Vladimir L. Kharitonov

Lyapunov-Krasovskii approach to the robust stability analysis of time-delay systems

In this paper, a procedure for construction of quadratic Lyapunov-Krasovskii functionals for linear time-delay systems is proposed. It is shown that these functionals admit a quadratic low bound. The functionals are used to derive robust stability conditions.

Robust stability analysis of time delay systems: A survey

Abstract In this survey some recent contributions to stability and robust stability analysis of linear time delay systems with parameter uncertainty are discussed. The main aim of the paper is to discuss some basic techniques used for deriving tractable stability and robust stability conditions. The reference list presents rather a small portion of the large number of publications in this field.

Exponential estimates for retarded time-delay systems: an LMI approach

Exponential estimates and sufficient conditions for the exponential stability of linear time delay systems are given. The proof make use of Lyapunov-Krasovskii functionals and the conditions are expressed in terms of linear matrix inequalities.

Robust stability of time-delay systems

There are two fundamental results available when we study stability of a polynomial family that is described by convex polytope in the coefficient space: the edge theorem and the theory based on the concept of convex directions. Many known results can be explained with these two results. This paper deals with a generalization of this line of research to the case of quasipolynomials that are entire functions which include both degree of the independent variable and exponential functions. The main objects of the paper are the developing of the concept of convex directions for quasipolynomials and exploiting this concept for construction of testing sets for quasipolynomial families. One of the primary sources of motivation for the class of problems considered in this paper is derived from process control. A typical problem formulation almost always includes a delay element in each subsystem process block. When we interconnect a number of such blocks in a feedback system, the study of robust stability involves quasipolynomials of the sort considered in this paper. >

On delay-dependent stability conditions ☆

It is quite common in stability analysis of time-delay systems to make a special transformation of the system under investigation in order to obtain stability conditions which depend on the values of the delays. In this note we discuss additional conditions for stability and robust stability of the transformed system which do not appear when the original system is considered.

Exponential estimates for time delay systems

Abstract In this paper, we demonstrate how Lyapunov–Krasovskii functionals can be used to obtain exponential bounds for the solutions of time-invariant linear delay systems.

Static output feedback stabilization: necessary conditions for multiple delay controllers

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.

Exponential estimates for neutral time delay systems: an LMI approach

Exponential estimates and sufficient conditions for the exponential stability of linear neutral time delay systems are given. The estimates are obtained for the case of known parameters as well as the uncertain case, including uncertainties in the difference term. The proof is based on Lyapunov-Krasovskii functionals, and the conditions are expressed in terms of linear matrix inequalities (LMIs).

Lyapunov functionals and Lyapunov matrices for neutral type time delay systems: a single delay case

Quadratic Lyapunov functionals with a given time derivative for the case of neutral type time delay systems have been presented in Rodriguez et al. (“Robust stability of neutral systems: a Lyapunov–Krasovskii constructive approach”, International Journal of Robust and Nonlinear Control, 14, pp. 1345–1358, 2004). In this contribution a new form of the functionals is proposed. The functionals now do not include the time derivative of the system state. This modification provides new quadratic bounds for the functionals and makes them useful in computation of exponential estimates for solutions of the systems, as well as in calculation of the robustness bounds. Special attention has also been paid to the Lyapunov matrices which define the functionals. A new definition of the matrices is given, and their properties are analysed in detail.Quadratic Lyapunov functionals with a given time derivative for the case of neutral type time delay systems have been presented in Rodriguez et al. (“Robust stability of neutral systems: a Lyapunov–Krasovskii constructive approach”, International Journal of Robust and Nonlinear Control, 14, pp. 1345–1358, 2004). In this contribution a new form of the functionals is proposed. The functionals now do not include the time derivative of the system state. This modification provides new quadratic bounds for the functionals and makes them useful in computation of exponential estimates for solutions of the systems, as well as in calculation of the robustness bounds. Special attention has also been paid to the Lyapunov matrices which define the functionals. A new definition of the matrices is given, and their properties are analysed in detail.

On delay-dependent stability conditions for time-varying systems

In this paper, some recent results on additional dynamics for transformed time-delay systems are extended to the case of time-varying systems. Special equations which describe these dynamics are derived. Additional restrictions on stability and robust stability imposed by the transformations are obtained.