Reduced Order Methods for Large Scale Riccati Equations
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Solving the linear quadratic regulator (LQR) problem for partial differential equa- tions (PDEs) leads to many computational challenges. The primary challenge comes from the fact that discretization methods for PDEs typically lead to very large sys- tems of differential or differential algebraic equations. These systems are used to form algebraic Riccati equations involving high rank matrices. Although we restrict our attention to control problems with small numbers of control inputs, we allow for po- tentially high order control outputs. Problems with this structure appear in a number of practical applications yet no suitable algorithm exists. We propose and analyze so- lution strategies based on applying model order reduction methods to Chandrasekhar equations, Lyapunov/Sylvester equations, or combinations of these equations. Our nu- merical examples illustrate improvements in computational time up to several orders of magnitude over standard tools (when these tools can be used). We also present exam- ples that cannot be solved using existing methods. These cases are motivated by flow control problems that are solved by computing feedback controllers for the linearized system.
- Doctoral Dissertations