Stability Analysis of Linear Fractional Order Neutral System with Multiple Delays by Algebraic Approach
In this paper, we study the stability of n-dimensional linear fractional neutral differential equation with time delays. By using the Laplace transform, we introduce a characteristic equation for the above system with multiple time delays. We discover that if all roots of the characteristic equation have negative parts, then the equilibrium of the above linear system with fractional order is Lyapunov globally asymptotical stable if the equilibrium exist that is almost the same as that of classical differential equations. An example is provided to show the effectiveness of the approach presented in this paper.
[1] M.-W. Spong, A theorem on neutral delay systems Original Research
Article Systems & Control Letters, 6(1985): 291-294.
[2] Y. He, M. Wu, J.-H. She, G.-P. Liu, Delay-dependent robust stability
criteria for uncertain neutral systems with mixed delays, Systems &
Control Letters, 51(2004): 57-65.
[3] C.-H. Lien, K.-W. Yu, Y.-J. Chung, Y.-F. Lin, L.-Y. Chung, J.-D. Chen,
Exponential stability analysis for uncertain switched neutral systems with
interval-time-varying state delay, Nonlinear Analysis: Hybrid Systems,
3(2009):334-342.
[4] L.-L. Xiong, S.-M. Zhong, M. Ye, S.-L. Wu, New stability and stabilization
for switched neutral control systems Original Research Article
Chaos, Solitons & Fractals, 42(2009):1800-1811.
[5] X.-G. Liu, M. Wu, Ralph Martin, M.-L. Tang, Stability analysis for
neutral systems with mixed delays Original Research Article Journal of
Computational and Applied Mathematics, 202(2007):478-497.
[6] L.-L. Xiong, S.-M. Zhong, J.-K. Tian, Novel robust stability criteria of
uncertain neutral systems with discrete and distributed delays, Chaos,
Solitons & Fractals, 40(2009):771-777.
[7] W.-J. Xiong , J.-L. Liang. Novel stability criteria for neutral systems with
multiple time delays. Chaos,solitions & Fractals, 32(2007): 1735-1741.
[8] H. Li, H.-B. Li, S.-M. Zhong. Some new simple stability criteria of linear
neutral systems with a single delay. Journal of Computational and Applied
Mathematics, 200(2007):441-447.
[9] W.-H. Deng, C.-P. Li, J.-H. Lv, Stability analysis of linear fractional
differential system with multiple time delays, Nonlinear Dynamics,
48(2007):409-416.
[10] Y.-Q. Chen, H.S. Ahn, I. Podlubny, Robust stability check of fractional
order linear time invariant systems with interval uncertainties. Signal
Process, 86(2006):2611-2618.
[11] H.-S. Ahn, Y.-Q. Chen, I. Podlubny, Robust stability test of a class
of linear time-invariant interval fractional-order system using Lyapunov
inequality. Appl.Math Comput, 187(2007):27-34.
[12] Moze M, Sabatier J. LMI tools for stability analysis of fractional
systems. In: Proceedings ASME 2005 international design engineering
technical conferences & computers and information in engineering conference,
Paper DETC2005-85182, Long Beach, CA, USA, September 24-
28, 2005.
[13] Petras I, Chen Y, Vinagre BM. Robust stability test for interval fractional
order linear systems. In: Blondel VD, Megretski A, editors. Unsolved
problems in the mathematics of systems and control, vols. 208210.
Princeton, NJ: Princeton University Press; 2004 [Chapter 6.5].
[14] Petras I, Chen Y, Vinagre BM, Podlubny I. Stability of linear time invariant
systems with interval fractional orders and interval coefficients. In:
Proceedings of the international conference on computation cybernetics
(ICCC04), Vienna Technical University, Vienna, Austria, 8/30-9/1; 2005.
p. 14.
[15] R. Hotzel, Some stability conditions for fractional delay systems. J Math
Syst Estimat Control, 8(1998):1-9.
[16] C. Hwang, Y.C. Cheng, A numerical algorithm for stability testing of
fractional delay systems, Automatica, 42 (2006): 825-831.
[17] D. Matignon, Stability properties for generalized fractional differential
systems, ESAIM: Proc. 5 (1998): 145-158.
[18] P. Ostalczyk, Nyquist characteristics of a fractional order integrator,
Journal of Fractional Calculus, 19(2001):67-78.
[19] Mohammad Saleh Tavazoei, Mohammad Haeri, Sadegh Bolouki, Milad
Siami, Stability preservation analysis for frequency-based methods in numerical
simulation of fractional order systems, Siam Journal of Numerical
Analysis, 47(2008): 321-338.
[20] 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(2010): 152-158.
