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CLC number: TP311

On-line Access: 2018-04-08

Received: 2016-09-11

Revision Accepted: 2017-01-04

Crosschecked: 2018-02-15

Cited: 0

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Citations:  Bibtex RefMan EndNote GB/T7714


Samir Ladaci


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Frontiers of Information Technology & Electronic Engineering  2018 Vol.19 No.2 P.180-191


Bifurcation-based fractional-order PIλDμ controller design approach for nonlinear chaotic systems

Author(s):  Karima Rabah, Samir Ladaci, Mohamed Lashab

Affiliation(s):  Signal Processing Laboratory, Department of Electronics, University of Mentouri, Constantine 25000, Algeria; more

Corresponding email(s):   samir_ladaci@yahoo.fr

Key Words:  Fractional order system, Bifurcation diagram, Fractional PIλDμ, Multi-scroll Chen chaotic system, Genesio-Tesi chaotic system

Karima Rabah, Samir Ladaci, Mohamed Lashab. Bifurcation-based fractional-order PIλDμ controller design approach for nonlinear chaotic systems[J]. Frontiers of Information Technology & Electronic Engineering, 2018, 19(2): 180-191.

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%A Mohamed Lashab
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T1 - Bifurcation-based fractional-order PIλDμ controller design approach for nonlinear chaotic systems
A1 - Karima Rabah
A1 - Samir Ladaci
A1 - Mohamed Lashab
J0 - Frontiers of Information Technology & Electronic Engineering
VL - 19
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SP - 180
EP - 191
%@ 2095-9184
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PB - Zhejiang University Press & Springer
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DOI - 10.1631/FITEE.1601543

We propose a novel approach called the robust fractional-order proportional-integral-derivative (FOPID) controller, to stabilize a perturbed nonlinear chaotic system on one of its unstable fixed points. The stability analysis of the nonlinear chaotic system is made based on the proportional-integral-derivative actions using the bifurcation diagram. We extract an initial set of controller parameters, which are subsequently optimized using a quadratic criterion. The integral and derivative fractional orders are also identified by this quadratic criterion. By applying numerical simulations on two nonlinear systems, namely the multi-scroll Chen system and the Genesio-Tesi system, we show that the fractional PIλDμ controller provides the best closed-loop system performance in stabilizing the unstable fixed points, even in the presence of random perturbation.




Darkslateblue:Affiliate; Royal Blue:Author; Turquoise:Article


[1]Berns D, Moiola J, Chen G, 1998. Feedback control of limit cycle amplitude from a frequency domain approach. Automatica, 34(12):1567-1573.

[2]Bouafoura M, Braiek N, 2010. PIλDμ controller design for integer and fractional plants using piecewise orthogonal functions. Commun Nonlinear Sci Numer Simulat, 15:1267-1278.

[3]Boukabou A, Chebbah A, Belmahboul A, 2007. Stabilizing unstable periodic orbits of the multi-scroll Chua&x2019;s attractor. Nonlinear Anal Model Contr, 12(4):469-477.

[4]Charef A, Assabaa M, Ladaci S, et al., 2013. Fractional order adaptive controller for stabilized systems via high-gain feedback. IET Contr Theory Appl, 7(6):822-828.

[5]Chen D, Zhao W, Sprott J, et al., 2013. Application of Takagi-Sugeno fuzzy model to a class of chaotic synchronization and anti-synchronization. Nonlinear Dyn, 73(3):1495-1505.

[6]Chen D, Zhang R, Liu X, et al., 2014. Fractional order Lyapunov stability theorem and its applications in synchronization of complex dynamical networks. Commun Nonlinear Sci Numer Simulat, 19:4105-4121.

[7]Chen D, Liu Y, Ma X, et al., 2012. Control of a class of fractional-order chaotic systems via sliding mode. Nonlinear Dyn, 67:893-901.

[8]Chen L, Zhao T, Li W, et al., 2016. Bifurcation control of bounded noise excited Duffing oscillator by a weakly fractional-order pIλDμ feedback controller. Nonlinear Dyn, 83(1):529-539.

[9]Chen Z, Yuan X, Ji B, et al., 2014. Design of a fractional order PID controller for hydraulic turbine regulating system using chaotic non-dominated sorting genetic algorithm II. Energy Convers Manage, 84:390-404.

[10]Colonius F, Grne L, 2002. Dynamics, Bifurcations, and Control.

[11]Dadras S, Momeni H, 2009. Control uncertain Genesio-Tesi chaotic system: adaptive sliding mode approach. Chaos, Solitons & Fractals, 42:3140-3146.

[12]Delavari H, Ghaderi R, Ranjbar N, et al., 2010. Adaptive fractional PID controller for robot manipulator. 4th Int Workshop on Fractional Differentiation and its Applications, Badajoz, Spain, p.1-7.

[13]Diethlem K, 2003. Efficient solution of multi-term fractional differential equations using P(EC)mE methods. Computing, 71:305-319.

[14]Ditto W, 1996. Applications of chaos in biology and medicine. Chaos Chang Nat Sci Med, 376:175-202.

[15]Faieghi MR, Naderi M, Jalali AA, 2011. Design of fractional-order PID for ship roll motion control using chaos embedded PSO algorithm. 2nd Int Conf on Control, Instrumentation, and Automation, p.606-610.

[16]Faieghi M, Delavari H, 2012. Chaos in fractional-order Genesio-Tesi system and its synchronization. Commun Nonlinear Sci Numer Simulat, 17:731-741.

