Lizette Zietsman's Home Page
Spring 2017: MATH 4446.
Current interests involve the development and analysis of numerical methods for solving optimal control problems where the dynamics are described by partial differential equations; for example, fundamental fluid flows. This includes applications such as the design, optimization and control of energy efficient buildings. Commercial buildings are responsible for 40% of the energy consumption and greenhouse gas emissions worldwide and significantly exceed those of all transportation combined. Reducing energy consumption of commercial buildings can have a tremendous impact on energy cost and greenhouse gas emission.
Recent results have shown that by considering the whole building as an integrated system and applying modern estimation and control techniques tothis system, one can achieve greater efficiencies than obtained by optimizing individual building components such as lighting and HVAC. This approach leads to models that are complex, multi-scale, highly uncertain dynamic systems with wide varieties of disturbances. Keeping the physics in the problem for as long as possible and only discretizing at the last stage lead to more accurate models, but also extremely large systems that result in numerous computational challenges.
These numerical challenges include computational algorithms for the optimal placement of sensors and actuators that maximize observability and controllability. In addition, the discretization of the optimal control problem leads to extremely large systems of finite dimensional Riccati equations. In order to generate practical algorithms, we take advantage of the underlying infinite dimensional partial differential equation structure. This includes studying the sensitivity of the algebraic Riccati equation, mesh-independence of Newton methods used to solve the Riccati equation, as well as the use of adaptive methods to compute the optimal feedback gain. Furthermore, in designing numerical methods for control problems, special attention should be paid to properties that will guarantee that the approximation scheme yield strong convergence of the operators. The approximation method used in the generating the Riccati equation must also be selected to ensure that the finite dimensional equations are well conditioned and additional features like mesh-independence are obtained.
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