Convergence of coprime factor perturbations for robust stabilization of Oseen systems

Jan Heiland

doi:10.3934/mcrf.2021043

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2022, Volume 12, Issue 3: 747-761. Doi: 10.3934/mcrf.2021043

This issue Previous Article Optimal control of transverse vibration of a moving string with time-varying lengths Next Article Asymptotic gain results for attractors of semilinear systems

Convergence of coprime factor perturbations for robust stabilization of Oseen systems

Jan Heiland^,

Max Planck Institute for Dynamics of, Complex Technical Systems and OVGU Magdeburg, 39106 Magdeburg, Germany

^* Corresponding author: Jan Heiland
^* Corresponding author: Jan Heiland

Received: August 31, 2019

Revised: January 31, 2021

Early access: September 2021

Published: July 2022

This article was recruited by Andrii Mironchenko

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Abstract

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.

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Mathematics Subject Classification: Primary: 93B52, 35Q30; Secondary: 93C73.

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