left[ \begin{matrix} 0.1 \\ 0 \\ \end{matrix} \right], \quad \ {{C}_{z1}}=\left[ \begin{matrix} 0.2 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0.1 \\ \end{matrix} \right] \\ & {{B}_{21}}=\left[ \begin{matrix} 1 & 1 \\ 0 & 1 \\ 0 & 1 \\ \end{matrix} \right], \quad {{H}_{a1}}=\left[ \begin{matrix} 0.1 \\ 0.1 \\ 0.1 \\ \end{matrix} \right],\quad {{H}_{c1}}=\left[ \begin{matrix} 0.1 \\ 0 \\ 0.1 \\ \end{matrix} \right] \\ & {{H}_{a2}}=\left[ \begin{matrix} 0.1 \\ 0 \\ 0.1 \\ \end{matrix} \right], \quad \ {{H}_{c2}}=\left[ \begin{matrix} 0.1 \\ 0 \\ 0.5 \\ \end{matrix} \right],\quad {{D}_{32}}=\left[ \begin{matrix} 0.1 \\ 0 \\ 0.1 \\ \end{matrix} \right] \\ & E={{\left[ \begin{matrix} 0.1 \\ 0.1 \\ 0.1 \\ \end{matrix} \right]}^{T}},{{A}_{2}}=\left[ \begin{matrix} 1 & -0.38 & 0 \\ -0.2 & 0 & 0.21 \\ 0.1 & 0 & -0.55 \\ \end{matrix} \right] \\ & {{B}_{12}}=\left[ \begin{matrix} 1 & 0 \\ 1 & 1 \\ 0 & 1 \\ \end{matrix} \right],\quad {{A}_{d2}}=\left[ \begin{matrix} 0 & 0.01 & -0.01 \\ 0.02 & 0.03 & 0 \\ 0.04 & 0.05 & -0.1 \\ \end{matrix} \right] \\ & {{D}_{12}}=\left[ \begin{matrix} 0.1 \\ 0 \\ 0.1 \\ \end{matrix} \right],\quad {{C}_{z2}}=\left[ \begin{matrix} 0.1 & 0 & 0 \\ 0.2 & 0 & 0.2 \\ 0 & 0.1 & 0.2 \\ \end{matrix} \right] \\ & {{B}_{22}}=\left[ \begin{matrix} 1 & 0 \\ 0 & 1 \\ 1 & 1 \\ \end{matrix} \right]. \\ \end{align} $
Assume that the time-varying communication delays satisfy $2 \leq\tau_m\leq 6$ $(m=1, 2)$ and
$ \begin{align*} & \bar{\alpha}_1={E}\{\alpha_1(k)\}=0.8, \quad\bar{\alpha}_2={E}\{\alpha_2(k)\}=0.6 \\[1mm] & \bar{\omega}_1={E}\{\omega_1(k)\}=0.4, \quad \bar{\omega}_2={E}\{\omega_2(k)\}=0.6. \end{align*} $
Assume also that the probabilistic density functions of $\beta_1$ and $\beta_2$ in $[0 \quad 1]$ are read as
$ \begin{align} q_1(s_1)=\begin{cases} 0,&s_1=0 \\ 0.1,&s_2=0.5 \\ 0.9,&s_3=1 \end{cases}, \quad &q_2(s_2)=\begin{cases} 0,& s_2=0\\ 0.2,&s_2=0.5 \\ 0.8,&s_3=1 \end{cases}. \end{align} $
(23) The membership functions are described as
$ \begin{align} &h_1=\begin{cases} 1,&x_0(1)=0 \\ \left|\dfrac{\sin(x_0(1))}{x_0(1)}\right|,&\hbox{else} \end{cases} \nonumber\\& h_2=1-h_1. \end{align} $
(24) Now, we are to design a dynamic-output feedback paralleled controller in the form of (6) such that (7) is exponentially stable with a given $H_\infty$ norm bound $\gamma$. In the example, we assume $\gamma=0.9$ and obtain the desired $H_\infty$ controller parameters as follows
$ \begin{align} & {{A}_{k1}}=\left[ \begin{matrix} -0.0127 & -0.0083 & -0.0317 \\ 0.0229 & 0.0149 & 0.0221 \\ -0.0588 & -0.0429 & -0.0654 \\ \end{matrix} \right] \\ & {{A}_{k2}}=\left[ \begin{matrix} -0.1365 & -0.1296 & -0.0570 \\ -0.0107 & -0.0095 & 0.0239 \\ -0.0125 & -0.0129 & -0.0260 \\ \end{matrix} \right] \\ & {{B}_{k1}}=\left[ \begin{matrix} -0.3236 & 0.1389 \\ 0.0291 & -0.0043 \\ -0.3077 & 0.1867 \\ \end{matrix} \right] \\ & {{B}_{k2}}=\left[ \begin{matrix} 0.1664 & 0.0834 \\ 0.1374 & -0.0712 \\ -0.4340 & 0.5688 \\ \end{matrix} \right] \\ & {{C}_{k1}}=\left[ \begin{matrix} 0.1355 & 0.0856 & 0.1789 \\ 0.0311 & 0.0209 & 0.0372 \\ \end{matrix} \right] \\ & {{C}_{k2}}=\left[ \begin{matrix} 0.0110 & 0.0464 & 0.0731 \\ 0.0832 & 0.0622 & 0.0502 \\ \end{matrix} \right]. \\ \end{align} $
We take the initial conditions $ x_0=[1 \quad 0 \quad-1]^T$, $x_{c0}$ $=$ $[0 \quad 0 \quad 0]^T $ for the simulation purpose and let external disturbance $v(k)=0$. Fig. 2 depicts the state responses for the uncontrolled fuzzy systems, which are unstable. We can see the fact that the closed-loop fuzzy systems are exponentially stable from the Fig. 3.
