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The numerical minimization of the spectral abscissa function of linear time-invariant time-delay systems, an established approach to compute stabilizing controllers with a fixed structure, often gives rise to minima characterized by active characteristic roots with multiplicity higher than one. At the same time, recent theoretical results reveal situations where the so-called multiplicity induced dominancy property holds, i.e. a sufficiently high multiplicity implies that the root is dominant. Using an integrative approach, combining analytical characterizations, computation of characteristic roots and numerical optimization, a complete characterization of the stabilizability of second-order systems with input delays is provided, for both state feedback and delayed output feedback. The level sets of the minimal achievable spectral abscissa are also characterized. These results shed light on the complex relations between (configurations involving) multiple roots, the property of being dominant roots and the property of corresponding to (local/global) minimizers of the spectral abscissa function. [ABSTRACT FROM AUTHOR]

This work revisits recent results on maximal multiplicity induced-dominancy for spectral values in reduced-order time-delay systems and extends it to the general class of second-order retarded differential equations. A parametric multiplicity-induced-dominancy property is characterized, allowing to a delayed stabilizing design with reduced complexity. As a matter of fact, the approach is merely a delayed-output-feedback where the candidates' delays and gains result from the manifold defining the maximal multiplicity of a real spectral value, then, the dominancy is shown using the argument principle. Sensitivity of the control design with respect to the parameters uncertainties/variation is discussed. Various reduced order examples illustrate the applicative perspectives of the approach. [ABSTRACT FROM AUTHOR]

In this paper, the output feedback stabilization of an underactuated chain of scalar hyperbolic systems coupled at one end with a finite-dimensional system is addressed. As encountered in practical applications, only the first PDE subsystem is actuated. The measurements are collocated with the actuation. The proposed approach uses a recursive dynamics interconnection framework. More precisely, we first estimate (delayed) values of the states at each subsystem's boundaries. Then, we design a state-predictor that gives access to the boundary states' present and future values. It becomes finally possible to design a state-observer for the entire system. Combining this observer with a stabilizing state-feedback law, we can recursively design an output-feedback control law that stabilizes the whole chain. Some illustrative examples complete the presentation. [ABSTRACT FROM AUTHOR]

This article presents basic concepts and recent research directions about the stability of sampled-data systems with aperiodic sampling. We focus mainly on the stability problem for systems with arbitrary time-varying sampling intervals which has been addressed in several areas of research in Control Theory. Systems with aperiodic sampling can be seen as time-delay systems, hybrid systems, Input/Output interconnections, discrete-time systems with time-varying parameters, etc. The goal of the article is to provide a structural overview of the progress made on the stability analysis problem. Without being exhaustive, which would be neither possible nor useful, we try to bring together results from diverse communities and present them in a unified manner. For each of the existing approaches, the basic concepts, fundamental results, converse stability theorems (when available), and relations with the other approaches are discussed in detail. Results concerning extensions of Lyapunov and frequency domain methods for systems with aperiodic sampling are recalled, as they allow to derive constructive stability conditions. Furthermore, numerical criteria are presented while indicating the sources of conservatism, the problems that remain open and the possible directions of improvement. At last, some emerging research directions, such as the design of stabilizing sampling sequences, are briefly discussed. [ABSTRACT FROM AUTHOR]

The PID control is favored in controlling industrial processes for its ease of implementation. In this paper we present an analytical study, with an aim at stability robustness, tracking performance, and pole placement of first-order unstable plants under PID control. We employ a multiplicity-induced dominancy (MID) strategy to design and tune PID controller gains, which is shown to possess desirable properties that ensure specified decay rate of the closed-loop response and the stability robustness of the closed-loop system in spite of variations in the delay parameter. We also study the tradeoff between delay robustness and tracking performance of PID controllers. We show that under the constraint of steady-state tracking, the maximal delay robustness range can be computed by solving a unimodal pseudo-concave optimization problem. [ABSTRACT FROM AUTHOR]

In most of the numerical examples of time-delay systems proposed in the literature, the number of unstable characteristic roots remains positive before and after a multiple critical imaginary root (CIR) appears (as the delay, seen as a parameter, increases). This fact may lead to some misunderstandings: (i) A multiple CIR may at most affect the instability degree; (ii) It cannot cause any stability reversals (stability transitions from instability to stability). As far as we know, whether the appearance of a multiple CIR can induce stability is still unclear (in fact, when a CIR generates a stability reversal has not been specifically investigated). In this paper, we provide a finer analysis of stability reversals and some new insights into the classification: the link between the multiplicity of a CIR and the asymptotic behavior with the stabilizing effect. Based on these results, we present an example illustrating that a multiple CIR's asymptotic behavior is able to cause a stability reversal. To the best of the authors' knowledge, such an example is a novelty in the literature on time-delay systems. [ABSTRACT FROM AUTHOR]