Optimal Mismatched Disturbance Rejection Control for Continuous-Time Uncontrollable Systems

This article focuses on optimal mismatched disturbance rejection control for linear continuous-time uncontrollable systems. Different from previous studies that solely focuses on controllable systems, this research broadens its scope by relaxing the requirement for system controllability. It addresses the problem of mismatched disturbance rejection control in uncontrollable systems by reframing it as a linear quadratic tracking problem. This transformation is achieved through the introduction of a novel quadratic performance index, enabling the regulated state to accurately track the reference trajectory while minimizing the impact of disturbances. The necessary and sufficient conditions for the solvability and the disturbance rejection controller are obtained by solving a forward–backward differential equation over a finite horizon. A sufficient condition for system stability is obtained over an infinite horizon under detectable condition. In addition, the stability analysis of the system is presented in combined with generalized extended state observer. This article details our novel approach for transforming disturbance rejection into a linear quadratic tracking problem. The effectiveness and feasibility of the proposed method are demonstrated through numerical example and experimental results from the permanent magnet synchronous motor servo system.