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Fractional Approaches in Path Tracking Design (or Motion Control): Prefiltering, Shaping, and Flatness

  • Chapter
Fractional Order Differentiation and Robust Control Design

Part of the book series: Intelligent Systems, Control and Automation: Science and Engineering ((ISCA,volume 77))

  • 2076 Accesses

Abstract

Automatic control has a long history in engineering. At the end of the seventeenth century, Hooke introduced a system of balls rotating around an axis in which the velocity was proportional to the velocity of the windmill: the greater the ball velocity, the larger the gap from the axis activating the windmill sails in order to reduce the velocity. During the industrial revolution, Watt adapted a ball regulator for steam engines: the greater the ball velocity, the wider the opening of a valve that released the steam. By lowering the boiler pressure, the velocity could be reduced. The main problem was to maintain a constant speed despite load variations. In 1868, the physicist Maxwell published the first mathematical analysis explaining some of the behaviors observed on the regulators employed at that time. That was the beginning of several studies on stability to which the mathematicians Hurwitz and Routh were the main contributors. After theoretical and technological developments, it became possible to handle linear and nonlinear multivariable systems thanks to the important contributions of Bellman in dynamical programming (Brassard and Bratley 1996), Kalman (1960) in filtering and linear quadratic control and Pontryagin et al. (1962) in optimal control. Their contributions still feed research in control theory today.

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Notes

  1. 1.

    A noetherian ring is a ring dotted with the addition and multiplication compatible with the addition in the distributivity sense

  2. 2.

    For an n-tuple \( \tau =1 \sec \), t 0

  3. 3.

    By taking t 2, we come back to a physical case where two masses are coupled with a spring, the whole being piloted by a force u

  4. 4.

    \( GL_m (\mathrm{R} \lfloor\text{D}^{v}\rfloor) \) is the group of unimodular square \( \text{D}^{v} \)-polynomial matrices, i.e. the set of invertible square \( \text{D}^{v} \)-polynomial matrices whose inverse is also a \( \text{D}^{v} \)-polynomial matrix.

References

  • Ayadi M (2002) Contributions à la commande des systèmes linéaires plats de dimension finie. Institut National Polytechnique de Toulouse

    Google Scholar 

  • Battaglia J-L, Le Lay L, Batsale J-C, Oustaloup A, Cois O (2000) Heat flux estimation through inverted non integer identification models. Int J Therm Sci 39:374–389

    Article  Google Scholar 

  • Battaglia J-L, Cois O, Puigsegur L, Oustaloup A (2001) Solving an inverse heat conduction problem using a non-integer identified model. Int J Heat Mass Transf 44(14):2671–2680

    Article  MATH  Google Scholar 

  • Bindel R, Nitsche R, Rothfuß R, Zeitz M (2000) Flachheitsbasierte Regelung eines hydraulischen Antriebs mit zwei Ventilen für einen Groß manipulator. Automatisierungstechnik 48:124–131

    Article  Google Scholar 

  • Bitauld L, Fliess M, Levine J (1997) A flatness based control synthesis of linear systems and application to windshield wipers. European Control Conference ECC'97, Kos, Greece, July 2–5, 2007

    Google Scholar 

  • Bobrow JE, Dubowsky S, Gibson JS (1985) Time-optimal control of robotic manipulators along specified paths. Int J Robot Res 4(3):3–17

    Article  Google Scholar 

  • Bourlès H, Fliess M (1997) Finite poles and zeros of linear systems: an intrinsic approach. Int J Control 68:897–922

    Article  MATH  Google Scholar 

  • Brassard G, Bratley P (1996) Fundamentals of algorithmics. Prentice Hall, Upper Saddle River

    Google Scholar 

  • Cois O (2002) Systèmes linéaires non entiers et identification par modèle non entier: application en thermique. Ph.D. thesis, Université Bordeaux 1, Bordeaux

    Google Scholar 

  • D'Andrea-Novel B, Cohen de Lara M (1993) Commande linéaire de systèmes dynamiques. Ed. Masson

