L ₂-Gain Control for T-S Fuzzy Systems Over an Event-Triggered Communication Network Using Delay Decomposition and Deviation Bounds of Membership Functions

Abstract

This paper is concerned with networked $L_2$-gain control for a T-S fuzzy system over an event-triggered communication network. Taking an event-triggered communication scheme and network-induced delays into consideration, the networked control system is described by an asynchronous T-S fuzzy system with an interval time-varying delay. Due to variation characteristic of network-induced delays, the interval is decomposed into N subintervals and the jumping among these subintervals is governed by a Markov chain. A new relaxation method, which fully utilizes the convexity of normalized membership functions and the deviation bounds of asynchronous normalized membership functions, is proposed and a stochastic Lyapunov–Krasovskii functional is constructed to derive some delay-dependent criteria on $L_2$-gain performance analysis and controller design of the asynchronous T-S fuzzy system. An illustrative example is provided to show that the proposed criteria are of less conservatism and less computational complexity than some existing results, and are effective in achieving a prescribed $L_2$-gain performance.

Appendices

Appendix 1: Proof of Theorem 1

For the system (11), define a new process $\{(x_t,r(t)),\ t\!\ge \!0\}$ by $\{x_t(s)\!=\!x(t+s),\ -\tau (t,r(t))\!\le \! s\!\le \!0\}$. Now we construct a stochastic Lyapunov–Krasovskii functional (LKF)

$$V(x_t,r(t),t)=\sum _{i=1}^4V_i(x_t,r(t),t)\quad t\in \Omega _{l,k},$$

(26)

where

$$\begin{aligned} V_1(x_t,r(t),t)&=V_1(x_t)=x^T(t)Px(t), \\ V_2(x_t,r(t),t)&=V_2(x_t,t)=\int _{t-\tau _m}^tx^T(s)Q_1x(s){\mathrm{d}}s+\int _{t-\tau _M}^tx^T(s)Q_2x(s){\mathrm{d}}s, \\ V_3(x_t,r(t),t)&=V_3(x_t,t)=\int _{-\tau _M}^{-\tau _m}\int _{t+\theta }^t\dot{x}^T(s)R_1\dot{x}(s){\mathrm{d}}s{\mathrm {d}}\theta + \int _{-\tau _m}^0\int _{t+\theta }^t\dot{x}^T(s)R_2\dot{x}(s){\mathrm{d}}s{\mathrm{d}}\theta ,\\ V_4(x_t,r(t),t)&=(\tau _M\!-\!\tau (t,r(t)))\{[x^T(t)\!-\!x^T(s_k)]W_1[x(t)\!-\!x(s_k)]\!+\!\int _{s_k}^t\dot{x}^T(s)W_2\dot{x}(s){\mathrm{d}}s\}, \end{aligned}$$

$P\!>\!0, Q_i\!>\!0,R_i\!>\!0$ , and $W_i\!>\!0 \ (i\!=\!1,2)$, $s_k\!=\!i_k^lh\!+\!\tau _{t_k+l}$, and $x_t\!=\!x(t\!+\!\theta ), \ \theta \!\in \![-\tau _M,0]$ is an element of Banach space ${\mathcal {C}}([-\tau _M,0],R^n)$ of continuous functions from $[-\tau _M,0]$ to $R^n$.

First, $W_1,W_2$-dependent terms in the LKF (26) are discontinuous at updating instants $\{i_k^lh\!+\!\tau _{t_k+l}\}_{k=0, l=0}^{\infty ,N_k},$ and they are non-negative just before $i_k^lh\!+\!\tau _{t_k+l}$ and become zero just after $i_k^lh\!+\!\tau _{t_k+l}$; the other terms are continuous, i.e., $\lim _{t\rightarrow (i_k^lh+\tau _{t_k+l})^{-}}V_4(x_t,r(t),t)\!\ge \!0$ and $\lim _{t\rightarrow (i_k^lh+\tau _{t_k+l})^{+}}V_4(x_t,r(t),t)\!=\!0$. This implies $V(s_k^+)\!\le \!V(s_k^-)$.

