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Dynamics and chaos control in nonlinear electrostatic transducers
Chaos, Solitons and Fractals 21 (2004) 1093–1108 www.elsevier.com/locate/chaos
Dynamics and chaos control in nonlinear electrostatic transducers F.M. Moukam Kakmeni a
a,*
, S. Bowong b, C. Tchawoua a, E. Kaptouom
c
Laboratoire de Mecanique, Departement de Physique, Faculte des sciences, Universite de Yaounde I, B.P. 812 Yaounde, Cameroon b INRIA Lorraine, Projet Conge & University of Metz, I.S.G.M.P., B^at A, 57045 Metz Cedex 01, France c Departement des Genies Industriel et Mecanique, Ecole Nationale Superieure Polytechnique Universite de Yaounde I, B.P. 8390 Yaounde, Cameroon Accepted 4 December 2003
Abstract In this paper, we analyze the dynamics of a system consisting of two coupled nonlinearly Duffing oscillators, obtained from a nonlinear electrostatic device which is a prototype of emitters and receivers in communication engineering. Inverse or backward period doubling cascades and sudden transition to chaos are observed. A sliding mode controller is applied to control the electrostatic transducers system. The sliding surface used is one dimension higher than the traditional surface and guarantees its passage through the initial states of the controlled system. By means of the design of sliding mode dynamics characteristics, the controlled system performance is arbitrarily determined by assigning the switching gain of the sliding mode dynamics. Therefore, using the characteristic of this sliding mode we aim to design a controller that can meet the desired specification and use less control energy by comparing with the result in the current literature. The results show that the proposed controller can steer electrostatic transducers to the desired reference trajectory without chattering phenomenon and abrupt state change. Ó 2004 Elsevier Ltd. All rights reserved.
1. Introduction Many investigations have considered various models of dynamical systems in the study of mechanical vibrations [1– 10]. Among these, good models candidate of nonlinearly coupled oscillators have particular interest due to their application in various physical systems such as electromechanical, chemical and biological systems. The linear superposition principle which is valid for linear differential equations is no longer valid for those systems. A physical consequence is that when the given system admits an oscillatory motion, the frequency of the oscillation is in general amplitude-dependent, particularly, this can have drastic consequence in the case of forced and damped nonlinear oscillators leading to nonlinear resonances and jump (hysteresis) phenomenon for low strengths of nonlinearity parameters. Such behaviors can be analyzed using various approximation methods [2–5]. Very recently, using the method of multiple scale, Rajasekar et al. [4] obtain a set of autonomous equations for the amplitudes and phases of a two coupled Duffing–van der Pol oscillators with a nonlinear coupling. Also, the stability boundaries of fixed points of the approximated equations are obtained using the Routh–Hurwitz criterion and the stability boundaries are drawn in various parameters space. The influence of the nonlinear coupling strength on the response dynamics have been studied.
* Corresponding author. Permanent address: Universite de Yaounde I, B.P. 8329 Yaounde, Cameroon. Tel: +237-976-22-29; fax: +237-231-95-84. E-mail addresses: [email protected] (F.M. Moukam Kakmeni), [email protected] (S. Bowong), [email protected] (C. Tchawoua).
0960-0779/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2003.12.087
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On the other hand, Chaos is an interesting phenomenon of nonlinear systems. A deterministic chaotic system has some remarkable characteristics, such as system evolution sensitive to the change of in initial condition, broad spectrum of Fourier transform, and fractal properties of the motion in phase space [6]. Due to its powerful applications in various field of sciences including communications, controlling these complex chaotic dynamics for engineering applications has emerged as a new and attractive field and has developed many profound theories and methodologies to date [9–29]. The chaos suppression problem is generally considered as the stabilization of the chaotic system in a periodic orbit or equilibrium point. In this way the phenomena of chaos control, which, along with other approaches have been widely addressed in the literature as well as his broad range of potential applications that goes from life sciences and earth science to the problems facing physicists, engineers and applied mathematicians. Although, the robust tracking control problem for nonlinear chaotic systems, due to its difficulty and its practical interest, is one of the most challenging of the control theory, it has never been addressed for the electrostatic transducers system. In this paper, we consider a nonlinear electrostatic transducers model described by the following set of two nonlinearly coupled differential equations: (
€q þ k0q q_ þ x2q q þ c0q q3 a0q x b0q xq ¼ 0 €x þ k0x x_ þ x2x x þ c0x x3 a0x q b0x q2 ¼ F cos xt
ð1Þ
where the dot denotes time derivative, q and x are the coordinates of both oscillators, k0q and k0x are the viscous damping coefficients, xq and xx are the natural frequencies and, c0q and c0x the nonlinear coefficients. a0q , b0q , a0x and b0x characterized the coupling between the oscillators. The model (1) can be found in a electrostatic microphone and loudspeaker which consist of a mechanical part (described here by x) connected capacitively to an R–L–C circuit whose capacitance can be controlled by acoustic waves (for instance harmonic acoustic waves). The interest on such devices is justified by the fact that with the revival of the electret old idea, electrostatic microphones are widely used for various types of technological applications such as monoscope tubes for TV set signals, cassette recorders devices and evidently telephone devices [3,9,10]. Two points are considered: (i) analyze the oscillatory states in our model and study through numerical simulation some bifurcation structures and transition to chaos; and (ii) carry out the problem of how to design a feedback controller to drive the chaotic trajectories of system (1) to an inherent periodic orbit of the system. Previously, [3,9] investigated the dynamic and the problem of chaotic behaviour of the model without the first Duffing term c0q q3 . After, we completed the study by designing an estimated state feedback controller to drive a chaotic trajectory to an inherent periodic orbit of the system [13]. Recently, we go further with the work to understand how coupling typical nonlinear systems respond to two external periodic forces and exhibits different dynamical resonance and dynamical transition involving chaos. Throughout this note, we extend the work to nonlinearly coupled Duffing equation. The cubic term c0q q3 in the first equation of (1) is a new term introduced to described the nonlinearity of Duffing type in the condenser. This would result from an eventual high charge of the capacitor. The content of the paper is arranged as follows: in Section 2, an analysis is presented for perturbated solutions and the hysteresis phenomena. In Section 3, we report a numerical study, the periodic and chaotic behaviors of the coupled model will be presented and the rote the system follow to the chaotic state. Section 4 is devoted to the control of chaos using an estimated state feedback controller based on sliding mode control. Simulation results are presented to validate the proposed control scheme. We end our investigation in Section 5 with a brief conclusion.
2. Perturbated solutions An approximate analytical solution of Eq. (1) is obtained by assuming that the coefficients k0i , c0i , a0i , b0i , i ¼ q, x and F are small. This smallness can be characterized using a single coefficient e (where e is a small parameter, i.e., e 1) as a scaling factor. Thus we set k0i ¼ eki , c0i ¼ ci , a0i ¼ eai , b0i ¼ ebi and F ¼ eF0 . In general we consider qðtÞ and xðtÞ in the form qðtÞ ¼ q0 ðT0 ; T1 Þ þ q1 ðT0 ; T1 Þ þ xðtÞ ¼ x0 ðT0 ; T1 Þ þ x1 ðT0 ; T1 Þ þ
ð2Þ
where T0 ¼ t is a fast scale and T1 ¼ et is a slow scale, characterizing the modulation in the amplitude and phase caused by the nonlinearity, damping and resonances. Substituting Eq. (2) into Eq. (1) and equating coefficients of the same power of e, we obtain
F.M. Moukam Kakmeni et al. / Chaos, Solitons and Fractals 21 (2004) 1093–1108
D20 q0 þ x2q q0 ¼ 0 D20 x0 þ x2x x0 ¼ 0 D20 q1 þ x2q q1 ¼ aq x0 þ bq x0 q0 2D0 D1 q0 kq D0 q0 cq q30 D20 x1 þ x2x x1 ¼ ax q0 þ bx q20 2D0 D1 x0 kx D0 x0 cx x30 þ F0 cos xT0
1095
ð3Þ
ð4Þ
where Dn ¼ oTon and Tn ¼ en t with n ¼ 0; 1; 2; 3; . . . It is convenient to express the solution of Eq. (3) in the complex form q0 ¼ AðT1 Þeixq T0 þ cc x0 ¼ BðT1 Þeixx T0 þ cc
ð5Þ
where Ôcc’ stands for the complex conjugate term and i2 ¼ 1. AðT1 Þ and BðT1 Þ are arbitrary complex functions of T1 at this level of approximation. It is determined by imposing solvability conditions at next levels of approximation. Substituting Eq. (5) into Eq. (4), it comes that D20 q1 þ x2q q1 ¼ ðikq xq A 2ixq A0 3cq A2 AÞeixq T0 þ aq Beixx T0 þ bq ½ABeiðxq þxx ÞT0 þ ABeiðxx xq ÞT0 cq A3 e3ixq T0 þ cc F0 D20 x1 þ x2x x1 ¼ ðikx xx B 2ixx B0 3cx B2 BÞeixx T0 þ ax Aeixq T0 þ bx A2 e2ixq T0 þ bx AA cx B3 e3ixx T0 þ eixT0 þ cc 2 ð6Þ where A (respectively, B) is the complex conjugate of A (respectively B) and (0 ) denotes the time derivative with respect T1 . Depending on the order of the nonlinearity in the system, various secular terms can be produce with the commensurate relationships between frequencies. These commensurable relationships of frequencies can cause the corresponding mode to be strongly (respectively, weakly) coupled and internal (respectively, external) resonance is said to exist. Accordingly, in our dynamical system, three situations can be analyzed. In Sections 2.1 and 2.2 we consider the case that the electric oscillator enters in resonance with the external excitation and we study in this condition the internal nonresonant and resonant cases. 2.1. External resonance Here we analyze the case where xx is different of xq and 2xq . To express quantitatively the nearness of x to xx , we introduce the detuning parameter r defined by x ¼ xx þ er. In this case, none of the nonlinear coupling terms produces a secular term. The condition for the elimination of the secular term in Eq. (6) become ikq xq A 2ixq A0 3cq A2 A ¼ 0 F0 ikx xx B 2ixx B0 3cx B2 B þ eirT1 ¼ 0 2
