Optimal variable-order fractional PID controllers for dynamical systems. (English) Zbl 1392.49033

Dabiri, A.; Moghaddam, B. P.; Machado, J. A. Tenreiro

J. Comput. Appl. Math. 339, 40-48 (2018).

Summary: This paper studies the design of variable-order fractional proportional-integral-derivative (VFPID) controllers for linear dynamical systems. For this purpose, a technique to discretize fractional differential equations with variable-order operators is proposed. The resulting model and a particle swarm optimization algorithm are used to search for the optimal parameters of the VFPID controllers. Three examples illustrate the performance of the closed-loop system under the action of the VFPID and compare it with the results obtained by means of the PID and fractional PID (FPID). Furthermore, time-dependent parameters are also tested, showing that the VFPID yields a superior performance.

MSC:

93B51	Design techniques (robust design, computer-aided design, etc.)
93C05	Linear systems in control theory
93B35	Sensitivity (robustness)
34A08	Fractional ordinary differential equations
26A33	Fractional derivatives and integrals
49J27	Existence theories for problems in abstract spaces
37M05	Simulation of dynamical systems
97N50	Interpolation and approximation (educational aspects)
34H05	Control problems involving ordinary differential equations

Keywords:

variable-order fractional calculus; variable-order proportional-integral-derivative (PID) controller; robust control; optimal control; particle swarm optimization

Software:

DFOC; sysdfod

References:

