Quantized Feedback Adaptive Reliable $H_{\infty }$ Control for Linear Time-Varying Delayed Systems

Abstract

Based on an improved $H_\infty$ performance index, the method of designing a reliable adaptive $H_\infty$ controller with quantized state is addressed for the time-varying delayed system in this paper. On the basis of online estimates of actuator faults, the controller parameters are updated automatically to compensate the influence of actuator faults on the system while the desired improved $H_\infty$ performance is preserved. A Lyapunov function candidate is constructed to prove that the closed-loop system is asymptotically stable. And the existing sufficient conditions of the controller are proved to be less conservative. The gains of the controller and the parameters of the adaptive law are co-designed and obtained in terms of solutions to a set of linear matrix inequalities. Finally, two numerical examples are given to illustrate that the proposed method is more effective than the previous methods for time-varying delayed systems.

Appendix

1.1 Appendix 1: The proof of Theorem 1

Proof

Consider the Lyapunov function candidate,

$$\begin{aligned} V(t)= & {} x^\mathrm{{T}} (t)Px(t) + \int _{t - \tau (t)}^t {x^\mathrm{{T}} (s)R_1 x(s)\mathrm{{d}}s} + \int _{t - h}^t {x^\mathrm{{T}} (s)R_2 x(s)\mathrm{{d}}s}\\&\,+\frac{h}{{2\lambda _{\mathrm{{max}}} (Q)}}\int _{ - h}^0 {\int _{t + \theta }^t {\dot{x}^\mathrm{{T}} (s)Q\dot{x}(s)\mathrm{{d}}s\mathrm{{d}}\theta } } + \sum \limits _{i = 1}^m {\frac{{\tilde{\rho }_i^2 (t)}}{{l_i }}}, \end{aligned}$$

where positive matrices $P, R_1, R_2$ and Q are to be determined, and $l_i>0$ is a given constant.

According to Lemma 1 [30], in the case of system failure, the derivative of V(t) along the system (10) is described as follows,

$$\begin{aligned}&\dot{V}(t) + z^\mathrm{{T}} (t)z(t) - \gamma _f^2 w^\mathrm{{T}} (t)w(t) - h^2 w^\mathrm{{T}} (t)B_2^\mathrm{{T}} B_2 w(t)\\&\quad \le \dot{V}(t) + z^\mathrm{{T}} (t)z(t) - \gamma _f^2 w^\mathrm{{T}} (t)w(t) - h^2 w^\mathrm{{T}} (t)B_2^\mathrm{{T}} B_2 w(t)\\&\qquad -2[f(x(t))-\triangle _{\sigma 0}x(t)]^\mathrm {T}S[f(x(t))-\triangle _{\sigma 1}x(t)]\\&\quad \le \eta _1^{\mathrm {T}} (t)\varOmega \eta _1 (t)-\frac{h}{2\lambda _\mathrm{{max}} (Q)}\int _{t-h}^t {\dot{x}^{\mathrm {T}}(s)Q\dot{x}(s)\text{ d }s}+\sum \limits _{i=1}^m {\frac{2\tilde{\rho }_i (t)\dot{\tilde{\rho }}_i (t)}{l_i}}\\&\qquad +2\{x^\mathrm {T}(t)P[B_1K_a(\widetilde{\rho }(t))+B_1\widetilde{\rho }(t)K_b(\hat{\rho }(t))]f(x(t))\}\\&\qquad +2\{x^\mathrm {T}(t)P[B_1M_a(\widetilde{\rho }(t))+B_1\widetilde{\rho }(t)M_b(\hat{\rho }(t))]f(x(t-\tau (t)))\}\\&\qquad +z^\mathrm {T}(t)z(t)-\gamma _f^2 w^{\mathrm {T}}(t)w(t)-h^2w^{\mathrm {T}}(t)B_2^{\mathrm {T}} B_2 w(t), \end{aligned}$$

