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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 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]

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]