On finite-time stabilization of a class of nonlinear time-delay systems: Implicit Lyapunov-Razumikhin approach

Abstract

Theorems on Implicit Lyapunov-Razumikhin functions (ILRF) for asymptotic, exponential, finite-time and nearly fixed-time stability analysis of nonlinear time-delay systems are presented. Based on these results, finite-time stabilization of a special class of such systems is addressed. These systems are represented by a chain of integrators with a timedelay term multiplied by a function of instantaneous state vector. Possible explicit restriction on nonlinear time-delay terms is discussed. Simple procedure of control parameters calculation is given in terms of linear matrix inequalities (LMIs). Some aspects of digital implementations of the presented nonlinear control law are touched upon. Theoretical results are illustrated by numerical simulations.

Finite-Time Stability Tools for Control and Estimation

“…As we can conclude from the results of this section, FTS is not a natural type of behavior of time-delay systems, since the influence of past values of the state may block a finite-time settling of the trajectories at the origin, then additional structural conditions are needed. Control design approaches for finite-time stabilization of time-delay systems can be found in Polyakov et al, 2015b;Nekhoroshikh et al, 2020), and the simplest ways of obtaining accelerated regulation in this class of systems is by using prediction techniques to compensate the delays as in (Karafyllis, 2006), the theory of homogeneity (Zimenko et al, 2017;Zimenko et al, 2019) or the domination approach (Wang et al, 2020b).…”

Hyperexponential and Fixed-Time Stability of Time-Delay Systems: Lyapunov–Razumikhin Method

“…• Extension of the results presented in [3], [14] to hyperexponential and fixed-time stability analysis and stabilization. Due to the complexity of the required analysis, two Lyapunov-Razumikhin functions are introduced.…”

“…Unfortunately, control design of nonlinear systems is a complicated task even for linear plants, e.g., a chain of integrators [1]. To overcome this difficulty the so-called Implicit Lyapunov method has been proposed for delay-free [12], [16] and time-delay systems [14], [15], [17]. This approach provides a constructive way for the control synthesis: it allows control parameters to be found as the solution of linear matrix inequalities (LMIs).…”

“…• Introduction of a wider class of time-delay systems compared to the ones given in [14] for which the "necessary" condition on finite-time stabilization [4], [13] is fulfilled.…”