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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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\textit{A. Pisano} et al., Int. J. Robust Nonlinear Control 20, No. 18, 2045--2056 (2010; Zbl 1207.93079)

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### References:

[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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