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Technical Brief

# $H∞$ 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 8092, Switzerland
e-mail: lhewing@ethz.ch

S. Leonhardt, B. J. E. Misgeld

Philips Chair for Medical Information Technology (MedIT),
RWTH Aachen University,
Aachen 52074, Germany

P. Apkarian

Control Systems Department,
Onera,
Toulouse 31055, France

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received August 23, 2016; final manuscript received February 8, 2017; published online June 5, 2017. Assoc. Editor: Ryozo Nagamune.

J. Dyn. Sys., Meas., Control 139(9), 094502 (Jun 05, 2017) (8 pages) Paper No: DS-16-1413; 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 $H∞$ 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 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 $H∞$ constraint. In two examples, this method is shown to be more reliable and displays little change in the achieved $H∞$ norm compared to the unconstrained design, making it a promising tool for passivity-based controller design.

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## Figures

Fig. 1

Multi-objective generalized plant configuration

Fig. 2

Block diagram representation of the multi-objective Cayley transformation

Fig. 3

Block diagram of a simple H∞ disturbance rejection problem

Fig. 4

Singular value plot of the H∞ channel with different controllers

Fig. 5

Bode plot of the closed-loop frequency response Tcl(jω)=y(jω)/d(jω)

Fig. 6

Simple series elastic actuator (SEA) model consisting of motor inertia J, viscous friction coefficient b, and ideal spring constant k

Fig. 7

Augmented plant for controller design of simple SEA

Fig. 8

Nyquist plot of the impedance transfer function for simple SEA example with either classically designed H∞ controller or positive real constraint Cayley approach H∞ controller

Fig. 10

Reference tracking step response for simple SEA example with either classically designed H∞ controller or positive real constraint Cayley approach H∞ controller

Fig. 9

Bode plot of the controller transfer function for both classic H∞ controller and positive real constraint Cayley approach H∞ controller

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