0
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

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.

FIGURES IN THIS ARTICLE
<>
Copyright © 2017 by ASME
Your Session has timed out. Please sign back in to continue.

References

Ortega, R. , Loría, A. , Nicklasson, P. J. , and Sira-Ramirez, H. , 2013, Passivity-Based Control of Euler-Lagrange Systems: Mechanical, Electrical, and Electromechanical Applications, Springer Science & Business Media, Springer-Verlag London, U.K.
Vallery, H. , Veneman, J. , van Asseldonk, E. , Ekkelenkamp, R. , Buss, M. , and van Der Kooij, H. , 2008, “ Compliant Actuation of Rehabilitation Robots,” IEEE Rob. Autom. Mag., 15(3), pp. 60–69. [CrossRef]
Kottenstette, N. , and Antsaklis, P. , 2010, “ Relationships Between Positive Real, Passive Dissipative, & Positive Systems,” American Control Conference (ACC), pp. 409–416.
Forbes, J. R. , 2013, “ Dual Approaches to Strictly Positive Real Controller Synthesis With a H2 Performance Using Linear Matrix Inequalities,” Int. J. Robust Nonlinear Control, 23(8), pp. 903–918. [CrossRef]
Bernussou, J. , Geromel, J. , and Oliveira, M. , 1999, “ On Strict Positive Real Systems Design: Guaranteed Cost and Robustness Issues,” Syst. Control Lett., 36(2), pp. 135–141. [CrossRef]
Wang, S. , Zhang, G. , and Liu, W. , 2010, “ Dissipative Analysis and Control for Discrete-Time State-Space Symmetric Systems,” 29th Chinese Control Conference (CCC), pp. 2010–2015.
Chapel, J. , and Su, R. , 1991, “ Attaining Impedance Control Objectives Using H Design Methods,” IEEE International Conference on Robotics and Automation (ROBOT), pp. 1482–1487.
Scherer, C. , Gahinet, P. , and Chilali, M. , 1997, “ Multi-Objective Output-Feedback Control via LMI Optimization,” IEEE Trans. Autom. Control, 42(7), pp. 896–911. [CrossRef]
Safonov, M. , Jonckherre, E. , Vermaj, M. , and Limebeer, D. , 1987, “ Synthesis of Positive Real Multivariable Feedback Systems,” Int. J. Control, 45(3), pp. 817–842. [CrossRef]
Bao, J. , and Lee, P. L. , 2007, Process Control: The Passive Systems Approach, Springer-Verlag, London.
Bao, J. , Lee, P. L. , Wang, F. , and Zhou, W. , 1998, “ New Robust Stability Criterion and Robust Controller Synthesis,” Int. J. Robust Nonlinear Control, 8(1), pp. 49–59. [CrossRef]
Brogliato, B. , Lozano, R. , Maschke, B. , and Egeland, O. , 2007, Dissipative Systems Analysis and Control: Theory and Applications, Springer-Verlag, London.
Gahinet, P. , and Apkarian, P. , 1994, “ A Linear Matrix Inequality Approach to H Control,” Int. J. Robust Nonlinear Control, 4(4), pp. 421–448. [CrossRef]
Boyd, S. , El Ghaoui, L. , Feron, E. , and Balakrishnan, V. , 1994, Linear Matrix Inequalities in System and Control Theory, SIAM, University City Science Center, Philadelphia, PA.
Anderson, B. D. O. , and Vongpanitlerd, S. , 2006, Network Analysis and Synthesis: A Modern Systems Theory Approach, Dover Publications, Mineola, NY.
Scherer, C. , and Weiland, S. , 2005, “ Linear Matrix Inequalities in Control,” Lecture Notes, Delft University of Technology, Delft, The Netherlands.
Boyd, S. , and Vandenberghe, L. , 2004, Convex Optimization, Cambridge University Press, Cambridge, UK.
Apkarian, P. , and Noll, D. , 2006, “ Non-Smooth H Synthesis,” IEEE Trans. Autom. Control, 51(1), pp. 71–86. [CrossRef]
Apkarian, P. , and Noll, D. , 2006, “ Non=Smooth Optimization for Multidisk H Synthesis,” Eur. J. Control, 12(3), pp. 229–244. [CrossRef]
Blondel, V. , and Tsitsiklis, J. N. , 1997, “ NP-Hardness of Some Linear Control Design Problems,” SIAM J. Control Optim., 35(6), pp. 2118–2127. [CrossRef]
MathWorks, 2015, “ Robust Control Toolbox R2015b,” MathWorks, Natick, MA.
Löfberg, J. , 2004, “ YALMIP: A Toolbox for Modeling and Optimization in MATLAB,” CACSD Conference, pp. 284–289.
Sturm, J. , 1999, “ Using SeDuMi 1.02, A Matlab Toolbox for Optimization Over Symmetric Cones,” Optim. Methods Software, 11, pp. 625–653. [CrossRef]
Vanderborght, B. , Albu-Schäffer, A. , Bicchi, A. , Burdet, E. , Caldwell, D. , Carloni, R. , Catalano, M. , Eiberger, O. , Friedl, W. , Ganesh, G. , Garabini, M. , Grebenstein, M. , Grioli, G. , Haddadin, S. , Hoppner, H. , Jafari, A. , Laffranchi, M. , Lefeber, D. , Petit, F. , Stramigioli, S. , Tsagarakis, N. , Van Damme, M. , Van Ham, R. , Visser, L. C. , and Wolf, S. , 2013, “ Variable Impedance Actuators: A Review,” Rob. Auton. Syst., 61(12), pp. 1601–1614. [CrossRef]
Hindi, H. A. , Hassibi, B. , and Boyd, S. P. , 1998, “ Multi-Objective H2/H-Optimal Control via Finite Dimensional q-Parametrization and Linear Matrix Inequalities,” American Control Conference (ACC), Vol. 5, IEEE, Philadelphia, PA, pp. 3244–3249.
Scherer, C. W. , 2000, “ An Efficient Solution to Multi-Objective Control Problems With LMI Objectives,” Syst. Control Lett., 40(1), pp. 43–57. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

Multi-objective generalized plant configuration

Grahic Jump Location
Fig. 4

Singular value plot of the H∞ channel with different controllers

Grahic Jump Location
Fig. 2

Block diagram representation of the multi-objective Cayley transformation

Grahic Jump Location
Fig. 7

Augmented plant for controller design of simple SEA

Grahic Jump Location
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

Grahic Jump Location
Fig. 10

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

Grahic Jump Location
Fig. 9

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

Grahic Jump Location
Fig. 3

Block diagram of a simple H∞ disturbance rejection problem

Grahic Jump Location
Fig. 6

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

Grahic Jump Location
Fig. 5

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

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In