A Distributed Robust Two-Time-Scale Switched Algorithm for Constrained Aggregative Games

Robust Nash equilibrium (NE) seeking for a constrained aggregative game is explored. Our setting has the following three features: 1) local feasibility sets are considered; 2) each player has first-order dynamics that are influenced by unknown time-varying disturbances and unmodeled terms; 3) the communication topology among players may be influenced by attacks. The existence of unknown disturbances results in the trajectories of the players going out of the feasible sets even though the initial values start from inside. To make the ideas clear, we first use an example that exhibits the three features to introduce the problem formulation to be studied. Subsequently, a two-time-scale distributed, projected algorithm is proposed. The simulation results are also shown. To further illustrate the effectiveness of the proposed algorithm through theoretical analysis, such an algorithm is generalized into a two-time-scale hybrid system by using an average dwell-time automaton and a time-ratio monitor to impose restrictions on the frequency and relative activation time of the attacks. To analyze stability of such a hybrid system, the reduced system and the boundary layer system are presented and corresponding Lyapunov functions are constructed. Correspondingly, uniform (uniform global) asymptotic stability is obtained under some mild assumptions. Also, some results are given for systems without attacks. Finally, two aggregative games are used to show the effectiveness of our algorithm and theoretical analysis.