    Google Scholar 

  • Davidson DW, Cole RH (1951) Dielectric relaxation in glycerol, propylene glycol and n-propanol. J Chem Phys 19(12):1484–1490

    Article  Google Scholar 

  • De Luca A, Lanari L, Oriolo G (1991) A sensitivity approach to optimal spline robot trajectories. IEEE Trans Autom Control 27(3):535–539

    Google Scholar 

  • Dombre E, Khalil W (1988) Modélisation et commande des robots. Editions Hermès, Paris

    Google Scholar 

  • Dubois N (2000) Systèmes plats. Ph.D. thesis, Laboratoire du STIX, Ecole Polytechnique, Paris

    Google Scholar 

  • Fliess M (1990) Some basic structural properties of generalized linear systems. Syst Control Lett 15:391–396

    Article  MATH  MathSciNet  Google Scholar 

  • Fliess M (1992) A remark on Willem's trajectory characterization of linear controllability. Syst Control Lett 19:43–45

    Article  MATH  MathSciNet  Google Scholar 

  • Fliess M (1994) Une interprétation algébrique de la transformation de Laplace et des matrices de transfert. Linear Algebra Appl 203–204:429–442

    Article  MathSciNet  Google Scholar 

  • Fliess M (2000) Variations sur la notion de contrôlabilité. Journée Société Mathématique de France

    Google Scholar 

  • Fliess M, Marquez R (2000) Continuous-time linear predictive control and flatness; a module-theoretic setting with examples. Int J Control 73:606–623

    Article  MATH  MathSciNet  Google Scholar 

  • Fliess M, Lévine J, Martin P, Rouchon P (1992) Sur les systèmes non linéaires différentiellement plats. CR Acad Sci Paris I-315:619–624

    Google Scholar 

  • Fliess M, Lévine J, Martin P, Rouchon P (1993) Linéarisations par bouclage dynamique et transformations de Lie-Bäcklund. CR Acad Sci Paris I-317:981–986

    Google Scholar 

  • Fliess M, Lévine J, Martin P, Rouchon P (1995a) Flatness and defect of nonlinear systems: introductory theory and examples. Int J Control 61:1327–1361

    Article  MATH  Google Scholar 

  • Fliess M, Lévine J, Martin P, Rouchon P (1995b) Design of trajectory stabilizing feedback for driftless flat systems. 2nd European Control Conference ECC'95, pp 1882–1887

    Google Scholar 

  • Fliess M, Martin P, Petit N, Rouchon P (1998a) Commande de l'équation des télégraphistes et restauration active d'un signal. TRAIT SIGNAL 15:619–625

    MATH  Google Scholar 

  • Fliess M, Mounier H, Rouchon P, Rudolph J (1998b) Controlling the transient of a chemical reactor: a distributed parameter approach. CESA'98 IMACS Multiconferences

    Google Scholar 

  • Fliess M, Lévine J, Martin P, Rouchon P (1999) A Lie-Bäcklund approach to equivalence and flatness of nonlinear systems. IEEE Trans Autom Control 44:922–937

    Article  MATH  Google Scholar 

  • Gantmacher FR (1966) Théorie des matrices. Ed. Dunod, Paris

    Google Scholar 

  • Hamilton HJ (1946) Roots of equation by fractional iteration. Duke Math J 13:113–123

    Article  MATH  MathSciNet  Google Scholar 

  • Hilbert D (1912) Über den Begriff der Klasse von Differentialgleichungen. Math Ann 73:95–108

    Article  MATH  MathSciNet  Google Scholar 

  • Horowitz I (1992) Quantitative feedback design theory (QFT). QFT publications, Boulder

    Google Scholar 

  • Ilchmann A (1985) Time-varying linear systems and invariants of system equivalence. Int J Control 42:759–790

