Finite-Time Stability of Continuous Autonomous Systems

Abstract

Abstract. Finite-time stability is defined for equilibria of continuous but non-Lipschitzian autonomous systems. Continuity, Lipschitz continuity, and Hölder continuity of the settling-time function are studied and illustrated with several examples. Lyapunov and converse Lyapunov results involving scalar differential inequalities are given for finite-time stability. It is shown that the regularity properties of the Lyapunov function and those of the settling-time function are related. Consequently, converse Lyapunov results can only assure the existence of continuous Lyapunov functions. Finally, the sensitivity of finite-time-stable systems to perturbations is investigated.Key words. stability, finite-time stability, non-Lipschitzian dynamics AMS subject classifications. 34D99, 93D99PII. S03630129973213581. Introduction. The object of this paper is to provide a rigorous foundation for the theory of finite-time stability of continuous autonomous systems and motivate a closer examination of finite-time stability as a possible objective in control design.Classical optimal control theory provides several examples of systems that exhibit convergence to the equilibrium in finite time [17]. A well-known example is the double integrator with bang-bang time-optimal feedback control [2]. These examples typically involve dynamics that are discontinuous. Discontinuous dynamics, besides making a rigorous analysis difficult (see [9]), may also lead to chattering [10] or excite high frequency dynamics in applications involving flexible structures. Reference [8] considers finite-time stabilization using time-varying feedback controllers. However, it is well known that the stability analysis of time-varying systems is more complicated than that of autonomous systems. Therefore, with simplicity as well as applications in mind, we focus on continuous autonomous systems.Finite [21] suggests, based on a scalar example, that systems with finite-settling-time dynamics possess better disturbance rejection and robustness properties. However, no precise results exist for multidimensional systems. This paper attempts to fill these gaps.In section 2, we define finite-time stability for equilibria of continuous autonomous systems that have unique solutions in forward time. Continuity and forward uniqueness render the solutions continuous functions of the initial conditions, so that the

Fast Control Systems: Nonlinear Approach

“…Some results in this context can be discovered in the literature. In particular, fast stability of ODEs is represented by the notions of finite-time and fixed-time stabilities [50], [43], [20], [11], [6], [29], [26], [2], [14], [33], [39], but hyper exponential transitions are studied in [38] as fast behavior of time delay systems. Fast models described by partial differential equations may demonstrate the so-called finite-time extinction property [46], [19], [31], [10] also known as super stability [5], [13].…”

“…Definition 7 (Finite-time stability ( [43], [6])). The origin of the system (1) is said to be finite-time stable if it is Lyapunov stable in U(t 0 ) ∈ R n ,t 0 ∈ R and finite-time attractive : ∀x 0 ∈ U(t 0 ), ∃T = T (t 0 , x 0 ) ≥ 0 such that x t 0 ,x 0 (t) = 0, ∀t ≥ t 0 + T .…”

A distributed finite-time observer for linear systems

“…Theorem 1 ([21], [23], [24]): For the system (1) with r-homogeneous and continuous function f the following properties are equivalent:…”

Fault-tolerant control allocation with actuator dynamics: Finite-time control reconfiguration

“…The basic idea is to compensate the effect of the inputs u i (t) while simultaneously the inputũ tracks the desired actuator stateũ c in finite-time [1]. We set…”