Reduced-order observer design methods for both linear and nonlinear discrete-time descriptor systems based on the linear matrix inequality (LMI) approach are investigated. We conclude that the conditions under which a full-order observer exists can also guarantee the existence of a reduced-order observer. By choosing a special reduced-order observer gain matrix, a reduced-order unknown input observer is proposed for linear system with unknown inputs, and then an unknown input reconstruction is provided for some special cases. We also extend above results to the cases of nonlinear systems. Finally, three numerical comparative simulation examples are given to illustrate the effectiveness and merits of proposed methods.
Issue Section:
Technical Brief
Keywords:
Adaptive,
Control theory,
Nonlinear systems,
Uncertain systems ,
Optimal control,
Control,
Robust control
References
1.
Kautsky
, J.
, Nichols
, N. K.
, and Van Dooren
, P.
, 1985
, “Robust Pole Assignment in Linear State Feedback
,” Int. J. Control
, 41
(5
), pp. 1129
–1155
.10.1080/00207185089611882.
Brockett
, R. W.
, and Liberzon
, D.
, 2000
, “Quantized Feedback Stabilization of Linear Systems
,” IEEE Trans. Autom. Control
, 45
(7
), pp. 1279
–1289
.10.1109/9.8670213.
Ilic'-Spong
, M.
, Marino
, R.
, Peresada
, S. M.
, and Taylor
, D.
, 1987
, “Feedback Linearizing Control of Switched Reluctance Motors
,” IEEE Trans. Autom. Control
, 32
(5
), pp. 371
–379
.10.1109/TAC.1987.11046164.
Devasia
, S.
, and Paden
, B.
, 1994
, “Exact Output Tracking for Nonlinear Time-Varying Systems
,” 33rd IEEE
Conference on Decision and Control
, Vol. 3
, pp. 2346
–2355
.10.1109/CDC.1994.4114655.
Veillette
, R. J.
, 1995
, “Reliable Linear-Quadratic State-Feedback Control
,” Automatica
, 31
(1
), pp. 137
–143
.10.1016/0005-1098(94)E0045-J6.
Lunze
, J.
, and Lehmann
, D.
, 2010
, “A State-Feedback Approach to Event-Based Control
,” Automatica
, 46
(1
), pp. 211
–215
.10.1016/j.automatica.2009.10.0357.
Luenberger
, D. G.
, 1964
, “Observing the State of a Linear System
,” IEEE Trans. Mil. Electron.
, 8
(2
), pp. 74
–80
.10.1109/TME.1964.43231248.
Zhu
, F. L.
, and Ding
, X. H.
, 2007
, “The Design of Reduced-Order Observer for Systems With Monotone Nonlinearities
,” Acta Autom. Sin.
, 33
(12
), pp. 1290
–1293
.10.1360/aas-007-12909.
Zhang
, W.
, Su
, H. S.
, Wang
, H. W.
, and Han
, Z. Z.
, 2012
, “Full-Order and Reduced-Order Observers for One-Sided Lipschitz Nonlinear Systems Using Riccati Equations
,” Commun. Nonlinear Sci. Numer. Simul.
, 17
(12
), pp. 4968
–4977
.10.1016/j.cnsns.2012.05.02710.
Boutata
, D.
, Boutat-Baddasb
, L.
, and Darouachb
, M.
, 2012
, “A New Reduced-Order Observer Normal Form for Nonlinear Discrete Time Systems
,” Syst. Control Lett.
, 61
(10
), pp. 1003
–1008
.10.1016/j.sysconle.2012.07.00711.
Mehta
, A. J.
, Bandyopadhyay
, B.
, and Inoue
, A.
, 2010
, “Reduced-Order Observer Design for Servo System Using Duality to Discrete-Time Sliding-Surface Design
,” IEEE Trans. Ind. Electron.
, 57
(11
), pp. 3793
–3800
.10.1109/TIE.2010.204055512.
Wang
, H.
, and Daley
, S.
, 1996
, “Actuator Fault Diagnosis: An Adaptive Observer-Based Technique
,” IEEE Trans. Autom. Control
, 41
(7
), pp. 1073
–1078
.10.1109/9.50891913.
