Approximation and Feedback Control of Nonlinear Systems with Applications to Thermo-Fluids

Abstract

This dissertation studies approximation and feedback control for nonlinear systems, with an emphasis on high-dimensional thermo-fluid models arising from fluid dynamics and indoor-air applications. The central challenge is that many nonlinear control problems originate from partial differential equations whose discretizations are too large for direct optimization, analysis, or real-time feedback design. This work shows that such problems can be made tractable by utilizing approximation theory, model reduction, and nonlinear feedback synthesis. The first part of the dissertation develops theoretical results for offline value-function approximation in reproducing kernel Hilbert spaces. In a class of native spaces, explicit rates of convergence are derived for approximations generated within a policy-iteration framework. These results show that approximation accuracy depends not only on smoothness properties of the basis, but also on the geometric distribution of the centers defining the approximation space, thereby providing a rigorous connection between approximation theory and reinforcement learning-based control. The second part applies control-oriented model reduction and nonlinear feedback design to the fluidic pinball, a benchmark flow-control problem. Using interpolatory model order reduction, the full-order model is reduced from approximately 30,000 states to a reduced model of dimension r=80. A Quadratic--Quadratic Regulator (QQR) is then constructed and evaluated on the full-order model. For Re_D=30, the nonlinear controller achieves the target performance criteria faster than the corresponding linear controller, and for Re_D=50, it stabilizes the flow while the linear controller fails. The final part develops a reduced-order nonlinear control framework for buoyancy-driven indoor-air flows motivated by HVAC applications. A finite-element discretization of the Boussinesq equations produces a large-scale descriptor system with on the order of 5×10^4 degrees of freedom. After linearization, the system is reduced using the Iterative Rational Krylov Algorithm (IRKA) to a control-oriented model of dimension r=26. The reduced model is then used to construct nonlinear feedback through the QQR framework. Closed-loop simulations show improved transient regulation and a reduction in cumulative control cost relative to linear quadratic regulation. Taken together, these results demonstrate that mathematically grounded approximation and model order reduction can bridge the gap between high-fidelity nonlinear models and implementable feedback controllers. The dissertation contributes both new theory for approximation in nonlinear control and new computational methodologies for reduced-order feedback design in thermo-fluid systems.