Identification, stability analysis and control of linear time periodic systems via harmonic transfer functions

Abstract

Many important systems encountered in nature such as wind turbines, helicopter rotors, power networks or nonlinear systems which are linearized around periodic orbit can be modeled as linear time periodic (LTP) systems. Such systems have been analyzed and discussed from analytical viewpoint extensively in the literature. However, only a few method are available in the literature for the identi cation of LTP systems which utilize input/output measurements. Especially, due to obtaining analytical solutions for LTP systems are quite challenging, utilization of experimental data to identify, analyze and stabilize such systems may be preferable. To achieve this aim, the utilization of harmonic transfer functions (HTFs) of LTP systems can be quite helpful. In the rst part of this thesis, we aim to obtain harmonic transfer functions (HTFs) of LTP systems via data-driven approach by using only input and output data of the system. In this respect, we rst present the identi cation procedure of HTFs by using single cosine input signal with a speci c frequency. However, because of the fact that this method requires multiple experiments in order to cover desired frequency range, we propose a formula for the sum of cosine input signal including di erent frequencies which their output components do not coincide. Then, we present the prediction performance of the estimated HTFs by using single cosine and sum of cosine input signals according to analytical solution of HTFs. In the second part of the thesis, our goal is to utilize harmonic transfer functions in order to analyze and design controllers which stabilize and enhance the performance of LTP systems. In this regard, we implement well known Nyquist stability criterion which is based on eigenloci of HTFs. As an illustrative example, we consider the well-known (unstable) damped Mathieu equation and design P, PD and PID controllers by using obtained Nyquist diagram. Finally, for the unknown LTP systems whose state space model may not be available, we seek to design a novel methodology, where we can obtain Nyquist plots of unknown LTP systems via input-output data analysis using the concept of HTFs. Then, we design PD controllers for the unknown LTP system by using Nyquist diagram in order to enhance the performance and increase the robustness. We illustrate the performance results of these controllers in time domain simulations.