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Sliding mode control approaches to the robust regulation of linear multivariable fractional-order dynamics. (English) Zbl 1207.93079

Summary: Sliding mode control approaches are developed to stabilize a class of linear uncertain fractional-order dynamics. After making a suitable transformation that simplifies the sliding manifold design, two sliding mode control schemes are presented. The first one is based on the conventional discontinuous first-order sliding mode control technique. The second scheme is based on the chattering-free second-order sliding mode approach that leads to the same robust performance but using a continuous control action. Simple controller tuning formulas are constructively developed along the paper by a Lyapunov analysis. The simulation results confirm the expected performance.

MSC:

93D09 Robust stability
93D20 Asymptotic stability in control theory
34K37 Functional-differential equations with fractional derivatives
93C41 Control/observation systems with incomplete information
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[1] Riewe, Nonconservative Lagrangian and Hamiltonian mechanics, Physical Review E 53 (2) pp 1890– (1995)
[2] Atanacković, On distributed derivative model of a viscoelastic body, Comptes Rendus Mecanique 331 pp 687– (2003)
[3] Magin, Fractional Calculus in Bioengineering (2006)
[4] Sabatier, Advances in Fractional Calculus-Theoretical Developments and Applications in Physics and Engineering (2007) · Zbl 1116.00014
[5] Manabe, The non-integer integral and its application to control systems, Electrotechnical Journal of Japan 6 (3-4) pp 83– (1961)
[6] Oustaloup, The CRONE suspension, Control Engineering Practice 4 (8) pp 1101– (1996) · Zbl 1219.93073
[7] Podlubny, Fractional Differential Equations (1999)
[8] Podlubny, Fractional order systems and PI{\(\lambda\)}D{\(\mu\)}-controllers, IEEE Transactions on Automatic Control 44 (1) pp 208– (1999)
[9] Agrawal, Formulation of Euler-Lagrange equations for fractional variational problems, Journal of Mathematical Analysis and Applications 272 pp 368– (2002) · Zbl 1070.49013
[10] Agrawal, A general formulation and solution for fractional optimal control problems, Nonlinear Dynamics 38 pp 323– (2004) · Zbl 1121.70019
[11] Agrawal, A Hamiltonian formulation and a direct numerical scheme for fractional optimal control problems, Journal of Vibrations and Control 13 pp 1269– (2007) · Zbl 1182.70047
[12] Jeličić, Optimality conditions and a solution scheme for fractional optimal control problems, Structural and Multidisciplinary Optimization 38 pp 571– (2009)
[13] Hartley, Dynamics and control of initialized fractional-order systems, Nonlinear Dynamics 29 pp 201– (2002) · Zbl 1021.93019
[14] Vinagre, Using fractional order adjustment rules and fractional order reference models in model reference adaptive control, Nonlinear Dynamics 29 (1-4) pp 269– (2002) · Zbl 1031.93110
[15] Ladaci, On fractional adaptive control, Nonlinear Dynamics 43 (4) pp 365– (2006) · Zbl 1134.93356
[16] Hartley TT Lorenzo CF The vector linear fractional initialization problem 1999
[17] Chen, Robust controllability of interval fractional order linear time invariant system, Signal Processing 86 pp 2794– (2006)
[18] Chen, Robust stability check of fractional order linear time invariant systems with interval uncertainties, Signal Processing 86 pp 2611– (2006) · Zbl 1172.94385
[19] Tavazoei, A note on the stability of fractional order systems, Mathematics and Computers in Simulation 79 pp 1566– (2009) · Zbl 1168.34036
[20] Utkin, Sliding Modes in Control and Optimization (1992)
[21] Calderon, Seventh Portuguese Conference on Automatic Control (CONTROLO’2006) (2006)
[22] Efe, A fractional adaptation law for sliding mode control, International Journal of Adaptive Control and Signal Processing 22 pp 968– (2008) · Zbl 1241.93015
[23] Si-Ammour, A sliding mode control for linear fractional systems with input and state delays, Communications in Nonlinear Science and Numerical Simulation 14 (5) pp 2310– (2009) · Zbl 1221.93048
[24] Efe MO Fractional order sliding mode controller design for fractional order dynamic systems 2009
[25] Sira-Ramírez, Springer Lecture Notes in Control and Information Sciences, in: Modern Sliding Mode Control Theory. New Perspectives and Applications pp 215– (2008)
[26] Springer Lecture Notes in Control and Information Sciences, in: Modern Sliding Mode Control Theory. New Perspectives and Applications (2008)
[27] Levant, Construction principles of 2-sliding mode design, Automatica 43 (4) pp 576– (2007) · Zbl 1261.93027
[28] Emelyanov, Higher-order sliding modes in control systems, Differential Equations 29 (11) pp 1627– (1993)
[29] Emelyanov, Springer Communication and Control Engineering Series, in: Control Complex and Uncertain Systems: New Types of Feed Back (2000)
[30] Oldham, The Fractional Calculus (1974) · Zbl 0206.46601
[31] Kilbas, Theory and Applications of Fractional Differential Equations (2006) · Zbl 1138.26300
[32] Edwards, Sliding Mode Control: Theory And Applications (1998)
[33] Khalil, Nonlinear Systems (2002)
[34] Moreno JA Osorio M A Lyapunov approach to second-order sliding mode controllers and observers
[35] Petráš, Analogue Realizations of Fractional-Order Controllers (2002)
[36] Vinagre, Two direct Tustin discretization methods for fractional-order differentiator/integrator, Journal of the Franclin Institute 34 pp 349– (2003) · Zbl 1051.93031
[37] Barbosa, Time-domain design of fractional differintegrators using least squares, Signal Processing 86 pp 2567– (2006) · Zbl 1172.94364
[38] Atanacković, A diffusion wave equation with two fractional derivatives of different order, Journal of Physics A 40 pp 5319– (2007)
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