[21] Z.-X. Tai, X.-C. Wang, Controllability of fractional-order impulsive
neutral functional infinite delay integrodifferential systems in Banach
spaces, Applied Mathematics Letters, 22(2009):1760-1765.
[22] Y. Zhou, F. Jiao, J. Li, Existence and uniqueness for fractional neutral
differential equations with infinite delay, Nonlinear Analysis: Theory,
Methods & Applications, 71(2009): 3249-3256.
[23] Y. Zhou, F. Jiao, J. Li, Existence and uniqueness for p− type fractional
neutral differential equations, Nonlinear Analysis: Theory, Methods &
Applications, 71(2009):2724-2733.
[24] R.P. Agarwal, Y. Zhou, Y.-Y. He, Existence of fractional neutral functional
differential equations,
[25] Y. Zhou, F. Jiao, Computers and Mathematics with Applications,
59(2010):1095-1100.
[26] Y. Zhou, F. Jiao, Existence of mild solutions for fractional neutral
evolution equations, Computers and Mathematics with Applications,
59(2010):1063-1077.
[27] G.M. Mophou, G. M. N-Guerekata, Existence of mild solutions of some
semilinear neutral fractional functional evolution equations with infinite
delay, Applied Mathematics and Computation, 216(2010):61-69.
[28] Kamran Akbari Moornani, Mohammad Haeri, On robust stability of LTI
fractional-order delay systems of retarded and neutral type, Automatica
46 (2010) 362-368.
[29] A.-H. Lin, Y. Ren, N.-M. Xia, On neutral impulsive stochastic integrodifferential
equations with infinite delays via fractional operators, Mathematical
and Computer Modelling, 51(2010):413-424.
[30] G.-D. Hu, Some new simple stability criteria of neutral delay-differential
systems, Applied mathematics and computaiton, 80(1996):257-271.
[31] D.-Q. Cao, P. He, Stability criteria of linear neutral systems with a single
delay, Applied Mathematics and Computation, 148(2004):135-143.
[32] E.J. Muth, Transform Methods with Applications to Engineering and
Operations Research. Prentice-Hall, New Jersey, 1977.
[33] P. Lancaster, M. Tismenetsky, The Theory of Matrices, Academic Press,
Orlando, FL, 1985.
[1] M.-W. Spong, A theorem on neutral delay systems Original Research
Article Systems & Control Letters, 6(1985): 291-294.
[2] Y. He, M. Wu, J.-H. She, G.-P. Liu, Delay-dependent robust stability
criteria for uncertain neutral systems with mixed delays, Systems &
Control Letters, 51(2004): 57-65.
[3] C.-H. Lien, K.-W. Yu, Y.-J. Chung, Y.-F. Lin, L.-Y. Chung, J.-D. Chen,
Exponential stability analysis for uncertain switched neutral systems with
interval-time-varying state delay, Nonlinear Analysis: Hybrid Systems,
3(2009):334-342.
[4] L.-L. Xiong, S.-M. Zhong, M. Ye, S.-L. Wu, New stability and stabilization
for switched neutral control systems Original Research Article
Chaos, Solitons & Fractals, 42(2009):1800-1811.
[5] X.-G. Liu, M. Wu, Ralph Martin, M.-L. Tang, Stability analysis for
neutral systems with mixed delays Original Research Article Journal of
Computational and Applied Mathematics, 202(2007):478-497.
[6] L.-L. Xiong, S.-M. Zhong, J.-K. Tian, Novel robust stability criteria of
uncertain neutral systems with discrete and distributed delays, Chaos,
Solitons & Fractals, 40(2009):771-777.
[7] W.-J. Xiong , J.-L. Liang. Novel stability criteria for neutral systems with
multiple time delays. Chaos,solitions & Fractals, 32(2007): 1735-1741.
[8] H. Li, H.-B. Li, S.-M. Zhong. Some new simple stability criteria of linear
neutral systems with a single delay. Journal of Computational and Applied
Mathematics, 200(2007):441-447.
[9] W.-H. Deng, C.-P. Li, J.-H. Lv, Stability analysis of linear fractional
differential system with multiple time delays, Nonlinear Dynamics,
48(2007):409-416.
[10] Y.-Q. Chen, H.S. Ahn, I. Podlubny, Robust stability check of fractional
order linear time invariant systems with interval uncertainties. Signal
Process, 86(2006):2611-2618.
[11] H.-S. Ahn, Y.-Q. Chen, I. Podlubny, Robust stability test of a class
of linear time-invariant interval fractional-order system using Lyapunov
inequality. Appl.Math Comput, 187(2007):27-34.
