Convergence of coprime factor perturbations for robust stabilization of Oseen systems

Abstract

<p style='text-indent:20px;'>Linearization based controllers for incompressible flows have been proven to work in theory and in simulations. To realize such a controller numerically, the infinite dimensional system has to be linearized and discretized. The unavoidable consistency errors add a small but critical uncertainty to the controller model which will likely make it fail, especially when an observer is involved. Standard robust controller designs can compensate small uncertainties if they can be qualified as a coprime factor perturbation of the plant. We show that for the linearized Navier-Stokes equations, a linearization error can be expressed as a coprime factor perturbation and that this perturbation smoothly depends on the size of the linearization error. In particular, improving the linearization makes the perturbation smaller so that, eventually, standard robust controller will stabilize the system.</p>

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

“…In view of stabilization of incompressible Navier-Stokes equations by linear output-feedback controllers, the following considerations are relevant with respect to the estimate (17). It has been shown that an error in the linearization used for controller design, smoothly transfers to a coprime factor perturbation in the transfer function; see [7,30]. Accordingly, with increasing accuracy in the computation of the linearization, the difference becomes arbitrarily small such that, eventually, a robust controller based on a numerically computed linearization will be able to stabilize the system.…”

“…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].…”

“…The situation that the controller is based on inexact linearizations is relevant in applications and fits well into the presented framework; cf. [8,14,30]. The presented numerical studies are based on the popular setup of the two-dimensional wake behind a cylinder; see, for examples, [17,26,27,39,47].…”