Closed-form Solutions to Robust and Optimal Control Problems of Second-Order Systems

Abstract

This thesis presents a unified, structure-preserving framework for the optimal and robust control of second-order dynamical systems. Traditional numerical approaches to solving continuous-time algebraic Riccati equations (CAREs), computing coprime factorizations, and synthesizing robust controllers often lose physical insight and become computationally expensive for large-size systems. To overcome these challenges, we develop closed-form solutions and expressions for various control problems that offer high accuracy, scalability, and insight. We begin by proposing novel closed-form stabilizing and destabilizing solutions to the CARE for second-order systems, explicitly in terms of the physically insightful and structurally significant second-order system matrices. They outperform standard solvers in both speed and accuracy, especially for large-size systems. We show that engineering problems, such as the optimal control of vibrating structures, can be addressed more simply and efficiently using our solution. Moreover, this foundation enables closed-form solutions to linear time-varying (LTV) optimal control problems via decomposition into linear time-invariant (LTI) sub-problems. Next, by deriving the closed-form solution to the filtering Riccati equation, we obtain closed-form coprime factorizations for second-order systems. These are especially valuable when frequent recomputation is needed, as in reconfigurable system scenarios. Demonstrations include a coprime factor-based Youla-Kucera controller for satellite networks and mechanical systems, showing robust and high-performance behavior. Finally, we present closed-form and simplified robust control tools for undamped second-order systems. These include explicit expressions for robustness quantification and robust controller design in the presence of parametric uncertainties. Collectively, these contributions provide scalable, real-time implementable, and physically interpretable tools for the control of second-order dynamical systems.