×

Stability of discrete fractional order state-space systems. (English) Zbl 1229.93143

Summary: In this article, the stability problem for discrete-time fractional order systems is considered. The discrete-time fractional order state-space model introduced by the authors in earlier works is recalled in this context. The proposed stability definition is adopted from one used for infinite dimensional systems. Using this definition, the main stability result is presented in the form of a simple stability condition for the fractional order discrete state-space system. This is one of the first few attempts to give the stability conditions for this type of system. The condition presented is conservative1 the method gives only sufficient conditions, and the stability areas obtained when using it are smaller than those obtained from numerical solutions of the system. The relationship between the eigenvalues of the system matrix and the poles of the fractional-order system transfer function is also discussed. The main observation in this respect is that a set of L poles is related to every eigenvalue of the system matrix.

MSC:

93D99 Stability of control systems
93C55 Discrete-time control/observation systems
26A33 Fractional derivatives and integrals

Software:

CRONE
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bagley, R.L., AIAA Journal 21 pp 741– (1983) · Zbl 0514.73048
[2] Barbosa, R.S., Proceedings of the First IFAC Workshop on Fractional Differentation and its Applications
[3] Debeljković, D.L., Facta Universitatis, Series: Mechanical Engineering 1 (9) pp 1147– (2002)
[4] Dorčák, L., Proceedings of the 2002 International Carpathian Control Conference
[5] Dzieliński, A., Proceedings of the 2005 International Conference on Computational Intelligence for Modelling Control and Automation
[6] Hilfer, R., Application of Fractional Calculus in Physics (2000) · Zbl 1046.82009
[7] Matignon, D., Methods and Applications 5 pp 145– (1998)
[8] le Méhauté, A., Fractal Geometries: Theory and Applications (1991) · Zbl 0766.68142
[9] Oldham, K.B., The Fractional Calculus (1974) · Zbl 0292.26011
[10] Ostalczyk, P., International Journal of Systems Science 31 (12) pp 1551– (2000) · Zbl 1080.93592
[11] Oustaloup, A., Commande CRONE (1993) · Zbl 0864.93003
[12] Oustaloup, A., La Dérivation non Entiére (1995)
[13] Podlubny, I., Fractional Differential Equations (1999) · Zbl 0924.34008
[14] Sierociuk, D., International Journal of Applied Mathematics and Computer Science 16 (1) pp 101– (2006)
[15] Stiassnie, M., Applied Mathematical Modelling 3 pp 300– (1979) · Zbl 0419.73038
[16] Tenreiro Machado, J.A., Journal of Systems Analysis, Modelling, Simulation 27 pp 107– (1997) · Zbl 0875.93154
[17] Vinagre, B.M., 41st IEEE Conference on Decision and Control, Tutorial Workshop #2: Fractional Calculus Applications in Automatic Control and Robotics
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.