ð7Þ
A first-order approximation can be obtained by expressing A and B in real form using the polar representation 1 A ¼ aðT1 Þeiu1 ðT1 Þ 2 1 B ¼ bðT1 Þeiu2 ðT1 Þ 2
ð8Þ
where aðT1 Þ, bðT1 Þ, u1 ðT1 Þ and u2 ðT1 Þ are the amplitudes and phases of the fundamental frequency. Substituting Eq. (8) into Eq. (7), we thus obtain after separating real and imaginary parts the following set of coupled first-order differential equations for the amplitude and phase 3 2 c a a xq au01 ¼ 0 8 q kq xq a þ xq a0 ¼ 0 2
ð9Þ
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3 2 F0 cq b b xx bu02 cos h ¼ 0 2 8 kx F0 0 xx b þ xx b sin h ¼ 0 2 2
ð10Þ
where h ¼ rT1 u2 . In its general form, Eqs. (9) and (10) show that in the nonresonant case, both oscillators are uncoupled. Thus when there is no internal resonance, the first approximation is not influenced by the nonlinear coupling terms. The time evolution of the first oscillator corresponds to the amplitude of the autonomous Duffing oscillator and the second for the forced Duffing oscillator. The steady-state response must have a0 ¼ b0 ¼ 0 and h0 ¼ 0. Thus a ¼ 0 and u02 ¼ r. Eliminating h from Eq. (10), we obtain the frequency–response equation for amplitude of the second oscillator 9 2 6 c b ð3cx xx rÞb4 þ x2x ð4r2 þ k2x Þb2 F02 ¼ 0 16 x
ð11Þ
Solving this equation using Newton–Raphson algorithm, the response curves in term of the detuning parameter r and the external force F0 present the classical multivalued amplitudes and hence to a jump phenomenon as observe earlier [2,3]. 2.2. The internal resonance When there is an internal resonance, the nonlinear coupling terms produce secular terms and the solutions can differ drastically from those of no internal resonance. 2.2.1. First case of internal resonance To analyze the case of internal resonance, we introduce a new detuning parameter r1 according to xx ¼ 2xq er1 ;
x ¼ xx þ er:
ð12Þ
Substituting the above equation into Eq. (6), we obtain the new solvability conditions ikq xq A 2ixq A0 3cq A2 A þ bq ABeier1 T0 2ixx B0 ikx xx B 3cx B2 B þ bx A2 eier1 T0 þ
F0 ierT0 e ¼0 2
ð13Þ
Again, substituting Eq. (8) into Eq. (13) after some algebraic manipulations, we obtain the following new set of coupled and nonlinear first-order differential equations: 3 1 xq au01 cq a3 þ bq ab cos d0 ¼ 0 8 4 ð14Þ 1 1 kq xq a xq a0 þ bq ab sin d0 ¼ 0 2 4 and 3 1 F0 xx bu02 cx b3 þ bx a2 cos d0 þ sin d ¼ 0 2 8 4 1 1 F0 0 2 kx xx b xx b þ bx a sin d0 þ sin d ¼ 0 2 2 4 with d0 ¼ r1 T1 þ 2u1 u2 and d ¼ rT1 u2 . For thus, we analyze the steady-state solution (i.e., u02 ¼ r and u01 ¼ 12 ðr r1 Þ), Eqs. (14) and (15) become 1 3 2 1 a xq ðr r1 Þ cq a þ bq b cos d0 ¼ 0 2 8 4 1 1 a kq xq þ bq b sin d0 ¼ 0 2 4
ð15Þ
ð16Þ
F.M. Moukam Kakmeni et al. / Chaos, Solitons and Fractals 21 (2004) 1093–1108
3 1 F0 xx br cx b3 þ bx a2 cos d0 þ cos d ¼ 0 2 8 4 1 1 F0 2 kx xx b þ bx a sin d0 þ sin d ¼ 0 2 4 2
1097
ð17Þ
We find two possibilities. In the first, we suppose a ¼ 0 (see Eq. (16)), then Eq. (17) leads to the resonant equation (11) and it is essentially the solution of the linear problem. In the second, we suppose a 6¼ 0, eliminating d0 from Eq. (16) leads to " #1=2 2 2 3 2 1 2 2 c a xq ðr r1 Þ þ kq xq 4 ð18Þ b¼ bq 8 q 2 This expression is not realistic since b becomes infinite when bq is zero. 2.3. Second case of internal resonance The second case of internal resonance is concerned with xq ¼ xx þ er0 ;
x ¼ xx þ er
ð19Þ
Substituting in to Eq. (6), we obtain the new solvability conditions ikq xq A 2ixq A0 3cq A2 A þ aq Beir0 T1 ¼ 0 F0 2ixx B0 ikx xx B 3cx B2 B þ ax Aeir1 T1 þ eirT1 ¼ 0 2
ð20Þ
Substituting Eq. (8) into Eq. (20), after some algebraic manipulations, we obtain the following new set of coupled and nonlinear first-order differential equation: 3 1 xq au01 cq a3 þ aq b cos d1 ¼ 0 8 2 1 1 0 kq xq a xq a þ aq b sin d1 ¼ 0 2 2
ð21Þ
3 1 F0 xx bu02 cx b3 þ ax a cos d1 þ cos d ¼ 0 8 2 2 1 1 F0 0 kx xx b xx b xx b þ ax a sin d1 þ sin d ¼ 0 2 2 2
ð22Þ
with d1 ¼ r0 T1 þ 2u1 u2 and d ¼ rT1 u2 . For the steady-state solution (i.e., and u02 and u01 ¼ 12 ðr r0 ÞÞ, Eqs. (21) and (22) lead to the following amplitude equations:
3 4 c b xx rb2 8 x
2 2 ax 2 3 2 1 ax F2 cq a xq ðr r0 Þ a þ kx xx b2 kq xq a2 ¼ b2 0 8 4 aq aq 4
ð23Þ
where 4a2 b ¼ aq 2
"
3 2 c a xq ðr r0 Þ 8 q
2 þ
1 kq xq 2
2 # ð24Þ
These equations involve the coupling parameters ai thus the steady-state motions are coupled. In Fig. 1(a), a typical frequency–response graph is shown. It is notice that the curve presents two resonances points and the hysteresis phenomenon does not appear. As the amplitude of the external force increases the hysteresis phenomenon appears and for some values of r it is possible to have up seven values of the amplitude as presented in Fig. 1(b). For a given value of r, one can evaluate the effect of the term cq on the coupling response curve. We see from Fig. 2(a) and (b) that for r ¼ 0, the curve is a single branch with multivalue of amplitude for low values of cq . This can be seen in the bifurcation diagrams since small values of cq bring chaotic state in the system. For r 6¼ 0 this curve degenerates into tree branches. This behavior of the coupling-response curve shows that when the oscillators move
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Fig. 1. Frequency–response curve in the case of internal and external resonances a(r). The parameters are kq ¼ kx ¼ 0:1, cq ¼ 1, cx ¼ 0, aq ¼ ax ¼ 0:1, bq ¼ bq ¼ 0:2, xq ¼ xx ¼ x ¼ 1: (a) F ¼ 0:1 and (b) F ¼ 0:8.
Fig. 2. Amplitude response curve as a function of cq with the parameters of Fig. 1.
away from the exact external resonance point ðr ¼ 0Þ, their amplitude may be multivalued depending on the new cubic term of the system.
3. Chaotic behavior of the model In this section, we process numerically. The aim of the numerical study is to find the sensibility and some set of parameters which can lead to chaotic behavior. Among methods use for the identification of the type of motion we carried our study using the bifurcation diagrams. They provide a nice summary for the transition between different types of motion that can occur as one parameter of the system is varied. Figs. 3–6 present a set of sample bifurcation diagrams of our model with respect to F , x, and c0q . Other parameters are fixed as: k0q ¼ 0:1, a0q ¼ 0:4, c0q ¼ 0:1, b0q ¼ 0:4, xq ¼ 0:4, k0x ¼ 0:3, c0x ¼ 0:6, xx ¼ 0:1 and x ¼ 1. The plots represent the amplitude x of the second oscillator in a Poincare cross-section when one parameter is varied the other being fixed. In the bifurcation diagram of Fig. 3 where F is varied from 0 to 1.5, a discontinuous ‘‘jump’’ from a stable period-1 solution to a band of chaotic motion is evident for approximate F 0:55. For 0:55 < F < 0:75, we observe that the
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Fig. 3. Bifurcation diagrams of the electrostatic transducer model showing the transition order–chaos-order as F varied.
Fig. 4. A section of Fig. 3 showing detailed bifurcation and chaotic behaviors of the parameter interval between F ¼ 0:73 and F ¼ 0:83.
dynamic is almost chaotic with a very short window of period-1 orbit through intermittency. For F > 0:75, one can see that there exists reversed period-doubling bifurcations that bring back the dynamic to a period-1 state (see Fig. 4). As the value of F 0:85, is reaches, the system suddenly becomes chaotic and the inverse period doubling cascade starts at F 1:08 (Fig. 5) leading now to a period-1 orbit. Quasiperiodicity, mode-locking and inverse period doubling cascade are exemplified in Fig. 5 when x is varied from 0 to 0.6. As concerned the bifurcation diagrams of the nonlinearly coupled Duffing oscillator with respect to the new term c0q in the interval 0 < c0q < 0:2, the results obtained, shown in Fig. 6 present an overall behavior similar to that described in the preceding paragraph and depicted in Fig. 3. Here, the inverse period doubling cascade closely starting at c0q 0:102. A typical Poincare map corresponding to chaotic motion is presented in Fig. 7.
4. Chaos control Several kinds of interesting nonlinear dynamic behaviors of the system have been studied in previous sections. They have shown that the forced system exhibited both regular and chaotic motion. Usually, chaos is unwanted or undesirable.
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Fig. 5. Bifurcation diagrams of the electrostatic transducer for F ¼ 1:07 when x varied.
Fig. 6. Bifurcation diagrams of the electrostatic transducer system when F ¼ 1:07, x ¼ 1 and c0q varied.
In order to improve the performance of a dynamic system or avoid the chaotic phenomena, we need to control chaotic system to a regular target trajectory embedded in the chaotic attractor which is beneficial for working with a particular condition [11,13,14,17,18]. However, the practical chaotic systems may contain many types of uncertainties [12]. These uncertainties may cause chaotic perturbations to originally regular behavior, or induce additional chaos in originally chaotic but known behavior, generating unknown chaotic motion (Fig. 8). In this case, it would be desirable to have a feedback scheme to achieve control in spite of the system’s uncertainties. The aim of this section is to study the tracking control problem of the electrostatic transducer system with uncertainties based on the sliding mode design. A sliding mode control is a nonlinear control strategy requiring: (i) a switching manifold that prescribes the desired dynamics and (ii) a control law such that the system trajectory first reaches the manifold and then stays on it forever [17]. The proposed scheme comprises a sliding controller and an uncertainty estimator. The design algorithm for the control via estimation of the lumping uncertainties is the following: first, the uncertain terms are lumped in a nonlinear function in such a way that the lumping nonlinear function is interpreted as a variable state into an extended system. Then, a state estimator is designed to carry out the dynamic reconstruction of the extended variable state, and consequently, an estimated value of the uncertainties is obtained. Finally, a sliding controller is provided with the uncertainty estimate value. In this way, the control law counteracts the uncertain terms and the desired dynamics can be induced.
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Fig. 7. Typical Poincare maps of the electrostatic transducer system for chaotic behavior, i.e., F ¼ 1:07, x ¼ 1 and c0q ¼ 0:04.
Fig. 8. Typical phase diagrams of the electrostatic transducer system for chaotic behavior, i.e., F ¼ 1:07, x ¼ 1 and c0q ¼ 0:04.
