Harmonic transfer functions based controllers for linear time-periodic systems

The analysis, identification and control of periodic systems has gained increasing interest during the last few decades due to the increased use of dynamical systems that exhibit periodic motion. The vast majority of these studies focus on the analysis and control problem for a known state-space formulation of the linear time-periodic (LTP) system. On the other hand, there are also some studies that focus on data-driven identification of LTP systems with unknown state-space formulations. However, most of these methods provide numerical estimates for the harmonic transfer functions (HTFs) of an LTP system that are extremely difficult to work with during controller design. The goal of this paper is to provide a simple controller design methodology for unknown LTP systems by utilizing so-called HTFs estimates. To this end, we first build a mathematical basis of LTP controller design for known LTP systems using the Nyquist diagrams and analytically derived HTFs. We propose a new methodology to design P-, PD- and PID-type controllers for LTP systems using Nyquist diagrams and the eigenlocus of the HTFs. Having established the HTF-based controller design procedure, we extend our methodology to unknown LTP systems by presenting a new sum-of-cosine signal-based data-driven system identification method. We show that the proposed data-driven controller design method allows estimation of the HTFs and it provides simple tools for optimizing certain time-domain performance metrics. We provide numerical examples for both known and unknown LTP system cases to illustrate the performance of the proposed controller design methodology.