Linear Quadratic Control for Systems with Structured Uncertainty

The standard LQR design technique is extended to systems with parametric uncertainty in the open-loop "A" matrix. This design, called the robust LQR (RLQR), guarantees the stability of the uncertain system, and the same level of performance robustness as standard LQR designs. To determine the properties of the RLQR design, simulations are performed on various mass-spring systems, and compared to a mismatched LQR controller, designed on the "nominal" system. These simulations show the RLQR design first reduces the length of the uncertain springs to their equilibrium value, so as to mitigate their effect on the dynamics of the system, and then regulates the system to the zero position. Additional control variables increase the performance robustness of the design. Simulations also show that disturbances are attenuated, even better than in the mismatched LQR design. The RLQR design differs from the standard LQR design in that two additional terms are added to the standard LQR cost functional. The first is interpreted as a weighted sum of the uncertain stored potential energies of the springs. The second is equivalent to a "worst-case" disturbance in the direction of the parametric uncertainty. We then show these interpretations hold in general structural systems with uncertain stiffness and damping matrices. We show that we are guaranteed better performance robustness than the mismatched LQR design. The price we pay is less robustness to high-frequency unstructured uncertainty. Also, we show that the design is conservative with respect to stability robustness. Thesis Supervisor: Michael Athans, Professor of Electrical Engineering Thesis Co-supervisor: Andreas von Flotow, Associate Professor of Aeronautics and Astronautics Acknowledgments I would like to thank my thesis supervisor, Professor Michael Athans, for his help and guidance over these last two years. He has helped direct me in my work, and shown the importance of my results in the "big picture." Professor Athans has also helped teach me how to properly present my work, both in oral presentations and on paper. I would also like to thank my thesis co-supervisor, Professor Andreas von Flotow, for his willingness to dedicate his time to my work on short notice. There are many other people who have offered me their technical guidance during this research. These include Professor Armando Rodriguez, who has been both my advisor and my friend; Douglas MacMartin, who belped me understand power dissipation in structural systems, and Leonard Lublin, who has helped me understand various aspects of structural systems; Dr. Daniel Miller, who has helped with various proofs; Dr. Dragan Obradovic and Ignacio Diaz-Bobillo, who have helped me better understand control theory. I would also like to thank the various friends I have made over these past two years. I would especially like to thank my officemate, John Wissinger, who has kept me inpired about research. The completion of this thesis marks my twentieth consecutive year in school. Therefore, I would like to dedicate this thesis to the students in my family. This research was carried out at the MIT Space Engineering Research Center with support from NASA Grant NAGW-1335 from NASA, Washington, D.C., with Mr. Samuel Vexineri serving as technical monitor.