zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Robust synchronization of fractional-order unified chaotic systems via linear control. (English) Zbl 1238.93045
Summary: A new scheme for accomplishing synchronization between two fractional-order unified chaotic systems is proposed. The scheme does not require that the nonlinear dynamics of the synchronization error system must be eliminated. Moreover, the parameter of the systems does not have to be known. A controller is a linear feedback controller, which is simple in implementation. It is designed based on an LMI condition. The LMI condition guarantees that the synchronization between the slave system and the master system is achieved. Numerical simulations are performed to demonstrate the effectiveness of the proposed scheme.
MSC:
93C15Control systems governed by ODE
34A08Fractional differential equations
34H10Chaos control (ODE)
93A13Hierarchical systems
93B52Feedback control
37N35Dynamical systems in control
References:
[1]Podlubny, I.: Fractional differential equations, (1999)
[2]Cafagna, D.: Fractional calculus: a mathematical tool from the past for present engineers, IEEE industrial electronics magazine 1, 35-40 (2007)
[3]Heaviside, O.: Electromagnetic theory, (1971)
[4]Ichise, M.; Nagayanagi, Y.; Kojima, T.: An analog simulation of noninteger order transfer functions for analysis of electrode process, Journal of electroanalytical chemistry 33, 253-265 (1971)
[5]Sun, H. H.; Abdelwahad, A. A.; Onaral, B.: Linear approximation of transfer function with a pole of fractional order, IEEE transactions on automatic control 29, 441-444 (1984) · Zbl 0532.93025 · doi:10.1109/TAC.1984.1103551
[6]Koeller, R. C.: Application of fractional calculus to the theory of viscoelasticity, Journal of applied mechanics 51, 299-307 (1984) · Zbl 0544.73052 · doi:10.1115/1.3167616
[7]Ahmad, W. M.; Sprott, J. C.: Chaos in fractional-oder autonomous nonlinear systems, Chaos, solitons and fractals 16, 339-351 (2003) · Zbl 1033.37019 · doi:10.1016/S0960-0779(02)00438-1
[8]Gao, X.; Yu, J.: Chaos in the fractional order periodically forced complex Duffing’s oscillators, Chaos, solitons and fractals 26, 1125-1133 (2005)
[9]Ge, Z. -M.; Ou, C. -Y.: Chaos in a fractional order modified Duffing system, Chaos, solitons and fractals 34, 262-291 (2007) · Zbl 1132.37324 · doi:10.1016/j.chaos.2005.11.059
[10]Li, C. P.; Peng, G. J.: Chaos in Chen’s system with a fractional order, Chaos, solitons and fractals 20, 443-450 (2004) · Zbl 1060.37026 · doi:10.1016/j.chaos.2004.02.013
[11]Li, C.; Chen, G.: Chaos in the fractional order Chen system and its control, Chaos, solitons and fractals 22, 549-554 (2004) · Zbl 1069.37025 · doi:10.1016/j.chaos.2004.02.035
[12]Deng, W. H.; Li, C. P.: Chaos synchronization of the fractional Lü system, Physica A 353, 61-72 (2005)
[13]Deng, W.; Li, C.: The evolution of chaotic dynamics for fractional unified system, Physics letters A 372, 401-407 (2008) · Zbl 1217.37026 · doi:10.1016/j.physleta.2007.07.049
[14]Wu, X.; Li, J.; Chen, G.: Chaos in the fractional order unified system and its synchronization, Journal of the franklin institute 345, 392-401 (2008) · Zbl 1166.34030 · doi:10.1016/j.jfranklin.2007.11.003
[15]Daftardar-Gejji, V.; Bhalekar, S.: Chaos in fractional ordered Liu system, Computers and mathematics with applications 59, 1117-1127 (2010) · Zbl 1189.34081 · doi:10.1016/j.camwa.2009.07.003
[16]Wang, F.; Liu, C.: Synchronization of unified chaotic system based on passive control, Physica D 225, 55-60 (2008) · Zbl 1119.34332 · doi:10.1016/j.physd.2006.09.038
[17]Njah, A. N.; Vincent, U. E.: Synchronization and anti-synchronization of chaos in an extended bonhöffer–van der Pol oscillator using active control, Journal of sound and vibration 319, 41-49 (2009)
[18]Kuntanapreeda, S.: Chaos synchronization of unified chaotic systems via LMI, Physics letters A 373, 2837-2840 (2009) · Zbl 1233.93047 · doi:10.1016/j.physleta.2009.06.006
[19]Haeri, M.; Tavazoei, M. S.; Naseh, M. R.: Synchronization of uncertain chaotic systems using active sliding mode control, Chaos, solitons and fractals 33, 1230-1239 (2007) · Zbl 1138.93045 · doi:10.1016/j.chaos.2006.01.076
[20]Yu, Y.: Adaptive synchronization of a unified chaotic system, Chaos, solitons and fractals 36, 329-333 (2008) · Zbl 1141.93361 · doi:10.1016/j.chaos.2006.06.104
