On Time-Synchronized Stability and Control

Abstract

Previous research on finite-time control focuses on forcing a system state (vector) to converge within a certain time moment, regardless of how each state element converges. In the present work, we introduce a control problem with unique finite/fixed-time stability considerations, namely time-synchronized stability (TSS), where <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">at the same time</i> , all the system state elements converge to the origin, and fixed-TSS, where the upper bound of the synchronized settling time is invariant with any initial state. Accordingly, sufficient conditions for (fixed-) TSS are presented. On the basis of these formulations of the time-synchronized convergence property, the classical sign function, and also a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">norm-normalized sign function</i> , are first revisited. Then in terms of this notion of TSS, we investigate their differences with applications in control system design for first-order systems (to illustrate the key concepts and outcomes), paying special attention to their convergence performance. It is found that while both these sign functions contribute to system stability, nevertheless an important result can be drawn that norm-normalized sign functions help a system to additionally achieve TSS. Furthermore, we propose a fixed-time-synchronized sliding-mode controller for second-order systems; and we also consider the important related matters of singularity avoidance there. Finally, numerical simulations are conducted to present the (fixed-) time-synchronized features attained; and further explorations of the merits of the proposed (fixed-) TSS are described.