    Article  MATH  MathSciNet  Google Scholar 

  • Jallouli-Khlif R, Melchior P, Orsoni B, Derbel N, Oustaloup A (2007) Path tracking design based on fractional prefilter with acceleration input and comparison with preshaper approach. Fourth IEEE SSD'07, Hammamet, Tunisia, 19–22 Mar

    Google Scholar 

  • Jallouli-Khlif R, Melchior P, Levron F, Derbel N, Oustaloup A (2008) Impulse response of third generation CRONE control. 3rd IFAC FDA'08, Ankara, Turkey, 5–7 Nov

    Google Scholar 

  • Jallouli-Khlif R, Melchior P, Orsoni B, Derbel N, Oustaloup A (2009a) Path tracking design: comparison between Davidson-Cole prefilter with acceleration input and shaping approach. Trans Syst Signals Devices 4(1):91–104, Shaker Verlag Publisher

    Google Scholar 

  • Jallouli-Khlif R, Melchior P, Derbel N, Oustaloup A (2009b) Preshaping command inputs for third generation CRONE control: robustness study. 6th IEEE SSD'09, Djerba, Tunisia, 23–26 Mar

    Google Scholar 

  • Jallouli-Khlif R, Melchior P, Levron F, Derbel N, Oustaloup A (2010a) Analytical impulse response of third generation Crone control. New Trends in Nanotechnology and Fractional Calculus Applications, Springer-Verlag Ed., Chap 5 "Fractional Control Systems", pp 343–356

    Google Scholar 

  • Jallouli-Khlif R, Melchior P, Levron F, Derbel N, Oustaloup A (2010b) Initial values of third generation CRONE control responses. 4th IFAC FDA'10, Badajoz, Spain, 18–20 Oct

    Google Scholar 

  • Jallouli-Khlif R, Melchior P, Derbel N, Oustaloup A (2012) Robust path tracking by preshaping approach designed for third generation CRONE control. Int J Model Identif Control 15(2):125–133, Interscience Publishers

    Article  Google Scholar 

  • Kailath T (1980) Linear systems. Prentice Hall, Englewood Cliffs

    MATH  Google Scholar 

  • Kalman RE (1960) A new approach to linear filtering and prediction problems. Trans ASME J Basic Eng 82:35–45

    Article  Google Scholar 

  • Kalman RE (1963) Mathematical description of linear dynamical systems. J SIAM Control 1:152–192

    MATH  MathSciNet  Google Scholar 

  • Kalman RE (1969) Topics in mathematical system theory. McGraw-Hill, New York, pp 237–339

    MATH  Google Scholar 

  • Khalil W, Dombre E (1999) Modélisation, identification et commande des robots. Hermès Editions, Paris

    MATH  Google Scholar 

  • Kieffer J, Cahill AJ, James MR (1997) Robust and accurate time-optimal path-tracking control for robot manipulators. IEEE Trans Robot Autom 13(6):880–890

    Article  Google Scholar 

  • Kiss B, Lévine J, Lantos B (1999) Trajectory planning for dextrous manipulation with rolling contacts. 37th IEEE conference decision control, pp 2118–2119

    Google Scholar 

  • Laroche B (2000) Extension de la notion de platitude à des systèmes décrits par des équations aux dérivées partielles linéaires. Ph.D. thesis, Ecole Nationale Supérieure des Mines de Paris, Paris

    Google Scholar 

  • Laroche B, Martin P, Rouchon P (1998) Motion planning for a class of partial differential equations with boundary control. IEEE-CDC'98, pp 3494–3497

    Google Scholar 

  • Lavigne L (2003) Outils d'analyse et de synthèse des lois de commande robuste des systèmes dynamiques plats. Ph.D. thesis, Université Bordeaux 1, Bordeaux

    Google Scholar 

  • Le Mehaute A, Heliodore F, Oustaloup A (1991) Cole-Cole relaxation and CRONE relaxation. IMACS-IFAC symposium MCTS'91, Lille, France, May

    Google Scholar 

  • Lévine J, Nguyen DV (2003) Flat output characterization for linear systems using polynomial matrices. Syst Control Lett 48:69–75

