Computation of Imaginary Axis Eigenvalues and Critical Parameters for Neutral Time Delay Systems

Abstract

Abstract. In this paper, a procedure for the computation of the Lyapunov matrix of neutral type systems, with delays multiple of a basic one, is recalled: It consists in solving a boundary value problem for a delay free system of matrix equations. The important property that the elements of the spectrum of the delay system that are symmetric with respect to the imaginary axis belong to the spectrum of the delay free system is also established. This property is exploited for the determination of the critical values of the time delay system.

A survey of inequality techniques for stability analysis of time‐delay systems

“…The Jensen's double-integral inequality (34) was originally obtained from Schur complement. 1 It also can be obtained via Cauchy-Schwartz inequality ⟨p, p⟩ 2 ⟨1, 1⟩ 2 ≥ ⟨p, 1⟩ 2 ⟨p, 1⟩ 2 , with the inner product defined as follows…”

“…In frequency-domain approach, the eigenvalues of a system matrix are calculated to study the stability of time-delay systems. 12,[32][33][34][35][36][37] In time-domain approach, the Lyapunov-Krasovskii functional (LKF) is widely employed, along with the linear matrix inequality (LMI) technique. 38,39 There are several ways to reach a necessary and sufficient stability condition for linear systems with a constant time delay, for example, by constructing the complete LKF method.…”

“…There are two main approaches to stability analysis of time‐delay systems: frequency‐domain approach and time‐domain approach. In frequency‐domain approach, the eigenvalues of a system matrix are calculated to study the stability of time‐delay systems 12,32‐37 . In time‐domain approach, the Lyapunov–Krasovskii functional (LKF) is widely employed, along with the linear matrix inequality (LMI) technique 38,39 .…”

Vladimir Kharitonov: Achievements in research

“…The results on complete type functionals embodied in Kharitonov's monograph 62 allow the solution of several issues: analyze the robust stability with respect to time-invariant, time-varying or even nonlinear disturbances, 62,63,88,[108][109][110][111][112][113] determine the critical values of the parameters at which the stability is lost, 62,[114][115][116][117][118][119] compute the  2 norm, 84,85,120 estimate the domain of attraction, [121][122][123][124][125] compute the quadratic performance indices, 62,77,126 synthesize suboptimal control laws. [127][128][129][130][131][132] Another application is, of course, the stability analysis of individual systems based on Theorem 3.…”