Let us consider the electrostatic transducer system with an external control input u as 8 > > > x_ 1 ¼ x2 < x_ 2 ¼ k0q x2 x2q x1 c0q x31 þ a0q x3 þ b0q x1 x3 x_ 3 ¼ x4 > > > : x_ 4 ¼ k0x x4 x2 x3 c0q x3 þ a0x x1 þ b0x x2 þ F cos xt þ u x 3 1
ð25Þ
_ x3 ¼ x, x4 ¼ x_ and u the physical control input needed to be chosen. The control input u is added to where x1 ¼ q, x2 ¼ q, order or to guide the chaotic dynamics to meet our specific requirements. Without loss of the generality, we can assume that the system’s output is y ¼ x1 . Here, the control problem is to find a robust feedback controller such that the output y ¼ x1 tracks asymptotically the desired reference trajectory yd ¼ x1d as t > T for any initial time t0 P 0 and initial conditions ðx1 ð0Þ, x2 ð0Þ, x3 ð0Þ, x4 ð0ÞÞ 2 R4 in spite of modelling errors, parametric variations, perturbing external forces, noisy measurements, and nonmodeled actuator dynamics. 4.1. System standardization Suppose that we want to drive the system’s output to the periodic reference
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x1d ðtÞ ¼ e0 sin x0 t
ð26Þ
where e0 is a certain constant. In this way, the desired reference trajectory meeting Eq. (25) is given by 8 x1d ðtÞ ¼ e0 sin x0 t > > > > x2d ðtÞ ¼ e0 x0 cos x0 > > > < e ðx2 x2 Þ sin x0 tþk0q e0 x0 cos x0 tþc0q e30 sin3 x0 t x3d ðtÞ ¼ 0 q 0 a0q þe0 b0q sin x0 t > > e0 x0 ½x2q x20 b0q x3d cos x0 tk0q e0 x20 sin x0 tþ3c0q e30 sin2 x0 t cos x0 t > > x4d ¼ > > a0q þe0 b0q sin x0 t > : ur ¼ nðtÞ þ k0x x4d þ x2x x3d þ c0x x31d b0x x21d F cos xt
ð27Þ
with nðtÞ ¼
e0 x20 ðx2q x20 b0q x3d Þ sin x0 t k0q e0 x30 cos x0 t þ 6c0q e30 x20 sin2 x0 t cos2 x0 t 3c0q e30 x20 sin3 x0 t a0q þ e0 b0q sin x0 t a0q þ e0 b0q sin x0 t
It should be mentioned that the constant e0 is chosen in such a way that a0q þ e0 sin x0 t 6¼ 0 for all t P 0. Let the error states and the change of control law of the system be e1 ¼ x1 x1d
e2 ¼ x2 x2d
e3 ¼ x3 x3d
e4 ¼ x4 x4d
and
v ¼ u ud
Then the error state dynamic equations is 8 e_ 1 ¼ e2 > > < e_ 2 ¼ k0q e2 x2q e1 c0q ðe21 þ 3e1 x1d þ 3x21r Þe1 þ a0q e3 þ b0q ½e3 ðe1 þ x1d Þ þ e1 x3d > e_ 3 ¼ e4 > : e_ 4 ¼ k0x e4 x2x e3 c0x ðe23 þ 3e3 x3d þ 3x23r Þe3 þ a0x e1 þ b0x ðe1 þ 2x1d Þe1 þ v
ð28Þ
ð29Þ
To determine the sliding mode control law, the reformulation of the state space equation (29) into an extended controllable canonical form is required. To this end, the change of coordinates: 8 z1 ¼ e 1 > > > > z2 ¼ e 2 > < z3 ¼ k0q e2 x2q e1 c0q ðe21 þ 3e1 x1d þ 3x21r Þe1 þ a0q e3 þ b0q ½e3 ðe1 þ x1d Þ þ e1 x3d ð30Þ > > 2 > z ¼ k z x e 3c ðe þ x Þðe þ 1Þ þ a e þ b ½e ðe þ x Þ þ ðe þ x Þe > 4 0q 3 1 1 1d 1 0q 4 4 1 1d 3 3d 2 0q 0q q > : ¼ f ðe1 ; e2 ; e3 Þ þ ½a0q þ b0q ðe1 þ x1d Þe4 is applied, where for simplicity we define the function f to be everything in the definition of z4 above except the last term, which is ½a0q þ b0q ðe1 þ x1d Þe4 . Note that e4 appears only in this last term so that f depends only on e1 , e2 and e3 . The above mapping is actually a global diffeomorphism. Its inverse is likewise found by inspection to be 8 e1 ¼ z1 > > >e ¼ z > < 2 2 z3 þk0q z2 þx2q z1 þc0q ðz21 þ3z1 x1d þ3x21r Þz1 b0q z1 x3d ð31Þ e ¼ 3 a0q þb0q ðz1 þx1d Þ > > > 2 z þ3c ðz þx Þðz þ1Þb ðe þx Þz z þk z þx > 4 0q 3 2 1 1d 1 3 3d 2 0q 0q q : e4 ¼ a0q þb0q ðz1 þx1d Þ Thus, the dynamics of Eq. (29) can be transferred to a controllable canonical form as z_ i ¼ ziþ1 ; i ¼ 1; 2; 3 z_ 4 ¼ fðz; xd Þ þ mðz1 ; x1d Þv where fðz; xd Þ ¼
of of of e2 þ z3 þ e4 þ ½a0q þ b0q ðe1 þ x1d Þ½k0x e4 x2x e3 c0x ðe23 þ 3e3 x3d þ 3x23r Þe3 oe1 oe2 oe3 þ a0x e1 þ b0x ðe1 þ 2x1d Þe1
mðz1 ; x1d Þ ¼ ½a0q þ b0q ðz1 þ x1d Þ
ð32Þ
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1103
Now, let us assume the following: Assumption 1. Only x1 is available for feedback by online measurements. Assumption 2. The nonlinear terms fðz; xd Þ and mðz1 ; x1d Þ are uncertain. Assumption 3. mðz1 ; x1d Þ is bounded. However, an estimate ^mðz1 ; x1d Þ of mðz; xd Þ satisfying signð^mðz1 ; x1d ÞÞ ¼ signðmðz1 ; x1d ÞÞ: Some comments concerning the above assumptions should be noted: Assumption 1 is realistic because in most cases only one state is available for feedback. Assumption 2 is a practical situation because it is practically impossible to avoid tolerances in the real-world physical parameters. For example, the tolerance of the various electric elements in the nonlinear electrostatic transducer system will certainly lead to small but finite difference in the physical parameters and thus the operations of the device. Concerning Assumption 3, the knowledge of mðz1 ; x1d Þ is a restriction because in some cases the variable z1 change with the parameters variations. Now, let us define /ðz; xd Þ ¼ mðz; xd Þ ^mðz1 ; x1d Þ; g¼
4 X
Hðz; xd ; vÞ ¼ fðz; xd Þ þ /ðz; xd Þv;
z5 ¼ Hðz; xd ; vÞ þ ^mðz1 ; x1d Þv;
and ð33Þ
zkþ1 ok Hðz; xd ; vÞ þ /ðz; xd Þ_v þ z2 o1^mðz1 ; x1d Þ
k¼1
Thus, the system (32) can be rewritten as follows: 8 < z_ i ¼ ziþ1 ; i ¼ 1; . . . ; 4 z_ ¼ g þ ^mðz1 ; x1d Þ_v : 5 g_ ¼ Nðz; z5 ; g; xd ; v; v_ ; €v; tÞ
ð34Þ
where Nðz; z5 ; g; xd ; v; v_ ; €v; tÞ ¼
4 X
z24 ok z5
þ
k¼1
3 X
" zkþ2 ok z5 þ ðg þ ^mðz1 ; z1d ÞÞo4 z5 þ v_ /ðz; xd Þ þ ^mðz1 ; x1d Þ þ
k¼1
4 X
# ok /ðz; xd Þ
k¼1
þ /ðz; xd Þ€v with ok z5 ¼ oz5 =ozk and ok /ðz; xd Þ ¼ o/ðz; xd Þ, k ¼ 1; 2; 3; 4. The following must be pointed out. 1. System (34) is dynamically externally equivalent to system (32). This is for all differentiable input u 2 R. In fact, the manifold Wðz; z5 ; g; xd ; v; v_ ; tÞ ¼ g Hðz; xd ; vÞ is invariant under the trajectories of system (34). It suffices to prove that dWðz; z5 ; g; xd ; v; v_ ; tÞ=dt ¼ 0 along the trajectories of system (34). Then from the equality Wðz; z5 ; g; xd ; v; v_ ; tÞ ¼ 0 and the condition dWðz; z5 ; g; xd ; v; v_ ; tÞ=dt ¼ 0, one can take the first integral of the system (34) to get g ¼ Hðz; xd ; vÞ. When the first integral is backsubstituted in system (34), we obtain the solution of system (32). This implies that the solution zðtÞ 2 R4 of system (32) is the solution of the upper subsystem (34), hence, p:ðz; z5 ; gÞ ¼ z. Thus, the augmented state g provides the dynamics of the uncertain function Hðz; xd ; vÞ and, consequently, of the uncertain terms fðz; xd Þ and mðz1 ; x1d Þ. 2. System (34) is in cascade form. This means that when we taken actions to achieve lim z ! 0, the part Nðz; z5 ; g; xd ; v; v_ ; €v; tÞ ! Nð0; 0; g; xd ; v; v_ ; €v; tÞ ! 0 asymptotically for the so-called cascade character [21]. 4.2. Sliding mode controller design According to sliding mode control design, the sliding surface is defined as Z t X 5 S ¼ z5 z5 ð0Þ þ hð6jÞ kj zj ¼ 0 0
ð35Þ
j¼1
where z5 ð0Þ is the initial state of z5 . The sliding surface used in this paper is one dimension higher than the traditional sliding surface which guarantees that it passes through the initial states of the system being controlled. Equation (35) can also be formulated as
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F.M. Moukam Kakmeni et al. / Chaos, Solitons and Fractals 21 (2004) 1093–1108
z_ 5 ¼
5 X
hð6jÞ kj zj
ð36Þ
j¼1
Therefore the sliding mode dynamics (the desired dynamics) can be described as 8 ; i ¼ 1; . . . ; 4 < z_ i ¼ ziþ1P z_ 5 ¼ 5j¼1 hð6jÞ kj zj : g_ ¼ Nðz; z5 ; g; xd ; v; v_ ; €v; tÞ Or in a matrix equation form as z_ ¼ hK1 h MKh z g_ ¼ Nðz; z5 ; g; xd ; v; v_ ; €v; tÞ where z ¼ ðz1 ; z2 ; z3 ; z4 ; z5 ÞT , 2 0 1 0 0 6 0 0 1 0 6 6 0 0 0 1 6 4 0 0 0 0 k1 k2 k3 k4
3 0 0 7 7 0 7 7 1 5 k5
ð37Þ
ð38Þ
2
and
h 0 6 0 h2 6 Kh ¼ 6 60 0 40 0 0 0
0 0 h3 0 0
0 0 0 h4 0
3 0 07 7 07 7 05 5 h
with K1 h the inverse matrix of Kh . The design of kj , j ¼ 1; . . . ; 5 can be determined by choosing the eigenvalues of MðkÞ such that the corresponding characteristic polynomial equation Pk ðsÞ ¼ s5 þ k5 s4 þ k4 s3 þ k3 s2 þ k2 s þ k1 is Hurwitz. The reaching law can be chosen as S_ ¼ bS h sgnðSÞ
ð39Þ
where 0 6 b < 1, sgnð:Þ denotes the signum function and h > 0 can be considered as the switching gain to be determined such that the sliding condition is satisfied and sliding mode motion will occur. Combine Eqs. (35) and (39), we obtain S_ ¼ bS h sgnðSÞ ¼ g þ ^mðz1 ; x1d Þ_v þ
5 X
hð6jÞ kj zj
ð40Þ
j¼1
and if the initial condition vð0Þ ¼ 0, then the differential equation of control signal v can be obtained as " # 5 X 1 bS h sgnðSÞ g hð6jÞ kj zj v_ ¼ ^mðz1 ; x1d Þ j¼1 Therefore, the implemented control input can be described as !# Z t" 5 X 1 ð6jÞ bS h sgnðSÞ g h kj zj dt vðtÞ ¼ ^mðz1 ; x1d Þ 0 j¼1
ð41Þ
ð42Þ
with vð0Þ ¼ 0. Substituting the control law (41) into the extended system (34), the dynamics of closed-loop system can be described as 8 4 < z_ i ¼ ziþ1 ; i ¼ 1; . . . ; P z_ 5 ¼ bS h sgnðSÞ 5j¼1 h6j kj zj ð43Þ : g_ ¼ Nðz; z5 ; g; xd ; v; v_ ; €v; tÞ Here the large h is important for the realization of control, which is associated with the system information of the chaotic system. We can qualitatively analyze this question with Lyapunov stability theory as follows. Let the Lyapunov function of the system be 1 V ¼ S2 2
ð44Þ
F.M. Moukam Kakmeni et al. / Chaos, Solitons and Fractals 21 (2004) 1093–1108
The first derivative of V with respect to time is obtained as ! 5 X ð6jÞ _V ¼ S S_ ¼ S z_ 5 þ h kj zj ¼ S½bS h sgnðSÞ ¼ bS 2 h absðSÞ 6 absðSÞ½b absðSÞ h
1105
ð45Þ
j¼1
From Eq. (35), we known that S ¼ hðzÞ. From the boundness of chaotic attractor, we known that S is bounded. So large enough h will lead to V_ 6 0. Here we get the same conclusion as that in [19]. In many situations, condition V_ 6 0 can be satisfied by choosing a large enough switching gain h. The convergence of gðtÞ to zero follows from the fact that the closed-loop system (43) is in cascade form. If system uncertainties can be estimated more accurately, then the resulting control input will be more accurate. Nevertheless, the terms z5 and g are not available for feedback from on-line measurements (Assumption 2). Then the sliding feedback (42) is not physically realizable. An alternative is to use estimates of z5 , g and zi , i ¼ 1; . . . ; 4 in such a way that the main characteristics of the feedback can be retained. So we will use an observer to estimate z5 , g and zi , i ¼ 1; . . . ; 4. To this end, the following uncertainty estimator is proposed 8 < ^z_ i ¼ ^ziþ1 hi ci ð^z1 z1 Þ; i ¼ 1; . . . ; 4 ð46Þ ^z_ ¼ ^g þ ^mðz1 ; x1d Þ_v h5 c5 ð^z1 z1 Þ : _5 g^ ¼ h6 c6 ð^z1 z1 Þ where cj , j ¼ 1; . . . ; 6 are chosen such that the polynomial s6 þ c1 s5 þ c2 s4 þ c3 s3 þ c4 s2 þ c5 s þ c6 ¼ 0 is Hurwitz. Define the estimating error as: ~zi ¼ hð6iÞ ð^z1 z1 Þ, i ¼ 1; . . . ; 5 and ~z6 ¼ ^ g g. Then, we get the following system ~z_ ¼ hNðcÞ~z þ Xðz; z5 ; g; xd ; v; v_ ; €v; tÞ
