Design procedures are proposed for two different classes of observers for systems with unknown inputs. In the first approach, the state of the observed system is decomposed into known and unknown components. The unknown component is a projection, not necessarily orthogonal, of the whole state along the subspace in which the available state component resides. Then, a dynamical system to estimate the unknown component is constructed. Combining the output of the dynamical system, which estimates the unknown state component, with the available state information results in an observer that estimates the whole state. It is shown that some previously proposed observer architectures can be obtained using the projection operator approach presented in this paper. The second approach combines sliding modes and the second method of Lyapunov resulting in a nonlinear observer. The nonlinear component of the sliding mode observer forces the observation error into the sliding mode along a manifold in the observation error space. Design algorithms are given for both types of observers.
@article{bwmeta1.element.bwnjournal-article-amcv15i4p431bwm,
author = {Hui, Stefen and \.Zak, Stanis\l aw},
title = {Observer design for systems with unknown inputs},
journal = {International Journal of Applied Mathematics and Computer Science},
volume = {15},
year = {2005},
pages = {431-446},
zbl = {1127.93018},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-amcv15i4p431bwm}
}
Hui, Stefen; Żak, Stanisław. Observer design for systems with unknown inputs. International Journal of Applied Mathematics and Computer Science, Tome 15 (2005) pp. 431-446. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-amcv15i4p431bwm/
[000] Bhattacharyya S. P. (1978): Observer design for linear systems with unknown inputs. - IEEE Trans. Automat. Contr., Vol. AC-23, No. 3, pp. 483-484. | Zbl 0377.93025
[001] Chang S.-K., You W.-T., and Hsu P.-L. (1997): Design of general structured observers for linear systems with unknown inputs. - J. Franklin Instit., Vol. 334B, No. 2, pp. 213-232. | Zbl 0870.93019
[002] Chen J., Patton R. J., and Zhang H.-Y. (1996): Design of unknown input observers and robust fault detection filters. - Int. J. Contr., Vol. 63, No. 1, pp. 85-105. | Zbl 0844.93020
[003] Chen J. and Patton R. J. (1999): Robust Model-Based Fault Diagnosis for Dynamic Systems. - Boston: Kluwer. | Zbl 0920.93001
[004] Choi H. H. and Ro K.-S. (2005): LMI-based sliding-mode observer design method.- IEE Proc. Contr. Theory Appl., Vol. 152, No. 1, pp. 113-115.
[005] Corless M. and Tu J. (1998): State and input estimation for a class of uncertain systems. - Automatica, Vol. 34, No. 6, pp. 757-764. | Zbl 0932.93008
[006] Darouach M., Zasadzinski M., and Xu S. J. (1994): Full-order observers for linear systems with unknown inputs. - IEEE Trans. Automat. Contr., Vol. 39, No. 3, pp. 606-609. | Zbl 0813.93015
[007] Dawson D. M., Hu J., and Burg T. C. (1998): Nonlinear Control of Electric Machinery.- New York: Marcel Dekker.
[008] Edwards C. and Spurgeon S. K. (1998): Sliding Mode Control: Theory and Applications. - London: Taylor and Francis. | Zbl 0964.93019
[009] Edwards C., Spurgeon S. K., and Patton R. J. (2000): Sliding mode observers for fault detection and isolation. - Automatica, Vol. 36, No. 4, pp. 541-553. | Zbl 0968.93502
[010] Ellis G. (2002): Observers in Control Systems; A Practical Guide. - San Diego: Academic Press.
[011] Gantmacher F. R. (1990): The Theory of Matrices, Vol. 1, 2nd Ed.. - New York: Chelsea Publishing Company. | Zbl 0085.01001
[012] Ha Q. P., Trinh H., Nguyen H. T., and Tuan H. D. (2003): Dynamic output feedback sliding-mode control using pole placement and linear functional observers. - IEEE Trans. Industr. Electron., Vol. 50, No. 5, pp. 1030-1037.
[013] Hou M. and Muller P. C. (1992): Design of observers for linear systems with unknown inputs. - IEEE Trans. Automat. Contr., Vol. 37, No. 6, pp. 871-875. | Zbl 0775.93021
[014] Hou M., Pugh A. C. and Muller P. C. (1999): Disturbance decoupled functional observers. - IEEE Trans. Automat. Contr., Vol. 44, No. 2, pp. 382-386. | Zbl 0958.93017
[015] Hostetter G. and Meditch J. S. (1973): Observing systems with unmeasurable inputs.- IEEE Trans. Automat. Contr., Vol. AC-18, No. 3, pp. 307-308. | Zbl 0269.93008
[016] Hui S. and Żak S. H. (1990): Control and observation of uncertain systems: A variable structure systems approach, In: Control and Dynamic Systems; Advances in Theory and Applications, Vol. 34: Advances in Control Mechanics, Part 1 of 2 (C. T. Leondes, Ed.). - San Diego: Academic Press, pp. 175-204.
[017] Hui S. and Żak S. H. (1993): Low-order state estimators and compensators for dynamical systems with unknown inputs. - Syst. Contr. Lett., Vol. 21, No. 6, pp. 493-502. | Zbl 0794.93050
[018] Hui S. and Żak S. H. (2005): Low-order unknown input observers. -Proc. Amer. Contr. Conf., Portland, OR, pp. 4192-4197.
[019] Jiang B., Staroswiecki M., and Cocquempot V. (2004): Fault estimation in nonlinear uncertain systems using robust sliding-mode observers. - IEE Proc. Contr. Theory Appl., Vol. 151, No. 1, pp. 29-37.