    Article  MATH  Google Scholar 

  • Lévine J, Rémond B (2000) Flatness based control of an automatic clutch. MTNS-2000

    Google Scholar 

  • Lévine J, Lottin J, Ponsart JC (1996) A nonlinear approach to the control of magnetic bearings. IEEE Trans Control Syst Technol 4:524–544

    Article  Google Scholar 

  • Lin CS, Chang PR, Luh JYS (1983) Formulation and optimisation of cubic polynomial joint trajectories for industrial robots. IEEE Trans Autom Control 28(12):1066–1073

    Article  MATH  Google Scholar 

  • Malti R, Victor S, Oustaloup A, Garnier H (2008) An optimal instrumental variable method for continuous-time fractional model identification. The 17th IFAC World Congress (IFAC'08)

    Google Scholar 

  • Marquez R, Delaleau E (1999) Une application de la commande prédictive linéaire basée sur la platitude. Journées Doctorales d'Automatique'99, GdR MACS CNRS, pp 148–152

    Google Scholar 

  • Martin P, Devasia S, Peden B (1996) A different look at output feedback: control of a VTOL aircraft. Automatica 32:211–264

    Article  Google Scholar 

  • Matignon D, D'Andréa-Novel B (1996) Some results on controllability and observability of finite-dimensional fractional differential systems. IMACS 2:952–956

    Google Scholar 

  • Melchior P, Robin G, Lanusse P (1996) CRONE control of a XY cutting table. IEEE-SMC CESA'96 IMACS Multiconference, Lille, France, 9–12 July

    Google Scholar 

  • Melchior P, Robin G, L'Hostis S, Levron F (1998) Non integer order movement generation in path planning. IEEE-SMC CESA'98 IMACS Multiconference, Nabeul, Tunisia, 1–4 Apr

    Google Scholar 

  • Melchior P, Lanusse P, Dancla F, Cois O (1999) Valorisation de l'approche non entière par le logiciel CRONE. CETSIS-EEA'99, Montpellier, France

    Google Scholar 

  • Melchior P, Orsoni B, Badie Th, Robin G (2000a) Génération de consigne optimale par filtre à dérivée généralisée implicite: Application au véhicule électrique. IEEE CIFA'2000, Lille, France

    Google Scholar 

  • Melchior P, Orsoni B, Badie Th, Robin G, Oustaloup A (2000b) Non-integer motion control: application to an XY cutting table. 1st IFAC Conference on Mechatronic Systems, Darmstadt, Germany, 18–20 Sept

    Google Scholar 

  • Melchior P, Orsoni B, Oustaloup A (2001) Input/output fractional transfer function in path tracking design. International Conference SSD'2001, Hammamet, Tunisia, 27–30 Mar

    Google Scholar 

  • Melchior P, Orsoni B, Oustaloup A (2002) CAO en génération de mouvement: module "Fractional Path Tracking Design" de la boîte à outils "CRONE: Fractional Systems". 2ème IEEE conférence internationale Francophone d'Automatique (CIFA'2002), Nantes, France, 6–8 July

    Google Scholar 

  • Melchior P, Poty A, Levron F, Orsoni B, Oustaloup A (2003a) Motion control by preshaping: robustness study and application to the CRONE control. In: Proceedings of 2nd IEEE international conference on signals, systems, decision and information technology (SSD'03), Sousse, Tunisia

    Google Scholar 

  • Melchior P, Poty A, Sabatier J, Orsoni B, Oustaloup A (2003b) Modified ZV shaper for explicit fractional derivative systems: application to CRONE control. IMACS/IEEE-SMC Multiconference CESA'03, Lille, France, 9–11 July

    Google Scholar 

  • Melchior P, Poty A, Orsoni B, Oustaloup A (2003c) Preshaping command inputs for 2nd generation CRONE control: application on an instrumented DC motor bench. ASME DETC, first symposium on FDTA of the 19th conference on VIB, IDETC&CIEC, Chicago, Illinois, USA, 2–6 Sept

