Summary In this paper, we provide a new nonconservative upper bound for the settling time of a class of fixed‐time stable systems. To expose the value and the applicability of this result, we present four main contributions. First, we revisit the well‐known class of fixed‐time stable systems, to show the conservatism of the classical upper estimate of its settling time. Second, we provide the smallest constant that the uniformly upper bounds the settling time of any trajectory of the system under consideration. Third, introducing a slight modification of the previous class of fixed‐time systems, we propose a new predefined‐time convergent algorithm where the least upper bound of the settling time is set a priori as a parameter of the system. At last, we design a class of predefined‐time controllers for first‐ and second‐order systems based on the exposed stability analysis. Simulation results highlight the performance of the proposed scheme regarding settling time estimation compared to existing methods.

This paper introduces predefined-time stable dynamical systems which are a class of fixed-time stable dynamical systems with settling time as an explicit parameter that can be defined in advance. This concept allows for the design of observers and controllers for problems that require to fulfill hard time constraints. An example is encountered in the fault detection and isolation problem, where mode detection in a timely manner needs to be guaranteed in order to apply a recovery action. Furthermore, through the notion of strong predefined-time stability, the approach hereinafter presented permits to overcome the problem of overestimation of the convergence time bound encountered in previous methods for the analysis of finite-time stable systems, where the stabilization time is often an unbounded function of the initial conditions of the system. A Lyapunov analysis is provided together with a detailed discussion of the applications to consensus and first order sliding mode controller design. Finite-time stability, Sliding-mode control, Lyapunov stability, Robust control, Consensus.

Recently, there has been a great deal of attention in a class of finite-time stable dynamical systems, called fixed-time stable, that exhibit uniform convergence with respect to its initial condition, that is, there exists an upper bound for the settling-time (UBST) function, independent of the initial condition of the system. Of particular interest is the development of stabilizing controllers where the desired UBST can be selected a priori by the user since it allows the design of controllers to satisfy real-time constraints. Unfortunately, existing methodologies for the design of controllers for fixed-time stability exhibit the following drawbacks: on the one hand, in methods based on autonomous systems, either the UBST is unknown or its estimate is very conservative, leading to over-engineered solutions; on the other hand, in methods based on time-varying gains, the gain tends to infinity, which makes these methods unrealizable in practice. To bridge these gaps, we introduce a design methodology to stabilize a perturbed chain of integrators in a fixed-time, with the desired UBST that can be set arbitrarily tight. Our approach consists of redesigning autonomous stabilizing controllers by adding time-varying gains. However, unlike existing methods, we provide sufficient conditions such that the time-varying gain remains bounded, making our approach realizable in practice. K E Y W O R D Sfixed-time control, predefined-time control, predefined-time stabilization, prescribed-time control INTRODUCTIONRecently, 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 The fixed-time stability property is of great interest in the development of algorithms for scenarios where real-time constraints need to be satisfied. In fault detection, isolation, and recovery schemes, 20 failing to recover from the fault on time may lead to an unrecoverable mode. In missile guidance, 21 the impact time control guidance laws require stabilization in a desired time. 22,23 In hybrid dynamical systems, it is frequently required that the observer (respectively, controller) Int J Robust Nonlinear Control. 2020;30:3871-3885.wileyonlinelibrary.com/journal/rnc