In this paper, the fractionalized differentiating method is implemented to reduce commensurate fractional order models complexity. The prominent properties of this method are its simplicity and guarantee of preserving the stability of a specific class of fractional order models in their reduced counterparts. The presented reduction method is employed in simplifying complicated fractional order controllers to a fractional order PID (FOPID) controller and proposing tuning rules for its parameters adjustment. Finally, the efficiency of the FOPID tuning rule obtained based on the proposed reduction method is shown in the temperature control of a cutting process.
References
1.
2.
Westerlund
, S.
, and Ekstam
, L.
, 1994, “Capacitor Theory
,” IEEE Trans. Dielectr. Electr. Insul.
, 1
, pp. 826
–839
.3.
Darling
, R.
, and Newman
, J.
, 1997, “On the Short Behavior of Porous Intercalation Electrodes
,” J. Electrochem. Soc.
, 144
(9
), pp. 3057
–3063
.4.
Rossikhin
, Y. A.
, and Shitikova
, M. V.
, 1997, “Application of Fractional Derivatives to the Analysis of Damped Vibrations of Visco-Elastic Single Mass System
,” Acta. Mech.
, 120
, pp. 109
–125
.5.
Chen
, G.
, and Friedman
, G.
, 2005, “An RLC Interconnect Model Based on Fourier Analysis
,” IEEE Trans. Comput.-Aided Des.
, 24
(2
), pp. 170
–183
.6.
Oldham
, K. B.
, 2010, “Fractional Differential Equations in Electrochemistry
,” Adv. Eng. Software
, 41
(1
), pp. 9
–12
.7.
Battaglia
, J. L.
, Cois
, O.
, Puigsegura
, L.
, and Oustaloup
, A.
, 2001, “Solving an Inverse Heat Conduction Problem Using a Non-Integer Identified Model
,” Int. J. Heat Mass Transfer
, 44
(14
), pp. 2671
–2680
.8.
Cois
, O.
, Oustaloup
, A.
, Poinot
, T.
, and Battaglia
, J. L.
, 2001, “Fractional State Variable Filter for System Identification by Fractional Model
,” European Control Conference (ECC)
, Porto, Portugal.9.
Cois
, O.
, Oustaloup
, A.
, Battaglia
, E.
, and Battaglia
, J. L.
, 2002, “Non Integer Model From Modal Decomposition for Time Domain Identification
,” 41st IEEE CDC’2002 Tutorial Workshop 2
, Las Vegas, USA.10.
Sommacal
, L.
, Melchior
, P.
, and Oustaloup
, A.
, 2008, “Havriliak-Negami Function for Thermal System Identification
,” American Control Conference (ACC)
, Seattle, Washington, pp. 1316
–1321
.11.
Podlubny
, I.
, 1999, “Fractional-Order Systems and PIλDμ-Controllers
,” IEEE Trans. Autom. Control
, 44
(1
), pp. 208
–214
.12.
Monje
, C. A.
, Vinagre
, B. M.
, Feliu
, V.
, and Chen
, Y. Q.
, 2008, “Tuning and Auto Tuning of Fractional Order Controllers for Industry Applications
,” Control Eng. Pract.
, 16
(7
), pp. 798
–812
.13.
Gutman
, P.
, Mannerfelt
, C. F.
, and Molander
, P.
, 1982, “Contributions to the Model Reduction Problem
,” IEEE Trans. Autom. Control
, 27
(2
), pp. 454
–455
.14.
Podlubny
, I.
, 1999, Fractional Differential Equations
, Academic Press
, San Diego
.15.
Bonnet
, C.
, and Partington
, J. R.
, 2000, “Coprime Factorizations and Stability of Fractional Differential Systems
,” Syst. Control Lett.
, 41
(3
), pp. 167
–174
.16.
Matignon
, D.
, 1996, “Stability Results for Fractional Differential Equations With Applications to Control Processing
,” IEEE-SMC Proceedings of the Computational Engineering in Systems Applications
, IMACS, Lille, France, Vol. 2, pp. 963
–968
.17.
Tavakoli-Kakhki
, M.
, and Haeri
, M.
, 2009, “Model Reduction in Commensurate Fractional-Order Linear Systems
”, Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering
, Vol. 223, No. 4, pp. 493
–505
.18.
Tavakoli-Kakhki
, M.
, and Haeri
, M.
, 2010, “The Minimal State Space Realization for a Class of Fractional Order Transfer Functions
, SIAM J. Control Optim.
, 48
(7
), pp. 4317
–4326
.19.
Tavakoli-Kakhki
, M.
, Haeri
, M.
, and Tavazoei
, M. S.
, 2010, “Simple Fractional Order Model Structures and Their Applications in Control System Design
,” Eur. J. Control
, in press.20.
Hille
, E.
, 1973, Analytic Function Theory
, Chelsea
, New York
.21.
Lepschy
, A.
, and Viaro
, U.
, 1985, “Model Reduction for Control Systems With Restricted Complexity Controllers
,” J. Franklin Inst.
, 319
(6
), pp. 559
–567
.22.
Zhao
, C.
, Xue
, D.
, and Chen
, Y.
, 2005, “A Fractional Order PID Tuning Algorithm for a Class of Fractional Order Plants
,” IEEE International Conference on Mechatronics and Automation
, Niagara Falls, Canada, Vol. 1, pp. 216
–221
.23.
Valerio
, D.
, and da Costa
, J. S.
, 2006, “Tuning of Fractional PID Controllers With Ziegler-Nichols-Type Rules
,” Signal Process.
, 86
(10
), pp. 2771
–2784
.24.
Chen
, Y.
, Bhaskaran
, T.
, and Xue
, D.
, 2008, “Practical Tuning Rule Development for Fractional Order Proportional and Integral Controllers
,” J. Comput. Nonlinear Dyn.
, 3
(2
), p. 021403
.25.
Biswas
, A.
, Das
, S.
, Abraham
, A.
, and Dasgupta
, S.
, 2009, “Design of Fractional-Order PIλDμ Controllers With an Improved Differential Evolution
,” Eng. Applic. Artif. Intell.
, 22
(2
), pp. 343
–350
.26.
Petras
, I.
, Vinagre
, B. M.
, Dorcak
, V.
, and Feliu
, V.
, 2002, “Fractional Digital Control of a Heat Solid: Experimental Results
,” International Carpathian Control Conference (ICCC)
, Malenovice, Czech Republic.27.
Benchellal
, A.
, Poinot
, T.
, and Trigeassou
, J. C.
, 2006, “Approximation and Identification of Diffusive Interfaces by Fractional Models
,” Signal Process.
, 86
(10
), pp. 2712
–2727
.28.
Gabano
, J. D.
, and Poinot
, T.
, 2010, “Fractional Modeling and Identification of Thermal Systems
,” Signal Process.
, in press.29.
Sandvik
, A. B.
, 2010, General Turning
. Available online: http://www2.coromant. sandvik.com/coromant/pdf/Metalworking_Products_061/tech_a_1.pdf.30.
Rech
, J.
, Kusiakb
, A.
, and Battaglia
, J. L.
, 2004, “Tribological and Thermal Functions of Cutting Tool Coatings
,” Surf. Coat. Technol.
, 186
(3
), pp. 364
–371
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