[1]	Wang, J.; Zhou, Y.; Wei, W.; Xu, H., Nonlocal problems for fractional integrodifferential equations via fractional operators and optimal controls, Comput. Math. Appl., 62, 3, 1427-1441, (2011) · Zbl 1228.45015
[2]	Debbouche, A.; Baleanu, D.; Agarwal, R. P., Nonlocal nonlinear integrodifferential equations of fractional orders, Bound. Value Probl., 2012, 1, 78, (2012) · Zbl 1277.35337
[3]	Lizama, C., Solutions of two-term time fractional order differential equations with nonlocal initial conditions, Electron. J. Qual. Theory Differ. Equ., 82, 1-9, (2012) · Zbl 1340.34027
[4]	Debbouche, A.; Nieto, J. J., Relaxation in controlled systems described by fractional integro-differential equations with nonlocal control conditions, Electron. J. Differential Equations, 2015, 89, 1-18, (2015) · Zbl 1311.93039
[5]	Wu, Z.-H.; Debbouche, A.; Guirao, J.; Yang, X.-J., On local fractional {\scv} olterra integral equations in fractal heat transfer, Therm. Sci., 20, suppl. 3, 795-800, (2016)
[6]	Lizama, C.; Velasco, M. P., Weighted bounded solutions for a class of nonlinear fractional equations, Fract. Calc. Appl. Anal., 19, 4, (2016) · Zbl 1346.34012
[7]	Yang, X. J.; Machado, J. T.; Nieto, J. J., A new family of the local fractional pdes, Fund. Inform., 151, 1-4, 63-75, (2017) · Zbl 1386.35461
[8]	Agarwal, R. P.; Ahmad, B.; Nieto, J. J., Fractional differential equations with nonlocal (parametric type) anti-periodic boundary conditions, Filomat, 31, 5, 1207-1214, (2017)
[9]	Zhou, Y.; Peng, L., Weak solutions of the time-fractional {\scn} avier-{\scs} tokes equations and optimal control, Comput. Math. Appl., 73, 6, 1016-1027, (2017)
[10]	Baleanu, D.; Jajarmi, A.; Hajipour, M., A new formulation of the fractional optimal control problems involving Mittag-Leffler nonsingular kernel, J. Optim. Theory Appl., 175, 3, 718-737, (2017) · Zbl 1383.49030
[11]	Jajarmi, A.; Baleanu, D., Optimal control of nonlinear dynamical systems based on a new parallel eigenvalue decomposition approach, Optim. Control Appl. Methods, (2018) · Zbl 1391.93113
[12]	Machado, J. A.T., Analysis and design of fractional-order digital control systems, SAMS J. Syst. Anal. Model. Simul., 27, 107-122, (1997) · Zbl 0875.93154
[13]	Podlubny, I., Fractional-order systems and \(\text{PI}^\lambda \text{D}^\mu\)-controllers, IEEE Trans. Automat. Control, 44, 1, 208-214, (1999) · Zbl 1056.93542
[14]	Podlubny, I., Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, vol. 198, (1998), Academic Press
[15]	Barbosa, R. S.; Machado, J. A.T. M.; Ferreira, I. M., Tuning of {\scpid} controllers based on {\scb} ode’s ideal transfer function, Nonlinear Dynam., 38, 1, 305-321, (2004) · Zbl 1134.93334
[16]	Silva, M. F.; Machado, J. A.T., Fractional order PD joint control of legged robots, J. Vib. Control, 12, 12, 1483-1501, (2006) · Zbl 1182.70019
[17]	O’Dwyer, A., Handbook of {\scpi} and {\scpid} controller tuning rules, (2006), Imperial College Press
[18]	(Mansour, T., PID Control, Implementation and Tuning, (2011), InTech)
[19]	J.G. Ziegler, N.B. Nichols, Optimum settings for automatic controllers, 1942.
[20]	Åström, K. J.; Hägglund, T., Advanced PID control, (2006), ISA-The Instrumentation, Systems and Automation Society
[21]	Hang, C. C.; Åström, K. J.; Ho, W. K., Refinements of the {\scz} iegler-{\scn} ichols tuning formula, (IEE Proceedings D (Control Theory and Applications), Vol. 138, (1991), IET), 111-118
[22]	K.J. Åström, T. Hägglund, PID controllers: theory, design, and tuning, Vol. 2, ISA Research Triangle Park, NC, 1995.
[23]	Zhuang, M.; Atherton, D., Automatic tuning of optimum {\scpid} controllers, (IEE Proceedings D (Control Theory and Applications), Vol. 140, (1993), IET), 216-224 · Zbl 0775.93024
[24]	O’Dwyer, A., Handbook of {\scpi} and {\scpid} controller tuning rules, (2009), World Scientific
[25]	Ho, W. K.; Hang, C. C.; Cao, L. S., Tuning of {\scpid} controllers based on gain and phase margin specifications, Automatica, 31, 3, 497-502, (1995) · Zbl 0825.93598
[26]	Shah, P.; Agashe, S., Review of fractional PID controller, Mechatronics, 38, 29-41, (2016)
[27]	Valério, D.; da Costa, J. S., Tuning of fractional PID controllers with {\scz} iegler-{\scn} ichols-type rules, Signal Process., 86, 10, 2771-2784, (2006) · Zbl 1172.94496
[28]	Petras, I., Fractional-order nonlinear systems: modeling, analysis and simulation, (2011), Springer Science & Business Media · Zbl 1228.34002
[29]	Sabatier, J.; Agrawal, O. P.; Machado, J. T., Advances in fractional calculus, vol. 4, (2007), Springer
[30]	Machado, J. A.T.; Galhano, A. M.; Oliveira, A. M.; Tar, J. K., Optimal approximation of fractional derivatives through discrete-time fractions using genetic algorithms, Commun. Nonlinear Sci. Numer. Simul., 15, 3, 482-490, (2010)
[31]	Machado, J. A.T., Calculation of fractional derivatives of noisy data with genetic algorithms, Nonlinear Dynam., 57, 1-2, 253-260, (2008) · Zbl 1176.94016
[32]	Zhao, C.; Xue, D.; Chen, Y., A fractional order PID tuning algorithm for a class of fractional order plants, (IEEE International Conference Mechatronics and Automation, (2005), IEEE)
[33]	Vinagre, B.; Podlubny, I.; Dorcak, L.; Feliu, V., On fractional PID controllers: A frequency domain approach, IFAC Proc. Vol., 33, 4, 51-56, (2000)
