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This paper presents a systematic method to analyze the stability of systems with single delay in which the coefficient polynomials of the characteristic equation depend on the delay. Such systems often arise in, for example, life science and engineering systems. A method to analyze such systems was presented by Beretta and Kuang in a 2002 paper, but with some very restrictive assumptions. This paper extends their results to the general case with the exception of some degenerate cases. It is found that a much richer behavior is possible when the restrictive assumptions are removed. The interval of interest for the delay is partitioned into subintervals so that the magnitude condition generates a fixed number of frequencies as functions of the delay within each subinterval. The crossing conditions are expressed in a general form, and a simplified derivation for the first-order derivative criterion is obtained. Illustrative examples are also presented in this paper. [ABSTRACT FROM AUTHOR]

This article describes the stability crossing set for linear time-delay systems of arbitrary order with three delays. The crossing frequency set, which consists of all frequencies where a pair of zeros of the characteristic quasipolynomial may cross the imaginary axis, is partitioned to Grashof sets and Non-Grashof sets of various types. It was found that the general characteristics of the stability crossing set is completely determined by the partition structure of the crossing frequency set. With the exception of degenerate cases, this article provides a method of explicit and complete parameterization and geometric characterization of the stability crossing set of linear systems with three delays. With the well-known method of finding the crossing directions, this provides a powerful method of finding parameter regions of stable systems. Extension to systems with additional fixed delays is also discussed. [ABSTRACT FROM AUTHOR]

This technical note considers the problem of reducing the computational complexity associated with the Sum-of-Squares approach to stability analysis of time-delay systems. Specifically, this technical note considers systems with a large state-space but where delays affect only certain parts of the system. This yields a coefficient matrix of the delayed state with low rank—a common scenario in practice. The technical note uses the general framework of coupled differential-difference equations with delays in feedback channels. This framework includes systems of both the neutral and retarded-type. The approach is based on recent results which introduced a new Lyapunov–Krasovskii structure which was shown to be necessary and sufficient for stability of this class of systems. This technical note shows how exploiting the structure of the new functional can yield dramatic improvements in computational complexity. Numerical examples are given to illustrate this improvement. [ABSTRACT FROM AUTHOR]

This note is concerned with the stability problem of linear delay-differential systems of neutral type. A discretized Lyapunov functional approach is developed. The resulting stability criterion is formulated in the form of a linear matrix inequality. For nominal systems, the analytical results can be approached with fine discretization. Numerical examples show significant improvement over approaches in the literature. [ABSTRACT FROM AUTHOR]

Examines additional dynamics in transformed time-delay systems. Transformation of a system with single time delay to a system with distributed delay; Additional dynamics that can be characterized by appropriate additional eigenvalues; Stability criteria obtained.

Control systems often operate in the presence of delays, primarily due to the time it takes to acquire the information needed for decision-making, to create control decisions, and to execute these decisions, as shown in Figure 1. Systems with delays arise in engineering, biology, physics, operations research, and economics. In traffic-flow models, the drivers' delayed reactions, which combine sensing, perception, response, selection, and programming delays, must be considered [1]–[3]. These delays are critical in accounting for human behavior, analyzing traffic-flow stability, and designing collision-free traffic flow using adaptive cruise controllers [4]. [ABSTRACT FROM PUBLISHER]