[020] Kaczorek T. (1998): Vectors and Matrices in Control and Electrical Engineering, 2nd Ed.. - Warsaw: WNT, (in Polish).
[021] Koshkouei A. J. and Zinober A. S. I. (2004); Sliding mode state observation for non-linear systems. - Int. J. Contr., Vol. 77, No. 2, pp. 118-127. | Zbl 1049.93013
[022] Kudva P., Viswanadham N., and Ramakrishna A. (1980): Observers for linear systems with unknown inputs. - IEEE Trans. Automat. Contr., Vol. AC-25, No. 1, pp. 113-115. | Zbl 0443.93012
[023] Kurek J. E. (1983): The state vector reconstruction for linear systems with unknown inputs. - IEEE Trans. Automat. Contr., Vol. AC-28,No. 12, pp. 1120-1122. | Zbl 0538.93005
[024] Krzemiński S. and Kaczorek T. (2004): Perfect reduced-order unknown-input observer for standard systems. - Bull. Polish Acad. Sci., Techn. Sci., Vol. 52, No. 2, pp. 103-107. | Zbl 1140.93326
[025] Luenberger D. G. (1966): Observers for multivariable systems. - IEEE Trans. Automat. Contr., Vol. AC-11, No. 2, pp. 190-197.
[026] Luenberger D. G. (1971): An introduction to observers. - IEEE Trans. Automat. Contr., Vol. AC-16, No. 6, pp. 596-602.
[027] Luenberger D. G. (1979): Introduction to Dynamic Systems: Theory, Models, and Applications. - New York: Wiley. | Zbl 0458.93001
[028] Millerioux G. and Daafouz J. (2004): Unknown input observers for switched linear discrete time systems. - Proc. Amer. Contr. Conf., Boston, MA, pp. 5802-5805. | Zbl 1099.93520
[029] O'Reilly J. (1983): Observers for Linear Systems. - London: Academic Press.
[030] Robenack K. R and Lynch A. F. (2004): An efficient method for observer designwith approximately linear error dynamics. - Int. J. Contr., Vol. 77, No. 7, pp. 607-612. | Zbl 1062.93008
[031] Saif M. and Xiong Y. (2003): Sliding mode observers and their application in fault diagnosis, In: Fault Diagnosis and Fault Tolerance for Mechatronic Systems (F. Caccavale and L. Villani, Eds.). - Berlin: Springer, pp. 1-57. | Zbl 1043.93012
[032] Solsona J. A. and Valla M. I. (2003): Disturbance and nonlinear Luenberger observers for estimating mechanical variables in permanent magnet synchronous motors under mechanical parameters uncertainties. - IEEE Trans. Industr. Electron., Vol. 50, No. 4, pp. 717-725.
[033] Smith L. (1984): Linear Algebra, 2nd Ed.. - New York: Springer.
[034] Sundareswaran K. K., McLane P. J. and Bayoumi M. M. (1977): Observers for linear systems with arbitrary plant disturbances. - IEEE Trans. Automat. Contr., Vol. AC-22, No. 5, pp. 870-871. | Zbl 0361.93035
[035] Utkin V., Guldner J., and Shi J. (1999): Sliding Mode Control in Electromechanical Systems. - London: Taylor and Francis.
[036] Walcott B. L. and Żak S. H. (1987): State observation of nonlinear uncertain dynamical systems. - IEEE Trans. Automat. Contr., Vol. AC-32, No. 2, pp. 166-170. | Zbl 0618.93019
[037] Walcott B. L., Corless M. J., and Żak S. H. (1987): Comparative study of non-linear state-observation techniques. - Int. J. Contr., Vol. 45, No. 6, pp. 2109-2132. | Zbl 0627.93012
[038] Walcott B. L. and Żak S. H. (1988): Combined observer-controller synthesis for uncertain dynamical systems with applications. - IEEE Trans. Sys., Man Cybern., Vol. 18, No. 1, pp. 88-104. | Zbl 0652.93019
[039] Wang S.-H., Davison E. J. and Dorato P. (1975): Observing the states of systems with unmeasurable disturbances. - IEEE Trans. Automat. Contr., Vol. AC-20,No. 5, pp. 716-717. | Zbl 0318.93048
[040] Xiang J., Su H., and Chu J. (2005): On the design of Walcott-Zak sliding mode observer. - Proc. Amer. Contr. Conf. Portland, OR, pp. 2451-2456.
[041] Yang F. and Wilde R. W. (1988): Observers for linear systems with unknown inputs. - IEEE Trans. Automat. Contr., Vol. 33, No. 7, pp. 677-681. | Zbl 0646.93013
[042] Żak S. H. and Walcott B. L. (1990): State observation of nonlinear control systems via the method of Lyapunov, In: Deterministic Control of Uncertain Systems (A. S. I. Zinober, Ed.). - London: Peter Peregrinus Ltd,Ch. 16, pp. 333-350.
[043] Żak S. H. and Hui S. (1993): Output feedback variable structure controllers and state estimators for uncertain nonlinear dynamic systems. - IEE Proc. Contr. Theory Appl., Vol. 140, No. 1, pp. 41-50. | Zbl 0772.93016
[044] Żak S. H., Walcott B. L., and Hui S. (1993): Variable structure control and observation of nonlinear uncertain systems, In: Variable Structure Control for Robotics and Aerospace Applications (K.-K. D. Young, Ed.). - Amsterdam: Elsevier, Ch. 4, pp. 59-88.
[045] Żak S. H. (2003): Systems and Control. - New York: Oxford University Press.