ð47Þ
T
T
where ~z ¼ ð~z1 ; ~z2 ; ~z3 ; ~z4 ; ~z5 ; ~z6 Þ , Xðz; z5 ; g; xd ; v; v_ ; tÞ ¼ ½0; . . . ; 0; Nðz; z5 ; g; xd ; v; v_ ; €v; tÞ and N ðcÞ is as follows 2
c1 6 c2 6 6 c3 NðcÞ ¼ 6 6 c4 6 4 c5 c6
1 0 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
3 0 07 7 07 7 07 7 15 0
Since the trajectories qðtÞ and xðtÞ belong to some chaotic attractor, then Hðz; xd ; vÞ in Eq. (34) is a bounded function. Hence Nðz; z5 ; xd ; g; v; v_ ; €v; tÞ also is bounded function. Consequently, for a large value of the switching surface gain h, ~z ! 0 which implies that ^zj ! zj , j ¼ 1; . . . ; 4, ^z5 ! z5 and ^ g ! g. Then Eqs. (38) and (45) become S ¼ ^z5 ^z5 ð0Þ þ
Z
t 0
vðtÞ ¼
Z t" 0
5 X
hð5jÞ kj^zj ¼ 0
ð48Þ
j¼1
5 X 1 bS h sgnðSÞ ^g hð5jÞ kj^zj ^mð^z; x1d Þ j¼1
!# dt
ð49Þ
with vð0Þ ¼ 0. Note that since Hðz; xd ; vÞ is uncertain, the function Nðz; z5 ; xd ; g; v; v_ ; €v; tÞ correspondingly is unknown. Thus, such a term has been neglected in the construction of the observer (46). Note also that the sliding control law (49) only uses estimated values of the uncertain term Hðz; xd ; vÞ (by means ^z5 and g) and ^z. Thus, the robust asymptotic stability is given by the dynamic compensator (46) and the sliding control law (49). Switching high-gain can induce undesirable dynamics effects such as peaking phenomenon [24]. To diminish these effects, the control law can be modified by means of (Z " !# ) 5 t X 1 ðnjþ1Þ u ¼ Sat bS h sgnðSÞ ^g ð50Þ h Kj^zj dt ^mð^z; x1d Þ 0 j¼1 where Sat{.}: R5 ! S R5 , S is a bounded set [25].
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4.3. Numerical results In order to validate the performance of the proposed control law, we have performed simulations. The parameters of uncontrolled electrostatic transducer system are k0q ¼ 0:1, c0q ¼ 0:04, a0q ¼ 0:4, b0q ¼ 0:4, xq ¼ 0:4, k0x ¼ 0:3, c0x ¼ 0:6, xx ¼ 0:1, F ¼ 1:07 and x ¼ 1. The chaotic nature of the electrostatic transducer system is observed for the chosen initial conditions ðx1 ð0Þ; x2 ð0Þ; x3 ð0Þ; x4 ð0ÞÞ ¼ ð0; 0; 0; 0Þ in Fig. 6. The initial conditions for the system (29) is ½e1 ð0Þ; e2 ð0Þ; e3 ð0Þ; e4 ð0Þ ¼ ½0:1; 0:2; 0 and for the observer (46), ½^z1 ð0Þ; ^z2 ð0Þ; ^z3 ð0Þ; ^z4 ð0Þ; ^z5 ð0Þ; ^ gð0Þ ¼ ½0:1; 0; 0; 7:2. Let b ¼ 0:01 and h ¼ 20 and the eigenvalues corresponding to the sliding surface are )1 of which the coefficients of Hurwitz polynomial are ½k1 ; k2 ; k3 ; k4 ; k5 ¼ ½1; 5; 10; 10; 5; 1. The estimation constants was chosen as ½c1 ; c2 ; c3 ; c4 ; c5 ; c6 ¼ ½6; 15; 20; 15; 6; 1 so that the all of the roots of the polynomial s6 þ c1 s5 þ c2 s4 þ c3 s3 þ c4 s2 þ c5 s þ c6 ¼ 0 are located at )1. We have found appropriate to choose e0 ¼ 0:5. Fig. 9 shows the simulation results by applying the sliding controller (49) to the error system (29). Figs. 9(a), (b), (c) and (d) show the time evolutions of the states e1 ðtÞ, e2 ðtÞ, e3 ðtÞ and e4 ðtÞ, respectively. After a short transient, the positions tracking errors e1 ðtÞ and e3 ðtÞ and the velocities tracking errors e2 ðtÞ and e4 ðtÞ converge to zero which implies a fairly good tracking performance of the desired reference trajectory ðx1d ðtÞ; x2d ðtÞ; x3d ðtÞ; x4d ðtÞÞ in spite of model uncertainties and the control objective is attained. Fig. 10 presented the performance of the system for h ¼ 20. The corresponding control input is continuous, as shown in Fig. 10(a). The resulting control does not have an abrupt change and chattering phenomenon. Note that the control signal is equal to zero when the control is achieved. The sliding surface dynamics is depicted in Fig. 10(b).
Fig. 9. Tracking errors eðtÞ performed with h ¼ 20: (a) e1 ðtÞ; (b) e2 ðtÞ; (c) e3 ðtÞ and (d) e4 ðtÞ.
F.M. Moukam Kakmeni et al. / Chaos, Solitons and Fractals 21 (2004) 1093–1108
1107
Fig. 10. Performance of the system when h ¼ 20: (a) control signal and (b) sliding surface dynamics.
Fig. 11. Tracking error e1 ðtÞ for three different values of the switching gain h.
To illustrate the fact that an arbitrary convergence rate of the system can be prescribed, Fig. 11 presents the position of the tracking error e1 ðtÞ for three different values of the switching gain h. As expected, e1 ðtÞ converges to zero and the larger the value of h, the faster the convergence.
5. Concluding remarks In this paper, we have studied the dynamics of a nonlinear electrostatic transducer described by two coupled nonlinearly Duffing oscillators. Using the method of multiple scale, the well known hysteresis phenomena and the degeneressance of the curve when move away from the exact resonance point are observe. A nonlinear term has been added to the model as a chaos generating component. Bifurcation diagrams have been showing, sudden transitions to chaos and backward or inverse period doubling bifurcation. The control of the device described by our model equations in their chaotic states has been carried out using the sliding mode control. The activation of the sliding mode control governed by the desired state error z5 and its derivative
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z_ 5 . A high gain type controller is employed to steer the states q and x into a desired periodic reference signal, then a sliding mode controller is activated to regulate the system to the desired state. By means of the design of sliding mode dynamics characteristics, the controlled system performance can arbitrarily be determined by assigning the switching gain of the sliding mode dynamics. The results show that the proposed controller can stabilize chaotic motion of the electrostatic transducer system even the system has parameter mismatching and perturbing external forces. The control input in this study is continuous and has no abrupt increase in the desired state change. It provides a method that can achieve desired specification with less control energy by comparing against the results of other researches. In later work, it is hoped that the method will be useful for developing a practical synchronized chaotic electrostatic transducer models.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]
Rao SS. Mechanical vibration. New York: Addison Wiley; 1990. Nayfeh AH, Mook DT. Nonlinear oscillation. New York: Wiley; 1979. Woafo P, Fotsin HB, Chedjou JC. Dynamics of two nonlinearly coupled oscillators. Phys Scri 1998;57:195. Rajasekar S, Murali K. Resonance behaviour and jump phenomenon in two coupled Duffing–van der Pol oscillators. Chaos, Solitons & Fractals 2004;19:925. Nayfeh AH, Sanchez NE. Bifurcations in a forced softening Duffing oscillator. Int J Nonlinear Mech 1989;24:483. Moon FC. Chaotic vibrations. New York: Wiley; 1987. Koziowski J, Pralitz U, Lauterborn W. Bifurcation analysis of two coupled periodically driven Duffing oscillator. Phys Rev E 1995;51:1861. Pastor L, Perez-Garcia VM, Encinas-Sanz F, Guerra JM. Ordered and chaotic behavior of two coupled Van der Pol Oscillators. Phys Rev E 1993;48:171. Chembou Kouomou Y, Woafo P. Stability and Chaos control in electrostatic transducers. Phys Scri 2000;62:255. Sinclair IR. Sensors and transducers. second ed. London: Butterworth-Heinemann; 1992. Tsai HH, Fuh CC, Chang CN. A robust controller for chaotic systems under external excitation. Chaos, Soliton & Fractals 2002;14:627. Jun-Juh Y. Design of robust controllers for uncertain chaotic systems with nonlinear imputs. Chaos, Soliton & Fractals 2004;19:541. Moukam FK, Bowong S, Tchawoua C. Exponential stabilization of two nonlinearly coupled oscillators by an estimated state feedback. Phys Scri 2002;66:7. Kapitaniak T. Chaos for engineers (theory, applications and control). Berlin: Springer; 1998. Isidori A. Nonlinear control systems. third ed. Berlin: Springer-Verlag; 1995. Hassan KK. Nonlinear systems. second ed. Upper Saddle River, NJ: Prentice-Hall; 1995. Cheng D, Hong Y. Stability of ideal sliding dynamics. In: Cheng G, editor. Control theory, stochastic analysis and applications. Singapore: World Scientific; 1991. Edwards C, Tung PC. Sliding mode control theory and applications. London: Taylor & Francis. Yua H, Chua C, Chen C. Sliding controller design for chaotic systems. Int J Bifurcat Chaos 2000;5:1139. Kapitaniak T. Continuous control and synchronization in chaotic systems. Chaos, Soliton & Fractals 1995;6:237. Kapitaniak T, Thylwe KE. On transverse stability of synchronized chaotic attractors. Chaos, Soliton & Fractals 1996;7:1569. Lu Z, Shieh LS, Chen G. On robust control of uncertain chaotic systems: a sliding-mode synthesis via chaotic optimization. Chaos, Soliton & Fractals 2003;18:819. Kapitaniak T, Brindley J. Practical stability of chaotic attractors. Chaos, Soliton & Fractals 1998;9:43. Liao T, Huang N. Control and synchronization of discrete-time chaotic systems via variable structure control. Phys Lett A 1997;234:262. Femat R, Alvarez-Ramirez J, Fermandez-Anaya G. Adaptive synchronization of high-order chaotic systems: a feedback with loworder parametrization. Physica D 2000;139:231. Sira-Ramirez H. On the dynamical sliding mode control of nonlinear systems. Int J Bifurcat Chaos 1993;57:1039. Gao W. Variable structure control. IEEE Trans Ind Electron 1995;42:117. Sussman HJ, Kokotovic PV. The peaking phenomenon and the global stabilization of nonlinear systems. IEEE Trans Autom Contr 1991;36:461. Chua LO, Yang T, Zhong GQ, Wu CW. Adaptive synchronization of Chua oscillators. Int J Bifurcat Chaos 1996;6:189.
