This paper examines the gain-scheduling problem with a particular focus on controller interpolation with guaranteed stability of the nonlinear closed-loop system. For linear parameter varying model representations, a method of interpolating between controllers utilizing the Youla parametrization is proposed. Quadratic stability despite fast scheduling is guaranteed by construction, while the characteristics of individual controllers designed a priori are recovered at critical design points. Methods for reducing the state dimension of the interpolated controller are also given. The capability of the proposed approach to guarantee stability despite arbitrarily fast transitions leads naturally to application to switched linear systems. The efficacy of the method is demonstrated in simulation using a multi-input, multi-output, nonminimum-phase system, while interpolating between two controllers of different sizes and structures.
Issue Section:
Research Papers
1.
Leith
, D. J.
, and Leithead
, W. E.
, 2000, “Survey of Gain-Scheduling Analysis and Design
,” Int. J. Control
0020-7179, 73
, pp. 1001
–1025
.2.
Shamma
, J. S.
, and Athans
, M.
, 1991, “Gain Scheduling: Potential Hazards and Possible Remedies
,” Proceedings of the 1991 American Control Conference
, Vol. 1
, pp. 516
–521
.3.
Youla
, D. C.
, Bongiorno
, J. J.
, Jr., and Jabr
, H. A.
, 1976, “Modern Wiener-Hopf Design of Optimal Controllers Part I: The Single-Input-Output Case
,” IEEE Trans. Autom. Control
0018-9286, AC-21
, pp. 3
–13
.4.
Boyd
, S.
, and Yang
, Q.
, 1989, “Structured and Simultaneous Lyapunov Functions for System Stability Problems
,” Int. J. Control
0020-7179, 49
, pp. 2215
–2240
.5.
Johansson
, K. H.
, 2000, “Quadruple-Tank Process: A Multivariable Laboratory Process With an Adjustable Zero
,” IEEE Trans. Control Syst. Technol.
1063-6536, 8
, pp. 456
–465
.6.
Packard
, A.
, and Kantner
, M.
, 1996, “Gain Scheduling the LPV Way
,” Proceedings of the 35th IEEE Conference on Decision and Control
, Vol. 4
, pp. 3938
–3941
.7.
Apkarian
, P.
, and Gahinet
, P.
, 1995, “Convex Characterization of Gain-Scheduled H∞ Controllers
,” IEEE Trans. Autom. Control
0018-9286, 40
, pp. 853
–864
.8.
Stilwell
, D. J.
, and Rugh
, W. J.
, 2000, “Stability Preserving Interpolation Methods for the Synthesis of Gain Scheduled Controllers
,” Automatica
0005-1098, 36
, pp. 665
–671
.9.
Apkarian
, P.
, and Adams
, R. J.
, 1998, “Advanced Gain-Scheduling Techniques for Uncertain Systems
,” IEEE Trans. Control Syst. Technol.
1063-6536, 6
, pp. 21
–32
.10.
Guo
, D.
, and Rugh
, W. J.
, 1992, “An Approach to Gain Scheduling on Fast Variables
,” Proceedings of the 31st IEEE Conference on Decision and Control
, Vol. 1
, pp. 759
–763
.11.
Gahinet
, P.
, Apkarian
, P.
, and Chilali
, M.
, 1996, “Affine Parameter-Dependent Lyapunov Functions and Real Parametric Uncertainty
,” IEEE Trans. Autom. Control
0018-9286, 41
, pp. 436
–442
.12.
Rugh
, W. J.
, and Shamma
, J. S.
, 2000, “Research on Gain Scheduling
,” Automatica
0005-1098, 36
, pp. 1401
–1425
.13.
Shamma
, J. S.
, 1994, “Robust Stability With Time-Varying Structured Uncertainty: Automatic Control
,” IEEE Trans. Autom. Control
0018-9286, 39
, pp. 714
–724
.14.
Nichols
, R. A.
, Reichert
, R. T.
, and Rugh
, W. J.
, 1993, “Gain Scheduling for H-Infinity Controllers: A Flight Control Example
,” IEEE Trans. Control Syst. Technol.
1063-6536, 1
, pp. 69
–79
.15.
Reichert
, R. T.
, 1992, “Dynamic Scheduling of Modern-Robust-Control Autopilot Designs for Missiles
,” IEEE Control Syst. Mag.
0272-1708, 12
, pp. 35
–42
.16.
Hyde
, R. A.
, and Glover
, K.
, 1993, “Application of Scheduled H∞ Controllers to a VSTOL Aircraft
,” IEEE Trans. Autom. Control
0018-9286, 38
, pp. 1021
–1039
.17.
Murray-Smith
, R.
, and Johansen
, T.