    Google Scholar 

  • Melchior P, Poty A, Oustaloup A (2004) Motion control by ZV shaper synthesis extended for fractional systems and its application to CRONE control. Int J Nonlinear Dyn Chaos Eng Syst 38(1–4):401–416, Kluwer Academic Publishers

    MATH  Google Scholar 

  • Melchior P, Poty A, Oustaloup A (2005a) Path tracking design by frequency band-limited fractional differentiator prefilter. Fifth EUROMECH ENOC'05, Eindhoven, The Netherlands, 7–12 Aug

    Google Scholar 

  • Melchior P, Cugnet M, Sabatier J, Poty A, Oustaloup A (2005b) Flatness principle applied to the dynamic inversion of fractional systems. Third IEEE SSD'05, Sousse, Tunisia, 21–24 Mar

    Google Scholar 

  • Melchior P, Cugnet M, Sabatier J, Oustaloup A (2005c) Flatness control: application to a fractional thermal system. 20th ASME ASME IDETC/CIE'05, VIB, 2nd symposium on FDTA, Long Beach, California, USA, 26–28 Sept

    Google Scholar 

  • Melchior P, Poty A, Oustaloup A (2007a) Frequency band-limited fractional differentiator prefilter in path tracking design. Advances in fractional calculus: theoretical developments and applications in physics and engineering, Springer Verlag Ed., Chap. 7 Control, pp 477–492

    Google Scholar 

  • Melchior P, Jallouli-Khlif R, Orsoni B, Derbel N, Oustaloup A (2007b) Path tracking design based on jerk input fractional prefilter and jerk limited input shaper approaches. IEEE-ASME MESA'07 of the IDETC/CIE'07, Las Vegas, Nevada, USA, 4–7 Sept

    Google Scholar 

  • Melchior P, Cugnet M, Sabatier J, Poty A, Oustaloup A (2007c) Flatness control of a fractional thermal system. In: Advances in fractional calculus theoretical developments and applications in physics and engineering, Springer Verlag Ed, pp 493–509

    Google Scholar 

  • Melchior P, Inarn Ch, Oustaloup A (2009) Path tracking design by fractional prefilter extension to square MIMO systems. 7th MSNDC'09 of the 21th ASME IDETC/CIE'09, San Diego, USA, 30 Aug–2 Sept

    Google Scholar 

  • Melchior P, Yousfi N, Lanusse P, Derbel N, Oustaloup A (2012) Path tracking design based on frequency-band limited fractional differentiator with speed and acceleration input. 5th IFAC FDA'12, Hohai University, Nanjing, China, 14–17 May

    Google Scholar 

  • Melchior P, Yousfi N, Inarn Ch, Derbel N, Oustaloup A (2013) Path tracking design based on I/O Davidson-Cole type transfer function for square MIMO systems. Joint IFAC 5th SSSC'13, 11th TDS'13, 6th FDA'13, Grenoble, France, February 4–6

    Google Scholar 

  • Miller KS, Ross B (1993) An introduction to the fractional calculus and fractional differential equations. Wiley-Interscience Publication, New York

    MATH  Google Scholar 

  • Morio V (2009) Contribution au développement d'une loi de guidage autonome par platitude: Application à une mission de rentrée atmosphérique. Ph.D. thesis, Université Bordeaux 1, Bordeaux

    Google Scholar 

  • Mounier H, Rouchon P, Rudolph J (1997) Some examples of linear systems with delays. JESA-APII-RAIRO 31:911–925

    Google Scholar 

  • Mounier H, Rudolph J, Fliess M, Rouchon P (1998) Tracking control of a vibrating string with an interior mass viewed as a delay system. ESAIM: Control Optim Calc Var 3:315–321

    Article  MATH  MathSciNet  Google Scholar 

  • Nedler JA, Mead R (1965) A simplex method for function minimization. Comput J 7:308–313

