Critic learning control via zero-sum differential game for affine formation maneuver of multi-agent systems with cyber-attacks

The affine formation maneuver (AFM) control problem using the zero-sum differential game is studied for a second-order multi-agent system (MAS) against cyber-injection attacks on the actuators. First and foremost, the distributed control signals and the cyber-injected attack signals are viewed as two competing teams. Then, the leader-follower tracking error system is established where the followers aim to track the AFMs of the leaders. The leader-follower distributed optimal control policies and the cyber-attack attenuation policies are derived from the Nash equilibrium solution of the Hamilton-Jacobi-Isaac (HJI) equation. The solution of the HJI equation is estimated with the aid of the critic neural network. The weights of the critic system are learned online based on the adaptive dynamic programming (ADP) scheme. The uniformly ultimately bounded stability of the closed-loop system under the formulated zero-sum differential game control strategy has been proven by the Lyapunov stability theorem. A simulation example illustrates that the presented control method can attenuate cyber-injection attacks on the actuators and maintain the AFMs of the agents.