Second, we prove the stochastic stability for the system (11) with $\omega (t)=0$. Let ${\mathcal {L}}$ be the weak infinitesimal generator of $\{(x_t,r(t)),t\ge 0\}$, then for $r(t)=p\in {\mathcal {I}}$, we have

$$\begin{aligned} {\mathcal {L}}V=&2x^T(t)P\dot{x}(t)\!+\!x^T(t)Q_1x(t)\!\!-\!\!x^T(t\!\!-\!\!\tau _m)Q_1x(t\!\!-\!\!\tau _m)\!+\!x^T(t)Q_2x(t)\!\!-\!\!x^T(t\!\!-\!\!\tau _M)Q_2x(t\!\!-\!\!\tau _M) \\&+(\tau _M\!\!-\!\!\tau _m)\dot{x}^T(t)R_1\dot{x}(t)\;-\;\int _{t\!-\!\tau _M}^{t\!-\!\tau _m}\dot{x}^T(s)R_1\dot{x}(s){\mathrm{d}}s\!+\! \tau _m\dot{x}^T(t)R_2\dot{x}(t)\;-\;\int _{t\!-\!\tau _m}^t\dot{x}^T(s)R_2\dot{x}(s){\mathrm{d}}s \\&+(\tau _M\!-\!\tau (t,p))x^T(t)W_1\dot{x}(t)\!+\!(\tau _M\!-\!\tau (t,p))\dot{x}^T(t)W_1x(t)\!+\!(\tau (t,p)\!-\!\tau _M)x^T(s_k)W_1\dot{x}(t) \\&+(\tau (t,p)\!-\!\tau _M)\dot{x}^T(t)W_1x(s_k)\!+\!\tau _M\dot{x}^T(t)W_2\dot{x}(t)\!-\! \tau (t,p)\dot{x}^T(t)W_2\dot{x}(t) \\&-\sum _{q=1}^N\lambda _{pq}\tau (t,q)[x^T(t)\!-\!x^T(s_k)]W_1 [x(t)\!-\!x(s_k)]\!-\!\sum _{q=1}^N\lambda _{pq}\tau (t,q)\int _{s_k}^t\dot{x}^T(s)W_2\dot{x}(s){\mathrm{d}}s. \end{aligned}$$

For $r(t)\!=\!p$, $t\!\in \!\Omega _{l,k}$, $\tau (t,p)$ satisfies $\tau _m\!+\!(p\!-\!1)\sigma \!\le \!\tau (t,p)\!\le \!\tau _m\!+\!p\sigma$ and $\sum _{q=1}^N\lambda _{pq}=0$. Then we have

$$\sum _{q=1}^N\lambda _{pq}\tau (t,q)=\sigma \sum _{q=1}^Nq\lambda _{pq}.$$

(27)

For any appropriately dimensioned matrices $M_1$ and $M_2$, the following equality is true:

$$2\sum _{i=1}^r\sum _{j=1}^r\mu _i\mu _j^{k}(x^T(t)M_1^T\!+\!\dot{x}^T(t)M_2^T)[A_ix(t)\!+\!B_iK_jx(t\!-\!\tau (t,p))\!-\!B_iK_je_k(t\!-\!\tau (t,p))\!-\!\dot{x}(t)]\!=\!0.$$

(28)

One can use Lemmas 1 and 2 to obtain that

$$\begin{aligned}&\!-\!\int _{t-\tau _M}^{t-\tau _m}\dot{x}^T(s)R_1\dot{x}(s){\mathrm{d}}s \!\le \! -\frac{1}{\delta }\xi ^T(t)\left[ \begin{array}{l} (e_2\!-\!e_3) \\ (e_3\!-\!e_5) \\ \end{array} \right] ^T \left[ \begin{array}{ll} R_1 &{} S \\ *&{} R_1 \\ \end{array} \right] \left[ \begin{array}{l} (e_2\!-\!e_3) \\ (e_3\!-\!e_5)\\ \end{array} \right] \xi (t),\\&\!-\!\int _{t-\tau _m}^t\dot{x}^T(s)R_2\dot{x}(s){\mathrm{d}}s\!\le \! \xi ^T(t)[(e_2\!-\!e_1)^TE_1\!+\!E^T_1(e_2\!-\!e_1) \!+\!\tau _mE_1^TR_2^{-1}E_1]\xi (t),\\&\!-\!\int _{s_k}^{t}\dot{x}^T(s)W_2\dot{x}(s){\mathrm{d}}s\le \xi ^T(t)[(e_6\!-\!e_1)^TE_2+E_2^T(e_6\!-\!e_1)+(\tau (t,p)\!-\!\tau _m)E_2^TW_2^{-1}E_2]\xi (t), \end{aligned}$$

