Adaptive feedback linearizing control of proper orthogonal decomposition nonlinear flow models

This paper treats the question of feedback linearizing control of two-dimensional incompressible, unsteady wake flow. For definiteness, flow past a circular cylinder is considered, but the design approach presented here is applicable to other flow control problems. Two finite-dimensional lower order models based on proper orthogonal decomposition (POD) of dimension N with N actuators are considered. Models I and II are obtained using control function and penalty function methods, respectively. Control action can be achieved by a combination of suction, injection, and synthetic jets. For the design of controllers, it is assumed that the system matrices of the POD models are unknown. Nonlinear adaptive control systems for the two models are derived. For model I, nontrivial zero-error dynamics exists, which play a key role in the stability of the closed-loop system. But for model II, global adaptive trajectory control is achieved. In the closed-loop system, the mode amplitudes asymptotically follow the reference trajectories. Simulation results for a 4-mode POD model obtained using the penalty function method are presented. These results show that in the closed-loop system, unsteadiness in the mode amplitudes can be suppressed in spite of large uncertainties in the flow model.