    Article  Google Scholar 

  • Öhr J, Sternad M, Rönnbäck S (1998) H2-optimal anti-windup performance in SISO control systems. 4th SIAM conference on control and its Applications, Jacksonville, USA, May

    Google Scholar 

  • Orsoni B (2002) Dérivée généralisée en planification de trajectoire et génération de mouvement. Ph.D. thesis, University Bordeaux 1, France

    Google Scholar 

  • Orsoni B, Melchior P, Oustaloup A (2001) Davidson-Cole transfer function in path tracking design. In: Proceedings of 6th IEEE ECC'2001, Porto, Portugal, pp 1174–1179, 4–7 Sept

    Google Scholar 

  • Orsoni B, Melchior P, Badie T, Robin G, Oustaloup A (2002) Fractional motion control: application to an XY cutting table. Int J Nonlinear Dyn Chaos Eng Syst 29(1–4):297–314, Issue on "Fractional Order Calculus and its Applications", Kluwer Academic Publishers

    MATH  Google Scholar 

  • Oustaloup A (1991) La commande CRONE : Commande Robuste d'Ordre Non Entier. Editions Hermès, Paris

    MATH  Google Scholar 

  • Oustaloup A (1995) La dérivation non entière : théorie, synthèse et applications. Editions Hermès, Paris

    MATH  Google Scholar 

  • Oustaloup A, Melchior P, Lanusse P, Cois O, Dancla F (2000) The CRONE toolbox for Matlab. IEEE International Symposium on Computer-Aided Control-System Design, Anchorage, USA

    Google Scholar 

  • Petit N (2000) Platitude et planification de trajectoires pour certains systèmes à retards et E.D.P: applications en génie chimique. Ph.D. thesis, Ecole des Mines, Paris

    Google Scholar 

  • Petit N, Creff Y, Rouchon P (1997) δ-freeness of a class of linear delayed systems. 4th European Control Conference ECC'97

    Google Scholar 

  • Podlubny I (1999) Fractional differential equations. Academic, San Diego

    MATH  Google Scholar 

  • Pomet J-B (1995) Geometry in nonlinear control and differential inclusions. Banach Center Publications, Warsaw, pp 319–339

    Google Scholar 

  • Pontryagin LS, Boltayanskii VG, Gamkrelidze RV, Mishchenko EF (1962) The mathematical theory of optimal processes. Wiley, New York

    MATH  Google Scholar 

  • Poty A, Melchior P, Levron F, Orsoni B, Oustaloup A (2003a) Motion control by preshaping: extension for explicit generalized derivative systems. In: Proceedings of 2nd IEEE international conference on signals, systems, decision and information technology (SSD'03), Sousse, Tunisia

    Google Scholar 

  • Poty A, Melchior P, Orsoni B, Levron F, Oustaloup A (2003b) ZV and ZVD shapers for explicit fractional derivative systems. In: Proceedings of 11th IEEE-ICAR 2003, Coimbra, Portugal, 30 June–3 July

    Google Scholar 

  • Poty A, Melchior P, Levron F, Orsoni B, Oustaloup A (2003c) Preshaping command inputs for explicit fractional derivative systems: application to CRONE control. 7th IEEE ECC'03, University of Cambridge, UK, 1–4 Sept

    Google Scholar 

  • Poty A, Melchior P, Oustaloup A (2006a) Frequency band-limited fractional differentiator in path tracking design. 2nd IFAC workshop on "Fractional Differentiation and its Applications" (FDA'06), Porto, Portugal, 19–21 July

    Google Scholar 

  • Poty A, Melchior P, Levron F, Orsoni B, Oustaloup A (2006b) Motion control by preshaping: extension for explicit fractional derivative systems. Trans Syst Signals Devices 1(2):103–123, Shaker Publisher

    Google Scholar 

  • Rosenbrock HH (1970) Multivariable and state-space theory. Wiley, New-York

    MATH  Google Scholar 

  • Rothfuß R, Rudolph J, Zeitz M (1996) Flatness based control of a nonlinear chemical reactor model. Automatica 32:1433–1439

