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摘要: 传统基于肌电（Electromyography，EMG）的运动识别方法多是利用训练后的固定参数模型，分类已预先定义的有限个目标动作，但对肌肉疲劳导致的肌电变化，以及未定义的外部动作等干扰因素无能为力.针对这一问题，提出一种自更新混合分类模型（Self-update hybrid classification model，SUHC），该模型融合了用于排除外部动作干扰的一类支持向量机（Support vector machine，SVM），以及用于分类目标动作数据的多类线性判别分析（Linear discriminant analysis，LDA），并引入自更新机制以对抗肌电时变性干扰.通过手部动作识别实验验证提出方法的效果，在肌电大幅变化干扰下，SUHC的目标动作识别精度达到89%，对比传统的支持向量机、多层感知器（Multiple layer perceptron，MLP）和核线性判别分析（Kernel LDA，KLDA），提高了约18%，并且SUHC具备排除外部动作干扰能力，排除精度高达93%.Abstract: The traditional electromyography (EMG)-based motion recognition methods always classify a limited number of pre-defined target motions by using a trained parameter-fixed model. However, those methods have no ability to handle the interference factors, including changes of EMG signals caused by muscle fatigue and outlier motions undefined beforehand. For this problem, a self-update hybrid classification model (SUHC) is proposed. In the SUHC, the one-class support vector machine (SVM) that can reject outlier motions is combined to a multi-class linear discriminant analysis (LDA) that is used to recognize target motions; furthermore, a self-update mechanism is adopted to reduce the influence caused by EMG changes. The performance of the proposed method is verified by the experiment of EMG-based hand motion recognition. Under the interference that EMG signals vary greatly, the recognition accuracy of SUHC on target motions is about 89%, which is 18% higher than those of the normal SVM, multiple layer perceptron (MLP) and kernel LDA (KLDA); moreover, the SUHC has the ability to reject outlier motions with a 93% of rejection accuracy.
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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 2 肌肉收缩及静息状态下, 采集的连续sEMG
Fig. 2 The continuous sEMG signals sampled at the states of muscle contraction and resting
图 4 识别的5种手部动作以及手部休息姿势snz
Fig. 4 The five kinds of hand motions and the snooze state (snz) that were recognized in the paper
图 5 选取的肌肉及布置的采集电极，其中ch-$i$, $i = 1, 2, 3, 4$, 表示第$i$通道电极
Fig. 5 The selected muscles and the placement of electrodes, where ch-$i$ ($i = 1, 2, 3, 4$) represents the $i$th channel electrode
图 8 由10组测试样本数据计算的MAV占其标准值比
Fig. 8 The ratio of MAV with respect to the standard value calculated by using the ten sessions of test data
图 9 将Subject-1的第1组训练数据输入SUHC (仅给出该组数据的前2个循环), 实现模型在线更新
Fig. 9 Online updating the SUHC by inputting the first session of training data of Subject-1 into the model (only plot the first two cycles of the data)
图 10 无外部动作干扰情况下, 使用Subject-1的10组测试数据计算的分类精度
Fig. 10 In the case of no outlier-motion interference, the classification accuracies calculated by using the ten sessions of test data of Subject-1
图 11 使用Subject-1的1组测试数据获得的动作识别结果
Fig. 11 The motion recognition results obtained by using one session of test data of Subject-1
图 12 存在外部动作(rng)干扰情况下, 使用Subject-1的10组测试数据获得的识别结果
Fig. 12 In the case of outlier-motion (rng) interference, the recognition results obtained by using ten sessions of test data of Subject-1
图 13 使用Subject-1的测试数据, 计算的分类精度与排除精度的均值和标准差
Fig. 13 The means and standard deviations of classification and rejection accuracies computed by using the test data of Subject-1
表 1 使用所有测试者的10组测试样本计算的平均分类精度与标准差(%)
Table 1 The mean classification accuracies and standard deviations calculated by using the ten sessions of test data of all subjects (%)
SUHC SVM MLP KLDA 1 ${\bf{91.5\pm4.3}}$ $89.3\pm5.1$ $90.4\pm2.9$ $89.8\pm5.9$ 2 ${\bf{89.1\pm2.6}}$ $85.1\pm7.5$ $81.7\pm6.4$ $83.4\pm8.1$ 3 ${\bf{88.7\pm6.6}}$ $75.3\pm10.4$ $74.3\pm8.8$ $77.1\pm10.7$ 4 ${\bf{86.2\pm4.1}}$ $70.8\pm14.4$ $71.2\pm10.7$ $72.7\pm11.2$ 5 ${\bf{93.4\pm3.8}}$ $73.0\pm9.1$ $70.7\pm13.5$ $71.5\pm14.6$ 6 ${\bf{88.5\pm6.2}}$ $69.8\pm12.9$ $64.2\pm11.2$ $67.3\pm11.9$ 7 ${\bf{86.0\pm5.7}}$ $59.9\pm12.3$ $59.5\pm12.1$ $62.1\pm13.4$ 8 ${\bf{88.1\pm8.4}}$ $61.7\pm10.2$ $57.8\pm13.0$ $59.4\pm14.7$ 9 ${\bf{91.4\pm4.9}}$ $58.4\pm9.7$ $60.1\pm10.6$ $61.6\pm9.2$ 10 ${\bf{87.9\pm4.3}}$ $62.8\pm10.3$ $57.0\pm15.6$ $58.1\pm13.9$ $m\pm st$ ${\bf{89.1\pm5.1}}$ $70.6\pm10.2$ $68.7\pm10.5$ $70.3\pm11.4$ 
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表 2 使用所有测试者的测试数据计算的分类精度与排除精度的均值和标准差(%)
Table 2 The means and standard deviations of classification and rejection accuracies computed by using the test data of all subjects (%)
目标动作分类精度 外部动作排除精度 SUHC $90.4\pm4.4$ $93.5\pm3.0$ SVDD $82.1\pm8.7$ $91.9\pm5.2$ SVM $62.5\pm16.1$ $-$ MLP $60.8\pm18.4$ $-$ KLDA $60.1\pm15.7$ $-$
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