Finite-time stability and stabilization of time-delay systems

Abstract

International audienceFinite-time stability and stabilization of retarded-type functional differential equations are developed. First, a theoretical result on finite-time stability inspired by the theory of differential equations, using Lyapunov functionals, is given. As it may appear not easily usable in practice, we show how to obtain finite-time stabilization of linear systems with delays in the input by using an extension of Artstein's model reduction to nonlinear feedback. With this approach, we give an explicit finite-time controller for scalar linear systems and for the chain of integrators with delays in the input

Finite-Time Stability of Uncertain Nonlinear Systems with Time-Varying Delay

“…Definition 2 (see [30]). The equilibrium = 0 of system (4) is said to be finite-time stable if it is Lyapunov stable and finitetime convergent.…”

“…Although the finite-time stability problem of some kinds of systems with time delays is studied in [30,32], the parameter uncertainties and external disturbances are not taken into account in these papers. So our methods are more general compared with the existing results.…”

“…Reference [25] studied finite-time stability for continuous-time systems with uniqueness of solutions in forward time and obtained a necessary and sufficient condition for finite-time stability for this class of systems. Recently, huge efforts have been paid to the finite-time stability or stabilization problems for time-delay systems [29][30][31][32][33][34]. In [30], the authors provided a theoretical result on finitetime stability by using Lyapunov functional and showed how to obtain finite-time stabilization of linear systems with time delays.…”

“…Recently, huge efforts have been paid to the finite-time stability or stabilization problems for time-delay systems [29][30][31][32][33][34]. In [30], the authors provided a theoretical result on finitetime stability by using Lyapunov functional and showed how to obtain finite-time stabilization of linear systems with time delays. The authors in [32] investigated the finite-time stability and designed a finite-time ∞ controller for a class of nonlinear time-delay Hamiltonian systems.…”

“…Until now, there are rare results on the finite-time stability for nonlinear systems in the presence of time delays, parameter uncertainties, and external disturbances [43]. Note that the systems studied in [24,25] are without time delays, and those in [30,32] are with time delays but without parameter uncertainties and external disturbances.…”