Direct Method for Stability Analysis of Fractional Delay Systems
Keywords:fractional delay systems, stability windows, Root-Locus.
AbstractIn this paper, a direct method is presented to analyze the stability of fractional order systems with single and multiple commensurate time delays, against delay uncertainties.. It is shown that this method analytically reveals all possible stability windows exclusively in the parametric space of the time delay. Using the approach presented in this study, first, without using any approximation or substitution, the transcendental characteristic equation is converted to an algebraic one with some specific crossing points. The resulting algebraic equation also enables us to easily determine the delay dependency of the system stability and the sensitivities of crossing roots with respect to time delay. Then, an expression in terms of system parameters and imaginary root of the characteristic equation is derived for computing the delay margin .The number of unstable roots in each interval is calculated with the definition of root tendency on the boundary of each interval. Finally, the concept of stability is expressed as a function of delay. four illustrative examples are presented to confirm the proposed method results.
Stojanovic, S.B.;Debeljkovic,D.LJ.; Dimitrijevic, N. (2012); Stability of Discrete-Time Systems with Time-Varying Delay: Delay Decomposition Approach, Int J Comput Commun, ISSN 1841-9836, 7(4): 775-783.
Zhang, X.Z.; Wang,Y.N.; Yuan, X.F. (2010); H1 Robust T-S Fuzzy Design for Uncertain Nonlinear Systems with State Delays Based on Sliding Mode Control, Int J Comput Commun, ISSN 1841-9836, V(4):592-602.
Liu, C.L.; Liu,F. (2010); Consensus Problem of Second-order Dynamic Agents with Heterogeneous Input and Communication Delays, Int J Comput Commun, ISSN 1841-9836,V(3):325-335.
Pakzad, S.; Pakzad, M. A. (2011), Stability condition for discrete systems with multiple state delays, WSEAS Trans. on Systems and Control, 6(11), 417-426.
Walton, J KE.; Marshal, JE. (1987), Direct method for TDS stability analysis, IEE Proceeding Part D. 134, 101-107. http://dx.doi.org/10.1049/ip-d.1987.0018
Ozturk, N.; Uraz, A. (1985), An analysis stability test for a certain class of distributed parameter systems with delays, IEEE Trans. on Circuits and Systems, 34(4), 393-396. http://dx.doi.org/10.1109/TCS.1985.1085704
Jury, E.I.; Zeheb, E. (1986), On a stability test for a class of distributed system with delays, IEEE Trans. on Circuits and Systems, 37(10), 1027-1028. http://dx.doi.org/10.1109/TCS.1986.1085839
Bonnet, C.; Partington, J.R. (2002), Analysis of fractional delay systems of retarded and neutral type, Automatica, 38, (7), 1133-1138. http://dx.doi.org/10.1016/S0005-1098(01)00306-5
Buslowicz, M. (2008), Stability of linear continuous time fractional order systems with delays of the retarded type, Bull. Pol. Acad. Sci. Techn, 56, (4), 319-324.
Fioravanti, A.R . et al (2012), A numerical method for Stability windows and unstable root-locus calculation for linear fractional time-delay systems, Automatica, 135, (5), 18-27.
Hwang, C.; Cheng, Y.C. (2006), A numerical algorithm for stability testing of fractional delay systems, Automatica, 42(5), 825-831. http://dx.doi.org/10.1016/j.automatica.2006.01.008
Hwang, C.; Cheng, Y.C. (2005), A note on the use of the lambert w function in the stability analysis of time-delay systems, Automatica, 41(11), 1979-1985.
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