A Lyapunov-Like Characterization of Predefined-Time Stability

Abstract

This technical note studies Lyapunov-like conditions to ensure a class of dynamical systems to exhibit predefinedtime stability. The origin of a dynamical system is predefinedtime stable if it is fixed-time stable and an upper bound of the settling-time function can be arbitrarily chosen a priori through a suitable selection of the system parameters. We show that the studied Lyapunov-like conditions allow to demonstrate equivalence between previous Lyapunov theorems for predefinedtime stability for autonomous systems. Moreover, the obtained Lyapunov-like theorem is extended for analyzing the property of predefined-time ultimate boundedness with predefined bound, which is useful when analyzing uncertain dynamical systems. Therefore, the proposed results constitute a general framework for analyzing predefined-time stability, and they also unify a broad class of systems which present the predefined-time stability property. On the other hand, the proposed framework is used to design robust controllers for affine control systems, which induce predefined-time stability (predefined-time ultimate boundedness of the solutions) w.r.t. to some desired manifold. A simulation example is presented to show the behavior of a developed controller, especially regarding the settling time estimation.

Finite-time and fixed-time input-to-state stability: Explicit and implicit approaches

“…Since H r = 1 2 I 2 + 1−ν 2 ( 1 0 0 0 ) where I 2 is the identity matrix, H r → 1 2 I 2 as ν → 1 and 0 < P H r + H r P, so that ∂ V Q(V, x) < 0 for all (V, x) ∈ R + × R n \{0} and condition C4 is satisfied. Assuming additionally that P H r + H r P ≤ P and taking into account (from (17)…”

“…In the case of finite-time stability, the system's trajectories converge exactly to zero in a finite amount of time [4]; in the case of fixed-time stability exact convergence to the origin occurs in a maximum amount of time that is independent of the system's initial state [8,9,10]. Nonasymptotic convergence rates are a major feature in Sliding Mode Control [11,12,13,14] and some further developments in non-asymptotic convergence include a bound on the convergence time that is not only fixed but also arbitrarily selected [15,16,17], a better control performance when initial conditions are far away from the origin by separating low and high growing terms [18,19] and finite-time stable controllers with an enhancement of the domain of attraction for state-constrained systems [20]. Although systems with nonasymptotic rates of convergence may exhibit numerical inconsistencies [21] or lose some of its properties under discretization algorithms [22], recent advances in consistent discretization provide algorithms that overcome these issues [23,24].…”

Suboptimal closed‐form feedback control of input‐affine non‐linear systems

“…Under the SDRE framework, appropriate choice of the tuning matrices in (4) can ensure the convergence of the error to some desirable bounds, within a predefined time, as discussed in the later sections. The predefined‐time ultimate boundedness , which is based on the definition given in [26], is now formally defined as Definition 5 For a parameter vector of (3) and an arbitrarily selected constant , the solution of (3) is said to be locally predefined‐time ultimately bounded , if the origin is Lyapunov stable, for , and there exists a time such that the solution is satisfied for all , where is a ball of radius around the origin and is the settling time function which is bounded such that …”

On the design of nonautonomous fixed‐time controllers with a predefined upper bound of the settling time

“…Recently, there has been a great deal of attention in the control community on the analysis of a class of systems, known as fixed‐time stable systems, because they exhibit finite‐time convergence with an upper bound of the settling time (UBST) that is independent of the initial conditions of the system 1-5 . This effort has produced many contributions on algorithms with the fixed‐time convergence property, such as multiagent coordination, 6-9 distributed resource allocation, 10 synchronization of complex networks, 11,12 stabilizing controllers, 1,13-16 state observers, 17 and online differentiation algorithms 18,19 …”