In this paper, we build on the theory of semistability in probability to provide sufficient Lyapunov conditions for finite time and fixed time semistability in probability for discrete-time nonlinear stochastic dynamical systems. Specifically, we construct Lyapunov sufficient conditions for finite time semistability in probability using a Lyapunov difference operator which is a discrete-time analog of the infinitesimal generator for continuous-time Ito dynamical systems. In addition, we provide an upper bound for the average stochastic settling-time that captures the almost sure finite convergence behavior of the stochastic dynamical system. Furthermore, we develop a Lyapunov theorem for fixed time semistability in probability that guarantees the stochastic settling-time is almost surely uniformly bounded regardless of the system initial conditions. In particular, our Lyapunov theorem involves a Lyapunov difference satisfying an exponential inequality of the Lyapunov function that gives rise to a minimum guaranteed bound on the average stochastic settling-time characterized by the primary and secondary branches of the Lambert $W$ function. These results are then used to develop finite time and fixed time consensus protocols for discrete-time multiagent dynamical systems for achieving coordination tasks in finite and fixed time.
Lyapunov Theorems for Finite Time and Fixed Time Semistability of Discrete-Time Stochastic Systems
2023-03-04
1420499 byte
Conference paper
Electronic Resource
English
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