IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 57, NO. 4, APRIL 2010
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Chaos Control in Duffing System Using Impulsive Parametric Perturbations Alexander Jimenez-Triana, Wallace Kit-Sang Tang, Senior Member, IEEE, Guanrong Chen, Fellow, IEEE, and Alain Gauthier
Abstract—In this brief, a new scheme is developed for chaos control by applying periodic impulsive parametric perturbations. Based on Melnikov’s condition for the existence of chaos, it is mathematically proven that, in a neighborhood of a homoclinic orbit of the Duffing system, chaos can be suppressed. A sufficient condition is also established, serving as the design criterion for the amplitude and the width of the impulsive control signal. Finally, the control effect will clearly be demonstrated with simulation results. Index Terms—Chaos control, impulsive control, Melnikov’s method, parametric perturbation.
I. I NTRODUCTION
S
INCE THE LAST decade, chaos control has aroused much interest, and its applications have widely been explored in different fields [1]–[3]. Parametric perturbation is one of the major chaos control schemes for smooth systems. Periodic changes have been applied to some system parameters to eliminate, reduce, or generate homoclinic chaos [4]. A different approach implies the application of an external excitation. Following this approach, in [5] and [6], chaos is suppressed by changing the shape of the excitation, whereas, in [7], chaos is induced in an electronic circuit by applying an external excitation with time-varying frequency and phase. In order to analyze systems with a homoclinic orbit being slightly periodically perturbed, Melnikov’s method [8] has commonly been adopted. On the other hand, there are many real-world nonsmooth systems, including mechanical systems with dry friction [9] or impacts [10], electronic circuits with switching devices [11], etc. Recently, Melnikov’s method has been extended for a certain class of nonsmooth planar systems with a discontinuous surface being perturbed periodically. Particularly, Melnikov’s function Manuscript received August 4, 2009; revised October 28, 2009. Current version published April 21, 2010. This work was supported in part by a grant from the Research Grants Council of Hong Kong Special Administrative Region, China, CityU 120708. This paper was recommended by Associate Editor J. Lu. A. Jimenez-Triana is with the Department of Control Engineering, Universidad Distrital Francisco José de Caldas, Bogotá, Colombia and also with the Department of Electrical and Electronic Engineering, Universidad de los Andes, Bogotá, Colombia (e-mail:
[email protected]). W. Kit-Sang Tang and G. Chen are with the Department of Electronic Engineering, City University of Hong Kong, Kowloon, Hong Kong (e-mail:
[email protected],
[email protected]). A. Gauthier is with the Department of Electrical and Electronic Engineering, Universidad de los Andes, Bogotá, Colombia (e-mail: agauthie@uniandes. edu.co). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCSII.2010.2043464
is calculated in [9] for a planar system with a homoclinic orbit, which intersects the discontinuous surface in two points with no sliding allowed. Under certain assumptions, a more general approach has been used for planar and nonsmooth systems with or without sliding on the discontinuous surface [12], [13]. For the high-dimensional case illustrated in [14], a similar problem can be handled by the assumption of the existence of a homoclinic orbit transversally crossing the discontinuity surface. In this brief, Melnikov’s method has been adopted to justify the suppression of homoclinic chaos by a train of pulses periodically perturbing a parameter of a chaotic system. It is an extension of a recent paper [15], in which the perturbation is in-phased with the forcing signal in the Duffing system. However, due to the formulation in [15], a closed formula of the control is hard to obtain. Therefore, another kind of impulsive perturbation is suggested here, so that a closed form of the control signal can be derived. Moreover, this new control signal still preserves the ease for a practical implementation. The organization of this brief is given as follows. In Section II, the chaos control scheme using parametric perturbation is explained in detail. Based on Melnikov’s method, a mathematical proof is given in the same section to justify the effectiveness of this scheme. Simulations are then carried out, and the results are presented in Section III. Finally, conclusion is given in Section IV. II. C HAOS C ONTROL W ITH D ISCONTINUOUS PARAMETRIC P ERTURBATIONS In this brief, it is proposed to design a chaos control scheme for the Duffing system in a neighborhood of its homoclinic orbit, based on discontinuous parametric perturbations. A. Duffing System Consider a damped and forced Duffing oscillator expressed as follows: u˙ = v v˙ = u − u3 + ε (γ cos(ωt) − δv)
(1)
where u and v are state variables, ε, γ, ω, and δ > 0 are some positive constants, and t denotes the time. ε is assumed to be small such that the last term in the second state equation of (1) can be considered as a perturbation.
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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 57, NO. 4, APRIL 2010
As indicated in [8], there exist two homoclinic orbits for the unperturbed system of (1) when ε = 0; they are √ √ 0 = 2 sinh(t), − 2 sech(t) tanh(t) q+ 0 q−
=−
0 q+ .
