Convergent Data-Based Algorithms for Model Reduction and Feedback Control of PDEs

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In complex applications such as design and feedback control of energy efficient buildings, and wave or wind energy arrays, existing codes are often available for simulation but these codes are not always suitable for purposes such as model reduction or feedback control design. The outcome of this research is a suite of computationally efficient and provably convergent algorithms for grand challenge problems in PDE model reduction and feedback control. The proposed work focuses on two fundamental problem classes: (1) computing approximate solutions of infinite dimensional Lyapunov and Riccati equations used in feedback control design, and (2) algorithms for PDE balanced model reduction. This work utilizes specific solution data from PDE systems to produce provably convergent approximations for PDE model reduction and feedback control problems. Since the algorithms are data-based, the methods may use existing simulation codes with specialized discretization schemes and tools such as adaptive solvers, parallel algorithms, and multigrid methods to increase computational efficiency and accuracy. Since the algorithms can exploit such advanced simulation techniques, the algorithms can scale to treat highly complex grand challenge problems. The accurate solution of model reduction problems for partial differential equation systems is important for many applications including fast simulations, tractable optimization, feedback control design, and real time control implementation. Feedback control of PDEs has potential applications in many areas. For example, feedback control of air flow in commercial buildings could lead to dramatic reduction in energy consumption. In another area, feedback control of fluid-structure interactions in wave energy converters could lead to effective ocean wave energy extraction. The proposed work primarily considers the development and analysis of model reduction and feedback control algorithms for linear PDE systems, as extensions of these reduced models and feedback laws can be used to treat problems involving nonlinear PDEs. The algorithms can exploit advanced existing simulation codes, and therefore the completion of the proposed work will open the door for large advances in computational feedback control to challenging nonlinear PDE systems.