Abstract
The initial condition problem of fractional order systems is investigated in this paper. Firstly, a named aberration phenomenon is introduced, which reveals the nature of initial condition problem. And then, in order to interpret this odd phenomenon, definitions of the Riemann–Liouville derivative and Caputo derivative are revisited. As a result, the fractional order systems described by fractional differential equations and exact state-space models are linked much more closely, and it is found that relationship between these two kinds of models basically lies in the exact initial state distribution. The results also show the inborn defects of these two derivatives. Afterward, a practical method to estimate the exact initial states is studied naturally. At last, several simulation examples carefully illustrate the effectiveness of proposed method.
Similar content being viewed by others
References
Monje, C.A., Chen, Y.Q., Vinagre, B.M., Xue, D.Y., Feliu-Batlle, V.: Fractional-Order Systems and Controls: Fundamentals and Applications. Springer, London (2010)
Oustaloup, A.: Diversity and Non-integer Differentiation for System Dynamics. Wiley, London (2014)
Hartley, T.T., Lorenzo, C.F.: Fractional-order system identification based on continuous order-distributions. Sig. Process. 83(11), 2287–2300 (2003)
Victor, S., Malti, R., Garnier, H., Oustaloup, A.: Parameter and differentiation order estimation in fractional models. Automatica 49(4), 926–935 (2013)
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 Trans. Autom. Control 55(1), 152–158 (2010)
Aghababa, M.P.: A Lyapunov-based control scheme for robust stabilization of fractional chaotic systems. Nonlinear Dyn. 78(3), 2129–2140 (2014)
Wang, X.Y., He, Y.J.: Projective synchronization of fractional order chaotic system based on linear separation. Phys. Lett. A 372(4), 435–441 (2008)
Wang, X.Y., Song, J.M.: Synchronization of the fractional order hyperchaos Lorenz systems with activation feedback control. Commun. Nonlinear Sci. Numer. Simul. 14(8), 3351–3357 (2009)
Wang, X.Y., Zhang, X.P., Ma, C.: Modified projective synchronization of fractional-order chaotic systems via active sliding mode control. Nonlinear Dyn. 69(1–2), 511–517 (2012)
Luo, Y., Chen, Y.Q.: Stabilizing and robust fractional order pi controller synthesis for first order plus time delay systems. Automatica 48(9), 2159–2167 (2012)
Wei, Y.H., Chen, Y.Q., Liang, S., Wang, Y.: A novel algorithm on adaptive backstepping control of fractional order systems. Neurocomputing 165, 395–402 (2015)
Wang, X.Y., Wang, M.J.: Dynamic analysis of the fractional-order Liu system and its synchronization. Chaos Interdiscipl. J. Nonlinear Sci. 17(3), 033,106 (2007)
Wang, X.Y., He, Y.J., Wang, M.J.: Chaos control of a fractional order modified coupled dynamos system. Nonlinear Anal. Theory Methods Appl. 71(12), 6126–6134 (2009)
Luo, C., Wang, X.: Chaos in the fractional-order complex lorenz system and its synchronization. Nonlinear Dyn. 71(1–2), 241–257 (2013)
Hartley, T.T., Lorenzo, C.F., Trigeassou, J.C., Maamri, N.: Equivalence of history-function based and infinite-dimensional-state initializations for fractional-order operators. J. Comput. Nonlinear Dyn. 8(4), 041,014 (2013)
Fukunaga, M., Shimizu, N.: Role of prehistories in the initial value problems of fractional viscoelastic equations. Nonlinear Dyn. 38(1–4), 207–220 (2004)
Lorenzo, C.F., Hartley, T.T.: Variable order and distributed order fractional operators. Nonlinear Dyn. 29(1–4), 57–98 (2002)
Caputo, M.: Elasticità E Dissipazione. Zanichelli, Bologna (1969)
Hartley, T.T., Lorenzo, C.F.: Insights into the initialization of fractional order operators via Semi-Infinite lines. NASA TM-208407 (1998)
Hartley, T.T., Lorenzo, C.F.: Dynamics and control of initialized fractional-order systems. Nonlinear Dyn. 29(1–4), 201–233 (2002)
Sabatier, J., Merveillaut, M., Malti, R., Oustaloup, A.: How to impose physically coherent initial conditions to a fractional system? Commun. Nonlinear Sci. Numer. Simul. 15(5), 1318–1326 (2010)
Trigeassou, J.C., Maamri, N.: Initial conditions and initialization of linear fractional differential equations. Sig. Process. 91(3), 427–436 (2011)
Trigeassou, J.C., Maamri, N., Oustaloup, A.: Initialization of Riemann–Liouville and Caputo fractional derivatives. In: Proceedings of the ASME 2011 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, pp. 219–226. Washington, USA (2011)
Podlubny, I.: Fractional Differential Equations: An Introduction to Fraction Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Academic Press, San Diego (1999)
Du, M.L., Wang, Z.H.: Initialized fractional differential equations with Riemann–Liouville fractional-order derivative. Eur. Phys. J. Spec. Top. 193(1), 49–60 (2011)
Xu, Q.S.: Piezoelectric nanopositioning control using second-order discrete-time terminal sliding-mode strategy. IEEE Trans. Industr. Electron. 62(12), 7738–7748 (2015)
Xu, Q.S.: Digital integral terminal sliding mode predictive control of piezoelectric-driven motion system. IEEE Trans. Industr. Electron. 63(6), 3976–3984 (2016)
Trigeassou, J.C., Maamri, N., Sabatier, J., Oustaloup, A.: Transients of fractional-order integrator and derivatives. Signal Image Video Process. 6(3), 359–372 (2012)
Montseny, G.: Diffusive representation of pseudo-differential time-operators. In: Proceedings of the Fractional Differential System: Models. Methods and Applications, pp. 159–175. Toulouse, France (1998)
Liang, S., Peng, C., Liao, Z., Wang, Y.: State space approximation for general fractional order dynamic systems. Int. J. Syst. Sci. 45(10), 2203–2212 (2014)
Author information
Authors and Affiliations
Corresponding author
Additional information
This work is supported by the National Natural Science Foundation of China under Grant No. 61573332 and the Fundamental Research Funds for the Central Universities No. WK2100100028.
Rights and permissions
About this article
Cite this article
Du, B., Wei, Y., Liang, S. et al. Estimation of exact initial states of fractional order systems. Nonlinear Dyn 86, 2061–2070 (2016). https://doi.org/10.1007/s11071-016-3015-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-016-3015-7