(2)
0 is given in (2), and (∧) denotes the standard wedge where q+ product. Since (f ∧ g) = (f1 g2 − f2 g1 ), (9) can be expressed as
+∞ M (t0 ) = −∞
B. Chaos Control With Discontinuous Perturbations Our main result is enunciated in the following proposition: Proposition 1: Consider the Duffing’s system (1) in its chaotic regime, and apply a parametric perturbation on γ. One has
(3)
where F (t) is defined as a pulsing function with period T = 2π/ω, which is expressed as F (t) =
∞
F (τ ) ≈ FN (τ )
h(t) =
κ, 0,
h(t − nT )
−Δ + π2 ≤ t < Δ + π2 , Δ 1 otherwise.
It can be proven that chaos is suppressed if 4δ √ − γπω sech πω 2 3 2 κ< − ωΨ(Δ) Δω sech πω 2
for some integer N . Proof: Let ωt = τ ; (3) can be rewritten as ε x˙ = f (x) + g(x, τ ) ω
(5)
(6)
(7)
(8)
u x= v
v f1 ω f (x) = u−u ≡ 3 f2 ω
0 g1 g(x) = ≡ . (γ + F (τ )) cos(τ ) − δv g2
Consider a Melnikov’s function defined as follows: M (t0 ) =
+∞ 0 0 f q+ (t − t0 ) ∧ g q+ (t − t0 ), f dt −∞
4δ (μ(t0 ) − 1) 3
(12)
where
N πω πω (n − 1) sech (1 − n) Ψ(Δ) = sin(nΔ) 2 n=1 n 2
πω (n+1) − sin(nΔ) sec (1+ n) n 2
(11)
by truncating up to N terms. (Note that it can be proven that limN →∞ FN (τ ) = F (τ ).) Following the computation given in Appendix, one has
(4)
with
where
π κΔ 2κ + sin(nΔ) cos n τ − π nπ 2 n=1 N
=
M (t0 ) =
n=0
with
(10)
where FN (t) is the Fourier series approximation of F (t), i.e.,
u˙ = v v˙ = u − u3 + ε [(γ + F (t)) cos(ωt) − δv]
v 0 (τ −t0 ) (γ +FN (τ )) cos(τ )−δv 0 (τ −t0 ) dτ ω
(9)
√ πω 3 2 μ(t0 ) = sin(t0 ) + Δκωsech γπω sech 4δ 2 N πω × sin(t0 ) − κω ψn (t0 ) 2 n=1 ψn (t0 ) =
(13)
πω sin(nΔ) (n − 1) sech (1 − n) n 2 sin(nΔ) nπ − nt0 − (n + 1) × sin t0 + 2 n πω nπ (1 + n) sin t0 − + nt0 . (14) × sech 2 2
Referring to [15], the necessary condition for homoclinic chaos is to have sign change on M (t0 ) at some t0 . This also provides a design criterion for chaos suppression, i.e., if no sign change is found in M (t0 ) ∀ t0 , no chaos can be evolved. As given in (12), the sign of M (t0 ) keeps unaltered if μ(t0 ) < 1 for all t0 .
(15)
Since Δ is small, the maximum for μ(t0 ) can be found in t0 = π/2 with a good approximation. To prove it, it is noticed that (sin(nΔ)/n) ≈ Δ when nΔ is small, and the summation in (13) forms a telescopic series. After manipulating (13), one can simplify it to obtain √ πω 3 2 γπω sech sin(t0 ) + ΔκωN sech μ(t0 ) = 4δ 2 πω π N sin N t0 − (N − 1) × 2 2 πω (N + 1) sin + Δκω(N + 1) sech 2 π . (16) × (N + 1)t0 − N 2
JIMENEZ-TRIANA et al.: CHAOS CONTROL IN DUFFING SYSTEM USING PARAMETRIC PERTURBATION
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Since N Δ is small and sech(.) diminishes when N increases, the last two terms in this expression can be disregarded, and the maximum of (16) is found in (π/2). To achieve (15), the following relationship is to be established: πω πω π 3√2 = γπω sech + Δκω sech μ(t0 ) ≤ μ 2 4δ 2 2 − κωΨ(Δ) < 1 for all t0 (17) where N πω (n − 1) sech (1 − n) Ψ(Δ) = sin(nΔ) n 2 n=1
− sin(nΔ)
πω (n + 1) sech (1 + n) . (18) n 2
Similarly, (18) forms a telescopic series, and the denominator of the fraction in (6), after some manipulations, becomes
πω πω N − ωΨ(Δ) = Δω N Δω sech sech 2 N −1 2
πω N +1 (N + 1) > 0 (19) + Δω sech N 2
Fig. 1. Sufficient condition to suppress chaos in the Duffing system (3) by the parametric perturbation (4) and (5), where the parameters are γ = 0.27, ω = 1, and δ = 0.25.
which is positive. Referring to (17), we finally get
− γπω sech πω 2 . κ< − ωΨ(Δ) Δω sech πω 2 4δ √ 3 2
(20)
Hence, if (6) is satisfied, no homoclinical chaos exists, and the proof is completed. Remark 1: It can also be proven that κ < 0. To do so, we will check the sign of the numerator in (20). According to [8], the uncontrolled system (1) is chaotic if √ πω δ 3 2πω < sech (21) γ 4 2 and therefore πω 4δ √ − γπω sech