    Article  MATH  Google Scholar 

  • Rudolph J (2000) Randsteuerung von Wärmetauschern mit örtlichen verteilten Parametern: ein flachachheitsbasierter Zugang. Automatisierungstechnik

    Google Scholar 

  • Sabatier J, Melchior P, Oustaloup A (2003) Réalisation d'un banc d'essais thermique pour l'étude des systèmes non entiers. In: Proceedings of CETSIS'03, Toulouse, France, 13–14 Nov

    Google Scholar 

  • Samko SG, Kilbas AA, Marichev OI (1993) Fractional integrals and derivatives. Gordon and Breach Science Publishers, Amsterdam

    MATH  Google Scholar 

  • Shin KG, McKay ND (1985) Minimum-time control of robotic manipulators with geometric path constraints. IEEE Trans Autom Control 30(6):531–541

    Article  MATH  Google Scholar 

  • Shin KG, McKay ND (1987) Robust trajectory planning for robotic manipulators under payload uncertainties. IEEE Trans Autom Control 32(12):1044–1054

    Article  MATH  Google Scholar 

  • Singer NC (1989) Residual vibration reduction in computer controlled machines. Ph.D. thesis, Massachusetts Institute of Technology

    Google Scholar 

  • Singer NC, Seering WP (1990) Preshaping command inputs to reduce system vibration. J Dyn Syst MeasControl 112:76–82

    Article  Google Scholar 

  • Singhose W, Singer N, Seering W (1995) Comparison of command shaping methods for reducing residual vibration. In: Proceedings of 3rd European control conference, Rome, Italy, Sept

    Google Scholar 

  • Sontag ED (1998) Mathematical control theory, deterministic finite dimensional systems, 2nd edn. Springer, New York

    MATH  Google Scholar 

  • Van Nieuwstadt M, Rathinam M, Murray RM (1998) Flatness and absolute equivalence of nonlinear control systems. SIAM J Control Optim 36:1225–1239

    Article  MATH  MathSciNet  Google Scholar 

  • Victor S (2010) Identification par modèle non entier pour la poursuite robuste de trajectoire par platitude. Ph.D. thesis, Université Bordeaux 1, Bordeaux, France

    Google Scholar 

  • Victor S, Melchior P, Oustaloup A (2008a) Flatness principle extension to linear fractional MIMO systems: thermal application. 14th IEEE MELECON'08, Ajaccio, France, 5–7 May

    Google Scholar 

  • Victor S, Melchior P, Nelson-Gruel D, Oustaloup A (2008b) Flatness control for linear fractional MIMO systems: thermal application. 3rd IFAC FDA'08, Ankara, Turkey, 5–7 Nov

    Google Scholar 

  • Victor S, Malti R, Melchior P, Oustaloup A (2009) From system identification to path planning using fractional approach: a thermal application example. 7th MSNDC'09 of the 21st ASME IDETC/CIE'09, San Diego, USA, 30 Aug–2 Sept

    Google Scholar 

  • Victor S, Melchior P, Oustaloup A (2010) Robust path tracking using flatness for fractional linear MIMO systems: a thermal application. Comput Math Appl 59(5):1667–1678, Elsevier

    Article  MATH  MathSciNet  Google Scholar 

  • Victor S, Melchior P, Malti R, Oustaloup A (2011) Path tracking with flatness and CRONE control for fractional systems: thermal application. 18th IFAC World Congress'11, Milan, Italy, 28 Aug–2 Sept

    Google Scholar 

  • Victor S, Melchior P, Oustaloup A (2013) Computation of flat outputs for fractional systems: a thermal application. Joint IFAC 5th SSSC, 11th TDS, 6th FDA, Grenoble, France, 4–6 Feb

    Google Scholar 

  • Victor S, Melchior P, Levine J, Oustaloup A (2014) Flat output computation for fractional linear systems: application to a thermal system. 19th IFAC World Congress'14, Cape Town, South Africa, Aug

