Stabilization of Fractional Order Unified Chaotic Systems via Linear State Feedback Controller
01 Jan 2012-pp 85-94
TL;DR: This chapter deals with an asymptotical stability of the fractional-order unified chaotic systems and a systematic linear state feedback controller is gained to stabilize these systems.
Abstract: This chapter deals with an asymptotical stability of the fractional-order unified chaotic systems. A unified system is presented to show that three chaotic dynamics of the Lorenz, the Chen, and the Lu systems are in the same structure. Dynamics of the Lorenz, the Chen, and the Lu are categorized in a unified system with a same structure. This system will be distinguished different when an only relevant parameter α is accordingly tuned. A systematic linear state feedback controller is gained to stabilize these systems. This controller will be shown increasing the stability region with respect to their integer order counterpart. The stability region will be accordingly increased with respect to their integer order alternative. Simulation results are demonstrated for the Chen and the Lu fractional-order systems to illustrate the effectiveness of the proposed control scheme.
Citations
More filters
27 Jun 2012
TL;DR: Two synthesis approaches are proposed to stabilize the fractional order chaotic Lu system based on the Lyapunov technique which guarantees global asymptotically stability of the origin.
Abstract: In this paper, two synthesis approaches are proposed to stabilize the fractional order chaotic Lu system. In the first approach, the equilibrium point of the system is stabilized locally while the second approach is based on the Lyapunov technique which guarantees global asymptotically stability of the origin. Simulation results demonstrate validity and effectiveness of the proposed approaches.
Cites background from "Stabilization of Fractional Order U..."
...Lemma 1:[24] Consider nonlinear fractional order systems as: oD t q x = f(x) + u (7) We can locally asymptotically stabilize (7) with the following control signal:...
[...]
References
More filters
TL;DR: In this paper, a linear dissipative mechanism whose Q is almost frequency independent over large frequency ranges has been investigated by introducing fractional derivatives in the stressstrain relation, and a rigorous proof of the formulae to be used in obtaining the analytic expression of Q is given.
Abstract: Summary Laboratory experiments and field observations indicate that the Q of many non-ferromagnetic inorganic solids is almost frequency independent in the range 10-2-107 cis, although no single substance has been investigated over the entire frequency spectrum. One of the purposes of this investigation is to find the analytic expression for a linear dissipative mechanism whose Q is almost frequency independent over large frequency ranges. This will be obtained by introducing fractional derivatives in the stressstrain relation. Since the aim of this research is also to contribute to elucidating the dissipating mechanism in the Earth free modes, we shall treat the dissipation in the free, purely torsional, modes of a shell. The dissipation in a plane wave will also be treated. The theory is checked with the new values determined for the Q of spheroidal free modes of the Earth in the range between 10 and 5 min integrated with the Q of Rayleigh waves in the range between 5 and 0.6 min. Another check of the theory is made with the experimental values of the Q of the longitudinal waves in an aluminium rod in the range between lo-’ and 10-3s. In both checks the theory represents the observed phenomena very satisfactorily. The time derivative which enters the stress-strain relation in both cases is of order 0.15. The present paper is a generalized version of another (Caputo 1966b) in which an elementary definition of some differential operators was used. In this paper we give also a rigorous proof of the formulae to be used in obtaining the analytic expression of Q; moreover, we present two checks of the theory with experimental data. The present paper is a generalized version of another (Caputo 1966b) in which an elementary definition of some differential operators was used. In this paper we give also a rigorous proof of the formulae to be used in obtaining the analytic expression of Q; moreover, we present two checks of the theory with experimental data. In a homogeneous isotropic elastic field the elastic properties of the substance are specified by a description of the strains and stresses in a limited portion of the field since the strains and stresses are linearly related by two parameters which describe the elastic properties of the field. If the elastic field is not homogeneous nor isotropic the properties of the field are specified in a similar manner by a larger number of parameters which also depend on the position.
3,372 citations
"Stabilization of Fractional Order U..." refers background in this paper
...3) as unified chaotic system due to chaotic behavior for any α ∈ [0,1]....
[...]
...where x,y,z are the state variables and α ∈ [0,1] is a “homogeneity” parameter of the system....
[...]
..., α ∈ [0,1]....
[...]
...Among several definitions of fractional derivatives, the following Caputo-type definition is more popular with respect the rest (Caputo, 1967)....
