Expensive control of long-time averages using sum of squares and its application to a laminar wake flow

TL;DR

This work addresses reducing the long-time-average cost $\bar{\Phi}$ in nonlinear polynomial systems by replacing the intractable time-average optimization with the optimization of an upper bound $C$ through a Lyapunov-like function $V$, using sum-of-squares (SOS) and semidefinite programming (SDP). By assuming expensive control, the authors formulate a small-parameter expansion in $\epsilon$ that renders the joint optimization of the controller and the bound convex, solving a sequence of SDP problems to obtain polynomial controllers. The approach is demonstrated on a wake-flow problem: a full Navier–Stokes model at $Re=100$ is reduced to a 10-mode ROM via POD-Galerkin, and both linear and quadratic small-feedback controllers are designed to reduce the bound and, on the ROM, the actual long-time cost; DNS, while confirming qualitative trends, shows more modest gains due to reduced model fidelity. The study highlights the potential of SOS-based LTACC for polynomial systems and fluid flows, while also revealing limitations from ROM accuracy and computational cost that guide future improvements in higher-dimensional models. Overall, the paper provides a practical framework for convexified LTACC and demonstrates meaningful bound-based control in a canonical fluid-structure problem.

Abstract

The paper presents a nonlinear state-feedback control design approach for long-time average cost control, where the control effort is assumed to be expensive. The approach is based on sum-of-squares and semi-definite programming techniques. It is applicable to dynamical systems whose right-hand side is a polynomial function in the state variables and the controls. The key idea, first described but not implemented in (Chernyshenko et al., Phil. Trans. R. Soc. A, 372, 2014), is that the difficult problem of optimizing a cost function involving long-time averages is replaced by an optimization of the upper bound of the same average. As such, controller design requires the simultaneous optimization of both the control law and a tunable function, similar to a Lyapunov function. The present paper introduces a method resolving the well-known inherent non-convexity of this kind of optimization. The method is based on the formal assumption that the control is expensive, from which it follows that the optimal control is small. The resulting asymptotic optimization problems are convex. The derivation of all the polynomial coefficients in the controller is given in terms of the solvability conditions of state-dependent linear and bilinear inequalities. The proposed approach is applied to the problem of designing a full-information feedback controller that mitigates vortex shedding in the wake of a circular cylinder in the laminar regime via rotary oscillations. Control results on a reduced-order model of the actuated wake and in direct numerical simulation are reported.

Figures (6)

• Figure 1: Time-averaged costs $\overline{\Phi}_0$ and $\overline{\Phi}$ from closed-loop ROM simulation using linear controllers, and the asymptotics $C_{0,SOS}+\epsilon C_{1,SOS}$ of the upper bound for $\overline{\Phi}.$
• Figure 2: Time-averaged costs $\overline{\Phi}_0$ and $\overline{\Phi}$ from closed-loop ROM simulation using quadratic controllers, and the asymptotics $C_{0,SOS}+\epsilon C_{1,SOS}$ of the upper bound for $\overline{\Phi}.$
• Figure 3: Profile of the control input for the linear feedback controller with $\epsilon=0.025$.
• Figure 4: Profile of the control input for the quadratic feedback controller with $\epsilon=0.02$.
• Figure 5: Long-time averaged cost as a function of $\epsilon$ obtained by closed-loop direct numerical simulation. Results for linear (triangles) and quadratic (circles) controllers are reported.

Theorems & Definitions (4)

• Lemma 1
• Remark 1
• Theorem 1
• Remark 2