where

$$\begin{aligned} \eta _1^{\mathrm {T}} (t)= & {} [{\begin{array}{cccccc} {x^{\mathrm {T}}(t)} &{} {x^{\mathrm {T}}(t-\tau (t))} &{} {x^{\mathrm {T}}(t-h)} &{} {f^{\mathrm {T}}(x(t))} &{} {f^{\mathrm {T}}(x(t-\tau (t)))} &{}{w^{\mathrm {T}}(t)} \\ \end{array} }],\\ \varOmega= & {} \left[ {\begin{array}{cccccc} {\varOmega _{11} } &{} {\varOmega _{12} } &{} 0 &{} {\varOmega _{14}} &{} {\varOmega _{15}} &{} {\varOmega _{16}}\\ \mathrm{{*}} &{} {\varOmega _{22} } &{} 0 &{} {\varOmega _{24} } &{} {\varOmega _{25}} &{} {\varOmega _{26}}\\ \mathrm{{*}} &{} \mathrm{{*}} &{} { - R_2 } &{} 0 &{} 0 &{} 0\\ \mathrm{{*}} &{} \mathrm{{*}} &{} \mathrm{{*}} &{} {\varOmega _{44}} &{} {\varOmega _{45}} &{} {\varOmega _{46}}\\ \mathrm{{*}} &{} \mathrm{{*}} &{} \mathrm{{*}} &{} \mathrm{{*}} &{} {\varOmega _{55}} &{} {\varOmega _{56}}\\ \mathrm{{*}} &{} \mathrm{{*}} &{} \mathrm{{*}} &{} \mathrm{{*}} &{} \mathrm{{*}} &{} \Omega _{66} \\ \end{array}} \right] \\ \varOmega _{11}= & {} A_1 ^{\mathrm {T}}P+PA_1 +R_1 +R_2 +\frac{1}{2}h^2A_1^{\mathrm {T}}A_1-2\triangle _{\sigma 0}^\mathrm {T}S\triangle _{\sigma 1},\\ \varOmega _{12}= & {} PA_2\!+\!\frac{1}{2}h^2A_1^{\mathrm {T}}A_2 ,~ \varOmega _{14} =PM_1\!+\!\triangle _{\sigma 0}^\mathrm {T}S\!+\!\triangle _{\sigma 1}^\mathrm {T}S\!+\!\frac{1}{2}h^2A_1^{\mathrm {T}}B_1(I\!-\!\rho )K(\hat{\rho }(t)) ,\\ \varOmega _{15}= & {} PM_2+\frac{1}{2}h^2A_1^{\mathrm {T}}B_1(I-\rho )M(\hat{\rho }(t)),~ \varOmega _{16} \mathrm{{ = }}PB_2 + \frac{1}{2}h^2 A_1^\mathrm {T} B_2,\\ \varOmega _{22}= & {} -(1-h_1 )R_1+\frac{1}{2}h^2A_2^{\mathrm {T}}A_2-2\triangle _{\sigma 0}^\mathrm {T}S\triangle _{\sigma 1},\\ \varOmega _{24}= & {} \frac{1}{2}h^2A_2^{\mathrm {T}}B_1(I\!-\!\rho )K(\hat{\rho }(t)) , ~ \varOmega _{25} \!=\!\triangle _{\sigma 0}^\mathrm {T}S\!+\!\triangle _{\sigma 1}^\mathrm {T}S\!+\!\frac{1}{2}h^2A_2^{\mathrm {T}}B_1(I\!-\!\rho )M(\hat{\rho }(t)) , \\ \varOmega _{26}= & {} \frac{1}{2}h^2 A_2^\mathrm {T} B_2,~ \varOmega _{44} =-2S+\frac{1}{2}h^2[B_1(I-\rho )K(\hat{\rho }(t))]^{\mathrm {T}}B_1(I-\rho )K(\hat{\rho }(t)) , \\ \varOmega _{45}= & {} \frac{1}{2}h^2[B_1(I-\rho )K(\hat{\rho }(t))]^{\mathrm {T}}B_1(I-\rho )M(\hat{\rho }(t)) , \\ \varOmega _{46}= & {} \frac{1}{2}h^2[B_1(I-\rho )K(\hat{\rho }(t))]^{\mathrm {T}}B_2 , \\ \varOmega _{55}= & {} -2S+\frac{1}{2}h^2[B_1(I-\rho )M(\hat{\rho }(t))]^{\mathrm {T}}B_1(I-\rho )M(\hat{\rho }(t)) , \\ \varOmega _{56}= & {} \frac{1}{2}h^2[B_1(I-\rho )M(\hat{\rho }(t))]^{\mathrm {T}}B_2 ,~ \varOmega _{66}\,\mathrm{{=}}\,\frac{1}{2}h^2 B_2^\mathrm {T} B_2. \end{aligned}$$

