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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 τii= 1, 2,..., K. The characteristic quasipolynomial of such an equation is a multilinear function of e-Tis. 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.

This article presents the inverse of the kernel operator associated with the complete quadratic Lyapunov-Krasovskii functional for coupled differential-functional equations when the kernel operator is separable. Similar to the case of time-delay systems of retarded type, the inverse operator is instrumental in control synthesis. Unlike the power series expansion approach used in the previous literature, a direct algebraic method is used here. It is shown that the domain of definition of the infinitesimal generator is an invariant subspace of the inverse operator if it is an invariant subspace of the kernel operator. The process of control synthesis using the inverse operator is described, and a numerical example is presented using the sum-of-square formulation.

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. The characteristic quasipolynomial of such systems contains an exponential term with the sum of two delays (cross term) in its exponent. 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.

This paper focuses on the stability analysis of the closed-loop single-input-single-output (SISO) system subject to input delays and to PI controllers. A complete characterization of the crossing set, which consists of all frequencies where the number of unstable characteristic roots changes, is given in the parameter space defined by the controller. Next, we explicitly derive the PI controllers corresponding to each frequency in the crossing set. More precisely, we make a partition of the controller parameter space in regions where the number of unstable characteristic roots remains constant. Finally, we also present a methodology to compute the number of unstable roots in each region. Several illustrative examples end the presentation.

This article revisits some delay-dependent stability criteria previous discussed in the literature. A more systematic approach is taken, and different forms of stability criteria are developed. The form is more convenient to treat related uncertainty in different system matrices. The relationship between the stability criteria based on simple Lyapunov-Krasovskii functional method and on Razumikhin Theorem are more explicitly shown

This paper investigates the robust stability problem of time-delay systems with multiple passive uncertainties. A stability criterion is derived using the refined discretized Lyapunov functional method. The criterion is written in the form of a linear matrix inequality. Numerical examples are presented to illustrate the effectiveness of the method.

This article improves the formulation of discretized Lyapunov functional method for systems with distributed delays. The main idea is to apply Jensen's inequality and variable elimination in matrix inequalities. The resulting stability criterion is much simpler, and converges to the analytical result much faster. The new formulation is also applicable to a wider class of systems.