where $\xi (t)\!=\![x^T(t)\ \ x^T(t\!-\!\tau _m)\ \ x^T(t\!-\!\tau (t,p))\ \ e_k^T(t\!-\!\tau (t,p))\ \ x^T(t\!-\!\tau _M)\ \ x^T(s_k)\ \ \dot{x}^T(t)]^T$. Then for $t\!\in \!\Omega _{l,k}$, we have

$${\mathcal {L}}V\le \sum _{i=1}^r\sum _{j=1}^r\mu _i\mu _j^k\xi ^T(t)\Pi _{ij}^0(t)\xi (t),$$

(29)

where $\Pi _{ij}^0(t)\!=\!\Pi _{ij}^1(t)\!+\!\Pi _{ij}^2(t)\!+\!\Pi _{ij}^0\!+\!\tau _mE_1^TR_2^{-1}E_1$, $\Pi _{ij}^1(t)\!=\!(\tau _M\!-\!\tau (t,p))\Pi$, $\Pi _{ij}^2(t)\!=\!(\tau (t,p)\!-\!\tau _m)\sigma \sum _{q=1}^Nq\lambda _{pq}E_2^TW_2^{-1}E_2$. It can be seen from the inequality (12) that

$$\xi ^T(t)\left[ -\epsilon e_3^T\Phi e_3+\epsilon e_4^T\Phi e_3+\epsilon e_3^T\Phi e_4+ (1-\epsilon )e_4^T\Phi e_4\right] \xi (t)\le 0.$$

(30)

By using the S-Procedure technique, one can see that for all non-zero $\xi (t)$ satisfying (30), the inequality $\sum _{i=1}^r\sum _{j=1}^r\mu _i\mu _j^k\xi ^T(t)\Pi _{ij}^0(t)\xi (t)\!<\!0$ is equivalent to the existence of a scalar $\tilde{\delta }>0$ such that

$$\xi ^T(t)\sum _{i=1}^r\sum _{j=1}^r\mu _i\mu _j^k\left\{ \Pi _{ij}^0(t)-\tilde{\delta }\left[ -\epsilon e_3^T\Phi e_3+\epsilon e_4^T\Phi e_3+\epsilon e_3^T\Phi e_4+ (1-\epsilon )e_4^T\Phi e_4\right] \right\} \xi (t)<0.$$

(31)

If (31) holds, then the system (11) with $\omega (t)\!=\!0$ is stochastically stable.

Notice that $\Pi _{ij}^0(t)$ is a convex combination of $\Pi _{ij}^1(t)$ and $\Pi _{ij}^2(t)$ on $\tau (t,p)$, using the convex combination technique and Schur complement, the inequality (31) holds if the following inequalities are satisfied:

$$\sum _{i=1}^r\sum _{j=1}^r\mu _i\mu _j^k\Psi _{ij}^1=\left[ \begin{array}{ll} \sum _{i=1}^r\sum _{j=1}^r\mu _i\mu _j^k(\bar{\Pi }_{ij}^0+\delta \Pi ) &{}\quad \tau _mE_1^T\\ *&{}\quad -\tau _m R_2 \\ \end{array} \right] <0,$$

(32)

$$\begin{aligned}&\sum _{i=1}^r\sum _{j=1}^r\mu _i\mu _j^k\Psi _{ij}^2=\left[ \begin{array}{lll} \sum _{i=1}^r\sum _{j=1}^r\mu _i\mu _j^k\bar{\Pi }_{ij}^0 &{}\quad \tau _mE_1^T &{} \delta \sigma \sum _{q=1}^Nq\lambda _{pq}E_2^T\\ *&{}\quad -\tau _mR_2 &{}\quad 0\\ *&{}\quad *&{}\quad -\delta \sigma \sum _{q=1}^Nq\lambda _{pq}W_2\\ \end{array} \right] <0. \end{aligned}$$

(33)

