Reduction methods for feedback stabilization of fluid flows

This dissertation deals with reduction of two classes of dynamical systems in order to make them suitable for control design using linear systems theory. The first class consists of the large-dimensional systems governing fluid flows, for which we employ dimension reduction techniques for stabilization of unstable steady states. The second class consists of systems with a continuous symmetry, for which we use a symmetry reduction method for stabilization of unstable relative equilibria.;Numerical discretization of fluid flows results in a large system of equations of O(105--8), while linear systems tools are limited to dimensions of O(102--4). Model reduction has played an important role in making these tools available for flow control. An attractive method is the approximate balanced truncation, in which the governing equations (linearized about a steady state) are projected onto a small number (≤100) of dynamically important modes, and the resulting models accurately capture the input-output (actuation to sensing) behavior. A limitation is that this method is restricted to linearizations about stable steady states. In this work, we extend its applicability to unstable steady states, assuming a small number of unstable modes. The unstable dynamics is treated exactly while reduced models are obtained of the (large dimensional) stable dynamics. We show a theoretical equivalence between approximate balanced truncation and a system identification technique called eigensystem realization algorithm (ERA). We extend ERA to simulations and obtain an order-of-magnitude cost reduction over balanced truncation.;The reduction techniques are applied to a model problem of the two-dimensional flow past a at plate at a low Reynolds number and a large angle of attack. The natural (uncontrolled) flow is periodic vortex shedding, although there also exists an unstable steady state that we seek to stabilize. The control actuation is modeled using a localized body-force actuator close to the leading or trailing edge and velocities are measured at two near-wake sensor locations. We obtain reduced models of the input-output dynamics linearized about the unstable steady state and show that 20-30 order models accurately capture the full system dynamics. We use the models to develop sensor-based feedback controllers and include them in the full nonlinear simulations. Even though the models are valid in a local neighborhood of the steady state, we show that they are capable of suppressing the periodic vortex shedding, which is a nonlinear phenomenon.;We also consider systems with a continuous symmetry and use a template-based approach to reduce the equations to a frame in which the symmetry is factored out. Relative equilibria are steady states in the symmetry-reduced frame; an example is traveling waves in systems with translational symmetry. The control goal is to stabilize unstable relative equilibria and the control design is based on linearization of the reduced equations about these steady states. A systematic reconstruction procedure is provided to obtain the form of the controller in the original coordinates. The control design relies on standard linear systems tools and the controlled system retains the symmetry of the original system. The control is demonstrated using various examples, including stabilization of unstable traveling waves in the 1D Kuramoto-Sivashinsky equation.