Adaptive mechanism for synchronization of chaotic oscillators with interval time-delays

Abstract

This paper addresses the adaptive feedback controller design for the synchronization of chaotic systems with interval time-delays in their state vectors by exploiting a lower and an upper bound on time-delays. Simple control and adaptation laws are developed for chaos synchronization, and linear matrix inequalities (LMIs) are derived to ensure asymptotic convergence of the synchronization error between the master–slave systems, using the proposed feedback control strategy, by employing a novel treatment of Lyapunov–Krasovskii functional. Further, the proposed strategy is strengthened by exploiting $L_2$ stability against disturbances and perturbations and corresponding LMIs for robust adaptive controller synthesis are derived. Furthermore, a novel delay-range-dependent robust adaptive synchronization control approach for dealing with locally Lipschitz non-delayed and delayed nonlinearities in the dynamics of chaotic oscillators is provided by employing an additional adaptation law for the nonlinearities. A numerical simulation example is provided to illustrate effectiveness of the proposed synchronization approach.

Appendices

Appendix 1: Delay-dependent case of Theorem 2

Corollary 5.1

Consider the time-delay master–slave chaotic systems (1) and (4), the adaptive nonlinear controller (8) and the error dynamics (10) satisfying time-delay bounds in (2)–(3) with $\tau _1 =0$, $\tau _2 =\tau >0$, and $\mu =0$ and Assumption 1. Suppose there exist matrices $P=P^\mathrm{T}>0$, $R=R^\mathrm{T}>0$, $Q=Q^\mathrm{T}>0$, $X=X^\mathrm{T}>0$ and $W=W^\mathrm{T}>0$, a matrix M of appropriate dimensions and a scalar $\eta >0$ such that the LMI

$$\begin{aligned} \left[ {{\begin{array}{ccccccccc} {\Delta ^{(4)}}&{} {PA_d }&{} W&{} {PB}&{} {PB_d }&{} {PH}&{} I&{} 0&{} I \\ *&{} {-R}&{} {-W}&{} 0&{} 0&{} 0&{} 0&{} I&{} 0 \\ *&{} *&{} {-\frac{1}{\tau }Q}&{} 0&{} 0&{} 0&{} 0&{} 0&{} 0 \\ *&{} *&{} *&{} {-I}&{} 0&{} 0&{} 0&{} 0&{} 0 \\ *&{} *&{} *&{} *&{} {-I}&{} 0&{} 0&{} 0&{} 0 \\ *&{} *&{} *&{} *&{} *&{} {-\eta I}&{} 0&{} 0&{} 0 \\ *&{} *&{} *&{} *&{} *&{} *&{} {-\frac{1}{L_f^2 }I}&{} 0&{} 0 \\ *&{} *&{} *&{} *&{} *&{} *&{} *&{} {-\frac{1}{L_g^2 }I}&{} 0 \\ *&{} *&{} *&{} *&{} *&{} *&{} *&{} *&{} {-X} \\ \end{array} }} \right] <0 \end{aligned}$$

holds, where

$$\begin{aligned} \Delta ^{(4)}=A^\mathrm{T}P+PA+M+M^\mathrm{T}+R+\tau Q. \end{aligned}$$

If the controller feedback gains are taken to be $K=P^{-1}M$ and $\varepsilon =O^{n\times n}$ in (8) and the adaptive parameters are updated according to (12), then the synchronization error $e(t)$ is globally asymptotically stable at the origin under $d(t)=0$ and the $L_2$ gain from the disturbance $d(t)$ to the synchronization error $e(t)$ remains bounded under $d(t)\ne 0$.

In Corollary 5.1, a delay-dependent robust adaptive control outline for chaos synchronization is inferred from the main approach in Theorem 2 using $\tau _1 =0$, $\mu =0$ and $R_1 =R_2 =0$. Similar synchronization control results, using adaptive state feedback approach, based on the traditionalistic LK functionals, are available in the literature [28–30] and [32]. Indeed, an extrapolated LK functional arrangement is applied for derivation of the delay-range-dependent adaptive control laws in Theorems 1 and 2.

Appendix 2: Delay-dependent case of Theorem 3

Corollary 5.2

Consider the time-delay master–slave chaotic systems (1) and (8), the adaptive nonlinear controller (8) and the error dynamics (10), satisfying Assumption 1 in a local region, Assumption 2, and the delay bounds in (2)–(3) with $\tau _1 =0$, and $\tau _2 =\tau >0$. Suppose there exist matrices $P=P^\mathrm{T}>0$, $R=R^\mathrm{T}>0$, $Q=Q^\mathrm{T}>0$, $Q_g =Q_g^\mathrm{T} >0$, $X=X^\mathrm{T}>0$ and $W=W^\mathrm{T}>0$ , a matrix $K$ of appropriate dimensions and a scalar $\eta >0$ such that the LMI

$$\begin{aligned} \left[ {{\begin{array}{llllll} {\coprod \nolimits ^{(3)}}&{} {A_d }&{} W&{} 0&{} H&{} I \\ *&{} {-(1-\mu )R}&{} {-W}&{} 0&{} 0&{} 0 \\ *&{} *&{} {-\frac{1}{\tau }Q}&{} 0&{} 0&{} 0 \\ *&{} *&{} *&{} {B_d^\mathrm{T} B_d -(1-\mu )Q_g }&{} 0&{} 0 \\ *&{} *&{} *&{} *&{} {-\eta I}&{} 0 \\ *&{} *&{} *&{} *&{} *&{} {-X} \\ \end{array} }} \right] <0 \end{aligned}$$

holds, where

$$\begin{aligned} \coprod \nolimits ^{(3)}=A^\mathrm{T}+A+K+K^\mathrm{T}+R+\tau Q. \end{aligned}$$

If the adaptive gain of the controller is selected as $\varepsilon ={\text {diag}}(\varepsilon _1 ,\varepsilon _2 ,\ldots ,\varepsilon _n )$ and the adaptive parameters are updated according to (12) and (28), the synchronization error $e(t)$ is globally asymptotically stable at the origin under $d(t)=0$ and the $L_2$ gain from the disturbance $d(t)$ to the synchronization error $e(t)$ remains bounded under bounded disturbances satisfying $d(t)\ne 0$.

Corollary 5.2 establishes an adaptive delay-dependent control schema for attaining identical behavior of the chaotic oscillators (1) and (4) under disturbances and locally Lipschitz nonlinearities as a special result of the proposed approach in Theorem 3. It has been observed that the delay-dependent synchronization schemes, not unlike Corollary 5.2, rendering an adaptive dealing of both the non-delayed and the delayed locally Lipschitz nonlinear functions are deficient in the literature.