Mixed H₂/H_∞ Control for Nonlinear Stochastic Systems With Cooperative and Non-Cooperative Differential Game

This technical note discusses the design process of the cooperative differential game (CDG) with the mixed $H_{2}/H_{\infty }$ control to obtain Pareto solutions for nonlinear stochastic systems by adaptive dynamic programming (ADP). In order to obtain Pareto solutions, a non-cooperative differential game (NCDG) for nonlinear stochastic systems is also studied. It is worth mentioning that the mixed $H_{2}/H_{\infty }$ control can achieve the expected $H_{2}$ performance under the condition of the $H_{\infty }$ continuous attenuation. Firstly, the two-player nonzero-sum differential game for stochastic systems is given to obtain Nash equilibrium. Then, on this basis, a joint cost function is established, which is derived from Pareto strategies and the corresponding Hamiltonian function. Furthermore, two critic neural networks (NNs) are used to realize Nash equilibrium and Pareto equilibrium, respectively. Finally, a detailed algorithm and a numerical example are given.