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

S. Leonhardt

Philips Chair for Medical Information Technology (MedIT), RWTH Aachen University, Aachen, Germany

P. Apkarian

Control Systems Department, Onera, Toulouse, France

B.J.E. Misgeld

Philips Chair for Medical Information Technology (MedIT), RWTH Aachen University, Aachen, Germany

1Corresponding author.

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


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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