图 2 State evolution $x(k)$ of uncontrolled systems.Fig. 2 State evolution $x(k)$ of uncontrolled systems.
图 3 State evolution $x(k)$ of controlled systems.Fig. 3 State evolution $x(k)$ of controlled systems.In order to illustrate the disturbance-attenuation performance, we take the external disturbance
$ \begin{align*} v(k)= \begin{cases} 0.3,&20\leq k\leq 30 \\ -0.2,&50\leq k\leq 60 \\ 0,&\hbox{else}. \end{cases} \end{align*} $
Fig. 4 presents the controller-state evolution $x_c(k)$, Fig. 5 plots the state evolution of the controlled output $z(k)$, and Fig. 6 shows the output feedback controller. From Figs. 3$-$6, one can see that the convergence rate is rapid and effective. By the above simulation results, we can draw the conclusion that our theoretical analysis to the robust $H_\infty$ fuzzy-control problem is right completely.
Remark 2: The above simulation is performed on the basis of the software MATLAB 7.0 and the cone-complementarity linearization algorithm may takes several minutes because of choosing initial feasible set.
6. Conclusion
In this paper, we establish general networked systems model with multiple time-varying random communication delays and multiple missing measurements as weil as the random missing control and discuss its robust $H_\infty$ fuzzy-output feedback-control problem. The proposed system model includes parameter uncertainties, multiple stochastic time-varying delays, multiple missing measurements, and stochastic control input missing. The control strategy adopts the parallel distributed compensation. We obtain the sufficient conditions on the robustly exponential stability of the closed-loop T-S fuzzy-control system by using the CCL algorithm and the explicit expression of the desired controller parameters. An illustrative simulation example further shows that the fuzzy-control method to the proposed new control model is feasible and the new control model can be used for future applications. Whether to construct piecewise Lyapunov functions [8] to solve the proposed control model or not is an interesting topic and in active thought.
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图 3 融合反馈PID和前馈PDCCILC的控制结构
Fig. 3 Combined feedback PID and feedforward PDCCILC control structure
图 7 半圆轮廓跟踪均方根轮廓误差的仿真结果(RMS)
Fig. 7 RMS contour error versus iteration for the semi-circle contour in the simulations
图 8 半圆轮廓跟踪的稳态仿真结果
Fig. 8 Steady tracking results of the semi-circle contour in the simulations
图 9 抛物线轮廓跟踪的仿真均方根轮廓误差(RMS)
Fig. 9 RMS contour error versus iteration for the semi-circle contour in the simulations
图 10 抛物线轮廓跟踪的稳态仿真结果
Fig. 10 Steady tracking results of the parabolic contour in the simulations
图 11 螺旋线轮廓跟踪的仿真均方根轮廓误差(RMS)
Fig. 11 RMS contour error versus iteration for the spiral contour in the simulations
图 12 螺旋线轮廓跟踪的稳态仿真结果
Fig. 12 Steady tracking results of the spiral contour in the simulations
图 13 半圆轮廓跟踪的实验均方根轮廓误差(RMS)
Fig. 13 RMS contour error versus iteration for the semi-circle contour in the experiments
图 14 半圆轮廓跟踪的实验结果
Fig. 14 Tracking results of the semi-circle contour in the experiments
图 15 抛物线轮廓跟踪的实验均方根轮廓误差(RMS)
Fig. 15 RMS contour error versus iteration for the parabolic contour in the experiments
图 16 抛物线轮廓跟踪的实验结果
Fig. 16 Tracking results of the parabolic contour in the experiments
图 17 螺旋线轮廓跟踪的实验均方根轮廓误差(RMS)
Fig. 17 RMS contour error versus iteration for the spiral contour in the experiments
表 1 控制器参数
Table 1 Controller parameters
控制器 增益 ${K_p}$ ${K_i}$ ${K_d}$ PID 3 1 0 ILC 0.3 0.1 0.1 CCC 1 0.5 0 
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表 2 单调收敛的计算结果
Table 2 Computational results of monotonic convergence
控制器 $\bar \sigma \left(M \right)$的计算结果 半圆轮廓 抛物线轮廓 螺旋线轮廓 PDCCILC & PID 3 1 0 TDCCILC & PID 0.3 0.1 0.1 PDILC & PID 1 0.5 0 TDILC & PID 1 0.5 0
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表 3 四种控制器下的三种非线性轮廓跟踪结果实验统计数据(${\mu}$m)
Table 3 Experimental statistics of tracking performance (${\mu}$m)
稳态误差 半圆 抛物线 螺旋线 TDILC & PID RMS 3.405 3.637 7.566 MAX 8.501 8.334 17.429 TDCCILC & PID RMS 1.869 1.924 4.897 MAX 5.170 6.001 15.420 PDILC & PID RMS 1.344 1.197 2.992 MAX 4.348 4.696 11.106 PDCCILC & PID RMS 0.821 0.631 1.551 MAX 2.108 2.907 7.240
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