Asymptotic and Finite Time Semistability for Nonlinear Discrete-Time Systems with Application to Network Consensus

Semistability is the property whereby the solutions of a dynamical system converge to Lyapunov stable equilibrium points determined by the system initial conditions. In this paper, we build on the theories of semistability and finite-time semistability for continuous-time dynamical systems to develop a rigorous framework for discrete semistability and discrete finite-time semistability. Specifically, Lyapunov and converse Lyapunov theorems for semistability and finite time semistability are developed, and the regularity properties of the Lyapunov function establishing finite time semistability are shown to be related to the settling time function capturing the finite settling time behavior of the dynamical system. These results are then used to develop a general framework for designing semistable and finite time semistable consensus protocols for discrete dynamical networks for achieving multiagent coordination tasks asymptotically and in finite time. The proposed controller architectures involve the exchange of generalized energy state information between agents guaranteeing that the closed-loop dynamical network is semistable to an equipartitioned equilibrium.