Consistent Discretization of Finite-Time and Fixed-Time Stable Systems

Abstract

Algorithms of implicit discretization for generalized homogeneous systems having discontinuity only at the origin are developed. They are based on the transformation of the original system to an equivalent one which admits an implicit or a semi-implicit discretization schemes preserving the stability properties of the continuous-time system. Namely, the discretized model remains finite-time stable (in the case of negative homogeneity degree), and practically fixed-time stable (in the case of positive homogeneity degree). The theoretical results are supported with numerical examples. 1. Introduction. Discretization issues are important for a digital implementation of estimation and control algorithms. Construction of a consistent stable discretization is complex for essentially non-linear ordinary differential equations (ODEs), which do not satisfy some classical regularity assumptions. For example, the sliding mode algorithms are known to be difficult in practical realization [1], [2], [3] due to discontinuous (set-valued) nature, which may invoke chattering caused by the discretization. The mentioned papers have discovered that the implicit discretization technique is useful for practical implementation of non-smooth and discontinuous control and estimation algorithms. In particular, chattering suppression in both input and output, as well as a good closed-loop performance has been confirmed experimentally in [1], [4], [5]. Finite-time stability is a desirable property for many control and estimation algorithms [6], [7], [8], [9], [10], [11]. It means that system trajectories reach a stable equilibrium (or a set) in a finite time, in contrast to asymptotic stability allowing this only for the time tending to infinity. If the settling (reaching) time is globally bounded for all initial conditions then the origin is fixed-time stable (see, e.g. [12]). The corresponding ODE models do not satisfy Lipschitz condition (at least at the origin). In the general case, an application of the conventional implicit or explicit discretization schemes does not guarantee that finite-time or fixed-time stability properties will be preserved (see, e.g. [13], [14], [15]). The latter means that the discrete-time model may be inconsistent with the continuous-time one. However, the discretized systems may remain globally finite-time stable in some cases (see [1], [2], [16], [17]). The aim of this paper is to study systematically the problem of consistent discretization of the so-called generalized homogeneous non-linear systems. The discretized model is consistent if it preserves the stability property (e.g. exponential, finite-time or fixed-time stability) of the original continuous-time system. Homogeneity is a certain form of symmetry studied in systems and control theory [9], [18], [19], [20],[21], [22], [23]. The standard homogeneity (introduced originally by L. Euler in 17th century) is the symmetry of a mathematical object f (e.g. function, vector field, operator, etc) with respect to the uniform dilation of the argument x → λx, namely, f (λx) = λ 1+ν f (x), λ > 0