Adaptive event-triggered safety control for multiplayer mixed zero-sum game with partial inputs delay

This paper proposes an adaptive safety control method applicable to a multiplayer mixed zero-sum (MZS) game of nonlinear systems with partial inputs delay. Firstly, a framework is introduced involving N players, where player 1 and player N form a zero-sum (ZS) game, and player 1 and players 2 to N-1 form nonzero-sum (NZS) games, with some players experiencing time delays. Subsequently, the system's value function is augmented with a control barrier function (CBF) to ensure that the system's state remains within a safe region. Secondly, to approximate Nash equilibrium solutions, the study employs adaptive dynamic programming (ADP) and utilizes a critic-only neural network (NN) to approximate optimal solutions. Diverging from traditional time-trigger methods, computational and communication load reduction is achieved by introducing a state-related event trigger condition. The stability of the system is then meticulously analyzed using the Lyapunov theorem. Finally, to validate the effectiveness of the proposed method, the study provides a simulation example demonstrating its performance. In summary, this research introduces an efficient adaptive safety control method for addressing multiplayer MZS games with partial inputs delay, incorporating CBFs, ADP, and state-related event triggering.