Using the proposed relaxation method in Remark 3 and Lemma 3, if the LMIs (13)–(17) are satisfied, for $v\!=\!1,2$, we have $\sum _{i=1}^r\sum _{j=1}^r\mu _i\mu _j^k\Psi _{ij}^v\!<\!0$. It is easy to see that there exists a scalar $\alpha \!>\!0$, such that

$${\mathcal {L}}V\le -\alpha \Vert x(t)\Vert ^2\quad t\in \Omega _{l,k}.$$

Applying Dynkin’s formula, we can obtain

$${\mathcal {E}}\left\{ \int _0^t\Vert x(s)\Vert ^2{\mathrm{d}}s|(x(0),r(0),0)\right\} \le \frac{1}{\alpha }{\mathcal {E}}\left\{ V(x(0),r(0),0)\right\} ,$$

which implies that the closed-loop system (11) with $\omega (t)\!=\!0$ is stochastically stable.

Last, under the zero initial condition, we consider $L_2$-gain for the system (11) with $\omega (t)\!\ne \!0$. Employing the LKF (26), we have

$$\begin{aligned} {\mathcal {L}}V\le \sum _{i=1}^r\sum _{j=1}^r\mu _i\mu _j^k\left[ \! \begin{array}{ll} \xi ^T(t) &{}\quad \omega ^T(t) \\ \end{array}\! \right] \times \left[ \! \begin{array}{ll} \tilde{\Pi }_{ij}^0(t) &{}\quad \Gamma _i \\ *&{}\quad -\gamma ^2I \\ \end{array}\! \right] \left[ \! \begin{array}{l} \xi (t) \\ \omega (t) \\ \end{array}\! \right] \!-\!z^T(t)z(t)\!+\!\gamma ^2\omega ^T(t)\omega (t), \end{aligned}$$

(34)

where $\tilde{\Pi }_{ij}^0(t)\!=\!\Pi _{ij}^0(t)\!+\!e_1^TC_i^TC_ie_1$.

Using the convex combination technique and Schur complement, we have that

$$\begin{aligned} \sum _{i=1}^r\sum _{j=1}^r\mu _i\mu _j^k\left[ \begin{array}{ll} \tilde{\Pi }_{ij}^0(t)+\epsilon e_3^T\Phi e_3-\epsilon e_4^T\Phi e_3-\epsilon e_3^T\Phi e_4+ (\epsilon -1)e_4^T\Phi e_4 &{}\quad \Gamma _i \\ *&{}\quad -\gamma ^2I \\ \end{array} \right] <0 \end{aligned}$$

holds if the following inequalities are satisfied:

$$\sum _{i=1}^r\sum _{j=1}^r\mu _i\mu _j^k\tilde{\Sigma }_{ij}^1<0,$$

(35)

$$\sum _{i=1}^r\sum _{j=1}^r\mu _i\mu _j^k\tilde{\Sigma }_{ij}^2<0.$$

( 36)

Using the introduced free-weighting $H_i^v$ and the method mentioned in Remark 3, if the conditions (13)–(17) are satisfied, for $v=1,2$, we have