[17]Fayazi A, Rafsanjani H, 2011. Synchronization in the Genesio-Tesi and Coullet system using a fractional-order adaptive controller. 9th IEEE Int Conf on Control and Automation, p.889-894.

[18]Genesio R, Tesi A, 1992. A harmonic balance methods for the analysis of chaotic dynamics in nonlinear systems. Automatica, 28(3):531-548.

[19]Ghamati M, Balochian S, 2015. Design of adaptive sliding mode control for synchronization Genesio-Tesi chaotic system. Chaos, Solitons & Fractals, 75:111-117.

[20]Gholipour R, Khosravi A, Mojallali H, 2012. Intelligent back stepping control for Genesio-Tesi chaotic system using a chaotic particle swarm optimization algorithm. Int J Comput Electr Eng, 4(5):618-625.

[21]Hadef S, Boukabou A, 2014. Control of multi-scroll Chen system. J the Franklin Institute, 351:2728-2741.

[22]Harb A, Abdel-Jabbar N, 2003. Controlling Hopf bifurcation and chaos in a small power system. Chaos Solitons & Fractals, 18:1055-1063.

[23]Hosseinia S, Ghadri R, Ranjbar A, et al., 2010. Control of chaos via fractional order state feedback controller. New Trend in Nanotechnology and Fractional Calculus Applications, p.511-519.

[24]Ladaci S, Bensafia Y, 2016. Indirect fractional order pole assignment based adaptive control. Int J Eng Sci Technol, 19:518-530.

[25]Ladaci S, Charef A, 2006a. An adaptive fractional PIλDμ controller. 6th Int Symp on Tools and Methods of Competitive Engineering, Ljubljana, Slovenia, Avril 18-22, p.1533-1540.

[26]Ladaci S, Charef A, 2006b. On fractional adaptive control. Nonlinear Dyn, 43(4):365-378.

[27]Lamba P, Hudson J, 1987. Experiments on bifurcations to chaos in a forced chemical reactor. Chem Eng Sci, 42:1-8.

[28]Liu X, Shen X, Zhang H, 2012. Multi-scroll chatic and hyperchaotic attractors generated from Chen system. Int J Bifurcation Chaos, 22(2):1250033.

[29]Liu Y, Yang Q, 2010. Dynamics of a new Lorenz-like chaotic system. Nonlinear Anal Real World Appl, 11(4):2563-2572.

[30]Liu Y, 2012. Circuit implementation and finite-time synchronization of the 4D Rabinovich hyperchaotic system. Nonlinear Dyn, 67:89-96.

[31]Ma J, Wang C, Tang J, et al., 2009. Suppression of the spiral wave and turbulence in the excitability-modulated media. Int J Theor Phys, 48:150-157.

[32]Machado J, Galhano A, 2009. Approximating fractional derivatives in the perspective of system control. Nonlinear Dyn, 56:401-407.

[33]Machado J, 1997. Analysis and design of fractional-order digital control systems. SAMS, 27(2-3):107-122.

[34]Miller K, Ross B, 1993. An Introduction to the Fractional Calculus and Fractional Differential Equations.

[35]Necaibia A, Ladaci S, 2014. Self-tuning fractional order PIλDμ controller based on extremum seeking approach. Int J Autom Contr, 8(2):99-121.

[36]Oustaloup A, 1991. La commande CRONE (in French).

[37]Pan I, Das S, 2012. Chaotic multi-objective optimization based design of fractional order PIkDl controller in AVR system. Electr Power Energy Syst, 43:393-407.

[38]Park J, Kwon O, Lee S, 2008. LMI optimization approach to stabilization of Genesio-Tesi chaotic system via dynamic controller. Applied Math Comput, 196:200-206.

[39]Park J, 2007. Adaptive controller design for modified projective synchronization of Genesio-Tesi chaotic system with uncertain parameters. Chaos, Solitons & Fractals, 34(4):1154-1159.

[40]Podlubny I, 1999a. Fractional Differential Equations.

[41]Podlubny I, 1999b. Fractional-order systems and PIλDμ-controllers. IEEE Trans Autom Contr, 44(1):208-214.

[42]Rabah K, Ladaci S, Lashab M, 2015a. Stabilization of fractional Chen chaotic system by linear feedback control. 3rd IEEE Int Conf on Control, Engineering & Information Technology, Tlemcen, Algeria, p.1-5.

[43]Rabah K, Ladaci S, Lashab M, 2015b. State feedback with fractional PIλDμ control structure for Genesio-Tesi chaos stabilization. 16th IEEE Int Conf on Sciences and Techniques of Automatic Control & Computer Engineering, Monastir, Tunisia, p.328-333.

[44]Rabah K, Ladaci S, Lashab M, 2016. Stabilization of a Genesio-Tesi chaotic system using a fractional order PIλDμ regulator. Int J Sci Tech Autom Contr & Comput Eng, 10(1):2085-2090.

[45]Tang Y, Cui M, Hua C, et al., 2012. Optimum design of fractional order PIkDl controller for AVR system using chaotic ant swarm. Expert Systems with Applications, 39:6887-6896.

[46]Tavazoei M, Haeri M, 2008. Stabilization of unstable fixed points of chaotic fractional order systems by a state fractional PI controller. European J Contr, 3:247-257.

[47]Wang G, 2010. Stabilization and synchronization of Genesio-Tesi system via single variable feedback controller. Phys Lett A, 374:2831-2834.

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