Stability of Interval Fractional-order Systems with Order 0 < α < 1

In this paper, some brief sufficient conditions for the stability of FO-LTI systems dαx(t) dtα = Ax(t) with the fractional order are investigated when the matrix A and the fractional order α are uncertain or both α and A are uncertain, respectively. In addition, we also relate the stability of a fractional-order system with order 0 < α ≤ 1 to the stability of its equivalent fractional-order system with order 1 ≤ β < 2, the relationship between α and β is presented. Finally, a numeric experiment is given to demonstrate the effectiveness of our results.

• Hong Li
• Shou-ming Zhong
• Hou-biao Li

• Interval fractional-order systems
• linear matrix inequality (LMI)
• asymptotical stability.

[1] I.Podlubuy, Fractional differential equations. New York: Academic Press,
1999.
[2] B. O-Neill, R. Jeanloz, Geophys. Res. Lett. 17 (1990) 1477.
[3] J.J. Ita, L. Stixrude, J. Geophys. Res. 97 (1992) 6849.
[4] A.Oustaloup, X.Morean, M.Nouiuant, The CRONE suspension. Control
Eng.Pract. 4(8)(1996) 1101-1108.
[5] B.J.Lurie, Tunable TID controller. US Patent 5,371,630, December 6,
1944.
[6] I.Podlubuy,Fractional-order systems and PIλDu-controllers, IEEE
Trans. Automat. Control, 44(1)(1999) 208-214.
[7] D.Matignon, Stability results on fractional differential equations to control
processing, in:peocessings of Computational Engineering in Syatems and
Application Multiconference, vol.2, IMACS, IEEE-SMC, 1996, 963-968.
[8] Y.X.Sheng, J.G.Lu, Robust stability and stabilization of fractional-order
linear systems with nonlinear ncertain parameters:An LMI approach,
Chaos,Solitons and Fractals, 42(2009) 1163-1169.
[9] J.Sabatier, M.Moze, C.Farges, LMI stability conditions for fractional
order systems. Computers and Mathematics with Applications, 59(2010)
1594-1609.
[10] J.G.Lu, Y.Q.Chen, Robust stability and stabilization of fractional-order
interval systems with the Fractional Order ╬▒:The 0 < ╬▒ < 1 Case, IEEE
Transactions on Automatic Control, 55(1)(2010) 152-158.
[11] H.S.Ahn, Y.Q.Chen, I.Podlubny, Robust stability test of a class of linear
time-invariant interval fractional-order system using Lypunov inequality.
Applied Mathematics and Computation, 187(1)(2007) 27-34.
[12] J.C.Trigeassou, N.Maamri, J.Sabatier, A.Oustaloup, A Lyapunov approach
to the stability of fractional differential equations. Signal Processing,
91(2011) 437-445.
[13] Y.Sun,P.W.Nelson, A.G.Ulsoy,Survey on analysis of time delayed systems
via the Lambert W function. DCDIS A Supplement Advance in
Dynamical Systems, 14(S2)(2007) 296-301.
[14] Y.Sun, P.W.Nelson, A.G.Ulsoy, Delay differential equations via the matrix
lambert W function and bifurcation analysis: application to machine
tool chatter, Mathematic Biosciehces and Engineering, 4(2)(2007) 355-
368.
[15] M.S.Tavazoei, M.Haeri, A note on the stability of fractional order
systems, Mathematics and Computation in Simulation, 79(2009) 1566-
1576.
[16] I.Podlubny, Fractional Differential Equations, Academic Press, San
Diego, 1999.
[17] Z.Liao,C.Peng,W.Li,Y.Wang,Robust Stability Analysis for a class of
Fractional order Systems with Uncertain Parameters,Journal of the
Franklin Institute, 348(2011) 1101-1113.
[18] C.Farges,M.Moze,J.Sabatier,Pseudo-state Feedback Stabilization of
Commensurate Fractional order Systems,Automatica, 46(2010) 1730-
1734.
[19] J.G.Lu, G.R.Chen, Robust stability and stabilization of fractional-order
interval systems:An LMI approach, IEEE Transactions on Automatic
Control, 54(6)(2009) 1294-1299.