Dynamics and chaos control in nonlinear electrostatic transducers F.M. Moukam Kakmeni a
a,*
, S. Bowong b, C. Tchawoua a, E. Kaptouom
c
Laboratoire de Mecanique, Departement de Physique, Faculte des sciences, Universite de Yaounde I, B.P. 812 Yaounde, Cameroon b INRIA Lorraine, Projet Conge & University of Metz, I.S.G.M.P., B^at A, 57045 Metz Cedex 01, France c Departement des Genies Industriel et Mecanique, Ecole Nationale Superieure Polytechnique Universite de Yaounde I, B.P. 8390 Yaounde, Cameroon Accepted 4 December 2003
Abstract In this paper, we analyze the dynamics of a system consisting of two coupled nonlinearly Duffing oscillators, obtained from a nonlinear electrostatic device which is a prototype of emitters and receivers in communication engineering. Inverse or backward period doubling cascades and sudden transition to chaos are observed. A sliding mode controller is applied to control the electrostatic transducers system. The sliding surface used is one dimension higher than the traditional surface and guarantees its passage through the initial states of the controlled system. By means of the design of sliding mode dynamics characteristics, the controlled system performance is arbitrarily determined by assigning the switching gain of the sliding mode dynamics. Therefore, using the characteristic of this sliding mode we aim to design a controller that can meet the desired specification and use less control energy by comparing with the result in the current literature. The results show that the proposed controller can steer electrostatic transducers to the desired reference trajectory without chattering phenomenon and abrupt state change. Ó 2004 Elsevier Ltd. All rights reserved.
1. Introduction Many investigations have considered various models of dynamical systems in the study of mechanical vibrations [1– 10]. Among these, good models candidate of nonlinearly coupled oscillators have particular interest due to their application in various physical systems such as electromechanical, chemical and biological systems. The linear superposition principle which is valid for linear differential equations is no longer valid for those systems. A physical consequence is that when the given system admits an oscillatory motion, the frequency of the oscillation is in general amplitude-dependent, particularly, this can have drastic consequence in the case of forced and damped nonlinear oscillators leading to nonlinear resonances and jump (hysteresis) phenomenon for low strengths of nonlinearity parameters. Such behaviors can be analyzed using various approximation methods [2–5]. Very recently, using the method of multiple scale, Rajasekar et al. [4] obtain a set of autonomous equations for the amplitudes and phases of a two coupled Duffing–van der Pol oscillators with a nonlinear coupling. Also, the stability boundaries of fixed points of the approximated equations are obtained using the Routh–Hurwitz criterion and the stability boundaries are drawn in various parameters space. The influence of the nonlinear coupling strength on the response dynamics have been studied.
* Corresponding author. Permanent address: Universite de Yaounde I, B.P. 8329 Yaounde, Cameroon. Tel: +237-976-22-29; fax: +237-231-95-84. E-mail addresses: [email protected] (F.M. Moukam Kakmeni), [email protected] (S. Bowong), [email protected] (C. Tchawoua).
0960-0779/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2003.12.087
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On the other hand, Chaos is an interesting phenomenon of nonlinear systems. A deterministic chaotic system has some remarkable characteristics, such as system evolution sensitive to the change of in initial condition, broad spectrum of Fourier transform, and fractal properties of the motion in phase space [6]. Due to its powerful applications in various field of sciences including communications, controlling these complex chaotic dynamics for engineering applications has emerged as a new and attractive field and has developed many profound theories and methodologies to date [9–29]. The chaos suppression problem is generally considered as the stabilization of the chaotic system in a periodic orbit or equilibrium point. In this way the phenomena of chaos control, which, along with other approaches have been widely addressed in the literature as well as his broad range of potential applications that goes from life sciences and earth science to the problems facing physicists, engineers and applied mathematicians. Although, the robust tracking control problem for nonlinear chaotic systems, due to its difficulty and its practical interest, is one of the most challenging of the control theory, it has never been addressed for the electrostatic transducers system. In this paper, we consider a nonlinear electrostatic transducers model described by the following set of two nonlinearly coupled differential equations: (
€q þ k0q q_ þ x2q q þ c0q q3 a0q x b0q xq ¼ 0 €x þ k0x x_ þ x2x x þ c0x x3 a0x q b0x q2 ¼ F cos xt
ð1Þ
where the dot denotes time derivative, q and x are the coordinates of both oscillators, k0q and k0x are the viscous damping coefficients, xq and xx are the natural frequencies and, c0q and c0x the nonlinear coefficients. a0q , b0q , a0x and b0x characterized the coupling between the oscillators. The model (1) can be found in a electrostatic microphone and loudspeaker which consist of a mechanical part (described here by x) connected capacitively to an R–L–C circuit whose capacitance can be controlled by acoustic waves (for instance harmonic acoustic waves). The interest on such devices is justified by the fact that with the revival of the electret old idea, electrostatic microphones are widely used for various types of technological applications such as monoscope tubes for TV set signals, cassette recorders devices and evidently telephone devices [3,9,10]. Two points are considered: (i) analyze the oscillatory states in our model and study through numerical simulation some bifurcation structures and transition to chaos; and (ii) carry out the problem of how to design a feedback controller to drive the chaotic trajectories of system (1) to an inherent periodic orbit of the system. Previously, [3,9] investigated the dynamic and the problem of chaotic behaviour of the model without the first Duffing term c0q q3 . After, we completed the study by designing an estimated state feedback controller to drive a chaotic trajectory to an inherent periodic orbit of the system [13]. Recently, we go further with the work to understand how coupling typical nonlinear systems respond to two external periodic forces and exhibits different dynamical resonance and dynamical transition involving chaos. Throughout this note, we extend the work to nonlinearly coupled Duffing equation. The cubic term c0q q3 in the first equation of (1) is a new term introduced to described the nonlinearity of Duffing type in the condenser. This would result from an eventual high charge of the capacitor. The content of the paper is arranged as follows: in Section 2, an analysis is presented for perturbated solutions and the hysteresis phenomena. In Section 3, we report a numerical study, the periodic and chaotic behaviors of the coupled model will be presented and the rote the system follow to the chaotic state. Section 4 is devoted to the control of chaos using an estimated state feedback controller based on sliding mode control. Simulation results are presented to validate the proposed control scheme. We end our investigation in Section 5 with a brief conclusion.
