Differential-game for resource aware approximate optimal control of large-scale nonlinear systems with multiple players

In this paper, we propose a novel differential-game based neural network (NN) control architecture to solve an optimal control problem for a class of large-scale nonlinear systems involving N-players. We focus on optimizing the usage of the computational resources along with the system performance simultaneously. In particular, the N-players' control policies are desired to be designed such that they cooperatively optimize the large-scale system performance, and the sampling intervals for each player are desired to reduce the frequency of feedback execution. To develop a unified design framework that achieves both these objectives, we propose an optimal control problem by integrating both the design requirements, which leads to a multi-player differential-game. A solution to this problem is numerically obtained by solving the associated Hamilton-Jacobi (HJ) equation using event-driven approximate dynamic programming (E-ADP) and artificial NNs online and forward-in-time. We employ the critic neural networks to approximate the solution to the HJ equation, i.e., the optimal value function, with aperiodically available feedback information. Using the NN approximated value function, we design the control policies and the sampling schemes. Finally, the event-driven N-player system is remodeled as a hybrid dynamical system with impulsive weight update rules for analyzing its stability and convergence properties. The closed-loop practical stability of the system and Zeno free behavior of the sampling scheme are demonstrated using the Lyapunov method. Simulation results using a numerical example are also included to substantiate the analytical results.