Because

$$\begin{aligned} 2x^\mathrm {T}(t)PB_2w(t) \le \gamma _f^2 w^\mathrm {T}(t)w(t)+\frac{1}{\gamma _f^2 }x ^\mathrm {T}(t)PB_2B_2^\mathrm {T}Px(t) , \end{aligned}$$

and

$$\begin{aligned}&\frac{1}{2}h^2[\eta ^{\mathrm {T}}_2(t)\Delta ^\mathrm {T}B_2w(t)] \le \frac{1}{2}h^2w^\mathrm {T}(t)B_2B_2^\mathrm {T}w(t)\\&\quad \,+\frac{1}{2}h^2\eta _2^\mathrm {T}(t)\Delta ^\mathrm {T}\Delta \eta _2(t), \end{aligned}$$

where $[\eta ^{\mathrm {T}}_2(t)=[{\begin{array}{*{20}c} {x^{\mathrm {T}}(t)}&{x^{\mathrm {T}}(t-\tau (t))}&{f^\mathrm {T}(x(t))}&{f^\mathrm {T}(x(t-\tau (t)))} \end{array} }], ]$ $[\Delta =[ \displaystyle {A_1} \displaystyle {A_2} \displaystyle {B_1 (I-\rho )K(\hat{\rho }(t))} \displaystyle {B_1 (I-\rho )M(\hat{\rho }(t))}], ]$ then

$$\begin{aligned}&\dot{V}(t)+z^{\mathrm {T}}(t)z(t)-\gamma _f^2 w^{\mathrm {T}}(t)w(t)-h^2w^{\mathrm {T}}(t)B_2^{\mathrm {T}} B_2 w(t) \nonumber \\&\quad \le \eta ^{\mathrm {T}} (t)\varXi _1 \eta (t)-\frac{h}{2\lambda _\mathrm{{max}} (Q)}\int _{t-h}^t {\dot{x}^\mathrm{{T}}(s)Q\dot{x}(s)\text{ d }s}+\sum \limits _{i=1}^m {\frac{2\tilde{\rho }_i (t)\dot{\tilde{\rho }}_i (t)}{l_i}}\nonumber \\&\qquad +2\{x^\mathrm {T}(t)P[B_1K_a(\widetilde{\rho }(t))+B_1\widetilde{\rho }(t)K_b(\hat{\rho }(t))]f(x(t))\}\nonumber \\&\qquad +2\{x^\mathrm {T}(t)P[B_1M_a(\widetilde{\rho }(t))+B_1\widetilde{\rho }(t)M_b(\hat{\rho }(t))]f(x(t-\tau (t)))\}, \end{aligned}$$

(37)

where

$$\begin{aligned} \eta ^{\mathrm {T}}(t)=[{\begin{array}{ccccc} {x^{\mathrm {T}}(t)} &{} {x^{\mathrm {T}}(t-\tau (t))} &{} {x^{\mathrm {T}}(t-h)} &{} {f^\mathrm {T}(x(t))} &{} {f^\mathrm {T}(x(t-\tau (t)))}\\ \end{array} }]. \end{aligned}$$