2. Perturbated solutions An approximate analytical solution of Eq. (1) is obtained by assuming that the coefficients k0i , c0i , a0i , b0i , i ¼ q, x and F are small. This smallness can be characterized using a single coefficient e (where e is a small parameter, i.e., e 1) as a scaling factor. Thus we set k0i ¼ eki , c0i ¼ ci , a0i ¼ eai , b0i ¼ ebi and F ¼ eF0 . In general we consider qðtÞ and xðtÞ in the form qðtÞ ¼ q0 ðT0 ; T1 Þ þ q1 ðT0 ; T1 Þ þ xðtÞ ¼ x0 ðT0 ; T1 Þ þ x1 ðT0 ; T1 Þ þ
ð2Þ
where T0 ¼ t is a fast scale and T1 ¼ et is a slow scale, characterizing the modulation in the amplitude and phase caused by the nonlinearity, damping and resonances. Substituting Eq. (2) into Eq. (1) and equating coefficients of the same power of e, we obtain
F.M. Moukam Kakmeni et al. / Chaos, Solitons and Fractals 21 (2004) 1093–1108
D20 q0 þ x2q q0 ¼ 0 D20 x0 þ x2x x0 ¼ 0 D20 q1 þ x2q q1 ¼ aq x0 þ bq x0 q0 2D0 D1 q0 kq D0 q0 cq q30 D20 x1 þ x2x x1 ¼ ax q0 þ bx q20 2D0 D1 x0 kx D0 x0 cx x30 þ F0 cos xT0
1095
ð3Þ
ð4Þ
where Dn ¼ oTon and Tn ¼ en t with n ¼ 0; 1; 2; 3; . . . It is convenient to express the solution of Eq. (3) in the complex form q0 ¼ AðT1 Þeixq T0 þ cc x0 ¼ BðT1 Þeixx T0 þ cc
ð5Þ
where Ôcc’ stands for the complex conjugate term and i2 ¼ 1. AðT1 Þ and BðT1 Þ are arbitrary complex functions of T1 at this level of approximation. It is determined by imposing solvability conditions at next levels of approximation. Substituting Eq. (5) into Eq. (4), it comes that D20 q1 þ x2q q1 ¼ ðikq xq A 2ixq A0 3cq A2 AÞeixq T0 þ aq Beixx T0 þ bq ½ABeiðxq þxx ÞT0 þ ABeiðxx xq ÞT0 cq A3 e3ixq T0 þ cc F0 D20 x1 þ x2x x1 ¼ ðikx xx B 2ixx B0 3cx B2 BÞeixx T0 þ ax Aeixq T0 þ bx A2 e2ixq T0 þ bx AA cx B3 e3ixx T0 þ eixT0 þ cc 2 ð6Þ where A (respectively, B) is the complex conjugate of A (respectively B) and (0 ) denotes the time derivative with respect T1 . Depending on the order of the nonlinearity in the system, various secular terms can be produce with the commensurate relationships between frequencies. These commensurable relationships of frequencies can cause the corresponding mode to be strongly (respectively, weakly) coupled and internal (respectively, external) resonance is said to exist. Accordingly, in our dynamical system, three situations can be analyzed. In Sections 2.1 and 2.2 we consider the case that the electric oscillator enters in resonance with the external excitation and we study in this condition the internal nonresonant and resonant cases. 2.1. External resonance Here we analyze the case where xx is different of xq and 2xq . To express quantitatively the nearness of x to xx , we introduce the detuning parameter r defined by x ¼ xx þ er. In this case, none of the nonlinear coupling terms produces a secular term. The condition for the elimination of the secular term in Eq. (6) become ikq xq A 2ixq A0 3cq A2 A ¼ 0 F0 ikx xx B 2ixx B0 3cx B2 B þ eirT1 ¼ 0 2
ð7Þ
A first-order approximation can be obtained by expressing A and B in real form using the polar representation 1 A ¼ aðT1 Þeiu1 ðT1 Þ 2 1 B ¼ bðT1 Þeiu2 ðT1 Þ 2
ð8Þ
where aðT1 Þ, bðT1 Þ, u1 ðT1 Þ and u2 ðT1 Þ are the amplitudes and phases of the fundamental frequency. Substituting Eq. (8) into Eq. (7), we thus obtain after separating real and imaginary parts the following set of coupled first-order differential equations for the amplitude and phase 3 2 c a a xq au01 ¼ 0 8 q kq xq a þ xq a0 ¼ 0 2
ð9Þ
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3 2 F0 cq b b xx bu02 cos h ¼ 0 2 8 kx F0 0 xx b þ xx b sin h ¼ 0 2 2
ð10Þ
where h ¼ rT1 u2 . In its general form, Eqs. (9) and (10) show that in the nonresonant case, both oscillators are uncoupled. Thus when there is no internal resonance, the first approximation is not influenced by the nonlinear coupling terms. The time evolution of the first oscillator corresponds to the amplitude of the autonomous Duffing oscillator and the second for the forced Duffing oscillator. The steady-state response must have a0 ¼ b0 ¼ 0 and h0 ¼ 0. Thus a ¼ 0 and u02 ¼ r. Eliminating h from Eq. (10), we obtain the frequency–response equation for amplitude of the second oscillator 9 2 6 c b ð3cx xx rÞb4 þ x2x ð4r2 þ k2x Þb2 F02 ¼ 0 16 x
ð11Þ
Solving this equation using Newton–Raphson algorithm, the response curves in term of the detuning parameter r and the external force F0 present the classical multivalued amplitudes and hence to a jump phenomenon as observe earlier [2,3]. 2.2. The internal resonance When there is an internal resonance, the nonlinear coupling terms produce secular terms and the solutions can differ drastically from those of no internal resonance. 2.2.1. First case of internal resonance To analyze the case of internal resonance, we introduce a new detuning parameter r1 according to xx ¼ 2xq er1 ;
x ¼ xx þ er:
ð12Þ
Substituting the above equation into Eq. (6), we obtain the new solvability conditions ikq xq A 2ixq A0 3cq A2 A þ bq ABeier1 T0 2ixx B0 ikx xx B 3cx B2 B þ bx A2 eier1 T0 þ
F0 ierT0 e ¼0 2
ð13Þ
Again, substituting Eq. (8) into Eq. (13) after some algebraic manipulations, we obtain the following new set of coupled and nonlinear first-order differential equations: 3 1 xq au01 cq a3 þ bq ab cos d0 ¼ 0 8 4 ð14Þ 1 1 kq xq a xq a0 þ bq ab sin d0 ¼ 0 2 4 and 3 1 F0 xx bu02 cx b3 þ bx a2 cos d0 þ sin d ¼ 0 2 8 4 1 1 F0 0 2 kx xx b xx b þ bx a sin d0 þ sin d ¼ 0 2 2 4 with d0 ¼ r1 T1 þ 2u1 u2 and d ¼ rT1 u2 . For thus, we analyze the steady-state solution (i.e., u02 ¼ r and u01 ¼ 12 ðr r1 Þ), Eqs. (14) and (15) become 1 3 2 1 a xq ðr r1 Þ cq a þ bq b cos d0 ¼ 0 2 8 4 1 1 a kq xq þ bq b sin d0 ¼ 0 2 4
ð15Þ
ð16Þ
F.M. Moukam Kakmeni et al. / Chaos, Solitons and Fractals 21 (2004) 1093–1108
3 1 F0 xx br cx b3 þ bx a2 cos d0 þ cos d ¼ 0 2 8 4 1 1 F0 2 kx xx b þ bx a sin d0 þ sin d ¼ 0 2 4 2
1097
ð17Þ
We find two possibilities. In the first, we suppose a ¼ 0 (see Eq. (16)), then Eq. (17) leads to the resonant equation (11) and it is essentially the solution of the linear problem. In the second, we suppose a 6¼ 0, eliminating d0 from Eq. (16) leads to " #1=2 2 2 3 2 1 2 2 c a xq ðr r1 Þ þ kq xq 4 ð18Þ b¼ bq 8 q 2 This expression is not realistic since b becomes infinite when bq is zero. 2.3. Second case of internal resonance The second case of internal resonance is concerned with xq ¼ xx þ er0 ;
x ¼ xx þ er
ð19Þ
Substituting in to Eq. (6), we obtain the new solvability conditions ikq xq A 2ixq A0 3cq A2 A þ aq Beir0 T1 ¼ 0 F0 2ixx B0 ikx xx B 3cx B2 B þ ax Aeir1 T1 þ eirT1 ¼ 0 2
ð20Þ
Substituting Eq. (8) into Eq. (20), after some algebraic manipulations, we obtain the following new set of coupled and nonlinear first-order differential equation: 3 1 xq au01 cq a3 þ aq b cos d1 ¼ 0 8 2 1 1 0 kq xq a xq a þ aq b sin d1 ¼ 0 2 2
ð21Þ
3 1 F0 xx bu02 cx b3 þ ax a cos d1 þ cos d ¼ 0 8 2 2 1 1 F0 0 kx xx b xx b xx b þ ax a sin d1 þ sin d ¼ 0 2 2 2
ð22Þ
with d1 ¼ r0 T1 þ 2u1 u2 and d ¼ rT1 u2 . For the steady-state solution (i.e., and u02 and u01 ¼ 12 ðr r0 ÞÞ, Eqs. (21) and (22) lead to the following amplitude equations:
3 4 c b xx rb2 8 x
2 2 ax 2 3 2 1 ax F2 cq a xq ðr r0 Þ a þ kx xx b2 kq xq a2 ¼ b2 0 8 4 aq aq 4
ð23Þ
where 4a2 b ¼ aq 2
"
3 2 c a xq ðr r0 Þ 8 q
2 þ
1 kq xq 2
2 # ð24Þ
These equations involve the coupling parameters ai thus the steady-state motions are coupled. In Fig. 1(a), a typical frequency–response graph is shown. It is notice that the curve presents two resonances points and the hysteresis phenomenon does not appear. As the amplitude of the external force increases the hysteresis phenomenon appears and for some values of r it is possible to have up seven values of the amplitude as presented in Fig. 1(b). For a given value of r, one can evaluate the effect of the term cq on the coupling response curve. We see from Fig. 2(a) and (b) that for r ¼ 0, the curve is a single branch with multivalue of amplitude for low values of cq . This can be seen in the bifurcation diagrams since small values of cq bring chaotic state in the system. For r 6¼ 0 this curve degenerates into tree branches. This behavior of the coupling-response curve shows that when the oscillators move
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Fig. 1. Frequency–response curve in the case of internal and external resonances a(r). The parameters are kq ¼ kx ¼ 0:1, cq ¼ 1, cx ¼ 0, aq ¼ ax ¼ 0:1, bq ¼ bq ¼ 0:2, xq ¼ xx ¼ x ¼ 1: (a) F ¼ 0:1 and (b) F ¼ 0:8.
Fig. 2. Amplitude response curve as a function of cq with the parameters of Fig. 1.
away from the exact external resonance point ðr ¼ 0Þ, their amplitude may be multivalued depending on the new cubic term of the system.
3. Chaotic behavior of the model In this section, we process numerically. The aim of the numerical study is to find the sensibility and some set of parameters which can lead to chaotic behavior. Among methods use for the identification of the type of motion we carried our study using the bifurcation diagrams. They provide a nice summary for the transition between different types of motion that can occur as one parameter of the system is varied. Figs. 3–6 present a set of sample bifurcation diagrams of our model with respect to F , x, and c0q . Other parameters are fixed as: k0q ¼ 0:1, a0q ¼ 0:4, c0q ¼ 0:1, b0q ¼ 0:4, xq ¼ 0:4, k0x ¼ 0:3, c0x ¼ 0:6, xx ¼ 0:1 and x ¼ 1. The plots represent the amplitude x of the second oscillator in a Poincare cross-section when one parameter is varied the other being fixed. In the bifurcation diagram of Fig. 3 where F is varied from 0 to 1.5, a discontinuous ‘‘jump’’ from a stable period-1 solution to a band of chaotic motion is evident for approximate F 0:55. For 0:55 < F < 0:75, we observe that the
F.M. Moukam Kakmeni et al. / Chaos, Solitons and Fractals 21 (2004) 1093–1108
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Fig. 3. Bifurcation diagrams of the electrostatic transducer model showing the transition order–chaos-order as F varied.
Fig. 4. A section of Fig. 3 showing detailed bifurcation and chaotic behaviors of the parameter interval between F ¼ 0:73 and F ¼ 0:83.
dynamic is almost chaotic with a very short window of period-1 orbit through intermittency. For F > 0:75, one can see that there exists reversed period-doubling bifurcations that bring back the dynamic to a period-1 state (see Fig. 4). As the value of F 0:85, is reaches, the system suddenly becomes chaotic and the inverse period doubling cascade starts at F 1:08 (Fig. 5) leading now to a period-1 orbit. Quasiperiodicity, mode-locking and inverse period doubling cascade are exemplified in Fig. 5 when x is varied from 0 to 0.6. As concerned the bifurcation diagrams of the nonlinearly coupled Duffing oscillator with respect to the new term c0q in the interval 0 < c0q < 0:2, the results obtained, shown in Fig. 6 present an overall behavior similar to that described in the preceding paragraph and depicted in Fig. 3. Here, the inverse period doubling cascade closely starting at c0q 0:102. A typical Poincare map corresponding to chaotic motion is presented in Fig. 7.
4. Chaos control Several kinds of interesting nonlinear dynamic behaviors of the system have been studied in previous sections. They have shown that the forced system exhibited both regular and chaotic motion. Usually, chaos is unwanted or undesirable.
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Fig. 5. Bifurcation diagrams of the electrostatic transducer for F ¼ 1:07 when x varied.
Fig. 6. Bifurcation diagrams of the electrostatic transducer system when F ¼ 1:07, x ¼ 1 and c0q varied.
In order to improve the performance of a dynamic system or avoid the chaotic phenomena, we need to control chaotic system to a regular target trajectory embedded in the chaotic attractor which is beneficial for working with a particular condition [11,13,14,17,18]. However, the practical chaotic systems may contain many types of uncertainties [12]. These uncertainties may cause chaotic perturbations to originally regular behavior, or induce additional chaos in originally chaotic but known behavior, generating unknown chaotic motion (Fig. 8). In this case, it would be desirable to have a feedback scheme to achieve control in spite of the system’s uncertainties. The aim of this section is to study the tracking control problem of the electrostatic transducer system with uncertainties based on the sliding mode design. A sliding mode control is a nonlinear control strategy requiring: (i) a switching manifold that prescribes the desired dynamics and (ii) a control law such that the system trajectory first reaches the manifold and then stays on it forever [17]. The proposed scheme comprises a sliding controller and an uncertainty estimator. The design algorithm for the control via estimation of the lumping uncertainties is the following: first, the uncertain terms are lumped in a nonlinear function in such a way that the lumping nonlinear function is interpreted as a variable state into an extended system. Then, a state estimator is designed to carry out the dynamic reconstruction of the extended variable state, and consequently, an estimated value of the uncertainties is obtained. Finally, a sliding controller is provided with the uncertainty estimate value. In this way, the control law counteracts the uncertain terms and the desired dynamics can be induced.