, 1997, Multiple Model Approaches to Nonlinear Modelling and Control
, Taylor & Francis
, Bristol, PA
.18.
Hunt
, K. J.
, Johansen
, T. A.
, Kalkkuhl
, J.
, Fritz
, H.
, and Gottsche
, Th.
, 2000, “Speed Control Design for an Experimental Vehicle Using a Generalized Gain Scheduling Approach
,” IEEE Trans. Control Syst. Technol.
1063-6536, 8
, pp. 381
–395
.19.
Johansen
, T. A.
, Hunt
, K. J.
, and Petersen
, I.
, 2000, “Gain-Scheduled Control of a Solar Power Plant
,” Control Eng. Pract.
0967-0661, 8
, pp. 1011
–1022
.20.
Zhang
, R.
, Alleyne
, A.
, and Carter
, D. E.
, 2005, “Generalized Multivariable Gain Scheduling With Robust Stability Analysis
,” ASME J. Dyn. Syst., Meas., Control
0022-0434, 127
, pp. 668
–687
.21.
Niemann
, H.
, and Stoustrup
, J.
, 1999, “An Architecture for Implementation of Multivariable Controllers
,” Proceedings of the 1999 American Control Conference
, Vol. 6
, pp. 4029
–4033
.22.
Niemann
, H.
, and Stoustrup
, J.
, 1999, “Gain Scheduling Using the Youla Parameterization
,” The 38th IEEE Conference on Decision and Control
, Vol. 3
, pp. 2306
–2311
.23.
Rasmussen
, B. P.
, and Alleyne
, A.
, “Gain Scheduled Control of an Air Conditioning System Using the Youla Parameterization
,” IEEE Trans. Control Syst. Technol.
1063-6536, accepted.24.
Stilwell
, D. J.
, 2001, “State-Space Interpolation for a Gain-Scheduled Autopilot
,” J. Guid. Control Dyn.
0731-5090, 24
, pp. 460
–465
.25.
Niemann
, H.
, 2003, “Dual Youla Parameterisation
,” IEE Proc.: Control Theory Appl.
1350-2379, 150
, pp. 493
–497
.26.
Tay
, T. -T.
, Mareels
, I.
, and Moore
, J. B.
, 1998, High Performance Control
, Birkhauser
, Ann Arbor, MI
.27.
Stilwell
, D. J.
, 1999, “J-Q Interpolation for Gain Scheduled Controllers
,” Proceedings of the 1999 Conference on Decision and Control
, Vol. 1
, pp. 749
–754
.28.
Stoustrup
, J.
, and Niemann
, H.
, 1997, “Starting Up Unstable Multivariable Controllers Safely
,” Proceedings of the 36th IEEE Conference on Decision and Control
, Vol. 2
, pp. 1437
–1438
.29.
Stilwell
, D. J.
, and Rugh
, W. J.
, 1999, “Interpolation of Observer State Feedback Controllers for Gain Scheduling
,” IEEE Trans. Autom. Control
0018-9286, 44
, pp. 1225
–1229
.30.
Hespanha
, J. P.
, and Morse
, A. S.
, 2002, “Switching Between Stabilizing Controllers
,” Automatica
0005-1098, 38
, pp. 1905
–1917
.31.
Niemann
, H.
, Stoustrup
, J.
, and Abrahamsen
, R. B.
, 2004, “Switching Between Multivariable Controllers
,” Opt. Control Appl. Methods
0143-2087, 25
, pp. 51
–66
.32.
Xie
, W.
, and Eisaka
, T.
, 2004, “Design of LPV Control Systems Based on Youla Parameterisation
,” IEE Proc.: Control Theory Appl.
1350-2379, 151
, pp. 465
–472
.33.
Bendtsen
, J. D.
, Stoustrup
, J.
, and Trangbaek
, K.
, 2005, “Bumpless Transfer Between Observer-Based Gain Scheduled Controllers
,” Int. J. Control
0020-7179, 78
, pp. 491
–504
.34.
Khalil
, H. K.
, 1996, Nonlinear Systems
, Prentice-Hall
, Upper Saddle River, NJ
.35.
Boyd
, S.
, El Ghaoui
, L.
, Feron
, E.
, and Balakrishnan
, V.
, 1994, Linear Matrix Inequalities in System and Control Theory
, SIAM
, Philadelphia, PA
.36.
Zhou
, K.
, Doyle
, J. C.
, and Glover
, K.
, 1996, Robust and Optimal Control
, Prentice-Hall
, Englewood Cliffs, NJ
.37.
Liberzon
, D.
, 2003, Switching in Systems and Control
, Birkhäuser
, Boston, MA
.Copyright © 2010
by American Society of Mechanical Engineers
You do not currently have access to this content.
