A Koopman operator approach for the pitch stabilization of a hydrofoil in an unsteady flow field

The control of swimming robots presents several challenges, in large part due to the complex fluid-structure interaction. Low fidelity simplified formulas for drag and lift force lead to control amenable models, but do not capture key physics that can play an especially important role in the swimming of small-scale robots with limited actuation. Higher fidelity models of the fluid-structure interaction lead to nonlinear high dimensional control systems for which solution methods are not obvious. We propose the use of the Koopman operator in developing a linear representation for both the complex fluid structure interaction as well as actuation effects. As a test case for this framework we address the problem of stabilizing the pitching oscillations of a hydrofoil that is hinged in a simulated unsteady free stream flow. The actuator for the hydrofoil is an internal reaction wheel which presents an integral saturation constraint. Using the Koopman operator, the lifted control system is used to formulate a constrained optimal control problem which we solve using model predictive control. The framework proposed in this paper can potentially be extended to design a combination of data-driven and physics-based control algorithms for swimming robots.