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SUMMARYThe issue of exponential stability analysis of continuous‐time switched singular systems consisting of a family of stable and unstable subsystems with time‐varying delay is investigated in this paper. It is very difficult to analyze the stability of such systems because of the existence of time‐delay and unstable subsystems. In this regard, on the basis of the free‐weighting matrix approach, by constructing the new Lyapunov‐like Krasovskii functional, and using the average dwell‐time approach, delay‐dependent sufficient conditions are derived and formulated in terms of LMIs to check the exponential stability of such systems. This paper also highlights the relationship between the average dwell‐time of the switched singular time‐delay system, its stability, exponential convergence rate of differential states, and algebraic states. Finally, a numerical example is given to confirm the analytical results and illustrate the effectiveness of the proposed strategy. Copyright © 2012 John Wiley & Sons, Ltd.

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Summary In this work, the problem of stability analysis for a class of singular hybrid switched and impulsive system (HSIS) is addressed. Corresponding to each subsystem, a hybrid switched and impulsive controller is designed and then the exponential stability property of the proposed singular HSIS is discussed for linear and nonlinear cases. Because switched systems without impulses are a special case of HSISs, the results are also given to switched system with synchronous and asynchronous controllers. The obtained results apply to control singular systems, and the introduced theorems allow knowing how the control must be designed. Two numerical examples are given to show the effectiveness of the proposed approaches. At first, by using MATLAB® software, the proposed method is applied to a class of physiological processes of endocrine disruptor diethylstilbestrol models to illustrate the effectiveness of the results obtained here for the linear case. Thereafter, another numerical example is provided to support the presented theoretical results for the nonlinear case.

SummaryThis paper presents an approach to the stability analysis of a class of nonlinear interconnected continuous‐time singular systems with arbitrary switching signals. This class of interconnected subsystems consists of unknown but bounded state delay and nonlinear terms, and each subsystem can be globally stable, unstable, or locally stable. By constructing a new Lyapunov‐like Krasovskii functional, sufficient conditions are derived and formulated to check the asymptotic (exponential) stability of such systems with arbitrary switching signals. Then, some new general criteria for asymptotic (exponential) stability with average dwell‐time switching signals are also established. The theoretical developments are demonstrated by two numerical simulations. Copyright © 2014 John Wiley & Sons, Ltd.

This article presents a robust adaptive controller for chaos synchronization using the Szász–Mirakyan operator as a universal approximator. In accordance with the universal approximation theorem, the Szász–Mirakyan operator, an extended version of the Bernstein polynomial, can approximate uncertainties, including unmodeled dynamics and external disturbances. This fact is completely discussed in this article. It is shown that using the Szász–Mirakyan operator as basis functions and tuning the polynomial coefficients by the adaptive laws calculated in the stability analysis, uniformly ultimately bounded stability can be assured. Performance evaluation has also been carried out to confirm the satisfactory performance of transient response of the controller. Numerical simulations on the Duffing–Holmes oscillator are provided in order to demonstrate the effectiveness of this approach.

SummaryThe issues of stability analysis of a class of continuous uncertain switched singular time‐delay systems consisting of discrete and distributed delays is investigated in this paper. Based on the LMIs, constructing the new extended Lyapunov‐like Krasovskii functional, and the average dwell‐time approach, delay‐dependent sufficient conditions are derived to check the stability of such systems. By solving the corresponding LMIs, the average dwell‐time and switching signal condition are obtained. This paper also highlights the relationship between the average dwell‐time of the switched singular time‐delay system, exponential rate of differential, and algebraic states. Two numerical examples show the effectiveness of the proposed design method. Copyright © 2013 John Wiley & Sons, Ltd.