# Variable structure observer design for a class of uncertain systems with a time-varying delay

International Journal of Applied Mathematics and Computer Science (2012)

- Volume: 22, Issue: 3, page 575-583
- ISSN: 1641-876X

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topWen-Jeng Liu. "Variable structure observer design for a class of uncertain systems with a time-varying delay." International Journal of Applied Mathematics and Computer Science 22.3 (2012): 575-583. <http://eudml.org/doc/244057>.

@article{Wen2012,

abstract = {Design of a state observer is an important issue in control systems and signal processing. It is well known that it is difficult to obtain the desired properties of state feedback control if some or all of the system states cannot be directly measured. Moreover, the existence of a lumped perturbation and/or a time delay usually reduces the system performance or even produces an instability in the closed-loop system. Therefore, in this paper, a new Variable Structure Observer (VSO) is proposed for a class of uncertain systems subjected to a time varying delay and a lumped perturbation. Based on the strictly positive real concept, the stability of the equivalent error system is verified. Based on the generalized matrix inverse approach, the global reaching condition of the sliding mode of the error system is guaranteed. Also, the proposed variable structure observer will be shown to possess the invariance property in relation to the lumped perturbation, as the traditional variable structure controller does. Furthermore, two illustrative examples with a series of computer simulation studies are given to demonstrate the effectiveness of the proposed design method.},

author = {Wen-Jeng Liu},

journal = {International Journal of Applied Mathematics and Computer Science},

keywords = {variable structure observer; invariance property; generalized matrix inverse approach},

language = {eng},

number = {3},

pages = {575-583},

title = {Variable structure observer design for a class of uncertain systems with a time-varying delay},

url = {http://eudml.org/doc/244057},

volume = {22},

year = {2012},

}

TY - JOUR

AU - Wen-Jeng Liu

TI - Variable structure observer design for a class of uncertain systems with a time-varying delay

JO - International Journal of Applied Mathematics and Computer Science

PY - 2012

VL - 22

IS - 3

SP - 575

EP - 583

AB - Design of a state observer is an important issue in control systems and signal processing. It is well known that it is difficult to obtain the desired properties of state feedback control if some or all of the system states cannot be directly measured. Moreover, the existence of a lumped perturbation and/or a time delay usually reduces the system performance or even produces an instability in the closed-loop system. Therefore, in this paper, a new Variable Structure Observer (VSO) is proposed for a class of uncertain systems subjected to a time varying delay and a lumped perturbation. Based on the strictly positive real concept, the stability of the equivalent error system is verified. Based on the generalized matrix inverse approach, the global reaching condition of the sliding mode of the error system is guaranteed. Also, the proposed variable structure observer will be shown to possess the invariance property in relation to the lumped perturbation, as the traditional variable structure controller does. Furthermore, two illustrative examples with a series of computer simulation studies are given to demonstrate the effectiveness of the proposed design method.

LA - eng

KW - variable structure observer; invariance property; generalized matrix inverse approach

UR - http://eudml.org/doc/244057

ER -

## References

top- DeCarlo, R.A., Żak, S.H. and Matthews, G.P. (1988). Variable structure control of nonlinear multivariable systems: A tutorial, Proceedings of the IEEE 76(3): 212-232.
- Hua, C., Wang, Q. and Guan, X. (2008). Memoryless state feedback controller design for time delay systems with matched uncertain nonlinearities, IEEE Transactions on Automatic Control 53(3): 801-807.
- Hung, J.Y., Gao, W.B. and Hung, J.C. (1993). Variable structure control: A survey, IEEE Transactions on Industrial Electronic 40(1): 2-22.
- Karafyllis I. and Kravaris, C. (2007). On the observer problem for discrete-time control systems, IEEE Transactions on Automatic Control 52(1): 12-25.
- Khalil, H.K. (1996). Nonlinear Systems, Prentice-Hall, Upper Saddle River, NJ.
- Liu, P.L. (2005). Delay-dependent asymptotic stabilization for uncertain time-delay systems with saturating actuators, International Journal of Applied Mathematics and Computer Science 15(1): 45-51. Zbl1083.93046
- Liu, W.J. (2004). Design of the observer feedback gain for twodimensional discrete systems, IEEE Signal Processing Letters 11(4): 413-415.
- Liu, W.J., Shyu, K.K. and Hsu, K.C. (2009). Sliding mode observer design for a class of uncertain systems, 2009 CACS International Automatic Control Conference, Taipei, Taiwan.
- Luenberger, D. (1971). An introduction to observers, IEEE Transactions on Automatic Control 16(6): 596-602.
- Nijmeijer, H. and Fossen, T.I.(1999). New Directions in Nonlinear Observer Design, Springer-Verlag, London. Zbl0915.00063
- O'Reilly, J. (1983). Observer for Linear Systems, Academic Press, New York, NY.
- Röbenack, K. and Lynch, A.F. (2006). Observer design using a partial nonlinear observer canonical form, International Journal of Applied Mathematics and Computer Science 16(3): 333-343. Zbl1136.93313
- Shyu, K.K., Liu, W.J. and Hsu, K.C. (2005). Design of largescale time-delayed systems with dead-zone input via variable structure control, Automatica 41(7): 1239-1246. Zbl1080.93003
- Slotine, J.J.E., Hedrick, J.K. and Misawa, E.A. (1987). On sliding observers for non-linear systems, ASME Journal of Dynamic Systems, Measurement, and Control 109: 245-252. Zbl0661.93011
- Spurgeon, S.K. (2008). Sliding mode observers: A survey, International Journal of Systems Science 39(8): 751-764. Zbl1283.93066
- Sun, X.M., Wang, W., Liu, G.P. and Zhao, J. (2008). Stability analysis for linear switched systems with time-varying delay, IEEE Transactions on Systems, Man, and Cybernetics. Part B: Cybernetics, 38(2): 528-533.
- Unel, M., Sabanovic, A., Yilmaz, B. and Dogan, E. (2008). Visual motion and structure estimation using sliding mode observers, International Journal of Systems Science 39(2): 149-161. Zbl1283.93067
- Utkin, V.I. (1992). Sliding Modes in Control and Optimization, Springer-Verlag, Berlin/Heidelberg. Zbl0748.93044
- Walcott, B.L. and Żak, S.H. (1987). State observation of nonlinear uncertain dynamical systems, IEEE Transactions on Automatic Control 32(2): 166-170. Zbl0618.93019
- Xiang, Z., Wang, R. and Chen, Q. (2010). Fault tolerant control of switched nonlinear systems with time delay under asynchronous switching, International Journal of Applied Mathematics and Computer Science 20(3): 497-506, DOI: 10.2478/v10006-010-0036-0. Zbl1211.93075
- Yan, X.G. and Edwards, C. (2007). Nonlinear robust fault reconstruction and estimation using a sliding mode observer, Automatica 43(9): 1605-1614. Zbl1128.93389
- Żak, S.H. and Hui, S. (1993). Output feedback variable structure controllers and state estimators for uncertain/nonlinear dynamic systems, IEE Proceedings D 140(2): 41-50. Zbl0772.93016

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