# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Robust stability analysis for a class of fractional order systems with uncertain parameters. (English) Zbl 1222.93171
Summary: The research of robust stability for fractional order linear time-invariant (FO-LTI) interval systems with uncertain parameters has become a hot issue. In this paper, it is the first time to consider robust stability of uncertain parameters FO-LTI interval systems, which have deterministic linear coupling relationship between fractional order and other model parameters. Linear matrix inequalities (LMI) methods are used, and a criterion for checking asymptotical stability of this class of systems is presented. One numerical illustrative example is given to verify the correctness of the conclusions.

##### MSC:
 93D09 Robust stability of control systems 93D20 Asymptotic stability of control systems
Full Text:
##### References:
 [1] Torvik, P. J.; Bagley, R. L.: On the appearance of the fractional derivative in the behavior of real materials, Journal of applied mechanics 51, No. 2, 294-298 (1984) · Zbl 1203.74022 · doi:10.1115/1.3167615 [2] De Espíndola, J. J.; Bavastri, C. A.; Lopes, E. M. O.: On the passive control of vibrations with viscoelastic dynamic absorbers of ordinary and pendulum types, Journal of the franklin institute 347, No. 1, 102-115 (2010) · Zbl 1298.74119 [3] Wu, X.; Li, J.; Chen, G.: Chaos in the fractional order unified system and its synchronization, Journal of the franklin institute 345, No. 4, 392-401 (2008) · Zbl 1166.34030 · doi:10.1016/j.jfranklin.2007.11.003 [4] Jumarie, G.: Fractional multiple birth-death processes with birth probabilities ${\lambda}$i({$\Delta$}t)${\alpha}+$o(({$\Delta$}t){$\alpha$}), Journal of the franklin institute 347, No. 10, 1797-1813 (2010) · Zbl 1225.60141 · doi:10.1016/j.jfranklin.2010.09.004 [5] Carpinteri, A.; Mainardi, F.: Fractals and fractional calculus in continuum mechanics, (1997) · Zbl 0917.73004 [6] Zhu, C. X.; Zou, Y.: Summary of research on fractional-order control, Control and decision 24, No. 2, 161-169 (2009) · Zbl 1199.93002 [7] Podlubny, I.: Fractional differential equations, mathematics in science and engineering, (1999) · Zbl 0924.34008 [8] Monje, C. M.; Chen, Y. Q.; Vinagre, B. M.; Xue, D.; Feliu, V.: Fractional-order systems and controls--fundamentals and applications, advanced industrial control series, (2010) · Zbl 1211.93002 [9] Sheng, H.; Li, Y.; Chen, Y. Q.: Application of numerical inverse Laplace transform algorithms in fractional calculus, Journal of the franklin institute 348, No. 2, 315-330 (2011) · Zbl 1210.65201 · doi:10.1016/j.jfranklin.2010.11.009 [10] Çenesiz, Y.; Keskin, Y.; Kurnaz, A.: The solution of the bagley--torvik equation with the generalized Taylor collocation method, Journal of the franklin institute 347, No. 2, 452-466 (2010) · Zbl 1188.65107 · doi:10.1016/j.jfranklin.2009.10.007 [11] Podlubny, I.: Fractional-order systems and $PI{\lambda}$D${\mu}$-controllers, IEEE transactions on automatic control 44, No. 1, 208-214 (1999) · Zbl 1056.93542 · doi:10.1109/9.739144 [12] Podlubny, I.; Petráš, I.; Vinagre, B. M.; O’leary, P.; Dorčák, L’.: Analogue realizations of fractional-order controllers, Nonlinear dynamics 29, 281-296 (2002) · Zbl 1041.93022 · doi:10.1023/A:1016556604320 [13] D. Matignon, Stability result on fractional differential equations with applications to control processing, in: Proceedings of the IMACS-SMC, Lille, France, 1996, 963--968. [14] D. Matignon, Stability properties for generalized fractional differential systems, in: Proceedings of the Colloquium FDS’98: Fractional Differential Systems: Models, Methods and Applications, vol. 5, Paris, 1998, pp. 145--158. · Zbl 0920.34010 · doi:10.1051/proc:1998004 · http://www.edpsciences.org/articles/proc/Vol.5/contents.htm [15] D. Matignon, Generalized fractional differential and difference equations: stability properties and modeling issues, in: Proceedings of the Mathematical Theory of Networks and Systems symposium (MTNS’98), Padova, Italy, July 1998, 503--506. [16] A. Oustaloup, P. Melchior, The great principles of the CRONE control, in: Proceedings of the Systems, Man and Cybernetics. Systems Engineering in the Service of Humans, 1993, 118--129. [17] A. Oustaloup, M. Bansard, First generation CRONE control, in: Proceedings of the Systems, Man and Cybernetics. Systems Engineering in the Service of Humans, 1993, 130--135. [18] A. Oustaloup, B. Mathieu, P. Lanusse, Second generation CRONE control, in: Proceedings of the Systems, Man and Cybernetics. Systems Engineering in the Service of Humans, 1993, 136--142. [19] A. Oustaloup, B. Bluteau, M. Nouillant, First generation scalar CRONE control: application to a two DOF manipulator and comparison with non linear decoupling control, in: Proceedings of the Systems, Man and Cybernetics. Systems Engineering in the Service of Humans, 1993, 453--458. [20] Vinagre, B. M.; Chen, Y. Q.; Petráš, I.: Two direct tustin discretization methods for fractional-order differentiator/integrator, Journal of the franklin institute 340, No. 5, 349-362 (2003) · Zbl 1051.93031 · doi:10.1016/j.jfranklin.2003.08.001 [21] Wang, Z. B.; Cao, G. Y.; Zhu, X. J.: Stability conditions and criteria for fractional order linear time-invariant systems, Control theory applications 21, No. 6, 922-926 (2004) · Zbl 1092.34545 [22] Wang, Z. B.; Cao, G. Y.; Zhu, X. J.: Research on the internal and external stability of fractional order linear systems, Control and decision 19, No. 10, 1171-1174 (2004) [23] Hu, J. B.; Han, Y.; Zhao, L. D.: A novel stability theorem for fractional systems and its applying in synchronizing fractional chaotic system based on back-stepping approach, Acta physica sinica 58, No. 4, 2235-2239 (2009) · Zbl 1199.37062 [24] Chen, Y. Q.; Moore, K. L.: Analytical stability bound for delayed second-order systems with repeating poles using Lambert function W, Automatica 38, No. 5, 891-895 (2002) · Zbl 1020.93019 · doi:10.1016/S0005-1098(01)00264-3 [25] M. Moze, J. Sabatier, A. Oustaloup, LMI tools for stability analysis of fractional systems, in: Proceedings of the International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Long Beach, California, USA, 2005, pp. 1611--1619. [26] Sabatier, J.; Moze, M.; Farges, C.: LMI stability conditions for fractional order systems, Computers mathematics with applications 59, No. 5, 1594-1609 (2010) · Zbl 1189.34020 · doi:10.1016/j.camwa.2009.08.003 [27] Ahn, H. S.; Chen, Y. Q.; Podlubny, I.: Robust stability test of a class of linear time-invariant interval fractional-order system using Lyapunov inequality, Applied mathematics and computation 187, No. 1, 27-34 (2007) · Zbl 1123.93074 · doi:10.1016/j.amc.2006.08.099 [28] Chen, Y. Q.; Ahn, H. S.; Podlubny, I.: Robust stability check of fractional order linear time invariant systems with interval uncertainties, Signal processing 86, No. 10, 2611-2618 (2006) · Zbl 1172.94385 · doi:10.1016/j.sigpro.2006.02.011 [29] Ahn, H. S.; Chen, Y. Q.: Necessary and sufficient stability condition of fractional-order interval linear systems, Automatica 44, No. 11, 2985-2988 (2008) · Zbl 1152.93455 · doi:10.1016/j.automatica.2008.07.003 [30] Lu, J. G.; Chen, G. R.: Robust stability and stabilization of fractional-order interval systems: an LMI approach, IEEE transactions on automatic control 54, No. 6, 1294-1299 (2009) [31] Lu, J. G.; Chen, Y. Q.: Robust stability and stabilization of fractional-order interval systems with the fractional order ${\alpha}$: the $0{\alpha}1$ case, IEEE transactions on automatic control 55, No. 1, 152-158 (2010)