Optimal tracking control for linear systems with persistent disturbance

The optimal tracking control by dynamic compensation for linear time-invariant system with disturbance signal is considered. Firstly, in light of combining the system with disturbance system and expected output system, this optimal tracking control problem can be transformed into the standard linear-quadratic (LQ) optimal control problem. Then a dynamic compensator with a proper dynamic order is given such that the closed-loop system is asymptotically stable and its associated Lyapunov equation has a symmetric positive-definite solution, and more the quadratic performance index is derived to be a simple expression related to the symmetric positive-definite solution and the initial value of the closed-loop system. In order to solve the optimal control problem for the system, the proposed Lyapunov equation is transformed into a Bilinear Matrix Inequality (BMI) and a corresponding path-following algorithm to minimize the quadratic performance index is proposed in which an optimal dynamic compensator can be obtained. Finally, a numerical example is provided to demonstrate the effectiveness and feasibility of the proposed results.