Adaptive Tracking Games for Coupled Stochastic Linear Multi-Agent Systems: Stability, Optimality and Robustness

Abstract

Distributed adaptive tracking-type games are investigated for a class of coupled stochastic linear multi-agent systems with uncertainties of unknown structure parameters, external stochastic disturbances, unmodeled dynamics, and unknown agents' interactions. The control goal is to make the states of all the agents converge to a desired function of the population state average (PSA). Due to the fact that only local information is available for each agent, the control is distributed. For the time-invariant parameter case, the extended least-squares algorithm, Nash certainty equivalence (NCE) principle, and certainty equivalence (CE) principle are used to estimate the unknown parameters and the PSA term, and to design adaptive control, respectively. Under some mild conditions, it is shown that the closed-loop system is almost surely uniformly stable with respect to the population number N; the estimate for the PSA term is strongly consistent; the adaptive control is almost surely an asymptotic Nash equilibrium. When the dynamics of each agent contains time-varying parameters and unmodeled dynamics, the projected least mean square (LMS) algorithm, NCE principle, and CE principle are adopted to estimate the unknown time-varying parameters, and the unknown PSA term, and to design robust adaptive control, respectively. In addition to stability of the closed-loop system and consistency of the PSA estimate, the control law is shown to be robust Nash equilibrium with respect to the unmodeled dynamics, the variation of the unknown parameters, and the external disturbances. Two numerical examples are given to illustrate the methods and results of this paper.