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This article presents the inverse of the kernel operator associated with the complete quadratic Lyapunov-Krasovskii functional for coupled differential-functional equations when the kernel operator is separable. Similar to the case of time-delay systems of retarded type, the inverse operator is instrumental in control synthesis. Unlike the power series expansion approach used in the previous literature, a direct algebraic method is used here. It is shown that the domain of definition of the infinitesimal generator is an invariant subspace of the inverse operator if it is an invariant subspace of the kernel operator. The process of control synthesis using the inverse operator is described, and a numerical example is presented using the sum-of-square formulation.

This paper reviews some subtleties in time-delay systems of neutral type that are believed to be of particular relevance in practice. Both traditional formulation and the coupled differential-difference equation formulation are used. The discontinuity of the spectrum as a function of delays is discussed. Conditions to guarantee stability under small parameter variations are given. A number of subjects that have been discussed in the literature, often using different methods, are reviewed to illustrate some fundamental concepts. These include systems with small delays, the sensitivity of Smith predictor to small delay mismatch, and the discrete implementation of distributed-delay feedback control. The framework prsented in this paper makes it possible to provide simpler formulation and strengthen, generalize, or provide alternative interpretation of the existing results.

International audience

International audience; This chapter is devoted to the stability problem of time-delay systems using time-domain approach. Some basic concepts of time-delay systems are introduced. Then some simple Lyapunov-Krasovskii funtionals complete Quadratic Lyapunov-Krasovskii functional and discretization scheme are introduced with connections and extent of conservatism compared. The issue of time-varying delays are also discussed. The concept of Razumikhin Theorem is introduced. An alternative model of coupled difference-differential equations and its stability problem are also introduced. © Springer-Verlag London Limited 2006.

International audience; In the mathematical description of a physical or biological process, it is a common practice \0 assume that the future behavior of Ihe process considered depends only on the present slate, and therefore can be described by a finite sct of ordinary diffe­ rential equations. This is satisfactory for a large class of practical systems. However. the existence of lime-delay elements, such as material or infonnation transport, of­ tcn renders such description unsatisfactory in accounting for important behaviors of many practical systems. Indeed. due largely to the current lack of effective metho­ dology for analysis and control design for such systems, the lime-delay elements arc often either neglected or poorly approximated, which frequently results in analysis and simulation of insufficient accuracy, which in turns leads to poor performance of the systems designed. Indeed, it has been demonstrated in the area of automatic control that a relatively small delay may lead to instability or significantly deteriora­ ted perfonnances for the corresponding closed-loop systems.

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The problem of transforming a class of linear time-varying continuous time systems into controllable and observable block companion canonical forms is considered. In terms of system block controllability (observability) matrix, this paper generalizes the results of Shieh et al. [3] and provides systematic and straightforward algorithms for obtaining block companion canonical forms. An example is provided to illustrate this transformation technique.