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This paper gives a broad overview of the stability and control of time-delay systems. Emphasis is on the more recent progress and engineering applications. Examples of practical problems, mathematical descriptions, stability and performance analysis, and feedback control are discussed.

For the general linear scalar time-delay systems of arbitrary order with two delays, this article provides a detailed study on the stability crossing curves consisting of all the delays such that the characteristic quasipolynomial has at least one imaginary zero. The crossing set, consisting of all the frequencies corresponding to all the points in the stability crossing curves, are expressed in terms of simple inequality constraints and can be easily identified from the gain response curves of the coefficient transfer functions of the delay terms. This crossing set forms a finite number of intervals of finite length. The corresponding stability crossing curves form a series of smooth curves except at the points corresponding to multiple zeros and a number of other degenerate cases. These curves may be closed curves, open ended curves, and spiral-like curves oriented horizontally, vertically, or diagonally. The category of curves are determined by which constraints are violated at the two ends of the corresponding intervals of the crossing set. The directions in which the zeros cross the imaginary axis are explicitly expressed. An algorithm may be devised to calculate the maximum delay deviation without changing the number of right half plane zeros of the characteristic quasipolynomial (and preservation of stability as a special case).

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SUMMARYThis article presents a Lyapunov-Krasovskii formulation of scaled small gain problem for systems described by coupled differential-difference equations. This problem includes H ∞ problem with blockdiagonal uncertainty as a special case. A discretization may be applied to reduce the conditions into linear matrix inequalities. As an application, the stability problem of systems with time-varying delays is transformed into the scaled small gain problem through a process of either one-term approximation or two-term approximation. The cases of time-varying delays with and without derivative upper-bound are compared. Finally, it is shown that similar conditions can also be obtained by a direct Lyapunov-Krasovskii functional method for coupled differential-functional equations. Numerical examples are presented to illustrate the effectiveness of the method in tackling systems with time-varying delays.

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.