Identification and Control of Linear Time-Periodic Systems via Harmonic Transfer Functions

Abstract

The increased need for accurately modeling the input–output characteristics of linear time-periodic (LTP) systems necessitates novel identification and control algorithms as well as new test benches for their experimental validation. This thesis introduces a simple-to-build test bench for experimental identification and control of LTP systems both with time-invariant and time-varying controllers. We mechanically coupled the shafts of two DC motors and fed back the angular velocity to the second motor with a time-periodic modulation. This allowed us to imitate a time-periodic load for the first DC motor, thereby yielding an experimental LTP system plant. We used Matlab/Simulink target hardware support to implement the entire software in Simulink, which greatly simplifies input design, data collection, and analysis as compared to embedded programming. We used constant-frequency sinusoidal signals for the data collection on the experimental platform. Subsequently, we estimated the harmonic transfer functions and identified the parameters of a state-space model of the proposed LTP system. We then designed a time-periodic controller in order to regulate the output of the LTP system. We proposed a method for obtaining a linear time-periodic LQR controller in order to capture the time-varying dynamics of the systems. In this research, we also employed the Mathieu equation as a representative example of a Linear Time Periodic (LTP) system for simulation purposes. We analyzed the equation both theoretically and using data-driven methods, and verified the results by comparing the output predictions from both methods. In previous research, most controllers used for analyzing the Mathieu equation were linear time-invariant. In contrast, in this thesis, we proposed the implementation of time-periodic controllers. Following this, we employed linear quadratic regulator (LQR) controllers in LTI and LTP forms to control the system and evaluated their performance through simulations in MATLAB/Simulink. We compared metrics such as root mean squared error (RMSE) and system characteristics, including overshoot and settling time. The effects of linear time-invariant (LTI) and linear time-periodic (LTP) type controllers on the dynamics of an LTP system have been investigated.