According to Lemma 1 [7] and Lemma 2 [26], assuming $\displaystyle {0<\frac{\tau (t)}{h-\tau (t)}<1}$, we have

$$\begin{aligned}&h\int _{t-h}^t {\dot{x}^{\mathrm {T}}(s)Q\dot{x}(s)\text{ d }s}\\&\quad = h\int _{t-\tau (t)}^t {\dot{x}^{\mathrm {T}}(s)Q\dot{x}(s)\text{ d }s} +h\int _{t-h}^{t-\tau (t)} {\dot{x}^{\mathrm {T}}(s)Q\dot{x}(s)\text{ d }s} \\&\quad \ge \eta ^{\mathrm {T}} (t)\varXi _2 \eta (t)+\eta ^{\mathrm {T}} (t)\varXi _3 \eta (t)+\frac{\tau (t)}{h-\tau (t)}\eta ^{\mathrm {T}} (t)\varXi _2 \eta (t)\\&\qquad \,+\left( 1-\frac{\tau (t)}{h-\tau (t)}\right) \eta ^{\mathrm {T}} (t)\varXi _3 \eta (t). \end{aligned}$$

So (37) can be rewritten as

$$\begin{aligned}&\dot{V}(t) + z^\mathrm{{T}} (t)z(t) - \gamma _f^2 w^\mathrm{{T}} (t)w(t) - h^2 w^\mathrm{{T}} (t)B_2^\mathrm{{T}} B_2 w(t) \nonumber \\&\quad \le \eta ^\mathrm{{T}} (t)\varXi _1 \eta (t) + \sum \limits _{i = 1}^m {\frac{{2\tilde{\rho }_i (t)\dot{\tilde{ \rho }}_i (t)}}{{l_i }}}\nonumber \\&\qquad - \frac{1}{{2\lambda _\mathrm{{max}} (Q)}}[\eta ^{\mathrm {T}} (t)\varXi _2 \eta (t)+\eta ^{\mathrm {T}} (t)\varXi _3 \eta (t)\nonumber \\&\qquad +\,\frac{\tau (t)}{h-\tau (t)}\eta ^{\mathrm {T}} (t)\varXi _2 \eta (t)+\left( 1-\frac{\tau (t)}{h-\tau (t)}\right) \eta ^{\mathrm {T}} (t)\varXi _3 \eta (t)]\nonumber \\&\qquad +\,2\{x^\mathrm {T}(t)P[B_1K_a(\widetilde{\rho }(t))+B_1\widetilde{\rho }(t)K_b(\hat{\rho }(t))]f(x(t))\}\nonumber \\&\qquad +\,2\{x^\mathrm {T}(t)P[B_1M_a(\widetilde{\rho }(t))+B_1\widetilde{\rho }(t)M_b(\hat{\rho }(t))]f(x(t-\tau (t)))\}. \end{aligned}$$

(38)

According to the adaptive law (15), it is easy to know

$$\begin{aligned} 2\sum \limits _{i=1}^m {\tilde{\rho }_i (t)M_{3i} } +\sum \limits _{i=1}^m {\frac{2\tilde{\rho }_i (t)\dot{\tilde{\rho }}_i (t)}{l_i }} \le 0, \end{aligned}$$

(39)

it is equivalent to

$$\begin{aligned}&\sum \limits _{i = 1}^m {\frac{{2\tilde{\rho }_i (t)\dot{\tilde{ \rho }}_i (t)}}{{l_i }}} +2\{x^\mathrm {T}(t)P[B_1K_a(\widetilde{\rho }(t))+B_1\widetilde{\rho }(t)K_b(\hat{\rho }(t))]f(x(t))\}\\&\quad \,+2\{x^\mathrm {T}(t)P[B_1M_a(\widetilde{\rho }(t))+B_1\widetilde{\rho }(t)M_b(\hat{\rho }(t))]f(x(t-\tau (t)))\} \le 0. \end{aligned}$$

Substituting (39) into (38), it follows

$$\begin{aligned}&\dot{V}(t)+z^{\mathrm {T}}(t)z(t)-\gamma _f ^2w^{\mathrm {T}}(t)w(t)-h^2w^{\mathrm {T}}(t)B_2^{\mathrm {T}} B_2 w(t) \\&\quad \le \eta ^{\mathrm {T}}(t)\varXi _1 \eta (t)-\frac{1}{2\lambda _{\text{ max }} (Q)}[\eta ^{\mathrm {T}} (t)\varXi _2 \eta (t)+\eta ^{\mathrm {T}} (t)\varXi _3 \eta (t)\\&\qquad +\frac{\tau (t)}{h-\tau (t)}\eta ^{\mathrm {T}} (t)\varXi _2 \eta (t)+\left( 1-\frac{\tau (t)}{h-\tau (t)}\right) \eta ^{\mathrm {T}} (t)\varXi _3 \eta (t)]. \end{aligned}$$

