About global asymptotic stability of dynamic systems with time lags and uncertainties within polytopes

Abstract

Purpose

This purpose of this paper is to discuss a linear fractional representation (LFR) of parameter‐dependent systems which are linear in the parameters but uncertain, being eventually time‐varying real‐rational nonlinear parameterizations, and dynamics with constant point delays.

Design/methodology/approach

The formulation is made in terms of Lyapunov's second method whereby the Lyapunov function candidate is confirmed to be a Lyapunov function by testing a finite number of linear‐matrix inequalities when the uncertain parameter vector, which might be time‐varying, lies within a known polytope which characterizes the uncertainties. The tests are performed only on the set of vertices associated with polytopes.

Findings

Sufficient conditions for global asymptotic stability are obtained. Conditions constraining the system to be slowly time‐varying around a stable nominal parameterization are not imposed in order to guarantee the stability.

Research limitations/implications

The formulation is applied to a class of systems whose uncertainties might be parameterized through time‐varying real‐rational nonlinear parameterizations and which include point‐delayed dynamics with constant delays. However, such a class includes certain classes of neural networks with delays, systems with switched parameterizations and systems whose uncertain dynamics evolve arbitrarily in regions defined by known polytopes.

Practical implications

The stability tests are less involved than usual for time‐varying systems since only a finite number of them is necessary to investigate the stability.

Originality/value

LFR descriptions of linear time‐varying systems are extended to a wide class of systems with constant point delays. Also, the real‐rational nonlinear parameterizations of the uncertainties are admitted in both the delay‐free and delayed dynamics.

Citation

De la Sen, M. (2007), "About global asymptotic stability of dynamic systems with time lags and uncertainties within polytopes", Engineering Computations, Vol. 24 No. 6, pp. 564-571. https://doi.org/10.1108/02644400710774770