Optimal and Robust Control for Systems with Second Order Structure

Abstract

This work aims to investigate the utilization of the structured system matrices (mass, stiff- ness, damping) present in second order systems expressed in the first order and their appli- cations to problems in optimal control and robust control. These structured system matrices may - in certain cases - be symmetric, diagonally dominant, or positive definite. These prop- erties can be leveraged to obtain improvements in computational efficiency and accuracy, which is the core of this dissertation. Three methods are introduced in this work that exploit the structured system matrices in the context of the algebraic Riccati equation (ARE), ap- plied to optimal and robust control problems. As matrix sizes increase, traditional methods for solving the ARE becomes computationally expensive, due to the eigendecompositions in- volved in Schur/subspace methods. This work focuses on algorithmic solutions to the ARE that do not involve the eigendecompositions of the 2n ×2n system matrices, by leveraging properties of the mass, stiffness and damping elements. An algorithmic solution to the ARE is presented first, which is applicable to second order sys- tems with diagonally dominant system matrices (which may be asymmetric). This method is shown to improve computational efficiency for large systems. Next, the Newton-Kleinman algorithm is utilized in conjunction with the second order system matrices to develop a mod- ified Newton-Kleinman method tailored towards second order systems with positive definite mass, stiffness and damping matrices, in the context of the H-infnity control design prob- lem. This algorithm is shown to have an analytic proof of convergence. An application of this method is demonstrated via a data-driven controller that also minimizes the entropy of the closed loop system. Finally, the discrete time algebraic Riccati equation (DARE) is considered, and a modified discrete Newton-Kleinman algorithm for systems with a second order structure is introduced, that is shown to reduce computational expense as compared to Schur-based methods.