F.M. Moukam Kakmeni et al. / Chaos, Solitons and Fractals 21 (2004) 1093–1108
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Fig. 7. Typical Poincare maps of the electrostatic transducer system for chaotic behavior, i.e., F ¼ 1:07, x ¼ 1 and c0q ¼ 0:04.
Fig. 8. Typical phase diagrams of the electrostatic transducer system for chaotic behavior, i.e., F ¼ 1:07, x ¼ 1 and c0q ¼ 0:04.
Let us consider the electrostatic transducer system with an external control input u as 8 > > > x_ 1 ¼ x2 < x_ 2 ¼ k0q x2 x2q x1 c0q x31 þ a0q x3 þ b0q x1 x3 x_ 3 ¼ x4 > > > : x_ 4 ¼ k0x x4 x2 x3 c0q x3 þ a0x x1 þ b0x x2 þ F cos xt þ u x 3 1
ð25Þ
_ x3 ¼ x, x4 ¼ x_ and u the physical control input needed to be chosen. The control input u is added to where x1 ¼ q, x2 ¼ q, order or to guide the chaotic dynamics to meet our specific requirements. Without loss of the generality, we can assume that the system’s output is y ¼ x1 . Here, the control problem is to find a robust feedback controller such that the output y ¼ x1 tracks asymptotically the desired reference trajectory yd ¼ x1d as t > T for any initial time t0 P 0 and initial conditions ðx1 ð0Þ, x2 ð0Þ, x3 ð0Þ, x4 ð0ÞÞ 2 R4 in spite of modelling errors, parametric variations, perturbing external forces, noisy measurements, and nonmodeled actuator dynamics. 4.1. System standardization Suppose that we want to drive the system’s output to the periodic reference
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x1d ðtÞ ¼ e0 sin x0 t
ð26Þ
where e0 is a certain constant. In this way, the desired reference trajectory meeting Eq. (25) is given by 8 x1d ðtÞ ¼ e0 sin x0 t > > > > x2d ðtÞ ¼ e0 x0 cos x0 > > > < e ðx2 x2 Þ sin x0 tþk0q e0 x0 cos x0 tþc0q e30 sin3 x0 t x3d ðtÞ ¼ 0 q 0 a0q þe0 b0q sin x0 t > > e0 x0 ½x2q x20 b0q x3d cos x0 tk0q e0 x20 sin x0 tþ3c0q e30 sin2 x0 t cos x0 t > > x4d ¼ > > a0q þe0 b0q sin x0 t > : ur ¼ nðtÞ þ k0x x4d þ x2x x3d þ c0x x31d b0x x21d F cos xt
ð27Þ
with nðtÞ ¼
e0 x20 ðx2q x20 b0q x3d Þ sin x0 t k0q e0 x30 cos x0 t þ 6c0q e30 x20 sin2 x0 t cos2 x0 t 3c0q e30 x20 sin3 x0 t a0q þ e0 b0q sin x0 t a0q þ e0 b0q sin x0 t
It should be mentioned that the constant e0 is chosen in such a way that a0q þ e0 sin x0 t 6¼ 0 for all t P 0. Let the error states and the change of control law of the system be e1 ¼ x1 x1d
e2 ¼ x2 x2d
e3 ¼ x3 x3d
e4 ¼ x4 x4d
and
v ¼ u ud
Then the error state dynamic equations is 8 e_ 1 ¼ e2 > > < e_ 2 ¼ k0q e2 x2q e1 c0q ðe21 þ 3e1 x1d þ 3x21r Þe1 þ a0q e3 þ b0q ½e3 ðe1 þ x1d Þ þ e1 x3d > e_ 3 ¼ e4 > : e_ 4 ¼ k0x e4 x2x e3 c0x ðe23 þ 3e3 x3d þ 3x23r Þe3 þ a0x e1 þ b0x ðe1 þ 2x1d Þe1 þ v
ð28Þ
ð29Þ
To determine the sliding mode control law, the reformulation of the state space equation (29) into an extended controllable canonical form is required. To this end, the change of coordinates: 8 z1 ¼ e 1 > > > > z2 ¼ e 2 > < z3 ¼ k0q e2 x2q e1 c0q ðe21 þ 3e1 x1d þ 3x21r Þe1 þ a0q e3 þ b0q ½e3 ðe1 þ x1d Þ þ e1 x3d ð30Þ > > 2 > z ¼ k z x e 3c ðe þ x Þðe þ 1Þ þ a e þ b ½e ðe þ x Þ þ ðe þ x Þe > 4 0q 3 1 1 1d 1 0q 4 4 1 1d 3 3d 2 0q 0q q > : ¼ f ðe1 ; e2 ; e3 Þ þ ½a0q þ b0q ðe1 þ x1d Þe4 is applied, where for simplicity we define the function f to be everything in the definition of z4 above except the last term, which is ½a0q þ b0q ðe1 þ x1d Þe4 . Note that e4 appears only in this last term so that f depends only on e1 , e2 and e3 . The above mapping is actually a global diffeomorphism. Its inverse is likewise found by inspection to be 8 e1 ¼ z1 > > >e ¼ z > < 2 2 z3 þk0q z2 þx2q z1 þc0q ðz21 þ3z1 x1d þ3x21r Þz1 b0q z1 x3d ð31Þ e ¼ 3 a0q þb0q ðz1 þx1d Þ > > > 2 z þ3c ðz þx Þðz þ1Þb ðe þx Þz z þk z þx > 4 0q 3 2 1 1d 1 3 3d 2 0q 0q q : e4 ¼ a0q þb0q ðz1 þx1d Þ Thus, the dynamics of Eq. (29) can be transferred to a controllable canonical form as z_ i ¼ ziþ1 ; i ¼ 1; 2; 3 z_ 4 ¼ fðz; xd Þ þ mðz1 ; x1d Þv where fðz; xd Þ ¼
of of of e2 þ z3 þ e4 þ ½a0q þ b0q ðe1 þ x1d Þ½k0x e4 x2x e3 c0x ðe23 þ 3e3 x3d þ 3x23r Þe3 oe1 oe2 oe3 þ a0x e1 þ b0x ðe1 þ 2x1d Þe1
mðz1 ; x1d Þ ¼ ½a0q þ b0q ðz1 þ x1d Þ
ð32Þ
F.M. Moukam Kakmeni et al. / Chaos, Solitons and Fractals 21 (2004) 1093–1108
1103
Now, let us assume the following: Assumption 1. Only x1 is available for feedback by online measurements. Assumption 2. The nonlinear terms fðz; xd Þ and mðz1 ; x1d Þ are uncertain. Assumption 3. mðz1 ; x1d Þ is bounded. However, an estimate ^mðz1 ; x1d Þ of mðz; xd Þ satisfying signð^mðz1 ; x1d ÞÞ ¼ signðmðz1 ; x1d ÞÞ: Some comments concerning the above assumptions should be noted: Assumption 1 is realistic because in most cases only one state is available for feedback. Assumption 2 is a practical situation because it is practically impossible to avoid tolerances in the real-world physical parameters. For example, the tolerance of the various electric elements in the nonlinear electrostatic transducer system will certainly lead to small but finite difference in the physical parameters and thus the operations of the device. Concerning Assumption 3, the knowledge of mðz1 ; x1d Þ is a restriction because in some cases the variable z1 change with the parameters variations. Now, let us define /ðz; xd Þ ¼ mðz; xd Þ ^mðz1 ; x1d Þ; g¼
4 X
Hðz; xd ; vÞ ¼ fðz; xd Þ þ /ðz; xd Þv;
z5 ¼ Hðz; xd ; vÞ þ ^mðz1 ; x1d Þv;
and ð33Þ
zkþ1 ok Hðz; xd ; vÞ þ /ðz; xd Þ_v þ z2 o1^mðz1 ; x1d Þ
k¼1
Thus, the system (32) can be rewritten as follows: 8 < z_ i ¼ ziþ1 ; i ¼ 1; . . . ; 4 z_ ¼ g þ ^mðz1 ; x1d Þ_v : 5 g_ ¼ Nðz; z5 ; g; xd ; v; v_ ; €v; tÞ
ð34Þ
where Nðz; z5 ; g; xd ; v; v_ ; €v; tÞ ¼
4 X
z24 ok z5
þ
k¼1
3 X
" zkþ2 ok z5 þ ðg þ ^mðz1 ; z1d ÞÞo4 z5 þ v_ /ðz; xd Þ þ ^mðz1 ; x1d Þ þ
k¼1
4 X
# ok /ðz; xd Þ
k¼1
þ /ðz; xd Þ€v with ok z5 ¼ oz5 =ozk and ok /ðz; xd Þ ¼ o/ðz; xd Þ, k ¼ 1; 2; 3; 4. The following must be pointed out. 1. System (34) is dynamically externally equivalent to system (32). This is for all differentiable input u 2 R. In fact, the manifold Wðz; z5 ; g; xd ; v; v_ ; tÞ ¼ g Hðz; xd ; vÞ is invariant under the trajectories of system (34). It suffices to prove that dWðz; z5 ; g; xd ; v; v_ ; tÞ=dt ¼ 0 along the trajectories of system (34). Then from the equality Wðz; z5 ; g; xd ; v; v_ ; tÞ ¼ 0 and the condition dWðz; z5 ; g; xd ; v; v_ ; tÞ=dt ¼ 0, one can take the first integral of the system (34) to get g ¼ Hðz; xd ; vÞ. When the first integral is backsubstituted in system (34), we obtain the solution of system (32). This implies that the solution zðtÞ 2 R4 of system (32) is the solution of the upper subsystem (34), hence, p:ðz; z5 ; gÞ ¼ z. Thus, the augmented state g provides the dynamics of the uncertain function Hðz; xd ; vÞ and, consequently, of the uncertain terms fðz; xd Þ and mðz1 ; x1d Þ. 2. System (34) is in cascade form. This means that when we taken actions to achieve lim z ! 0, the part Nðz; z5 ; g; xd ; v; v_ ; €v; tÞ ! Nð0; 0; g; xd ; v; v_ ; €v; tÞ ! 0 asymptotically for the so-called cascade character [21]. 4.2. Sliding mode controller design According to sliding mode control design, the sliding surface is defined as Z t X 5 S ¼ z5 z5 ð0Þ þ hð6jÞ kj zj ¼ 0 0
ð35Þ
j¼1
where z5 ð0Þ is the initial state of z5 . The sliding surface used in this paper is one dimension higher than the traditional sliding surface which guarantees that it passes through the initial states of the system being controlled. Equation (35) can also be formulated as
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z_ 5 ¼
5 X
hð6jÞ kj zj
ð36Þ
j¼1
Therefore the sliding mode dynamics (the desired dynamics) can be described as 8 ; i ¼ 1; . . . ; 4 < z_ i ¼ ziþ1P z_ 5 ¼ 5j¼1 hð6jÞ kj zj : g_ ¼ Nðz; z5 ; g; xd ; v; v_ ; €v; tÞ Or in a matrix equation form as z_ ¼ hK1 h MKh z g_ ¼ Nðz; z5 ; g; xd ; v; v_ ; €v; tÞ where z ¼ ðz1 ; z2 ; z3 ; z4 ; z5 ÞT , 2 0 1 0 0 6 0 0 1 0 6 6 0 0 0 1 6 4 0 0 0 0 k1 k2 k3 k4
3 0 0 7 7 0 7 7 1 5 k5
ð37Þ
ð38Þ
2
and
h 0 6 0 h2 6 Kh ¼ 6 60 0 40 0 0 0
0 0 h3 0 0
0 0 0 h4 0
3 0 07 7 07 7 05 5 h
with K1 h the inverse matrix of Kh . The design of kj , j ¼ 1; . . . ; 5 can be determined by choosing the eigenvalues of MðkÞ such that the corresponding characteristic polynomial equation Pk ðsÞ ¼ s5 þ k5 s4 þ k4 s3 þ k3 s2 þ k2 s þ k1 is Hurwitz. The reaching law can be chosen as S_ ¼ bS h sgnðSÞ
ð39Þ
where 0 6 b < 1, sgnð:Þ denotes the signum function and h > 0 can be considered as the switching gain to be determined such that the sliding condition is satisfied and sliding mode motion will occur. Combine Eqs. (35) and (39), we obtain S_ ¼ bS h sgnðSÞ ¼ g þ ^mðz1 ; x1d Þ_v þ
5 X
hð6jÞ kj zj
ð40Þ
j¼1
