This project proposed 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.

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