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This article developed a new method to described the set of stabilizing PID control. The method is based on D‐parameterization with natural description of the set. It was found that the stability crossing surface is a ruled surface that is completely determined by a curve known as discriminant. The discriminant is divided into sectors at the cusps. Corresponding to the sectors, the stability crossing surface is divided into positive and negative patches. A systematic study is conducted to identify the regions with a fixed number of right half‐plane characteristic roots. The crossing directions of characteristic roots for positive patches and negative patches are also studied. As a result, a systematic method is developed to identify the regions of PID parameter such that the system is stabilized. [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: This article studies the stability crossing set of linear time-invariant systems with two scalar-delay channels. This study is crucial to the complete stability analysis along the idea of D-subdivision method. The characteristic quasipolynomial of such systems contains an exponential term with the sum of two delays in its exponent (cross term). A complete parameterization and geometric characterization of the stability crossing set is conducted. It was found instrumental to relate it to an associated quasipolynomial without such a cross term through an elimination process. However, a spurious stability crossing set may arise in this process. A revised parameterization method is derived that automatically eliminates such a spurious stability crossing set. [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]