Fractional order adaptive high-gain controllers for a class of linear systems. (English) Zbl 1221.93128
Summary: We show that a fractional adaptive controller based on high gain output feedback can always be found to stabilize any given linear, time-invariant, minimum phase, siso systems of relative degree one. We generalize the stability theorem of integer order controllers to the fractional order case, and we introduce a new tuning parameter for the performance behaviour of the controlled plant. A simulation example is given to illustrate the effectiveness of the proposed algorithm.
MSC:
| 93C40 | Adaptive control/observation systems |
| 34A08 | Fractional ordinary differential equations |
| 93B52 | Feedback control |
References:
| [1] | Allgöwer, F.; Ashman, J.; Ilchmann, A., High-gain adaptive \(λ\)-tracking for nonlinear systems, Automatica, 33, 5, 881-888 (1997) · Zbl 0876.93054 |
| [2] | Chen, Y. Q.; Moore, K. L., Analytical stability bound for a class of delayed fractional-order dynamic systems, Nonlinear Dynam, 29, 191-200 (2002) · Zbl 1020.34064 |
| [3] | Corless, M., First order adaptive controllers for systems which are stabilizable via high gain output feedback. First order adaptive controllers for systems which are stabilizable via high gain output feedback, Analysis and control of nonlinear systems (1988), Elsevier Science Publishers B.V.: Elsevier Science Publishers B.V. North-Holland, p. 13-6 · Zbl 0673.93045 |
| [4] | Fan, J. C.; Kobayashi, T., A simple adaptive pi controller for linear systems with constant disturbances, IEEE Trans Automat Control, 43, 5, 733-736 (1998) · Zbl 0925.93845 |
| [5] | Hartley, T. T.; Lorenzo, C. F., Dynamics and control of initialized fractional order systems, Nonlinear Dynam, 29, 201-233 (2002) · Zbl 1021.93019 |
| [6] | Hwang, C.; Cheng, Y. C., A numerical algorithm for stability testing of fractional delay systems, Automatica, 42, 825-831 (2006) · Zbl 1137.93375 |
| [7] | Ilchmann, A., Non-identifier-based high-gain adaptive control. Non-identifier-based high-gain adaptive control, Lecture notes in control and information sciences, no. 189 (1993), Springer-Verlag: Springer-Verlag Berlin, Germany · Zbl 0786.93059 |
| [8] | Ilchmann, A., Universal adaptive stabilization of nonlinear systems, Dynam Control, 7, 199-213 (1997) · Zbl 0888.93054 |
| [9] | Ilchmann, A.; Ryan, E. P., On tracking and disturbance rejection by adaptive control, Syst Control Lett, 52, 137-147 (2004) · Zbl 1157.93425 |
| [10] | Khusainov, T. D., Stability analysis of a linear-fractional delay system, Diff Equat, 37, 1184-1188 (2001) · Zbl 1006.34068 |
| [11] | Ladaci S, Charef A. MIT adaptive rule with fractional integration. In: Proceedings CESA’2003 IMACS multiconference computational engineering in systems applications, Lille, France, July 9-11, 2003.; Ladaci S, Charef A. MIT adaptive rule with fractional integration. In: Proceedings CESA’2003 IMACS multiconference computational engineering in systems applications, Lille, France, July 9-11, 2003. · Zbl 1134.93356 |
| [12] | Ladaci, S.; Charef, A., On fractional adaptive control, Nonlinear Dynam, 43, 4, 365-378 (2006) · Zbl 1134.93356 |
| [13] | Ladaci S, Charef A. An adaptive fractional \(PI^{ Λ; }^{ Μ; } \); Ladaci S, Charef A. An adaptive fractional \(PI^{ Λ; }^{ Μ; } \) · Zbl 1134.93356 |
| [14] | Lazarević, M. P., Finite time stability analysis of \(PD^α\) fractional control of robotic time-delay systems, Mech Res Commun, 33, 269-279 (2006) · Zbl 1192.70008 |
| [15] | Matignon D. Stability result on fractional differential equations with applications to control processing. In: IMACS-SMC proceeding, Lille, France, 1996, p. 963-8.; Matignon D. Stability result on fractional differential equations with applications to control processing. In: IMACS-SMC proceeding, Lille, France, 1996, p. 963-8. |
| [16] | Momani, S.; Hadid, S. B., Lyapounov stability solutions of fractional integrodifferential equations, Int J Math Math Sci, 47, 2503-2507 (2004) · Zbl 1074.45006 |
| [17] | Oldham, K. B.; Spanier, J., The fractional calculus (1974), Academic Press: Academic Press New York · Zbl 0428.26004 |
| [18] | PetrአI, Chen YQ, Vinagre BM. A robust stability test procedure for a class of uncertain LTI fractional order systems. In: Proceedings of the international carpathian conference ICCC’2002, Malenovice, Czech Republic, May 27-30, 2002, p. 247-51.; PetrአI, Chen YQ, Vinagre BM. A robust stability test procedure for a class of uncertain LTI fractional order systems. In: Proceedings of the international carpathian conference ICCC’2002, Malenovice, Czech Republic, May 27-30, 2002, p. 247-51. |
| [19] | Srivastava, H. M.; Saxena, R. K., Operators of fractional integration and their applications, Appl Math Comput, 118, 1-52 (2001) · Zbl 1022.26012 |
| [20] | Vinagre, B. M.; Petráš, I.; Podlubny, I.; Chen, Y. Q., Using fractional order adjustment rules and fractional order reference models in model-reference adaptive control, Nonlinear Dynam, 29, 269-279 (2002) · Zbl 1031.93110 |
| [21] | Vinagre BM, PetráŠ, I, Podlubny I, Chen YQ. Stability of fractional-order model reference adaptive control. In: Proceedings of the MTNS’2002 (Open problems book), Notre Dame, USA, August 12-16, 2002, p. 118-21.; Vinagre BM, PetráŠ, I, Podlubny I, Chen YQ. Stability of fractional-order model reference adaptive control. In: Proceedings of the MTNS’2002 (Open problems book), Notre Dame, USA, August 12-16, 2002, p. 118-21. |
| [22] | Willems, J. C.; Byrnes, C. I., Global adaptive stabilization in the absence of information on the sign of the high frequency gain. Global adaptive stabilization in the absence of information on the sign of the high frequency gain, Lecture notes in control and information sciences, 62 (1984), Springer-Verlag: Springer-Verlag Berlin, Germany, p. 9-57 · Zbl 0549.93043 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