$$\begin{aligned} \sum _{i=1}^r\sum _{j=1}^r\mu _i\mu _j^k\tilde{\Sigma }_{ij}^v \le&\sum _{i=1}^r\sum _{j\ne i}^r\mu _i\mu _j\left( \frac{\tilde{\Sigma }_{ij}^v\!\!-\!\!\tilde{\Sigma }_{ii}^v}{2}\right) \!+\!\sum _{i=1}^r\sum _{j\ne i}^r\mu _i^k\mu _j^k\left( \frac{\tilde{\Sigma }_{ij}^v\!\!-\!\!\tilde{\Sigma }_{ii}^v}{2}\right) \!\!+\!\!\sum _{i=1}^r\sum _{j\ne i}^r\mu _i\eta _j\left( \frac{\tilde{\Sigma }_{ij}^v\!\!+\!\!H_i^v}{2}\right) \\&+\sum _{i=1}^r\sum _{j\ne i}^r\eta _i\mu _j^k\left( \frac{\tilde{\Sigma }_{ij}^v\!+\!H_j^v}{2}\right) \!+\! \sum _{i=1}^r\left( \mu _i^k\!\!-\!\!\mu _i\right) ^2\left( -\frac{\tilde{\Sigma }_{ii}^v\!+\!H_i^v}{2}\right) \!+\!\sum _{i=1}^r\left( \mu _i\!\!+\!\mu _i^k\right) \frac{\tilde{\Sigma }_{ii}^v}{2} \\ =&\sum _{i=1}^r\mu _i\left[ \frac{\tilde{\Sigma }_{ii}^v}{2}\!+\!\sum _{k\ne i}^r\eta _k\left( \frac{\tilde{\Sigma }_{ik}^v\!+\!H_i^v}{2}\right) \right] \!+\!\sum _{i=1}^r\sum _{i<j\le r }\mu _i\mu _j\left( \frac{\Pi _{ij}^v\!+\!\Pi _{ji}^v}{2}\right) \\&+\sum _{i=1}^r\sum _{i<j\le r}\mu _i^k\mu _j^k\left( \frac{\Pi _{ij}^v\!+\!\Pi _{ji}^v}{2}\right) \!+\!\sum _{i=1}^r\mu _i^k\left[ \frac{\tilde{\Sigma }_{ii}^v}{2}\!+\!\sum _{k\ne i}^r\eta _k\left( \frac{\tilde{\Sigma }_{ki}^v\!+\!H_i^v}{2}\right) \right] \\&+\sum _{i=1}^r\left( \mu _i^k\!-\!\mu _i\right) ^2\left( -\frac{\tilde{\Sigma }_{ii}^v\!+\!H_i^v}{2}\right) <0, \end{aligned}$$

( 37)

which implies ${\mathcal {L}}V<-z^T(t)z(t)+\gamma ^2\omega ^T(t)\omega (t)$ for the system (11) with the constraint (12).

For $t\!\in \!\Omega _{l,k}$, it follows from (34) that

$${\mathcal {E}}\left\{ V(x_t,p,t)\right\} -{\mathcal {E}}\left\{ V(x_{s_k^+},p,s_k^+)\right\} \le -{\mathcal {E}}\left\{ \int _{s_k}^tz^T(s)z(s){\mathrm{d}}s\right\} + \gamma ^2\int _{s_k}^tw^T(s)w(s){\mathrm{d}}s,$$

since V(t) is non-increasing and positive definite in $[0,\infty )$, then under the zero initial condition, it is easy to obtain that ${\mathcal {E}}\left\{ \int _0^\infty z^T(s)z(s){\mathrm{d}}s\right\} \!\le \!\gamma ^2\int _0^\infty \omega ^T(s)\omega (s){\mathrm{d}}s$. Therefore, the closed-loop system (11) is stochastically stable with $\gamma$-disturbance attenuation. This completes the proof.

Appendix 2: Proof of Theorem 2

Define $M\!=\!M_1^{-1}\!=\!\beta M_2^{-1}, \ \bar{P}\!=\!M^TPM, \ \bar{S}\!=\!M^TSM,\ \bar{Q}_i\!=\!M^TQ_iM, \ \bar{R}_i\!=\!M^TR_iM, \ \bar{W}_i\!=\!M^TW_iM\ (i\!=\!1,2), \ \bar{E}_i\!=\!M^TE_iM\ (i\!=\!1,2), \Delta \!=$ diag$\{ M,M,M,M,M,M,M \}$, $\bar{H}_{i^1}\!\!=$ diag $\{ \Delta ^T, I,M^T\}$ $H_i^1$ diag $\{ \Delta ,I,M \},$ $\bar{H}_i^2=$ diag $\{ \Delta ^T,I,M^T,M^T \}$ $H_i^2$ diag $\{ \Delta ,I,M,M \}\ (i\!=\!1,2,\ldots ,r)$ , and $Y_j\!=\!K_jM(j\!=\!1,2,\ldots ,r)$. Then, pre- and postmultiply diag$\{M,M\}^T$ and its transpose to (13), diag$\{\Delta ^T,I,M^T\}$ and its transpose to (14)–(17) for $v\!=\!1$, diag$\{\Delta ^T, I, M^T,M^T\}$ and its transpose to (14)–(17) for $v\!=\!2$, respectively. Using the Schur complement, one can readily arrive at (18)–(22) from (13)–(17). This completes the proof.