    Google Scholar 

  • Vinagre BM, Feliu V (2002) Modeling and control of dynamic system using fractional calculus: application to electrochemical processes and flexible structures. In: Proceedings of 41st IEEE conference on decision and control, Las Vegas, vol 1, pp 214–239

    Google Scholar 

  • Yousfi N, Melchior P, Lanusse P, Derbel N, Oustaloup A (2011) Path tracking design by input/output Davidson-Cole transfer function. Eighth IEEE SSD'11, Sousse, Tunisia, 22–25 Mar

    Google Scholar 

  • Yousfi N, Melchior P, Rekik C, Derbel N, Oustaloup A (2012a) Comparison between H∞ and CRONE control combined with QFT approach to control multivariable systems in path tracking design. Int J Comput Appl 45(12):1–9, FCS Ed

    Google Scholar 

  • Yousfi N, Melchior P, Rekik C, Derbel N, Oustaloup A (2012b) Design of centralized CRONE controller combined with MIMO-QFT approach applied to non-square multivariable systems. Int J Comput Appl 45(16):6–14, FCS Ed

    Google Scholar 

  • Yousfi N, Melchior P, Rekik C, Derbel N, Oustaloup A (2012c) Path tracking design by fractional prefilter using a combined QFT/H∞ design for TDOF uncertain feedback systems. J Appl Nonlinear Dyn 1(3):239–261, L&H Scientific Publishing

    Article  MATH  Google Scholar 

  • Yousfi N, Melchior P, Lanusse P, Derbel N, Oustaloup A (2012d) Design of non-diagonal controller using MIMO-QFT approach combined with CRONE control: application to square system. 5th IFAC FDA'12, Hohai University, Nanjing, China, 14–17 May

    Google Scholar 

  • Yousfi N, Melchior P, Rekik C, Derbel N, Oustaloup A (2013a) Path tracking design based on Davidson-Cole prefilter using a centralized CRONE controller applied to multivariable systems. Nonlinear Dyn 71(4):701–712, Springer

    Article  MathSciNet  Google Scholar 

  • Yousfi N, Melchior P, Lanusse P, Derbel N, Oustaloup A (2013b) Decentralized CRONE control of nonsquare multivariable systems in path-tracking design. Nonlinear Dyn 76(1):447–457, Springer

    Article  MathSciNet  Google Scholar 

  • Yousfi N, Melchior P, Rekik C, Derbel N, Oustaloup A (2014a) Comparison between Davidson-Cole and frequency-band limited fractional differentiator I/O type transfer function with speed and acceleration inputs in path tracking design. J Appl Nonlinear Dyn 3(1):1–16, L&H Scientific Publishing

    Article  Google Scholar 

  • Yousfi N, Jallouli-Khlif R, Melchior P, Lanusse P, Derbel N, Oustaloup A (2014b) Path tracking design by Frequency Band Limited Fractional Differentiator prefilter: square MIMO systems. IEEE ICFDA'14, Catania, Italy, 23–25 June

    Google Scholar 

  • Zribi M, Sira-Ramiréz H, Ngai A (2001) Static and dynamic sliding mode control schemes for a PM stepper motor. Int J Control 74:103–117

    Article  MATH  Google Scholar 

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  1. IMS Laboratory - CNRS UMR 5218 – Bordeaux INP - Bordeaux University Bat A31 – 351 cours de la Libération, 33400, Cedex Talence, France

    P. Melchior & S. Victor

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  1. P. Melchior
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  2. S. Victor
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Correspondence to P. Melchior .

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Melchior, P., Victor, S. (2015). Fractional Approaches in Path Tracking Design (or Motion Control): Prefiltering, Shaping, and Flatness. In: Fractional Order Differentiation and Robust Control Design. Intelligent Systems, Control and Automation: Science and Engineering, vol 77. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-9807-5_5

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  • DOI: https://doi.org/10.1007/978-94-017-9807-5_5

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