Inverse linear-quadratic nonzero-sum differential games

Abstract: $ $This paper addresses the inverse problem for Linear-Quadratic (LQ) nonzero-sum $N$-player differential games, where the goal is to learn parameters of an unknown cost function for the game, called observed, given the demonstrated trajectories that are known to be generated by stationary linear feedback Nash equilibrium laws. Towards this end, using the demonstrated data, a synthesized game needs to be constructed, which is required to be equivalent to the observed game in the sense that the trajectories generated by the equilibrium feedback laws of the $N$ players in the synthesized game are the same as those demonstrated trajectories. We show a model-based algorithm that can accomplish this task using the given trajectories. We then extend this model-based algorithm to a model-free setting to solve the same problem in the case when the system's matrices are unknown. The algorithms combine both inverse optimal control and reinforcement learning methods making extensive use of gradient descent optimization for the latter. The analysis of the algorithm focuses on the proof of its convergence and stability. To further illustrate possible solution characterization, we show how to generate an infinite number of equivalent games, not requiring to run repeatedly the complete algorithm. Simulation results validate the effectiveness of the proposed algorithms.