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Design of robust fractional PID controller using triangular strip operational matrices. (English) Zbl 1328.93083

Summary: The present article proposes a simple tuning technique aimed to produce a robust non-integer order PID controller exhibiting the iso-damping property during the reparameterization of a plant. The required robustness property is achieved by letting the fractional PID control system to imitate the dynamics of a reference system having Bode’s ideal transfer function in its forward path. The objective of designing robust controller by tracking the dynamics of reference control system is defined mathematically as an \(H_{\infty}\)-optimal control problem. Fractional differential systems are transformed into algebraic equations by the use of triangular strip operational matrices. The \(H_{\infty}\)-optimal control problem is then changed to an \(\infty\)-norm minimization of a parameter \((K_{c},K_{I},K_{d}, \lambda, \mu)\) varying square matrix. Global optimization techniques, Luus-Jaakola direct search and particle swarm optimization are employed to find the optimum values of fractional PID controller parameters. The proposed method of control system design is implemented in heating furnace temperature control, automatic voltage regulator system and some integer and fractional order process models. Fractional PI, fractional PD and various versions of fractional PID controllers are designed as optimal controllers using the triangular strip operational matrix based control design method.

MSC:

93B35 Sensitivity (robustness)
26A33 Fractional derivatives and integrals
93C15 Control/observation systems governed by ordinary differential equations
93B36 \(H^\infty\)-control
93B51 Design techniques (robust design, computer-aided design, etc.)
93B52 Feedback control
49N05 Linear optimal control problems
90C31 Sensitivity, stability, parametric optimization

