Optimal Dynamic Controller Design for Linear Quadratic Tracking Problems

This article provides novel results on the infinite-horizon linear quadratic tracking (LQT) problem for continuous-time systems. In order to fulfill optimality for the tracker design, we introduce a new performance index to reformulate the LQT problem in terms of the minimal polynomial of the exosystem that generates the reference trajectory. We prove that, under routine assumptions, the obtained solution to the formulated LQT problem realizes both closed-loop stability and asymptotic tracking. It is worth pointing out that the proposed procedure does not impose any constraints on the stability of the exosystem. Besides, our method can determine both the optimal control gain and the optimal dynamic compensator, simultaneously. This is different from existing optimal tracker design based on internal model principle, which presets a nonoptimal dynamic compensator. Our theoretical results are corroborated by a simulation example.