Analysis and Real-time Implementation of State-dependent Riccati Equation Controlled Systems by Evrin

This study deals with one of the recently proposed methods for control of nonlinear dynamical systems, the State-Dependent Riccati Equation (SDRE) method. This method can be thought of as the nonlinear counterpart of LQR based control design, with the matrices A(x), B(x), Q(x) and R(x) being functions of the states. The greatest advantage offered by SDRE is the design flexibility of tuning the state and input penalty matrices Q(x) and R(x), respectively, as functions of states. This flexibility softens the classical state-error vs. input tradeoff seen in classical LQR control. Of the many open issues regarding SDRE, two are dealt with in this thesis: stability and real-time control implementation. The closed-loop system equations of an SDRE controlled system are usually not known explicitly, which makes stability analysis quite difficult. The SDRE controlled closed-loop system is known to be locally asymptotically stable, but how large this region of stability is is not known. In this work, analytical conditions are found to globally asymptotically stabilize second order nonlinear systems affine in the input. Comparison with feedback linearization is made, which reveals the advantages of SDRE. The theoretical results are demonstrated experimentally on a magnetically levitated ball tracking a square wave in real-time. A method for estimating the region of stability of SDRE controlled systems based on vector norms and overvaluing systems is also developed. This method requires knowledge of only the maximum and the minimum values of the feedback gains over a chosen domain in the state-space. Also, by this method, higher order systems can be reduced to tractable lower order ones at the expense of more conservative estimates. Finally, experimental real-time control using SDRE is addressed. The plant used is the Pendubot, a two-link underactuated robot, which is a highly nonlinear 4 th order system. The SDRE control is calculated by solving the Algebraic Riccati Equation online and the Pendubot is regulated at one of its unstable positions. The results show that the computational power requirement associated with SDRE real-time control is not very high. Simulation and experimental results reveal the advantages of SDRE control over LQR. v To My Father vi Acknowledgements First and foremost, I would like to thank my research advisor, Professor Andrew Alleyne, for his valuable guidance, patience, financial support and friendliness during my doctoral study. I thank my parents Nurşen Erdem and Ali nal Erdem for their ever-present support and encouragement during my studies. I am …