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[1] R.L. Bagley, P.J. Torvik, A theoretical basis for the application of fractional calculus. A brief story about the operators of generalized fractional calculus. J. Rheol. 27 (1983), 201-210.; · Zbl 0515.76012
[2] H.W. Bode, Network Analysis and Feedback Amplifier Design. Van Nostrand, New York (1945).;
[3] M.K. Bouafoura, N.B. Braiek, PIλDμ controller design for integer and fractional plants using piecewise orthogonal functions. Commun. Nonlinear Sci. Numer. Simulat. 15 (2010), 1267-1278.; · Zbl 1221.93073
[4] J.Y. Cao, B.G. Cao, Design of fractional order controller based on particle swarm optimization. International Journal of Control, Automation and Systems 4 (2006), 775-781.;
[5] R. Caponetto, G. Maione, A. Pisano, M.R. Rapaic, E. Usai, Analysis and shaping of the self-sustained oscillations in relay controlled fractional-order systems. Fract. Calc. Appl. Anal. 16, No 1 (2013), 93-108; DOI: 10.2478/s13540-013-0007-x; http://www.degruyter.com/view/j/fca.2013.16.issue-1/ issue-files/fca.2013.16.issue-1.xml.; · Zbl 1312.93072
[6] V. Feliu-Batlle, R. Rivas-Perez, L. Sanchez-Rodriguez, M.A. Ruiz- Torija, Robust fractional-order PI controller implemented on a laboratory hydraulic canal. J. Hydraul. Eng. 135 (2009), 271-282.;
[7] R. Hilfer, Applications of Fractional Calculus in Physics. World Scientific, New Jersey (2000).; · Zbl 0998.26002
[8] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006).; · Zbl 1092.45003
[9] P. Lanusse, J. Sabatier, PLC implementation of a CRONE controller.; · Zbl 1273.93002
[10] Fract. Calc. Appl. Anal. 14, No 4 (2011), 505-522; DOI: 10.2478/s13540-011-0031-7; http://www.degruyter.com/view/j/fca.2011.14.issue-4/issue-files/ fca.2011.14.issue-4.xml.;
[11] M. Li, D. Li, J. Wang, C. Zhao, Active disturbance rejection control for fractional-order system. ISA Transactions 52 (2013), 365-374.;
[12] B. Liao, R. Luus, Comparison of the Luus-Jaakola optimization procedure and the genetic algorithm. Eng. Optimiz. 37 (2005), 381-396.;
[13] Y. Luo, Y.Q. Chen, Fractional-order proportional derivative controller for robust motion control: Tuning procedure and validation. In: Proc. of American Control Conference, Hyatt Regency Riverfront, St. Louis, MO, USA (2009).;
[14] Y. Luo, Y.Q. Chen, Stabilizing and robust fractional order PI controller synthesis for first order plus time delay systems. Automatica 48 (2012), 2159-2167.; · Zbl 1257.93039
[15] M. Manfred, Z. Evanghelos, Robust Process Control. Prentice Hall Englewood Cliffs, New Jersey (1989).; · Zbl 0728.93031
[16] F. Merrikh-Bayat, M. Karimi-Ghartemani, Method for designing PID stabilizers for minimum-phase fractional-order systems. IET Control Theory Appl. 4 (2010), 61-70.;
[17] C.A. Monje, A.J. Calderon, B.M. Vinagre, Y. Chen, V. Feliu, On fractional PIλ Controllers: Some tuning rules for robustness to plant uncertainties. Nonlinear Dynam. 38 (2004), 369-381.; · Zbl 1134.93338
[18] C.A. Monje, Y.Q. Chen, B.M. Vinagre, D. Xue, V. Feliu, Fractionalorder Systems and Controls: Fundamentals and Applications. Springer- Verlag, London Limited, London (2010).; · Zbl 1211.93002
[19] C.A. Monje, B.M. Vinagre, Y.Q. Chen, V. Feliu, P. Lanusse, J.;
[20] Sabatier, Proposals for fractional PID tuning. In: 1st IFAC Workshop on Fractional Derivatives and Applications, Bordeaux, France (2004).;
[21] K.B. Oldham, J. Spanier, Fractional Calculus:Theory and Applications, Differentiation and Integration to Arbitrary Order. Academic Press, New York-London (1974).; · Zbl 0292.26011
[22] A. Oustaloup, La Commade CRONE: Commade Robuste d’Ordre Non Entier. Hermes, Paris (1991).; · Zbl 0864.93003
[23] I. Petras, Tuning and implementation methods for fractional-order controllers.; · Zbl 1269.93039
[24] Fract. Calc. Appl. Anal. 15, No 2 (2012), 282-303; DOI: 10.2478/s13540-012-0021-4; http://www.degruyter.com/view/j/fca.2012.15.issue-2/ issue-files/fca.2012.15.issue-2.xml.;
[25] I. Podlubny, Fractional-order systems and PID controllers. IEEE T.; · Zbl 1056.93542
[26] Automat. Contr. 44 (1999), 208-214.;
[27] I. Podlubny, Fractional Differential Equations. Academic Press, San Diego (1999).; · Zbl 0924.34008
[28] I. Podlubny, Matrix approach to discrete fractional calculus. Fract.; · Zbl 1030.26011
[29] Calc. Appl. Anal. 3 (2000), 359-386.;
[30] J. Robinson, Y. Rahmat-Samii, Particle swarm optimization in electromagnetics.;
[31] IEEE T. Antenn. Propag. 52 (2004), 397-407.;
[32] M.A. Sahib, A novel optimal PID plus second order derivative controller for AVR system. Engineering Science and Technology, an International Journal 18 (2015), 194-206.;
[33] M. Tabatabaei, M. Haeri, Design of fractional order proportional integral derivative controller based on moment matching and characteristic ratio assignment method. Proc. of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering 225 (2011), 1040-1053.;
[34] M. Talebpour, Y.M. Roshan, S. Mohseni, Developing robust FOPID controllers based on fuzzy set point weighting algorithm. Fract. Calc.; · Zbl 1198.26031
[35] Appl. Anal. 12 (2009), 373-390; http://www.math.bas.bg/∼fcaa.;
[36] Y. Tang, M. Cui, C. Hua, L. Li, Y. Yang, Optimum design of fractional order PIλDμ controller for AVR system using chaotic ant swarm. Expert Systems with Applications 39 (2012), 6887-6896.;
[37] M. Zamani, M. Karimi-Ghartemani, N. Sadati, FOPID controller design for robust performance using particl swarm optimization. Fract.; · Zbl 1141.93351
[38] Calc. Appl. Anal. 10 (2013), 169-188; http://www.math.bas.bg/∼fcaa.;
[39] C. Zhao, D. Xue, Y.Q. Chen, A fractional order PID tuning algorithm for a class of fractional order plants. In: Proc. of the IEEE: International Conference on Mechatronics & Automation, Niagara Falls, Canada (2005).;
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