Low-order state-feedback controller design for long-time average cost control of fluid flow systems: A sum-of-squares approach

This paper presents a novel state-feedback controller design approach for long-time average cost control of fluid flows. Due to the high dimensionality of fluid dynamical system, direct controller design renders to high-order state feedback that might not be applicable in practice. To resolve this, the original system is first transformed into a reduced-order uncertain system, where polynomial bounds for the involved uncertainties are evaluated analytically. Meanwhile, instead of minimizing the time-averaged cost itself, we use its upper bound as the objective function for controller design. Then, under the framework of sum-of-squares-based optimization, the control law and a tunable function similar to the Lyapunov function are optimized simultaneously. The inherent non-convexity of the optimization is overcome by assuming that the controller takes a small-feedback structure, which actually is a series in a small parameter with all the coefficients being finite-order polynomials of the reduced-order system state. The main characteristics of the induced controller lie in its low-order structure and robustness.