Yang
, J. Q.
, and Zhu
, F. L.
, 2013
, “Synchronization for Chaotic Systems and Chaos-Based Secure Communications Via Both Reduced-Order and Step-by-Step Sliding Mode Observers
,” Commun. Nonlinear Sci. Numer. Simul.
, 18
(4
), pp. 926
–937
.10.1016/j.cnsns.2012.09.00914.
Gao
, Z. W.
, and Ding
, S. X.
, 2007
, “Actuator Fault Robust Estimation and Fault-Tolerant Control for a Class of Nonlinear Descriptor Systems
,” Automatica
, 43
(5
), pp. 912
–920
.10.1016/j.automatica.2006.11.01815.
Duan
, G.
, 2010
, Analysis and Design of Descriptor Linear Systems
, Springer Science & Business Media
, New York.10.1007/978-1-4419-6397-016.
Abbaszadeh
, M.
, and Marquez
, H. J.
, 2012
, “A Generalized Framework for Robust Nonlinear H∞ Filtering of Lipschitz Descriptor Systems With Parametric and Nonlinear Uncertainties
,” Automatica
, 48
(5
), pp. 894
–900
.10.1016/j.automatica.2012.02.03317.
Shields
, D. N.
, 1997
, “Observer Design and Detection for Nonlinear Descriptor Systems
,” Int. J. Control
, 67
(2
), pp. 153
–168
.10.1080/00207179722423418.
Darouach
, M.
, and Boutayeb
, M.
, 1995
, “Design of Observers for Descriptor Systems
,” IEEE Trans. Autom. Control
, 40
(7
), pp. 1323
–1327
.10.1109/9.40046719.
Darouach
, M.
, and Boutat-Baddas
, L.
, 2008
, “Observers for a Class of Nonlinear Singular Systems
,” IEEE Trans. Autom. Control
, 53
(11
), pp. 2627
–2633
.10.1109/TAC.2008.200786820.
Yang
, C.
, Kong
, Q.
, and Zhang
, Q.
, 2013
, “Observer Design for a Class of Nonlinear Descriptor Systems
,” J. Franklin Inst.
, 350
(5
), pp. 1284
–1297
.10.1016/j.jfranklin.2012.04.00821.
Darouach
, M.
, 2012
, “On the Functional Observers for Linear Descriptor Systems
,” Syst. Control Lett.
, 61
(3
), pp. 427
–434
.10.1016/j.sysconle.2012.01.00622.
Yang
, C. Y.
, Zhang
, Q. L.
, and Chai
, T. Y.
, 2013
, “Nonlinear Observers for a Class of Nonlinear Descriptor Systems
,” Optim. Control Appl. Methods
, 34
(3
), pp. 348
–363
.10.1002/oca.202823.
Darouach
, M.
, Boutat-Baddas
, L.
, and Zerrougui
, M.
, 2013
, “H∞ Filter Design for a Class of Nonlinear Discrete-Time Singular Systems
,” Int. J. Control
, 86
(9
), pp. 1597
–1606
.10.1080/00207179.2013.79200024.
Xu
, S. Y.
, and Lam
, J.
, 2007
, “Reduced-Order H∞ Filtering for Singular Systems
,” Syst. Control Lett.
, 56
(1
), pp. 48
–57
.10.1016/j.sysconle.2006.07.01025.
Lu
, G. P.
, 2006
, “Full-Order and Reduced-Order Observers for Lipschitz Descriptor Systems: The Unified LMI Approach
,” IEEE Trans. Circuits Syst. II
, 53
(7
), pp. 563
–567
.10.1109/TCSII.2006.87533226.
Wang
, Z. H.
, Shen
, Y.
, Zhang
, X. L.
, and Wang
, Q.
, 2012
, “Observer Design for Discrete-Time Descriptor Systems: An LMI Approach
,” Syst. Control Lett.
, 61
(6
), pp. 683
–687
.10.1016/j.sysconle.2012.03.006Copyright © 2015 by ASME
You do not currently have access to this content.
