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Technical Brief

$\mathcal{H}_\infty$ optimal controller design with closed-loop positive real constraints

[+] Author and Article Information
L. Hewing

Institute for Dynamic Systems and Control (IDSC), ETH Zurich, Zurich, Switzerland
lhewing@ethz.ch

S. Leonhardt

Philips Chair for Medical Information Technology (MedIT), RWTH Aachen University, Aachen, Germany
Leonhardt@hia.rwth-aachen.de

P. Apkarian

Control Systems Department, Onera, Toulouse, France
pierre.apkarian@onera.fr

B.J.E. Misgeld

Philips Chair for Medical Information Technology (MedIT), RWTH Aachen University, Aachen, Germany
misgeld@hia.rwth-aachen.de

1Corresponding author.

ASME doi:10.1115/1.4036073 History: Received August 23, 2016; Revised February 08, 2017

Abstract

Positive real constraints on the closed-loop of linear systems guarantee stable interaction with arbitrary passive environments. Two such methods of $\mathcal{H}_\infty$ optimal controller synthesis subject to a positive real constraint are presented and demonstrated on numerical examples. The first approach is based on an established multi-objective optimal control framework using linear matrix inequalities (LMIs) and is shown to be overly restrictive and ultimately infeasible. The second method employs a sector transformation to substitute the positive real constraint with an equivalent $\mathcal{H}_\infty$ constraint. In two examples this method is shown to be more reliable and displays little change in the achieved $\mathcal{H}_\infty$ norm compared to the unconstrained design, making it a promising tool for passivity based controller design.

Copyright (c) 2017 by ASME
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