[21]Peng, C. -C.; Chen, C. -L.: Robust chaotic control of Lorenz system by backstepping design, Chaos, solitons and fractals 37, 598-608 (2008) · Zbl 1139.93317 · doi:10.1016/j.chaos.2006.09.057
[22]Wang, H.; Han, Z. -Z.; Xie, Q. -Y.; Zhang, W.: Finite-time chaos synchronization of unified chaotic system with uncertain parameters, Communications in nonlinear science and numerical simulation 14, 2239-2247 (2009)
[23]Lei, Y.; Yung, K. L.; Xu, Y.: Chaos synchronization and parameter estimation of single-degree-of-freedom oscillators via adaptive control, Journal of sound and vibration 329, 973-979 (2010)
[24]Sangpet, T.; Kuntanapreeda, S.: Adaptive synchronization of hyperchaotic systems via passivity feedback control with time-varying gains, Journal of sound and vibration 329, 2490-2496 (2010)
[25]Li, C. P.; Deng, W. H.; Xu, D.: Chaos synchronization of Chua system with a fractional order, Physica A 360, 171-185 (2006)
[26]Yan, J.; Li, C.: On chaos synchronization of fractional differential equations, Chaos, solitons and fractals 32, 725-735 (2007) · Zbl 1132.37308 · doi:10.1016/j.chaos.2005.11.062
[27]Bhalekar, S.; Daflardar-Gejji, V.: Synchronization of different fractional order chaotic systems using active control, Communications in nonlinear science and numerical simulation 15, 3536-3546 (2010) · Zbl 1222.94031 · doi:10.1016/j.cnsns.2009.12.016
[28]Odibat, Z.; Corson, N.; Aziz-Alaoui, M.; Bertelle, C.: Synchronization of chaotic fractional-order systems via linear control, International journal of bifurcation and chaos 20, 81-97 (2010) · Zbl 1183.34095 · doi:10.1142/S0218127410025429
[29]Asheghan, M. M.; Beheshti, M. T. H.; Tavazoei, M. S.: Robust synchronization of perturbed Chen’s fractional-order chaotic systems, Communications in nonlinear science and numerical simulation 16, 1044-1051 (2011) · Zbl 1221.34007 · doi:10.1016/j.cnsns.2010.05.024
[30]Xin, B.; Chen, T.; Liu, Y.: Projective synchronization of chaotic fractional-order energy resources demand-supply systems via linear control, Communications in nonlinear science and numerical simulation 16, 4479-4486 (2011) · Zbl 1222.93108 · doi:10.1016/j.cnsns.2011.01.021
[31]Taghvafard, H.; Erjaee, G. H.: Phase and anti-phase synchronization of fractional order chaotic systems via active control, Communications in nonlinear science and numerical simulation 16, 4079-4088 (2011) · Zbl 1221.65320 · doi:10.1016/j.cnsns.2011.02.015
[32]Cafagna, D.; Grassi, G.: Observer-based projective synchronization of fractional systems via a scalar signal: application to hyperchaotic Rössler systems, Nonlinear dynamics (2011)
[33]Zhang, R.; Yang, S.: Adaptive synchronization of fractional-order chaotic systems via a single driving variable, Nonlinear dynamics (2011)
[34]J.-Q. Xu, Adaptive synchronization of the fractional-order unified chaotic system with uncertain parameters, in: 30th Chinese Control Conference, CCC, 22–24 July 2011, pp. 2423–2428.
[35]Hosseinnia, S. H.; Ghaderi, R.; N., A. Ranjbar; Mahmoudian, M.; Momani, S.: Sliding mode synchronization of an uncertain fractional order chaotic system, Computers and mathematics with applications 59, 1637-1643 (2010) · Zbl 1189.34011 · doi:10.1016/j.camwa.2009.08.021
[36]Qi, D. -L.; Wang, Q.; Yang, J.: Comparison between two different sliding mode controllers for a fractional-order unified chaotic system, Chinese physics B 20, 100505-1-100505-9 (2011)
[37]Yin, C.; Zhong, S.; Chen, W.: Design of sliding mode controller for a class of fractional-order chaotic systems, Communications in nonlinear science and numerical simulation 17, 356-366 (2012)
[38]Monje, C. A.; Chen, Y. Q.; Vinagre, B. M.; Xue, D.; Feliu, V.: Fractional-order systems and controls: fundamentals and applications, (2010)
[39]Caponetto, R.; Dongola, G.; Fortuna, L.; Petráš, I.: Fractional order systems: modeling and control applications, (2010)
[40]Sabatier, J.; Moze, M.; Farges, C.: LMI stability conditions for fractional order systems, Computers and mathematics with applications 59, 1594-1609 (2010) · Zbl 1189.34020 · doi:10.1016/j.camwa.2009.08.003
[41]Lu, J. G.; Chen, Y. Q.: Robust stability and stabilization of fractional-order interval systems with the fractional order α: the 0α1 case, IEEE transactions on automatic control 55, 152-158 (2010)