Tracking problem under bounded disturbances. Algebraic synthesis method

We consider the problem of a zero-sum differential tracking game with a quadratic performance functional in which the plant subjected to uncontrolled disturbances is described by a nonlinear ordinary differential equation. The synthesis of optimal controls is known to necessitate online solving a scalar Bellman–Isaacs partial differential equation that contains information about the trajectory of the process to be monitored. The lack of information about this process over the entire control interval makes the synthesized controls unimplementable. An algebraic method is proposed for solving the Bellman-Isaacs equation, which contains the current value of the monitored process. As an illustration of the results obtained, we give the simulation of the behavior of a nonlinear system with two players with an open control horizon.