LQ Zero-Sum Differential Games for Stochastic Delay Systems and Application to Stability Control for Permanent Magnet Synchronous Motors

In this paper, we consider linear-quadratic zero-sum differential games for a class of stochastic systems with state delay and their application to the stability control problem of permanent magnet synchronous motors (PMSMs). First, we define a stabilization problem via a memoryless state feedback strategy and then solve the problem of minimizing the upper bound of the cost function using guaranteed cost control theory. Moreover, a saddle point equilibrium condition is established against external disturbance input. For this problem, the necessary conditions for the sub-optimality are established via stochastic cross-coupled matrix equations (SCCMEs) using the Karush-Kuhn-Tucker condition. Second, a practical example based on the stability control problem for the PMSM is presented to demonstrate the usefulness and effectiveness of the proposed method. The results demonstrate that the proposed strategy is effective even in the presence of stochastic noise, deterministic disturbance, and model uncertainties.