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This paper studies the migration of double imaginary roots of the systems’ characteristic equation when two parameters are subjected to small deviations. The proposed approach covers a wide range of models. Under the least degeneracy assumptions, we found that the local stability crossing curve has a cusp at the point that corresponds to the double root, and it divides the neighborhood of this point into an S-sector and a G-sector. When the parameters move into the G-sector, one of the roots moves to the right half-plane, and the other moves to the left half-plane. When the parameters move into the S-sector, both roots move either to the left half-plane or the right half-plane depending on the sign of a quantity that depends on the characteristic function and its derivatives up to the third order. [ABSTRACT FROM AUTHOR]

This article studies the strong stability of scalar difference equations of continuous time in which the delays are sums of a number of independent parameters τ i , i = 1 , 2 , … , K . The characteristic quasipolynomial of such an equation is a multilinear function of e − τ i s . It is known that the characteristic quasipolynomial of any difference equation set in the form of one-delay-per-scalar-channel (ODPSC) model is also in such a multilinear form. However, it is shown in this article that some multilinear forms of quasipolynomials are not characteristic quasipolynomials of any ODPSC difference equation set. The equivalence between local strong stability, the exponential stability of a fixed set of rationally independent delays, and the stability for all positive delays is shown, and relations with the structured singular value problem are presented. A procedure to determine strong stability in the special case of up to three independent delay parameters in finite steps is developed. This procedure means that the structured singular value problem in the case of up to three scalar complex uncertain blocks can be solved in finite steps. [ABSTRACT FROM AUTHOR]

Abstract: A new form of coupled differential–difference equations with one delay in each channel is proposed to model systems with multiple delays. This formulation has significant advantage over the traditional differential–difference equations or the prevailing form of coupled differential–difference equations with multiple delays. Fundamental solutions, general solutions, and the construction of the Lyapunov–Krasovskii functional are discussed. For systems with a large number of state variables with multiple low-dimensional delay elements, this formulation allows a drastic reduction of computational cost as compared to the traditional differential–difference equation formulation when the discretized Lyapunov–Krasovskii functional method is used. [Copyright &y& Elsevier]

Abstract: This article discusses the Lyapunov–Krasovskii functional approach for the stability problem of coupled differential-functional equations. Such systems include, as special cases, many types of time-delay systems, including the lossless propagation model, some neutral time-delay systems and singular time-delay systems. After the general stability theory, the special case of coupled differential-difference equations is discussed, and the necessity for the existence of quadratic Lyapunov–Krasovskii functional is established. Discretization is used to render the stability criterion to an LMI form. [Copyright &y& Elsevier]

Abstract: 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). [Copyright &y& Elsevier]