2.845

2023影响因子

(CJCR)

  • 中文核心
  • EI
  • 中国科技核心
  • Scopus
  • CSCD
  • 英国科学文摘

留言板

尊敬的读者、作者、审稿人, 关于本刊的投稿、审稿、编辑和出版的任何问题, 您可以本页添加留言。我们将尽快给您答复。谢谢您的支持!

姓名
邮箱
手机号码
标题
留言内容
验证码

多点测试的多模型机动目标跟踪算法

王伟 余玉揆

张必山, 马忠军, 杨美香. 既含有一般多个随机延迟以及多个测量丢失和随机控制丢失的鲁棒H∞模糊输出反馈控制. 自动化学报, 2017, 43(9): 1656-1664. doi: 10.16383/j.aas.2017.e150082
引用本文: 王伟, 余玉揆. 多点测试的多模型机动目标跟踪算法. 自动化学报, 2015, 41(6): 1201-1212. doi: 10.16383/j.aas.2015.c140471
Zhang Bishan, Ma Zhongjun, Yang Meixiang. Robust H∞ Fuzzy Output-feedback Control With Both General Multiple Probabilistic Delays and Multiple Missing Measurements and Random Missing Control. ACTA AUTOMATICA SINICA, 2017, 43(9): 1656-1664. doi: 10.16383/j.aas.2017.e150082
Citation: WANG Wei, YU Yu-Kui. Multi-try and Multi-model Particle Filter for Maneuvering Target Tracking. ACTA AUTOMATICA SINICA, 2015, 41(6): 1201-1212. doi: 10.16383/j.aas.2015.c140471

多点测试的多模型机动目标跟踪算法

doi: 10.16383/j.aas.2015.c140471
基金项目: 

新世纪优秀人才支持计划(NCET-11-0827),中央高校基本业务费专项资金(HEUCFX41308),中国博士后科学基金(2014M550182),黑龙江省博士后特别资助资金(LBH-TZ0410), 哈尔滨市科技创新人才(2013RFXXJ016)资助

详细信息
    作者简介:

    余玉揆 哈尔滨工程大学自动化学院博士研究生. 2011 年获得合肥工业大学自动化学院学士学位. 主要研究方向为非线性滤波, 光纤非线性, 光纤通信.E-mail: yykmaidou@gmail.com

    通讯作者:

    王伟 哈尔滨工程大学自动化学院教授. 2006 年获得哈尔滨工程大学自动化学院博士学位. 主要研究方向为高精度惯性器件与捷联惯性导航, 先进控制理论及应用. E-mail: wangwei407@hrbeu.edu.cn

Multi-try and Multi-model Particle Filter for Maneuvering Target Tracking

Funds: 

Supported by the New Century Excellent Talents Support Program (NCET-11-0827), Fundamental Research Funds for the Central Universities (HEUCFX41308), China Postdoctoral Science Foundation (2014M550182), Heilongjiang Postdoctoral Special Fund (LBH-TZ0410), and Innovation of Science and Technology Talents in Harbin (2013RFXXJ016)

  • 摘要: 针对机动目标跟踪问题,提出了一种多点测试多模型粒子滤波算法(Independence multi-try method, IMTM).整个算法分为两个阶段,第一阶段为利用多点测试(Multi-try method, MTM)结构从各模型产生的粒子中选取一个最优粒子,实现了模型间的交互;第二阶段为利用IMH (Independence Metropolis-Hastings)滤波算法对第一阶段产生的粒子进行取舍,完成整个状态估计.相对于传统的交互式多模型(Interacting multiple model, IMM)算法,该算法无需事先设定模型转移概率 矩阵且为整体并行结构,结构简单,能够充分地交互各模型之间的粒子,进而自动有效地调整各模型权值比重,降低了人为干扰.仿真表明,该算法能够有效地降低滤波峰值误差,整体跟踪精度较高,算法的实时性较好.
  • 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 $y$.

    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.

    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.

    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.

    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.

    图 4  Output feedback controller $x_c(k)$.
    Fig. 4  Output feedback controller $x_c(k)$.
    图 5  Controlled output $z(k)$.
    Fig. 5  Controlled output $z(k)$.
    图 6  Output feedback controller $u(k)$.
    Fig. 6  Output feedback controller $u(k)$.

    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.

    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.

