Control of a Class of High-Dimensional Nonlinear Oscillators: Application to Flow Stabilization

This article presents a methodology for designing linear time-invariant (LTI) controllers to stabilize high-dimensional nonlinear oscillator systems with an unstable equilibrium and a periodic or quasiperiodic attractor. The proposed approach is hybrid, combining ideas from classic model- based methods and more recent data-based approaches. The model-based component is aimed at guaranteeing the stability of the closed-loop system near the equilibrium, which is formulated using Youla parametrization. The data-based component tackles the nonlinearity and high-dimensionality of the system by utilizing simulation data in conjunction with derivative-free optimization to design LTI controllers. The approach results in a collection of LTI controllers that not only asymptotically stabilize the system near its equilibrium, but also drive the system from the attractor to the stabilized equilibrium. The efficacy of the method is demonstrated on a challenging example of a high-dimensional nonlinear oscillator from fluid mechanics: the incompressible flow over a 2-D open cavity at $\text {Re}=7500$ . Not only does it confirm the existence of simple LTI controllers stabilizing high-dimensional nonlinear dynamics in simulation, but it also shows the possibility of systematically finding solutions to this problem.