Discounted ${H_\infty }$ Tracking Control for Two-Time-Scale Systems With Application to Permanent Magnet Synchronous Motors

This article investigates ${H_\infty }$ tracking control problem with a discount factor for two-time-scale systems subject to disturbances. The ${H_\infty }$ controller design is reduced to solving a set of linear matrix inequalities by using Lyapunov functions with two-time-scale structure. Under the obtained controller, the closed-loop system can track the output of a nonasymptotically stable reference model and achieve the ${H_\infty }$ performance for the given attenuation level and discount factor. The stability of the closed-loop system under the developed controller is proven. Compared with the existing discounted tracking control methods, the proposed method is capable of avoiding the complicated process of calculating the upper bound of the discount factor to guarantee the stability of the closed-loop system. The effectiveness of our theoretical results is demonstrated through both simulation and experiment on speed control of permanent magnet synchronous motor.