On the design of nonautonomous fixed‐time controllers with a predefined upper bound of the settling time
David Gómez‐Gutiérrez
Abstract
Summary 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.