Robust Stabilization of Laminar Flows in Varying Flow Regimes

Abstract

The stabilization of laminar flows on the base of linearizations and feedback controllers has been the subject of many recent theoretical and computational studies. However, the applicability of the standard approaches is limited due to the inherent fragility of observer based controllers with respect to arbitrary small changes in the system. We show that a slight variation in the Reynolds number of a flow setup amounts to a coprime factor perturbation in the associated linear transfer function. Based on these findings, we argue that known concepts from robust control can be exploited to come up with an output feedback law that can stabilize the cylinder wake over the transition period from a stable to a stabilized regime.

Robust output-feedback stabilization for incompressible flows using low-dimensional $$\mathcal {H}_{\infty }$$-controllers

“…In the terms of the general system (2), the particular structural assumptions made for (9) mean that and and imply that Assumption 1 is fulfilled if (sE − A, B) is stabilizable and (sE − A, C) is detectable. Moreover, this form of B 1 , B 2 , C 1 and C 2 implies that and, from Proposition 1, that the existence of an admissible controller, which solves the suboptimal H ∞ -control problem (5) for the gain , is equivalent to the existence of symmetric positive semi-definite matrices X H ∞ and Y H ∞ , which solve such that the spectrum condition (7) holds and the matrix pencils…”

“…This rules out the standard linear quadratic Gaussian (LQG) design that has no guaranteed stability margin [21]. A general remedy is provided by H ∞ -controllers that, provably, can compensate for linearization errors [7,14,30], discretization errors [8,19], and truncation errors [37].…”

Convergence of coprime factor perturbations for robust stabilization of Oseen systems

“…for the two-dimensional input u = (u 1 , u 2 ). Note that, due to the evaluation of the trace,B is not bounded in L 2 (Ω; R 2 ) whereas B = ΠB is; see [4].…”

“…As for robust control, the Oseen linearization has been investigated in [2,12] in view of the operator H ∞ -Riccati equations. Separately, the need and applicability for H ∞ -robust controllers to account for discretization or linearization errors in flow control setups have been discussed in [4,5].…”

“…If (A, B, C) is exponentially detectable by L ∈ L(C k , X) thenN (s) = C(sI − A − LC) −1 B (15a) M (s) = I + C(sI − A − LC) −1 L (15b) define a (left) coprime factorization G = M −1 N of G over MA − (0).The class of transfer functions that possess coprime factorizations over MÂ − (0) (see[11, Def. 7.2.7]) is called the Callier-Desoer class and is well-suited for infinite dimensional linear time-invariant systems since, in particular, it contains transfer functions that are not rational as they occur, e.g., in the modeling of systems with delay.A perturbation δ A of A, that is still stabilized by LC, transfers to the coprime factors as follows; cp [4,. Thm.…”