[12] Moze M, Sabatier J. LMI tools for stability analysis of fractional
systems. In: Proceedings ASME 2005 international design engineering
technical conferences & computers and information in engineering conference,
Paper DETC2005-85182, Long Beach, CA, USA, September 24-
28, 2005.
[13] Petras I, Chen Y, Vinagre BM. Robust stability test for interval fractional
order linear systems. In: Blondel VD, Megretski A, editors. Unsolved
problems in the mathematics of systems and control, vols. 208210.
Princeton, NJ: Princeton University Press; 2004 [Chapter 6.5].
[14] Petras I, Chen Y, Vinagre BM, Podlubny I. Stability of linear time invariant
systems with interval fractional orders and interval coefficients. In:
Proceedings of the international conference on computation cybernetics
(ICCC04), Vienna Technical University, Vienna, Austria, 8/30-9/1; 2005.
p. 14.
[15] R. Hotzel, Some stability conditions for fractional delay systems. J Math
Syst Estimat Control, 8(1998):1-9.
[16] C. Hwang, Y.C. Cheng, A numerical algorithm for stability testing of
fractional delay systems, Automatica, 42 (2006): 825-831.
[17] D. Matignon, Stability properties for generalized fractional differential
systems, ESAIM: Proc. 5 (1998): 145-158.
[18] P. Ostalczyk, Nyquist characteristics of a fractional order integrator,
Journal of Fractional Calculus, 19(2001):67-78.
[19] Mohammad Saleh Tavazoei, Mohammad Haeri, Sadegh Bolouki, Milad
Siami, Stability preservation analysis for frequency-based methods in numerical
simulation of fractional order systems, Siam Journal of Numerical
Analysis, 47(2008): 321-338.
[20] 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(2010): 152-158.
[21] Z.-X. Tai, X.-C. Wang, Controllability of fractional-order impulsive
neutral functional infinite delay integrodifferential systems in Banach
spaces, Applied Mathematics Letters, 22(2009):1760-1765.
[22] Y. Zhou, F. Jiao, J. Li, Existence and uniqueness for fractional neutral
differential equations with infinite delay, Nonlinear Analysis: Theory,
Methods & Applications, 71(2009): 3249-3256.
[23] Y. Zhou, F. Jiao, J. Li, Existence and uniqueness for p− type fractional
neutral differential equations, Nonlinear Analysis: Theory, Methods &
Applications, 71(2009):2724-2733.
[24] R.P. Agarwal, Y. Zhou, Y.-Y. He, Existence of fractional neutral functional
differential equations,
[25] Y. Zhou, F. Jiao, Computers and Mathematics with Applications,
59(2010):1095-1100.
[26] Y. Zhou, F. Jiao, Existence of mild solutions for fractional neutral
evolution equations, Computers and Mathematics with Applications,
59(2010):1063-1077.
[27] G.M. Mophou, G. M. N-Guerekata, Existence of mild solutions of some
semilinear neutral fractional functional evolution equations with infinite
delay, Applied Mathematics and Computation, 216(2010):61-69.
[28] Kamran Akbari Moornani, Mohammad Haeri, On robust stability of LTI
fractional-order delay systems of retarded and neutral type, Automatica
46 (2010) 362-368.
[29] A.-H. Lin, Y. Ren, N.-M. Xia, On neutral impulsive stochastic integrodifferential
equations with infinite delays via fractional operators, Mathematical
and Computer Modelling, 51(2010):413-424.
[30] G.-D. Hu, Some new simple stability criteria of neutral delay-differential
systems, Applied mathematics and computaiton, 80(1996):257-271.
[31] D.-Q. Cao, P. He, Stability criteria of linear neutral systems with a single
delay, Applied Mathematics and Computation, 148(2004):135-143.
[32] E.J. Muth, Transform Methods with Applications to Engineering and
Operations Research. Prentice-Hall, New Jersey, 1977.
[33] P. Lancaster, M. Tismenetsky, The Theory of Matrices, Academic Press,
Orlando, FL, 1985.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:53786", author = "Lianglin Xiong and Yun Zhao and Tao Jiang", title = "Stability Analysis of Linear Fractional Order Neutral System with Multiple Delays by Algebraic Approach", abstract = "In this paper, we study the stability of n-dimensional linear fractional neutral differential equation with time delays. By using the Laplace transform, we introduce a characteristic equation for the above system with multiple time delays. We discover that if all roots of the characteristic equation have negative parts, then the equilibrium of the above linear system with fractional order is Lyapunov globally asymptotical stable if the equilibrium exist that is almost the same as that of classical differential equations. An example is provided to show the effectiveness of the approach presented in this paper.
", keywords = "Fractional neutral differential equation, Laplace transform, characteristic equation.", volume = "5", number = "4", pages = "575-4", }
