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Summary: The control problem of a general class of perturbed linear invariant time‐delay systems within the Sliding Mode Control framework is addressed. We introduce a novel class of sliding manifolds with a structure not previously reported in the literature, which constructively emerges from complete type functionals of the corresponding nominal time‐delay systems and depends on the whole state of the system. A fundamental conclusion derived from this proposal is that every time‐delay system with matched perturbations, whose associated nominal system is exponentially stabilizable, is robustly asymptotically stabilizable. In fact, the introduced sliding manifold enables to construct discontinuous and continuous sliding mode control‐based algorithms that mitigate different classes of perturbations. [ABSTRACT FROM AUTHOR]

Summary: We present novel theoretical concepts for linear time‐periodic systems with multiple delays, which are closely related to the spectral properties and Lyapunov matrices. At the basis of the main results is the associated dual system, constructed by transposition of the systems matrices and affine transformations of their arguments. We introduce, for the first time, the concepts of the ℋ2 norm and the dual Lyapunov matrix of periodic systems with delays. We show that the primal and dual system have the same ℋ2 norm, characterized by primal and dual delay Lyapunov equations, which extend the well‐known results for time‐invariant systems with delays, and periodic systems without delays. Having at hand the pair of primal‐dual Lyapunov matrices, along with some energy interpretations, allow us to generalize the concept of position balancing and explore its potential for model reduction. The obtained results are illustrated by several examples, including the delayed Mathieu equation. [ABSTRACT FROM AUTHOR]

This note addresses the study of input to state stability of nonlinear time-delay systems that are linearizable in the first approximation. It is established that, under mild conditions, the so-called complete type functional of the associated linear systems with delay is a Local Input to State Lyapunov–Krasovskii functional. Moreover, this complete type functional, which is fully defined by the delay Lyapunov matrix whenever the linear system is stable, allows the effective computation of the gains characterizing the input to state stability properties. An example illustrates the presented results. • Complete type functionals are used to study ISS of a class of time-delay systems. • A constructive approach for studying Local Input to State Stable systems is provided. • The approach allows the computation of the ISS gains. [ABSTRACT FROM AUTHOR]

Exponential stability necessary conditions for linear periodic time-delay systems are presented. They are obtained with the help of new properties of the Lyapunov matrix in the framework of Lyapunov-Krasvoskii functionals of complete type. An academic example illustrates our results. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

An overview of stability conditions in terms of the Lyapunov matrix for time-delay systems is presented. The main results and proofs are presented in details for the case of systems with multiple delays. The state of the art, ongoing research and potential extensions to other classes of delay systems are discussed. • Lyapunov necessary and sufficient stability theorems in the form of a positivity condition of the functional. • Stability tests expressed in terms of the system Lyapunov matrix. • Stability criteria that require a finite number of mathematical operations. • A general framework applicable to the stability analysis of linear systems with delays. [ABSTRACT FROM AUTHOR]

A new procedure for computing the delay Lyapunov matrix for periodic time-delay systems that is based on the numerical solution of a partial differential equations (PDE) system is presented. The introduction of a new set of boundary conditions that are satisfied by the PDE system allows us to propose a new methodology for computing the initial conditions required by the implemented numerical scheme. The potential of the presented results is demonstrated by obtaining robust stability conditions depending on the delay Lyapunov matrix with respect to the system parameters, the delay and the frequency. The theoretical results are applied to the widely known delayed Mathieu equation. [ABSTRACT FROM AUTHOR]