PL EN


Preferences help
enabled [disable] Abstract
Number of results
Journal
2013 | 11 | 6 | 855-862
Article title

General formula for stability testing of linear systems with fractional-delay characteristic equation

Content
Title variants
Languages of publication
EN
Abstracts
EN
In some applications (especially in the filed of control theory) the characteristic equation of system contains fractional powers of the Laplace variable s possibly in combination with exponentials of fractional powers of s. The aim of this paper is to propose an easy-to-use and effective formula for bounded-input boundedoutput (BIBO) stability testing of a linear time-invariant system with fractional-delay characteristic equation in the general form of $$\Delta \left( s \right) = P_0 \left( s \right) + \sum\nolimits_{i = 1}^N {P_i \left( s \right)\exp ( - \zeta _i s^{\beta _i } ) = 0}$$, where P
i(s) (i = 0,...,N) are the so-called fractional-order polynomials and ξ
i and β
i are positive real constants. The proposed formula determines the number of the roots of such a characteristic equation in the right half-plane of the first Riemann sheet by applying Rouche’s theorem. Numerical simulations are also presented to confirm the efficiency of the proposed formula.
Publisher
Journal
Year
Volume
11
Issue
6
Pages
855-862
Physical description
Dates
published
1 - 6 - 2013
online
9 - 10 - 2013
References
  • [1] R. Hilfer, Applications of Fractional Calculus in Physics (World Scientific, New Jersey, 2000) http://dx.doi.org/10.1142/9789812817747[Crossref]
  • [2] R. L. Bagley, P. Torvik, J. Appl. Mech. 51, 294 (1984) http://dx.doi.org/10.1115/1.3167615[Crossref]
  • [3] V. G. Jenson, G. V. Jeffreys, Mathematical Methods in Chemical Engineering, 2nd edition (Academic Press, New York, 1977)
  • [4] N. Nakris, M. C. Constantinous, J. Struct. Eng. 117, 2708 (1991) http://dx.doi.org/10.1061/(ASCE)0733-9445(1991)117:9(2708)[Crossref]
  • [5] G. Haneczok, M. Weller, Mat. Sci. Eng. A-Struct. 370, 209 (2004) http://dx.doi.org/10.1016/j.msea.2003.01.009[Crossref]
  • [6] I. Podlubny, Fractional Differential Equations (Academic Press, New York, 1999)
  • [7] B. Onaral, H. H. Sun, H. P. Schwan, In: Proceedigs of the 10th Northeast Bioengineering Conference, Mar. 1982, Hanover, NH, 46
  • [8] D. W. Davidson, R. H. Cole, Chem. Phys. 18, 1414 (1950) http://dx.doi.org/10.1063/1.1747492[Crossref]
  • [9] R. J. Schwartz, B. Friedland, Linear Systems (McGraw-Hill, New York, 1965)
  • [10] R. F. Curtain, H.J. Zwart, An Introduction to Infinite Dimensional Linear Systems Theory (Springer, Berlin, 1995) http://dx.doi.org/10.1007/978-1-4612-4224-6[Crossref]
  • [11] H. Zwart, Syst. Control Lett. 52, 247 (2004) http://dx.doi.org/10.1016/j.sysconle.2004.02.002[Crossref]
  • [12] T. Helie, D. Matignon, Signal Process. 86, 2516 (2006) http://dx.doi.org/10.1016/j.sigpro.2006.02.017[Crossref]
  • [13] D. Baleanu, K. Diethelm, E. Scalas, J.J. Trujillo, Fractional Calculus: Models and Numerical Methods (Series on Complexity, Nonlinearity and Chaos), (World Scientific, Singapore, 2012)
  • [14] K. Balachandran, J. Kokila, J. J. Trujillo, Comput. Math. Appl. 64, 3037 (2012) http://dx.doi.org/10.1016/j.camwa.2012.01.071[Crossref]
  • [15] S. Abbas, M. Benchohra, J. Nieto, Adv. Diff. Equ. 2011, 379876 (2011) http://dx.doi.org/10.1155/2011/379876[Crossref]
  • [16] A. Babakhani, D. Baleanu, Abstr. Appl. Anal. 2011, 391971 (2011) http://dx.doi.org/10.1155/2011/391971[Crossref]
  • [17] I. Podlubny, IEEE T. Automat. Contr. 44, 208 (1999) http://dx.doi.org/10.1109/9.739144[Crossref]
  • [18] F. Merrikh-Bayat, Can. J. Chem. Eng. 90, 1400 (2012) http://dx.doi.org/10.1002/cjce.21643[Crossref]
  • [19] C. Hwang, Y.-C. Cheng, Automatica 42, 825 (2006) http://dx.doi.org/10.1016/j.automatica.2006.01.008[Crossref]
  • [20] F. Jarad, T. Abdeljawad, D. Baleanu, Nonlinear Anal-Real 14, 780 (2013) http://dx.doi.org/10.1016/j.nonrwa.2012.08.001[Crossref]
  • [21] F. Jarad, T. Abdeljawad, D. Baleanu, K. Biçen, Abstr. Appl. Anal. 2012, 476581 (2012)
  • [22] H. Delavari, D. Baleanu, J. Sadati, Nonlinear Dynam. 67, 2433 (2012) http://dx.doi.org/10.1007/s11071-011-0157-5[Crossref]
  • [23] E. Kaslik, S. Sivasundaram, J. Comput. Appl. Math. 236, 4027 (2012) http://dx.doi.org/10.1016/j.cam.2012.03.010[Crossref]
  • [24] D. Matignon, ESAIM: Proceedings 5, 145 (1998) http://dx.doi.org/10.1051/proc:1998004[Crossref]
  • [25] M. Ikeda, S. Takahashi, Electron. Comm. Jpn. 160, 41 (1977)
  • [26] F. Merrikh-Bayat, M. Karimi-Ghartemani, ISA T. 48, 32 (2009) http://dx.doi.org/10.1016/j.isatra.2008.10.003[Crossref]
  • [27] F. Merrikh-Bayat, M. Karimi-Ghartemani, Math. Probl. Eng. 2008, DOI:10.1155/2008/419046 (2008) [Crossref]
  • [28] W. Gander, W. Gautschi, BIT 40, 84 (2000) http://dx.doi.org/10.1023/A:1022318402393[Crossref]
  • [29] L.F. Shampine, J. Comput. Appl. Math. 211, 131 (2008) http://dx.doi.org/10.1016/j.cam.2006.11.021[Crossref]
  • [30] J. Valsa, L. Brancik, Int. J. Numer. Model El. 11, 153 (1998) http://dx.doi.org/10.1002/(SICI)1099-1204(199805/06)11:3<153::AID-JNM299>3.0.CO;2-C[Crossref]
  • [31] N. Ozturk, A. Uraz, IEEE T. Automat. Contr. 29, 368 (1984) http://dx.doi.org/10.1109/TAC.1984.1103535[Crossref]
  • [32] N. Ozturk, A. Uraz, IEEE T. Circuits Syst. 32, 393 (1985) http://dx.doi.org/10.1109/TCS.1985.1085704[Crossref]
Document Type
Publication order reference
YADDA identifier
bwmeta1.element.-psjd-doi-10_2478_s11534-013-0226-0
Identifiers
JavaScript is turned off in your web browser. Turn it on to take full advantage of this site, then refresh the page.