[...]
...Among several definitions of fractional derivatives, the following Caputo-type definition [1] is more popular with respect the rest [17]....
[...]
TL;DR: In this article, a fractional-order PI/sup/spl lambda/D/sup /spl mu/controller with fractionalorder integrator and fractional order differentiator is proposed.
Abstract: Dynamic systems of an arbitrary real order (fractional-order systems) are considered. The concept of a fractional-order PI/sup /spl lambda//D/sup /spl mu//-controller, involving fractional-order integrator and fractional-order differentiator, is proposed. The Laplace transform formula for a new function of the Mittag-Leffler-type made it possible to obtain explicit analytical expressions for the unit-step and unit-impulse response of a linear fractional-order system with fractional-order controller for both open- and closed-loops. An example demonstrating the use of the obtained formulas and the advantages of the proposed PI/sup /spl lambda//D/sup /spl mu//-controllers is given.
2,479 citations
"Stabilization of Fractional Order U..." refers background in this paper
...Furthermore, fractional order controllers such as CRONE (Oustaloup et. al., 1996), TID (Lurie, 1994), fractional PID controller (Podlubny, 1999) and lead-lag compensator (Raynaud and Zerga Inoh, 2000) have been implemented to improve the performance and robustness of some closd l op control systems....
[...]
Proceedings Article•
01 Jan 1996
TL;DR: In this article, stability results for finite-dimensional linear fractional differential systems in state-space form are given for both internal and external stability, and the main qualitative result is that stabilities are guaranteed iff the roots of some polynomial lie outside the closed angular sector.
Abstract: In this paper, stability results of main concern for control theory are given for finite-dimensional linear fractional differential systems. For fractional differential systems in state-space form, both internal and external stabilities are investigated. For fractional differential systems in polynomial representation, external stability is thoroughly examined. Our main qualitative result is that stabilities are guaranteed iff the roots of some polynomial lie outside the closed angular sector |arg(σ)| ≤ απ/2, thus generalizing in a stupendous way the well-known results for the integer case α = 1.
1,604 citations
TL;DR: A unified chaotic system that contains the Lorenz and the Chen systems as two dual systems at the two extremes of its parameter spectrum and is chaotic over the entire spectrum of the key system parameter.
Abstract: This paper introduces a unified chaotic system that contains the Lorenz and the Chen systems as two dual systems at the two extremes of its parameter spectrum. The new system represents the continued transition from the Lorenz to the Chen system and is chaotic over the entire spectrum of the key system parameter. Dynamical behaviors of the unified system are investigated in somewhat detail.
806 citations
Additional excerpts
...[9] considered a kind of chaotic system which describes a class of unified form by: ⎧ ⎪⎨ ⎪⎩ dx dt = (25α+ 10)(y− x) dy dt = (28− 35α)x− xz+(29α− 1)y dz dt = xy− 8+α 3 z (7....
[...]
TL;DR: In this paper, the analytic expression of a linear dissipative mechanism whose Q is almost frequency independent over large frequency ranges was obtained by introducing fractional derivatives in the stress strain relation.
Abstract: Laboratory experiments and field observations indicate that tlie Q of many non ferromagnetic inorganic solids is almost frequency independent in the range 10' to 10~2 cps; although no single substance has been investigated over the entire frequency spectrum. One of the purposes of this investigation is to find the analytic expression of a linear dissipative mechanism whose Q is almost frequency independent over large frequency ranges. This will be obtained by introducing fractional derivatives in the stress strain relation. Since the aim of this research is to also contribute to elucidating the dissipating mechanism in the earth free modes, we shall treat the cases of dissipation in the free purely torsional modes of a shell and the purely radial vibration of a solid sphere. The theory is checked with the new values determined for the Q of the spheroidal free modes of the earth in the range between 10 and 5 minutes integrated with the Q of the Railegh waves in the range between 5 and 0.6 minutes. Another check of the theory is made with the experimental values of the Q of the longitudinal waves in an alluminimi rod, in the range between 10-5 and 10-3 seconds. In both clicks the theory represents the observed phenomena very satisfactory.
515 citations
"Stabilization of Fractional Order U..." refers methods in this paper
...Chen (Chen and Lu, 2002) considered that the parameter of the two unified chaotic systems is unknown and an adaptive controller was used to achieve synchronization based on Lyapunov stability theory....
[...]