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In this article, the tracking control problem for a class of nonlinear time‐delay systems is investigated. In particular, a new methodology for the design and the digital implementation of observer‐based tracking controllers is provided for a class of control‐affine nonlinear systems with state delays. First, a procedure for the design of continuous‐time observer‐based tracking controllers ensuring the global asymptotic stability of the corresponding closed‐loop tracking error system is provided for the considered class of systems. Then, sufficient conditions are provided for the existence of a suitably fast sampling and of an accurate quantization of the input/output channels such that the digital implementation of the proposed continuous‐time observer‐based tracking controller ensures the semi‐global practical stability property of the related sampled‐data quantized closed‐loop tracking error system, with arbitrarily small final target ball of the origin. Moreover, it is shown that, in the special case of delay‐free nonlinear systems, the sufficient conditions provided for the digital implementation of the proposed continuous‐time observer‐based tracking controller can be strongly relaxed. In the theory here developed, time‐varying sampling periods and nonuniform quantization of the input/output channels are taken into account. The proposed results are validated through examples concerning a class of neural networks systems and a class of time‐delay systems including, as a special case, a delay‐free actuated inverted pendulum. [ABSTRACT FROM AUTHOR]

The primary contribution of the paper is to show that the Lyapunov–Krasovskii conditions available in the literature, for checking the input-to-state stability property of systems described by neutral functional differential equations in Hale’s form, are also necessary. It is shown that the Lyapunov–Krasovskii conditions available in the literature are equivalent to the input-to-state stability property, for an enlarged class of neutral systems. Finally, a novel necessary and sufficient condition for the input-to-state stability property is provided. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we present a Lyapunov–Krasovskii methodology for checking the global asymptotic stability and the input-to-state stability of systems described by retarded functional differential equations coupled with continuous time difference equations, often referred to as special neutral systems, with un-matched and piece-wise continuous initial conditions. The methodology provides results in terms of the norm for the non differentiated variable and does not require the preliminary check of input-to-state stability for the difference part of the system. The Lyapunov conditions are given without involving, not even formally, the solution. [Copyright &y& Elsevier]

Abstract: The input-to-state stability of time-invariant systems described by coupled differential and difference equations with multiple noncommensurate and distributed time delays is investigated in this paper. Such equations include neutral functional differential equations in Hale’s form (which model, for instance, partial element equivalent circuits) and describe lossless propagation phenomena occurring in thermal, hydraulic and electrical engineering. A general methodology for systematically studying the input-to-state stability, by means of Liapunov–Krasovskii functionals, with respect to measurable and locally essentially bounded inputs, is provided. The technical problem concerning the absolute continuity of the functional evaluated at the solution has been studied and solved by introducing the hypothesis that the functional is locally Lipschitz. Computationally checkable LMI conditions are provided for the linear case. It is proved that a linear neutral system in Hale’s form with stable difference operator is input-to-state stable if and only if the trivial solution in the unforced case is asymptotically stable. A nonlinear example taken from the literature, concerning an electrical device, is reported, showing the effectiveness of the proposed methodology. [Copyright &y& Elsevier]

Abstract: In [Driver, R. D. (1962). Existence and stability of solutions of a delay-differential system. Archive for Rational Mechanics and Analysis 10, 401–426] a proper definition, not involving the solution, of the derivative of the Liapunov–Krasovskii functional for retarded functional differential equations with continuous right side is given and it is showed that this definition coincides with the non-constructive one given in Krasovskii [1956. On the application of the second method of A. M. Lyapunov to equations with time delays (in Russian). Prikladnaya Matematika i Mekhanika 20, 315–327] involving the solution, for functionals V which are locally Lipschitz (and not only continuous, as it is considered in most literature). In this paper, the result by Driver is extended to a general class of retarded functional differential equations coupled with continuous time difference equations with more general right sides, verifying the Carathéodory conditions. Such result is applied to build a new Liapunov–Krasovskii theorem for studying the input-to-state stability of time-invariant neutral functional differential equations with linear difference operator. An example taken from the literature, concerning transmission lines, is reported, showing the effectiveness of the methodology. [Copyright &y& Elsevier]