[34]	Machado, J. A.T., Optimal tuning of fractional controllers using genetic algorithms, Nonlinear Dynam., 62, 1-2, 447-452, (2010) · Zbl 1211.93070
[35]	Machado, J. A.T., Optimal controllers with complex order derivatives, J. Optim. Theory Appl., 156, 1, 2-12, (2012) · Zbl 1263.49031
[36]	Monje, C. A.; Vinagre, B. M.; Feliu, V.; Chen, Y., Tuning and auto-tuning of fractional order controllers for industry applications, Control Eng. Pract., 16, 7, 798-812, (2008)
[37]	Pires, E. J.S.; Machado, J. A.T.; de Moura Oliveira, P. B., Particle swarm optimization: dynamical analysis through fractional calculus, (Particle Swarm Optimization, (2009), InTech)
[38]	Pires, E. J.S.; Machado, J. A.T.; de Moura Oliveira, P. B., Fractional particle swarm optimization, (Mathematical Methods in Engineering, (2014), Springer), 47-56
[39]	Yang, X.-J.; Baleanu, D.; Srivastava, H., Local fractional similarity solution for the diffusion equation defined on {\scc} antor sets, Appl. Math. Lett., 47, 54-60, (2015)
[40]	Yang, X. J.; Machado, J. A.T.; Srivastava, H. M., A new numerical technique for solving the local fractional diffusion equation: two-dimensional extended differential transform approach, Appl. Math. Comput., 274, 143-151, (2016)
[41]	Sun, H. G.; Chen, W.; Chen, Y., Variable-order fractional differential operators in anomalous diffusion modeling, Physica A, 388, 21, 4586-4592, (2009)
[42]	Obembe, A. D.; Enamul Hossain, M.; Abu-Khamsin, S. A., Variable-order derivative time fractional diffusion model for heterogeneous porous media, J. Pet. Sci. Eng., 152, 391-405, (2017)
[43]	Samko, S. G.; Ross, B., Integration and differentiation to a variable fractional order, Integral Transforms Spec. Funct., 1, 4, 277-300, (1993) · Zbl 0820.26003
[44]	Ross, B.; Samko, S. G., Fractional integration operator of variable order in the {\sch}\(\ddot{\text{o}}\)lder spaces \(H \lambda(x)\), Int. J. Math. Math. Sci., 18, 4, 777-788, (1995)
[45]	Coimbra, C. F.M., Mechanics with variable-order differential operators, Ann. Phys., 12, 1112, 692-703, (2003) · Zbl 1103.26301
[46]	Yang, X. J.; Machado, J. A.T., A new fractional operator of variable order: application in the description of anomalous diffusion, Physica A, 481, 276-283, (2017)
[47]	Moghaddam, B. P.; Machado, J. A.T., Extended algorithms for approximating variable order fractional derivatives with applications, J. Sci. Comput., 1-24, (2016)
[48]	Moghaddam, B. P.; Machado, J. A.T., A computational approach for the solution of a class of variable-order fractional integro-differential equations with weakly singular kernels, Fract. Calc. Appl. Anal., 20, 4, 1023-1042, (2017) · Zbl 1376.65159
[49]	Moghaddam, B. P.; Machado, J. A.T., SM-algorithms for approximating the variable-order fractional derivative of high order, Fund. Inform., 151, 1-4, 293-311, (2017) · Zbl 1377.65031
[50]	Moghaddam, B.; Machado, J.; Behforooz, H., An integro quadratic spline approach for a class of variable-order fractional initial value problems, Chaos Solitons Fractals, 1-7, (2017) · Zbl 1422.65131
[51]	Moghaddam, B. P.; Yaghoobi, S.; Machado, J. A.T., An extended predictor-corrector algorithm for variable-order fractional delay differential equations, J. Comput. Nonlinear Dyn., 11, 6, 061001, (2016)
[52]	Jiang, W.; Liu, N., A numerical method for solving the time variable fractional order mobile-immobile advection-dispersion model, Appl. Numer. Math., 119, 18-32, (2017) · Zbl 1432.65155
[53]	Tavares, D.; Almeida, R.; Torres, D. F.M., Constrained fractional variational problems of variable order, IEEE/CAA J. Automat. Sinica, 4, 1, 80-88, (2017)
[54]	Tavares, D.; Almeida, R.; Torres, D. F., Caputo derivatives of fractional variable order: numerical approximations, Commun. Nonlinear Sci. Numer. Simul., 35, 69-87, (2016)
[55]	Almeida, R.; Pooseh, S.; Torres, D. F., Computational methods in the fractional calculus of variations, (2015), World Scientific Publishing Co Inc · Zbl 1322.49001
[56]	Sierociuk, D.; Macias, M., Comparison of variable fractional order PID controller for different types of variable order derivatives, (Proceedings of the 14th International Carpathian Control Conference, (ICCC), (2013), IEEE)
[57]	Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional integrals and derivatives: theory and applications, (1993), Gordon & Breach Sci. Publishers · Zbl 0818.26003
[58]	Sun, H.; Chen, W.; Wei, H.; Chen, Y., A comparative study of constant-order and variable-order fractional models in characterizing memory property of systems, Eur. Phys. J. Spec. Top., 193, 1, 185-192, (2011)
[59]	Ramirez, L. E.S.; Coimbra, C. F.M., On the selection and meaning of variable order operators for dynamic modeling, Int. J. Differ. Equ., 2010, 1-16, (2010) · Zbl 1207.34011
[60]	Gaing, Z. L., A particle swarm optimization approach for optimum design of PID controller in AVR system, IEEE Trans. Energy Convers., 19, 2, 384-391, (2004)
[61]	R. Hassan, B. Cohanim, O. De Weck, G. Venter, A comparison of particle swarm optimization and the genetic algorithm, in: 46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, 2005.
[62]	Chen, G. R.; Dong, X. N., On feedback control of chaotic continuous-time systems, IEEE Trans. Circuits Syst. I, 40, 9, 591-601, (1993) · Zbl 0800.93758