  • [1] Blom H A P, Bar-shalom Y. The interacting multiple model algorithm for systems with Markovian switching coefficients. IEEE Transactions on Automatic Control, 1988, 33(8): 780 -783
    [2] [2] Li X R, Jilkov V P. Survey of maneuvering target tracking, Part V. Multiple-model methods. IEEE Transactions on Aerospace and Electronic Systems, 2005, 41(4): 1255-1321
    [3] Shi Yong, Han Chong-Zhao. Adaptive UKF method with applications to target tracking. Acta Automatica Sinica, 2011, 37(6): 755-759(石勇, 韩崇昭. 自适应UKF算法在目标跟踪中的应用. 自动化学报, 2011, 37(6): 755-759)
    [4] [4] Doucet A, Johansen A M. A tutorial on particle filtering and smoothing: fifteen years later. Handbook of Nonlinear Filtering, 2009, 12: 656-704
    [5] [5] Foo P H, Ng G W. Combining the interacting multiple model method with particle filters for manoeuvring target tracking. IET Radar, Sonar and Navigation, 2011, 5(3): 234 -255
    [6] [6] Gordon N J, Maskell S, Kirubarajan T. Efficient particle filters for joint tracking and classification. In: Proceedings of the 2002 SPIE 4728: Signal and Data Processing of Small Targets. Orlando, FL, USA: SPIE, 2002. 439-449
    [7] [7] Hong S H, Shi Z G, Chen K S. Easy-hardware-implementation MMPF for maneuvering target tracking: algorithm and architecture. Journal of Signal Processing Systems, 2010, 61(3): 259-269
    [8] [8] McGinnity S, Irwin G W. Multiple model bootstrap filter for maneuvering target tracking. IEEE Transactions on Aerospace and Electronic Systems, 2000, 36(3): 1006-1012
    [9] [9] Yang N, Tian W F, Jin Z H. An interacting multiple model particle filter for manoeuvring target location. Measurement Science and Technology, 2006, 17(6): 1307-1311
    [10] Boers Y, Driessen J N. Interacting multiple model particle filter. IEE Proceedings Radar, Sonar, and Navigation, 2003, 150(5): 344-349
    [11] Sankaranarayanan A C, Chellappa R, Srivastava A. Algorithmic and architectural design methodology for particle filters in hardware. In: Proceedings of the 2005 IEEE International Conference on Computer Design: VLSI in Computers and Processors. San Jose, CA, USA: IEEE, 2005. 275- 280
    [12] Miao L F, Zhang J J, Chakrabarti C, Papandreou-Suppappola A. A new parallel implementation for particle filters and its application to adaptive waveform design. In: Proceedings of the 2010 IEEE Workshop on Signal Processing Systems (SIPS). San Francisco, CA: IEEE, 2010. 978(1): 19-24
    [13] Liu J S. Monte Carlo Strategies in Scientific Computing. New York: Springer Science and Business Media, 2004. 111 -123
    [14] Liu J S, Liang F M, Wong W H. The multiple-try method and local optimization in Metropolis sampling. Journal of the American Statistical Association, 2000, 95(449): 121- 134
    [15] Qin Z S, Liu J S. Multi-point Metropolis method with application to hybrid Monte Carlo. Journal of Computational Physics, 2001, 172(2): 827-840
    [16] Storvik G. On the flexibility of Metropolis-Hastings acceptance probabilities in auxiliary variable proposal generation. Scandinavian Journal of Statistics, 2011, 38(2): 342-358
    [17] Pandolfi S, Bartolucci F, Friel N. A generalization of the multiple-try Metropolis algorithm for Bayesian estimation and model selection. In: Proceedings of the 13th International Conference on Artificial Intelligence and Statistics. 2010. 581-588
    [18] Casarin R, Craiu R, Leisen F. Interacting multiple try algorithms with different proposal distributions. Statistics and Computing, 2013, 23(2): 185-200
    [19] Blom H A P, Bloem E A. Exact Bayesian and particle filtering of stochastic hybrid systems. IEEE Transactions on Aerospace and Electronic Systems, 2007, 43(1): 55-70
    [20] Sankaranarayanan A C, Srivastava A, Chellappa R. Algorithmic and architectural optimizations for computationally efficient particle filtering. IEEE Transactions on Image Processing, 2008, 17(5): 737-748
    [21] Arulampalam M S, Maskell S, Gordon N, Clapp T. A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking. IEEE Transactions on Signal Processing, 2002, 50(2): 174-188
    [22] Linder S P, Schell C. A non-Bayesian segmenting tracker for highly maneuvering targets. IEEE Transactions on Aerospace and Electronic Systems, 2005, 41(4): 1168-1177
    [23] Wang Xiao, Han Chong-Zhao. A multiple model particle filter for maneuvering target tracking based on composite sampling. Acta Automatica Sinica, 2013, 39(7): 1152-1156(王晓, 韩崇昭. 基于混合采样的多模型机动目标跟踪算法. 自动化学报, 2013, 39(7): 1152-1156)
  • 期刊类型引用(4)

    1. 练红海,肖伸平,罗毅平,周笔锋. 基于T-S模糊模型的采样系统鲁棒耗散控制. 自动化学报. 2022(11): 2852-2862 . 本站查看
    2. 顾晓清,倪彤光,张聪,戴臣超,王洪元. 结构辨识和参数优化协同学习的概率TSK模糊系统. 自动化学报. 2021(02): 349-362 . 本站查看
    3. 李军,黄卫剑,万文军,刘哲. 一种新型反馈控制器的研究与应用. 控制理论与应用. 2020(02): 411-422 . 百度学术
    4. 唐晓铭,邓梨,虞继敏,屈洪春. 基于区间二型T-S模糊模型的网络控制系统的输出反馈预测控制. 自动化学报. 2019(03): 604-616 . 本站查看

    其他类型引用(1)

  • 加载中
计量
  • 文章访问数:  2539
  • HTML全文浏览量:  100
  • PDF下载量:  1373
  • 被引次数: 5
出版历程
  • 收稿日期:  2014-07-01
  • 修回日期:  2014-12-31
  • 刊出日期:  2015-06-20

目录

/

返回文章
返回