Lyapunov matrix based necessary and sufficient stability condition by finite number of mathematical operations for retarded type systems

This paper investigates the stability of a linear finite-dimensional system interconnected to a single delay operator. From robust approaches, to derive delay-dependent frequency tests, a characterization of the delay behavior is required. Based on approximation methods, one describes the transported signal by lumped parameters. More precisely, by the use of the first Fourier-Legendre polynomials coefficients, we split the delay block into a finitedimensional part interconnected to a specific infinite-dimensional residual part. Two models are investigated with residuals related to two Fourier-Legendre remainders of the delayed transfer function. The main contribution is to highlight that the finite-dimensional models based on the first Legendre coefficients are proven to be related to Padé approximations and are recognized to be more and more accurate as the dimension increases. Interestingly, this modeling allows computing in an accurate manner the root locus of time-delay systems. Furthermore, as a by-product of this result, taking into account the infinite-dimensional remainders to keep track of the initial time-delay system, stability criteria are proposed by H∞ analysis. Considering both infinite-dimensional remainders as bounded delay-free uncertainties, the small-gain theorem provides a new sufficient condition of stability for retarded time-delay systems, which can be implemented as a delay-dependent frequency-sweeping test. Our results are illustrated on several academic examples.

Section: Illustration Of the Small Gain Theoremmentioning

confidence: 82%

Section: Illustration Of the Small Gain Theoremmentioning

confidence: 85%

Section: Augmented Time-delay Systemsmentioning

confidence: 99%

“…These two augmented time-delay systems (27), (29) have good properties to deal with spectrum analysis and assess stability of time-delay systems.…”

Section: X(t) Hz N (T)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Frequency delay‐dependent stability criterion for time‐delay systems thanks to Fourier–Legendre remainders

Bajodek

Gouaisbaut

Seuret

2021

Vladimir Kharitonov: Achievements in research

Egorov

2022

A survey of the main research results of Professor Kharitonov is provided. The following topics are considered: stability analysis of interval polynomials, quasi‐polynomials, and multivariate polynomials; Kalman's hypothesis on the generalization of the Lyapunov matrix equation; critical delay values, complete type functionals and prediction‐based control for time‐delay systems. The fundamentals of Kharitonov's work and the influence of his results on other researchers are discussed.

Stability tests for neutral‐type delay systems: A delay Lyapunov matrix and piecewise linear approximation of the functional argument approach

Portilla,

Alexandrova,

Mondié

2024

Self Cite

New necessary and sufficient stability tests for neutral‐type linear time delay systems are presented. They are based on functionals with prescribed derivative depending on the delay Lyapunov matrix and recent stability/instability theorems. A piecewise linear approximation of the initial function argument in the functional is employed. As a result, the functional is approximated via a quadratic form whose matrix provides stability criteria. This matrix involves the integrals based on the delay Lyapunov matrix which are handled via an efficient computational procedure. The main feature of these tests is that they are verified in a finite number of operations, which is achieved through rigorous bounding of the approximation error on a specific class of functions. Some examples which validate the obtained results are presented and discussed.