and if the initial condition vð0Þ ¼ 0, then the differential equation of control signal v can be obtained as " # 5 X 1 bS h sgnðSÞ g hð6jÞ kj zj v_ ¼ ^mðz1 ; x1d Þ j¼1 Therefore, the implemented control input can be described as !# Z t" 5 X 1 ð6jÞ bS h sgnðSÞ g h kj zj dt vðtÞ ¼ ^mðz1 ; x1d Þ 0 j¼1
ð41Þ
ð42Þ
with vð0Þ ¼ 0. Substituting the control law (41) into the extended system (34), the dynamics of closed-loop system can be described as 8 4 < z_ i ¼ ziþ1 ; i ¼ 1; . . . ; P z_ 5 ¼ bS h sgnðSÞ 5j¼1 h6j kj zj ð43Þ : g_ ¼ Nðz; z5 ; g; xd ; v; v_ ; €v; tÞ Here the large h is important for the realization of control, which is associated with the system information of the chaotic system. We can qualitatively analyze this question with Lyapunov stability theory as follows. Let the Lyapunov function of the system be 1 V ¼ S2 2
ð44Þ
F.M. Moukam Kakmeni et al. / Chaos, Solitons and Fractals 21 (2004) 1093–1108
The first derivative of V with respect to time is obtained as ! 5 X ð6jÞ _V ¼ S S_ ¼ S z_ 5 þ h kj zj ¼ S½bS h sgnðSÞ ¼ bS 2 h absðSÞ 6 absðSÞ½b absðSÞ h
1105
ð45Þ
j¼1
From Eq. (35), we known that S ¼ hðzÞ. From the boundness of chaotic attractor, we known that S is bounded. So large enough h will lead to V_ 6 0. Here we get the same conclusion as that in [19]. In many situations, condition V_ 6 0 can be satisfied by choosing a large enough switching gain h. The convergence of gðtÞ to zero follows from the fact that the closed-loop system (43) is in cascade form. If system uncertainties can be estimated more accurately, then the resulting control input will be more accurate. Nevertheless, the terms z5 and g are not available for feedback from on-line measurements (Assumption 2). Then the sliding feedback (42) is not physically realizable. An alternative is to use estimates of z5 , g and zi , i ¼ 1; . . . ; 4 in such a way that the main characteristics of the feedback can be retained. So we will use an observer to estimate z5 , g and zi , i ¼ 1; . . . ; 4. To this end, the following uncertainty estimator is proposed 8 < ^z_ i ¼ ^ziþ1 hi ci ð^z1 z1 Þ; i ¼ 1; . . . ; 4 ð46Þ ^z_ ¼ ^g þ ^mðz1 ; x1d Þ_v h5 c5 ð^z1 z1 Þ : _5 g^ ¼ h6 c6 ð^z1 z1 Þ where cj , j ¼ 1; . . . ; 6 are chosen such that the polynomial s6 þ c1 s5 þ c2 s4 þ c3 s3 þ c4 s2 þ c5 s þ c6 ¼ 0 is Hurwitz. Define the estimating error as: ~zi ¼ hð6iÞ ð^z1 z1 Þ, i ¼ 1; . . . ; 5 and ~z6 ¼ ^ g g. Then, we get the following system ~z_ ¼ hNðcÞ~z þ Xðz; z5 ; g; xd ; v; v_ ; €v; tÞ
ð47Þ
T
T
where ~z ¼ ð~z1 ; ~z2 ; ~z3 ; ~z4 ; ~z5 ; ~z6 Þ , Xðz; z5 ; g; xd ; v; v_ ; tÞ ¼ ½0; . . . ; 0; Nðz; z5 ; g; xd ; v; v_ ; €v; tÞ and N ðcÞ is as follows 2
c1 6 c2 6 6 c3 NðcÞ ¼ 6 6 c4 6 4 c5 c6
1 0 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
3 0 07 7 07 7 07 7 15 0
Since the trajectories qðtÞ and xðtÞ belong to some chaotic attractor, then Hðz; xd ; vÞ in Eq. (34) is a bounded function. Hence Nðz; z5 ; xd ; g; v; v_ ; €v; tÞ also is bounded function. Consequently, for a large value of the switching surface gain h, ~z ! 0 which implies that ^zj ! zj , j ¼ 1; . . . ; 4, ^z5 ! z5 and ^ g ! g. Then Eqs. (38) and (45) become S ¼ ^z5 ^z5 ð0Þ þ
Z
t 0
vðtÞ ¼
Z t" 0
5 X
hð5jÞ kj^zj ¼ 0
ð48Þ
j¼1
5 X 1 bS h sgnðSÞ ^g hð5jÞ kj^zj ^mð^z; x1d Þ j¼1
!# dt
ð49Þ
with vð0Þ ¼ 0. Note that since Hðz; xd ; vÞ is uncertain, the function Nðz; z5 ; xd ; g; v; v_ ; €v; tÞ correspondingly is unknown. Thus, such a term has been neglected in the construction of the observer (46). Note also that the sliding control law (49) only uses estimated values of the uncertain term Hðz; xd ; vÞ (by means ^z5 and g) and ^z. Thus, the robust asymptotic stability is given by the dynamic compensator (46) and the sliding control law (49). Switching high-gain can induce undesirable dynamics effects such as peaking phenomenon [24]. To diminish these effects, the control law can be modified by means of (Z " !# ) 5 t X 1 ðnjþ1Þ u ¼ Sat bS h sgnðSÞ ^g ð50Þ h Kj^zj dt ^mð^z; x1d Þ 0 j¼1 where Sat{.}: R5 ! S R5 , S is a bounded set [25].
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4.3. Numerical results In order to validate the performance of the proposed control law, we have performed simulations. The parameters of uncontrolled electrostatic transducer system are k0q ¼ 0:1, c0q ¼ 0:04, a0q ¼ 0:4, b0q ¼ 0:4, xq ¼ 0:4, k0x ¼ 0:3, c0x ¼ 0:6, xx ¼ 0:1, F ¼ 1:07 and x ¼ 1. The chaotic nature of the electrostatic transducer system is observed for the chosen initial conditions ðx1 ð0Þ; x2 ð0Þ; x3 ð0Þ; x4 ð0ÞÞ ¼ ð0; 0; 0; 0Þ in Fig. 6. The initial conditions for the system (29) is ½e1 ð0Þ; e2 ð0Þ; e3 ð0Þ; e4 ð0Þ ¼ ½0:1; 0:2; 0 and for the observer (46), ½^z1 ð0Þ; ^z2 ð0Þ; ^z3 ð0Þ; ^z4 ð0Þ; ^z5 ð0Þ; ^ gð0Þ ¼ ½0:1; 0; 0; 7:2. Let b ¼ 0:01 and h ¼ 20 and the eigenvalues corresponding to the sliding surface are )1 of which the coefficients of Hurwitz polynomial are ½k1 ; k2 ; k3 ; k4 ; k5 ¼ ½1; 5; 10; 10; 5; 1. The estimation constants was chosen as ½c1 ; c2 ; c3 ; c4 ; c5 ; c6 ¼ ½6; 15; 20; 15; 6; 1 so that the all of the roots of the polynomial s6 þ c1 s5 þ c2 s4 þ c3 s3 þ c4 s2 þ c5 s þ c6 ¼ 0 are located at )1. We have found appropriate to choose e0 ¼ 0:5. Fig. 9 shows the simulation results by applying the sliding controller (49) to the error system (29). Figs. 9(a), (b), (c) and (d) show the time evolutions of the states e1 ðtÞ, e2 ðtÞ, e3 ðtÞ and e4 ðtÞ, respectively. After a short transient, the positions tracking errors e1 ðtÞ and e3 ðtÞ and the velocities tracking errors e2 ðtÞ and e4 ðtÞ converge to zero which implies a fairly good tracking performance of the desired reference trajectory ðx1d ðtÞ; x2d ðtÞ; x3d ðtÞ; x4d ðtÞÞ in spite of model uncertainties and the control objective is attained. Fig. 10 presented the performance of the system for h ¼ 20. The corresponding control input is continuous, as shown in Fig. 10(a). The resulting control does not have an abrupt change and chattering phenomenon. Note that the control signal is equal to zero when the control is achieved. The sliding surface dynamics is depicted in Fig. 10(b).
Fig. 9. Tracking errors eðtÞ performed with h ¼ 20: (a) e1 ðtÞ; (b) e2 ðtÞ; (c) e3 ðtÞ and (d) e4 ðtÞ.
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Fig. 10. Performance of the system when h ¼ 20: (a) control signal and (b) sliding surface dynamics.
Fig. 11. Tracking error e1 ðtÞ for three different values of the switching gain h.
To illustrate the fact that an arbitrary convergence rate of the system can be prescribed, Fig. 11 presents the position of the tracking error e1 ðtÞ for three different values of the switching gain h. As expected, e1 ðtÞ converges to zero and the larger the value of h, the faster the convergence.
5. Concluding remarks In this paper, we have studied the dynamics of a nonlinear electrostatic transducer described by two coupled nonlinearly Duffing oscillators. Using the method of multiple scale, the well known hysteresis phenomena and the degeneressance of the curve when move away from the exact resonance point are observe. A nonlinear term has been added to the model as a chaos generating component. Bifurcation diagrams have been showing, sudden transitions to chaos and backward or inverse period doubling bifurcation. The control of the device described by our model equations in their chaotic states has been carried out using the sliding mode control. The activation of the sliding mode control governed by the desired state error z5 and its derivative
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z_ 5 . A high gain type controller is employed to steer the states q and x into a desired periodic reference signal, then a sliding mode controller is activated to regulate the system to the desired state. By means of the design of sliding mode dynamics characteristics, the controlled system performance can arbitrarily be determined by assigning the switching gain of the sliding mode dynamics. The results show that the proposed controller can stabilize chaotic motion of the electrostatic transducer system even the system has parameter mismatching and perturbing external forces. The control input in this study is continuous and has no abrupt increase in the desired state change. It provides a method that can achieve desired specification with less control energy by comparing against the results of other researches. In later work, it is hoped that the method will be useful for developing a practical synchronized chaotic electrostatic transducer models.
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