Sum-of-squares approach to feedback control of laminar wake flows

Abstract

In this paper a novel nonlinear feedback control design methodology for incompressible fluid flows aiming at the optimisation of long-time averages of flow quantities is presented. It applies to reduced-order finite-dimensional models of fluid flows, expressed as a set of first-order nonlinear ordinary differential equations with the right-hand side being a polynomial function in the state variables and in the controls. The key idea, first discussed in Chernyshenko et al. (Phil. Trans. R. Soc. Lond. A, vol. 372, 2014, 20130350), is that the difficulties of treating and optimising long-time averages of a cost are relaxed by using the upper/lower bounds of such averages as the objective function. In this setting, control design reduces to finding a feedback controller that optimises the bound, subject to a polynomial inequality constraint involving the cost function, the nonlinear system, the controller itself and a tunable polynomial function. A numerically tractable and efficient approach to the solution of such optimisation problems, based on sum-of-squares techniques and semidefinite programming, is proposed. To showcase the methodology, the mitigation of the fluctuation kinetic energy in the unsteady wake behind a circular cylinder in the laminar regime at Re = 100, via controlled angular motions of the surface, is numerically investigated. A compact reduced-order model that resolves the long-term behaviour of the fluid flow and the effects of actuation, is first derived using proper orthogonal decomposition and Galerkin projection. In a full-information setting, feedback controllers are then designed to reduce the long-time average of the resolved kinetic energy associated with the limit cycle. These controllers are then implemented in direct numerical simulations of the actuated flow. Control performance, total energy efficiency and the physical control mechanisms identified are analysed in detail. Key elements of the methodology, implications and future work are finally discussed.

Bounds for Deterministic and Stochastic Dynamical Systems using Sum-of-Squares Optimization

“…If the dimension of a system is small enough for SoS optimization to be computationally feasible, we expect that tight global bounds on deterministic averages can be obtained using the methods of §2. In fact, since the first draft of this work, the bounding techniques we have presented for deterministic systems have been successfully applied to the design of control systems for fluid flows [19,23]. With stochastic forcing that is not too weak, we expect that fairly tight bounds on stationary expectations also can be obtained using the methods of §4.…”

Modelling and feedback control of vortex shedding for drag reduction of a turbulent bluff body wake

“…The related limitation imposed by the waterbed constraint of (12), is one that applies in the context of the linear feedback-control methods employed in this work. It is perceivable that through the use of more complex nonlinear methods such as sum-of-squares optimisation (Chernyshenko et al, 2014;Lasagna et al, 2016), this limitation could to some extent be circumvented. Such approaches would require an accurate (nonlinear) model for the system, of which the models of section 3 may be a suitable example.…”

Systematic Search For Extreme and Singular Behavior in Some Fundamental Models of Fluid Mechanics

“…The auxiliary function V (u) used in (5.2) is related to the Lyapunov function employed in the study of nonlinear stability of fixed points (except that, unlike the Lyapunov function, it need not be positive semi-definite). Formulations based on auxiliary functions can also be used to obtain bounds on infinite-time and space averages of various quantities of interest leading to convex optimization problems analogous to (5.2) [82][83][84][85][86]. When the auxiliary function is fixed and quadratic whereas optimization is performed with respect to the form of a certain "background flow", this bounding framework reduces to the background method originally developed by Doering & Constantin [87] to obtain rigorous a priori bounds on energy dissipation in wall-bounded flows.…”