According to Lemma 2 [26], (18) and (19) hold imply that (40) holds,

$$\begin{aligned}&\left( \varXi _1 -\frac{1}{2\lambda _{\text{ max }} (Q)}\varXi _2-\frac{1}{2\lambda _{\text{ max }} (Q)}\varXi _3 \right) \nonumber \\&\quad \,-\frac{1}{2\lambda _{\text{ max }} (Q)}\left[ (1-\frac{\tau (t)}{h-\tau (t)})\varXi _2 +\frac{\tau (t)}{h-\tau (t)}\varXi _3\right] <0. \end{aligned}$$

(40)

From the above, for any nonzero vector $x\in R^n$, we have

$$\begin{aligned} \dot{V}(t)+z^{\mathrm {T}}(t)z(t)-\gamma _f^2 w^{\mathrm {T}}(t)w(t)-h^2w^{\mathrm {T}}(t)B_2^{\mathrm {T}} B_2 w(t)<0, \end{aligned}$$

(41)

which implies that $\dot{V}(t)<0$, when $w(t)=0$. So the states of the system (10) are asymptotically stable in the actuator fault case.

Integrate (41) from 0 to $\infty$ on both sides,

$$\begin{aligned}&V(\infty )-V(0)+\int _0^\infty {z^{\mathrm {T}}(t)z(t)\text{ d }t}\\&\quad \le \gamma _f^2 \int _0^\infty {w^{\mathrm {T}}(t)w(t)\text{ d }t} +h^2\int _0^\infty {w^{\mathrm {T}}(t)B_2^{\mathrm {T}} B_2 w(t)\text{ d }t} , \end{aligned}$$

which implies that (14) holds. The proof of the system (10) in the normal case is similar, so it is omitted here. $\square$

1.2 Appendix 2: The proof of Theorem 2

Proof

Let $P^{-1}=X, P^{-1}R_1 P^{-1} =E_1, P^{-1}R_2 P^{-1} = E_2, P^{-1}SP^{-1} =F_1, \mu P^{-1}QP^{-1} =F_2, K_0 P^{-1}=Y_0 , K_{ai} P^{-1} =Y_{ai}, K_{bi} P^{-1} =Y_{bi}, M_0 P^{-1} =N_0, M_{ai} P^{-1} =N_{ai}, M_{bi} P^{-1} =N_{bi}, i=1,\ldots ,m$.

Pre- and post-multiplying $\mathrm {diag}\{P^{-1},P^{-1},P^{-1},P^{-1},P^{-1}\}$ for (18), for any $\rho \in \{{\begin{array}{ccc} {\rho ^1} &{} \cdots &{} {\rho ^L} \\ \end{array} }\}$,

$$\begin{aligned}&\varPsi _{11} +\sum \limits _{i=1}^m {\hat{\rho }_iZ _i^1} +\left( \sum \limits _{i=1}^m {\hat{\rho }_iZ _i^1}\right) ^\mathrm {T}+\sum \limits _{i=1}^m{\sum \limits _{j=1}^m {\hat{\rho }_i\hat{\rho }_jZ _{ij}}}\nonumber \\&\quad \,+\left( U_0+\sum \limits _{i=1}^m{\hat{\rho }_iU_i}\right) ^\mathrm {T}\left( U_0+\sum \limits _{i=1}^m{\hat{\rho }_iU_i}\right) <0, \end{aligned}$$

(42)

According to Lemma 1 in literature [23], if the conditions (22) and (25) hold for any $\rho \in \{{\begin{array}{ccc} {\rho ^1} &{} \cdots &{} {\rho ^L} \\ \end{array} }\}$, then (42) holds. The proof of other inequalities is similar, so these are omitted here. $\square$