A Design Method of Optimal PID-Based Repetitive Control Systems

This paper is concerned with the design of a PID-based repetitive control strategy for a strictly proper plant with requirements of high-precision tracking performance for periodic signals and good rejecting performance for an external disturbance with a finite frequency range. The PID-based controller consists of a PID controller and a repetitive controller connected in series, and its control parameters are optimized to meet the performance requirements. First, utilizing Youla-Kucera Parameterization, the necessary and sufficient conditions for the existence of the PID-based repetitive controller are derived. Second, from the requirements of the control performance and the analysis of the sensitivity function, the description of the PID-based repetitive controller is deduced. Third, the design of PID control parameters depends on the stability of closed-loop system and the attenuation performance. By introducing an arbitrarily small relaxation variable and applying the H
_∞
control method, the design problem is transformed into a generalized eigenvalue minimization problem (GVEP) under linear matrix inequality (LMI) constraints. In addition, the largest cutoff frequency in the repetitive controller is also achieved by solving a GVEP. Finally, simulations are applied to examine the validity and efficacy of the proposed method.