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Summary This paper considers an asynchronous problem for sampled‐data control of switched linear systems, which are described as switched linear systems with an input delay. To handle the problem, this paper proposes a stability criterion for the systems by constructing a novel Lyapunov‐Krasovskii functional dependent not on system modes but on controller modes. The functional continuously remains when the system modes are switched but discontinuously changes whenever the controller mode moves to the current system mode at the sampling instants. Furthermore, the functional is allowed to increase or decrease up to a certain level when the functional discontinuously changes and to increase up to a certain level when the system modes and the controller modes are asynchronous. Based on the functional, this paper derives an average dwell time associated with the interval of samplings and the incremental level of the functional for guaranteeing the stability of the systems. A numerical example illustrates the validation of the proposed method.

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This paper is concerned with the stability analysis problems of discrete-time systems with time-varying delays using summation inequalities. In the literature focusing on the Lyapunov-Krasovskii approach, the Jensen integral/summation inequalities have played important roles to develop less conservative stability criteria and thus have been widely studied. Recently, the Jensen integral inequality was successfully generalized to the Bessel-Legendre inequalities constructed with arbitrary-order Legendre polynomials. It was also shown that general inequality contributes to the less conservatism of stability criteria. In the case of discrete-time systems, however, the Jensen summation inequality are hardly extensible to general ones since there have still not been general discrete orthogonal polynomials applicable to the developments of summation inequalities.Motivated by such observations, this paper proposes novel discrete orthogonal polynomials and then successfully derives general summation inequalities. The resulting summation inequalities are discrete-time counterparts of the Bessel-Legendre inequalities but are not based on the discrete Legendre polynomials. By developing hierarchical stability criteria based on the proposed summation inequalities, the effectiveness of the proposed approaches is demonstrated via three numerical examples for the stability analysis of discrete-time systems with time-varying delays. KEYWORDS discrete orthogonal polynomials, discrete-time system with delays, LMI, Lyapunov-Krasovskii stability theorem, stability analysis, summation inequality Int J Robust Nonlinear Control. 2019;29:473-491.wileyonlinelibrary.com/journal/rnc