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In this paper, we study the robust stability of uncertain time-delay systems. We consider uncertain quasi-polynomials whose coefficients may vary in a certain prescribed range. Our goal is to derive necessary and sufficient conditions for such uncertain quasi-polynomials to maintain stability independent of delay parameters. Our primary contributions are frequency-sweeping conditions for interval, diamond, and spherical quasi-polynomial families, which can be readily checked, requiring only the computation of two simple frequency-dependent functions. Additionally, we also obtain vertex- and edge-type results in the spirit of the Kharitonov approach known in robust stability analysis, showing that the stability of interval and diamond quasi-polynomials can be ascertained by checking the stability of certain special vertex and/or edge members in those families. Both type of results provide necessary and sufficient conditions for the quasi-polynomial families to be robustly stable independent of delay. Index Terms--Frequency-sweeping tests, robust stability, time-delay systems, uncertain quasi-polynomials, vertex tests.

In this note, we study the stability properties of linear neutral delay systems. We consider systems described by both neutral differential-difference and state-space equations, and we seek to determine the delay margin of such systems, that is, the largest range of delay values for which a neutral delay system may preserve its stability. In both cases, we show that the delay margin can be found by computing the eigenvalues and generalized eigenvalues of certain constant matrices, which can be executed efficiently and with high precision. The results extend previously known work on retarded systems, and demonstrate that similar stability tests exist for neutral systems; in particular, the tests require essentially the same amount of computation required for retarded systems. Index Terms--Delay margin, matrix pencil, neutral delay systems, stability, time delay.

We study the stability of a linear system with a point-wise, time-varying delay. We assume that the delay varies around a nominal value in a deterministic way and investigate the influence of this variation on stability. More precisely we are interested in characterizing situations where the time-varying delay system is stable, whereas the system with constant delay is unstable. Our approach consists of relating the stability properties of a system with a fast varying point-wise delay with these of a time-invariant system with a distributed delay. Then we can use frequency domain methods to analyze the problem and to derive stability criteria. The results are first illustrated with two theoretical examples. Then, we study a model of a variable speed rotating cutting tool. Based on the developed theory, we thereby provide both a theoretical explanation and a quantitative analysis tool for the beneficial effect of a variation of the machine speed on enhancing stability properties, which was reported in the literature. Index Terms--Delay, nonautonomous systems, stability of variable speed machines.

This note focuses on the static output feedback stabilization problem for a class of single-input-single-output systems when the control law includes multiple (distinct) delays. We are interested in giving necessary conditions for the existence of such stabilizing controllers. Illustrative examples (second-order system, chain of integrators, or chain of oscillators) are presented, and discussed. Index Terms--Asymptotic stability, delay, quasi-polynomial.

This note addresses the output feedback stabilization problem of a chain of integrators using multiple delays. We shall prove that either n distinct delays or a proportional+delay compensator with n--1 distinct delays are sufficient to stabilize a chain including n integrators. We present two different approaches. Both are constructive and rely on frequency-do-main techniques: on a derivative feedback approximation idea, and a pole placement idea, respectively. An illustrative example (triple integrator) is presented. Index Terms--Delay, integrator, interpolation, pole placement.

This note focuses on the stability of some class of delay systems, including uncertainty in the delays. More precisely, we are interested in guaranteeing the stability of perturbed delay systems by assuming the stability of the nominal system. If the delay perturbation is constant, necessary and sufficient conditions are derived in terms of generalized eigen-value distribution of some (finite-dimensional) constant matrix pencil. If the delay perturbation is time-varying, some sufficient stability conditions are derived using "exact" Lyapunov--Krasovskii functionals. Illustrative examples are also included. Index Terms--Delay, Lyapunov functionals, matrix pencils, uncertainty.

The problem of the global uniform asymptotic stabilization by bounded feedback of a chain of integrators with a delay in the input is solved. No limitation on the size of the delay is imposed. To validate the approach, a third-order example is presented. Index Terms--Bounded feedback, chain of integrators, time-delay systems.

This paper focuses on the design of [H.sub.[infinity]] memoryless state feedback controllers satisfying some [Alpha]-stability constraints on the closed-loop poles for a class of linear systems with delayed state, subject to disturbance inputs. A sufficient condition for feasibility is derived in terms of linear matrix inequalities. Furthermore, we also consider a particular optimization problem: the maximal bound allowed on time-delay such that the prescribed level on disturbance attenuation and the [Alpha]-stability constraints are still preserved. This problem is conversed into a linear matrix inequality (LMI) optimization problem and we give a suboptimal bound on the delay. Index Terms - [Alpha]-stability, delay systems, [H.sub.[infinity]] control, LMI approach, Lyapunov-Krasovskii functional.

This paper focuses on the problem of robust exponential stability of a class of uncertain systems described by functional differential equations with time-varying delays. The uncertainties are assumed to be continuous time-varying, nonlinear, and norm bounded. Sufficient conditions for robust exponential stability are given for both single and multiple delays cases. Index Terms - Exponential stability, robust stability, time-varying delay, uncertain systems.

This paper focuses on the problem of the control of a class of mechanical systems with a finite number of degrees-of-freedom, subject to unilateral constraints on the position. Roughly speaking, those systems are described by a set of ordinary differential equations that represent smooth dynamics, together with an algebraic inequality condition F(q) [greater than or equal to] 0 (where q is the vector of generalized coordinates) and an impact rule relating the interaction impulse and the velocity. Nonsmooth dynamics is at the core of the study of such systems. This implies one can suitably define solutions and stability concepts that fit with the considered model. Then, we discuss the closed-loop control problem, and we analyze various switching strategies. Index Terms - Closed-loop stability, mechanical systems, switching control, unilateral constraints.

A class of uncertain time-delay systems containing a saturating actuator is considered. These systems are characterized by delayed state equations (including a saturating actuator) with norm-bounded parameter uncertainty (possibly time varying) in the state and input matrices. The delay is assumed to be constant bounded but unknown. Using a Razumikhin approach for the stability of functional differential equations, upper bounds on the time delay are given such that the considered uncertain system is robustly stabilizable, in the case of constrained input, via memoryless state feedback control laws. These bounds are given in terms of solutions of appropriate finite dimensional Riccati equations.

This note presents some sufficient conditions for a linear system with (point or distributed) delay in the state to be passive. Passivity based (memoryless) stabilization of linear delay systems is also addressed. The prolapsed approach is based on a Lyapunov-Krasovskii functional construction. Numerical examples are also included. Index Terms--Delay, Lyapunov-Krasovskii, passivity.

This note focuses on the analysis of conservatism introduced due to some model transformations used in the control literature. This work is an extension of a previous paper by the ,same authors on additional dynamics. The additional dynamics of the second order and parameterized model transformations are examined. The neutral model transformation introduces additional constraints for stability, which can be analyzed in a very similar manner to additional dynamics. Index Terms--Delay, model transformation, stability.

In studying the stability of time delay systems, many published results use a transformation to transform a system with single time delay to a system with distributed delay. In this article, the inherent limitations of such approaches are studied. Specifically, it is shown that such a transformation incurs additional dynamics that can be characterized by appropriate additional eigenvalues. The critical delay values when such additional eigenvalues cross the imaginary axis can be explicitly calculated. If the smallest of such delays is less than the stability delay limit of the original system, then any stability criteria obtained using such transformation will be conservative. Some examples are also included. Index Terms--Razumikhin theorem, stability, time-delay systems.