Stability of piecewise affine systems with state-dependent delay, and application to congestion control

Abstract

Abstract-In this work, we consider the exponential stability of piecewise affine systems with time-and state-dependent delay, and delayed-state-dependent switching. The stability analysis is based on the use of Lyapunov-Krasovskii functionals, and is divided into two parts. First, global stability conditions are proposed in the case of systems with (state-independent) time-varying delay. Then, local stability conditions are derived in the case of systems with time-and state-dependent delay. In the latter case, estimations of the domain of attraction are also proposed. The theoretical results are applied to the congestion control problem, which can be modelled by such systems.

Stability analysis of a class of uncertain switched time‐delay systems with sliding modes

“…In the congestion control problem, automotive all‐wheel drive clutch systems, and some mechanical systems with some hydrodynamical phenomena such as high speed supercavitation vehicles (HSSVs), the time‐delayed states of the system appear in the dynamical equations and switching conditions, simultaneously . In HSSVs, the time delay is a function of vehicle velocity and consequently varies with any changes in velocity .…”

“…Recently, in some few works, some LMI stability conditions have been developed for the systems with time‐delayed state‐dependent switching conditions . In these works, the absorbing sliding surfaces in the state space have not been considered.…”

“…In the congestion control problem, automotive all-wheel drive clutch systems, and some mechanical systems with some hydrodynamical phenomena such as high speed supercavitation vehicles (HSSVs), the time-delayed states of the system appear in the dynamical equations and switching conditions, simultaneously. [14][15][16][17][18] In HSSVs, the time delay is a function of vehicle velocity and consequently varies with any changes in velocity. 16,17 Because the vehicle velocity mostly remains constant, the variation of velocity around its nominal value can be modeled as some additive uncertainties and time-varying delay in the simplified longitudinal dynamical model of the system.…”

Stabilization of switched affine systems with disturbed state‐dependent switching laws

“…Particular applications of sliding mode control to switched systems can be found in Sira‐Ramírez and Silva‐Ortigoza . Results on switched systems with delayed state‐dependent switching‐law design are also considered in Fiter and Fridman . Recently, a constructive method for the design of state‐dependent switching laws has been proposed in Hetel and Bernuau .…”

“…10 Results on switched systems with delayed state-dependent switching-law design are also considered in Fiter and Fridman. 23 Recently, a constructive method for the design of state-dependent switching laws has been proposed in Hetel and Bernuau. 8 This method uses a bilinear model 24,25 of the system and the existence of a stabilizing continuous control to derive a stabilizing switching law.…”