Robust fractional adaptive control based on the strictly Positive Realness Condition

Samir Ladaci ; Abdelfatah Charef ; Jean Jacques Loiseau

International Journal of Applied Mathematics and Computer Science, Tome 19 (2009), p. 69-76 / Harvested from The Polish Digital Mathematics Library

Résumé

This paper presents a new approach to robust adaptive control, using fractional order systems as parallel feedforward in the adaptation loop. The problem is that adaptive control systems may diverge when confronted with finite sensor and actuator dynamics, or with parasitic disturbances. One of the classical robust adaptive control solutions to these problems makes use of parallel feedforward and simplified adaptive controllers based on the concept of positive realness. The proposed control scheme is based on the Almost Strictly Positive Realness (ASPR) property of the plant. We show that this condition implies also robust stability in the case of fractional order controllers. An application to Model Reference Adaptive Control (MRAC) with a fractional order adaptation rule is provided with an implementable algorithm. A simulation example of a SISO robust adaptive control system illustrates the advantages of the proposed method in the presence of disturbances and noise.

Détail

Publié le : 2009-01-01

Zbl 1169.93349

EUDML-ID : urn:eudml:doc:207923

Bibliographie

[000] Anderson, B. D. and Vongpanitlerd, S. (1973). Network Analysis and Synthesis, Prentice-Hall, Englewood Cliffs, NJ.

[001] Åström, K. J. and Wittenmark, B. (1995). Adaptive Control, Addison-Wesley, Reading, MA.

[002] Bar-Kana, I. (1986). Positive realness in discrete-time adaptive control systems, International Journal of Systems Science 17(7): 1001-1006. | Zbl 0592.93035

[003] Bar-Kana, I. (1987). Parallel feedforward and simplified adaptive control, International Journal Adaptive Control and Signal Processing 1(2): 95-109. | Zbl 0736.93042

[004] Bar-Kana, I. (1989). On positive realness in multivariable stationary linear systems, Proceedings of the Conference on Information Sciences and Systems, Baltimore, MD, USA. | Zbl 0738.93040

[005] Bar-Kana, I. and Kaufman, H. (1985). Global stability and performance of a simplified adaptive algorithm, International Journal of Control 42(6): 1491-1505. | Zbl 0587.93045

[006] Brin, I. A. (1962). On the stability of certain systems with distributed and lumped parameters, Automation and Remote Control 23: 798-807. | Zbl 0122.32805

[007] Charef, A. (2006). Analogue realisation of fractional-order integrator, differentiator and fractional $PI^{λ}D^{μ}$ controller, IEE Proceedings-Control Theory and Applications 153(6): 714-720.

[008] Charef, A., Sun, H. H., Tsao, Y. Y. and Onaral, B. (1992). Fractal system as represented by singularity function, IEEE Transactions on Automatic Control 37(9): 1465-1470. | Zbl 0825.58027

[009] Desoer, C. A. and Vidyasagar, M. (1975). Feedback Systems: Input-Output Properties, Academic Press, New York, NY. | Zbl 0327.93009

[010] Ioannou, P. and Sun, J. (1996). Robust Adaptive Control, Prentice Hall, Englewood Cliffs, NJ. | Zbl 0839.93002

[011] Kwan, C., Dawson, D. M. and Lewis, F. L. (2001). Robust adaptive control of robots using neural network: Global stability, Asian Journal of Control 3(2): 111-121.

[012] Ladaci, S. and Charef, A. (2006). On fractional adaptive control, Nonlinear Dynamics 43(4): 365-378. | Zbl 1134.93356

[013] Ladaci, S., Loiseau, J. J. and Charef, A. (2008). Fractional order adaptive high-gain controllers for a class of linear systems, Communications in Nonlinear Science and Numerical Simulations 13(4): 707-714. | Zbl 1221.93128

[014] Ladaci, S. and Moulay, E. (2008). Lp-stability analysis of a class of nonlinear fractional differential equations, International Journal of Automation and Systems Engineering 2(1): 40-47.

[015] Landau, Y. D. (1979). Adaptive Control: The Model Reference Approach, Marcel Dekker, New York, NY. | Zbl 0475.93002

[016] Miller, K. S. and Ross, B. (1993). An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley Interscience, New York, NY. | Zbl 0789.26002

[017] Naceri, F. and Abida, L. (2003). A novel robust adaptive control algorithm for AC drives, Computers and Electrical Engineering 29: 523-534. | Zbl 1019.93036

[018] Oustaloup, A. (1991). La commande CRONE, Hermès, Paris, (in French).

[019] Oustaloup, A., Sabatier, J. and Moreau, X. (1998). From fractal robustness to the crone approach, ESAIM: Proceedings, Fractional Differential Systems: Models, Methods and Applications 5: 177-192. | Zbl 0918.93029

[020] Podlubny, I. (1999a). Fractional Differential Equations, Academic Press, New York, NY. | Zbl 0924.34008

[021] Podlubny, I. (1999b). Fractional order systems and PIλ Dμ controllers, IEEE Transactions on Automatic Control 44(1): 208-214. | Zbl 1056.93542

[022] Sabatier, J., Oustaloup, A., Iturricha, A. and Lanusse, P. (2002). Crone control: Principles and extension to time-variant plants with asymptotically constant coefficients, Nonlinear Dynamics 29: 363-385. | Zbl 1021.93014

[023] Shaked, U. (1977). The zero properties of linear passive systems, IEEE Transactions on Automatic Control 22(6): 973-976. | Zbl 0368.93012

[024] Sobel, K. and Kaufman, H. (1986). Direct model reference adaptive control for a class of MIMO systems, Control and Dynamic Systems 24: 973-976. | Zbl 0657.93035

[025] Sun, H. and Charef, A. (1990). Fractal system-A time domain approach, Annals of Biomedical Engineering 18: 597-621.

[026] Vinagre, B., Petras, I. and Chen, Y. (2002). Using fractional order adjustment rules and fractional order reference models in model-reference adaptive control, Nonlinear Dynamics 29: 269-279. | Zbl 1031.93110

[027] Zelmat, M. (2001). Commande Modale et Adaptative, OPU, Algiers.