Communications in Nonlinear Science and Numerical Simulation 11 (2006) 355–375 www.elsevier.com/locate/cnsns
Dynamics and chaos control of the self-sustained electromechanical device with and without discontinuity R. Yamapi
a,*
, S. Bowong
b
a
b
Laboratoire de Me´canique, Faculte´ de Sciences, Universite´ de Yaounde´ I, BP 812 Yaounde´, Cameroun Laboratoire de Mathe´matiques Applique´es, Faculte´ de Sciences, Universite´ de Douala, BP 24157 Douala, Cameroun Received 4 August 2004; received in revised form 17 September 2004; accepted 18 September 2004
Abstract In this paper, we consider the dynamics and chaos control of the self-sustained electromechanical device with and without discontinuity. The amplitude equations are derived in the general case using the harmonic balance method. The model without discontinuity is first considered. The effects of the amplitude of the parametric modulation and some particular coefficients are found in the response curves. The transition to chaotic behavior is found using numerical simulations of the equations of motion. We find that chaos appears in the model between the quasi-periodic and periodic orbits when the amplitude of the external excitation E0 vary. An adaptive Lyapunov control strategy enables us to drive the system from the chaotic states to a targeting periodic orbit. The effects of elasticity and damping on the dynamics of the selfsustained electromechanical system are also derived. Ó 2004 Published by Elsevier B.V. PACS: 05.45.Gg; 05.45.Ac; 04.45.Pq Keywords: Dynamics; Chaos control; Electromechanical system
*
Corresponding author. Tel.: +237 932 9376; fax: +237 222 6275. E-mail addresses:
[email protected] (R. Yamapi),
[email protected] (S. Bowong).
1007-5704/$ - see front matter Ó 2004 Published by Elsevier B.V. doi:10.1016/j.cnsns.2004.09.002
356 R. Yamapi, S. Bowong / Communications in Nonlinear Science and Numerical Simulation 11 (2006) 355–375
1. Introduction The dynamics of nonlinearly coupled parametrically excited systems (including Duffing, Rayleigh and Van der Pol oscillator) has received much attention over the last few years. Recent studies on these parametrically coupled nonlinear systems have shown various types of behavior [2,21,19,20,11,18,1,23]. Recently, Qinsheng [18] has considered the dynamical behavior of two coupled parametrically excited Van der Pol oscillators. Based on the averaging equations, the transition boundaries corresponding to different types of solutions were found and the periodic solutions may lose their stabilities via a generalized static bifurcation, which leads to stable quasi-periodic solutions, or via a generalized Hopf bifurcation, which leads to stable 3D tori. Bakri et al. [1] have considered the nonlinear dynamics of a one-mass system with two degree of freedom, nonlinearly coupled, with parametric excitation in one function in the external and parametric resonance. They derived conditions for stability of the trivial solution by using both the harmonic balance and the normal form of averaging method. They also showed that if this trivial solution become unstable, a periodic solution may emerge, there are also cases where the trivial solution is stable and co-exists with stable periodic solution. An attracting torus with large amplitudes by a Neimark–Sacker bifurcation was found when both the trivial solution and the periodic solution are unstable. In the context of coupled parametrically excited systems, we studied recently the dynamics of the forced parametric nonlinear electromechanical system [23] consisting of an electrical Duffing oscillator coupled magnetically and parametrically to a linear mechanical oscillator. Using respectively the method of harmonic balance and the Floquet theory [17], the frequency response and stability boundaries of harmonic oscillatory states have derived. Effects of the parametric modulation of the coupling coefficient on frequency response-curves and stability boundaries are analyzed. Various types of bifurcation structures were reported using numerical simulations of the equations of motion. We extend our study in this paper by considering the dynamics and chaos control of the self-sustained electromechanical system with parametric coupling and discontinuous parameters, which consists of an electrical Rayleigh oscillator coupled magnetically and parametrically to a linear mechanical oscillator. The variation of the coupling coefficients is considered for the engineering purpose. Two major problems are considered in this paper. We first analyze the behavior of the self-sustained electromechanical model without discontinuous parameters before taking into account the effects of the discontinuous parameters. The paper is organized as follows. After presenting the physical model and giving through harmonic balance calculations [17] the amplitudes equations of the harmonic oscillatory states in the next section, we consider in Section 3 the behavior of the model without discontinuous parameters. We find the behavior of the harmonic oscillatory states when the parameters of the system vary. Bifurcation structures and transitions to chaos are derived. An adaptive Lyapunov control strategy will be used to drive the self-sustained electromechanical device from the chaotic states to a desired targeting periodic orbit. In Section 4, the effects of discontinuity of elasticity and damping on the dynamics of the self-sustained electromechanical model are analyzed. We also find and analyze the response curves of the self-sustained electromechanical model when some particular parameters are varied. We conclude in the last section.
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2. Description of the model and harmonic balance calculations 2.1. Description of the physical model The self-sustained electromechanical system with discontinuous parameters we are dealing with is shown in Fig. 1 and is composed of an electrical part (Rayleigh oscillator) coupled magnetically and parametrically to a mechanical one governed by a linear mechanical oscillator (as we have described recently, see [23,24]). It creates a Laplace force in the mechanical part and the Lenz electromotive voltage in the electrical part. The electrical part of the system consists of a capacitor C, an inductor L, a nonlinear resistor NLR and a sinusoidal voltage source e(s) = v0 cos Xs (v0 and X being respectively the amplitude and frequency, and s the time), all connected in series. In the model, the nonlinear component is introduced through a resistor, so that the current–voltage characteristics of a resistor is also defined as " # 3 i i ; þ V R0 ¼ R0 i0 i0 i0 where R0 and i0 are respectively the normalization resistance and current. The presence of such a nonlinear resistor in our electromechanical system confers to the mechanism of self-sustained oscillations. This nonlinear resistor can be realized using a block consisting of two transistors [6] and has been used recently by Chedjou et al. [3]. The mechanical part is composed of a mobile beam which can move along the ~ z-axis on both sides. The rod T which has the similar motion is bound to a mobile beam with a spring.
Fig. 1. Schematic of a self-sustained electromechanical device with discontinuity.
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In certain circumstances, some parameters of the self-sustained electromechanical device can vary with time because of the functioning constraints. This is particularly the case for the parameters of the electromagnetic coupling: i.e., time variations of the magnetic field Bm and the region of electromagnetic action. We assume that the time variation is periodic with frequency 2X. So that, Bm and l can be explained as Bm ðtÞ ¼ Bm ð1 þ e cos 2XtÞ or lðtÞ ¼ lð1 þ e cos 2XtÞ: In this paper, only one form of this time variation is considered. It can easily be shown that the self-sustained electromechanical device is described by the following set of coupled differential equations " 2 # d2 q 1 dq dq q dz þ lBm ð1 þ cos 2XsÞ ¼ v0 cos Xs; þ L 2 R 1 2 ds ds C 0 ds i0 ds ð1Þ d2 z dq m 2 þ gðz; z_ Þ lBm ð1 þ cos 2XsÞ ¼ 0 ds ds with gðz; z_ Þ ¼
c1 z_ þ k 1 z c2 z_ þ k 2 z þ
for z 6 zc ; k 02 zc
for z > zc ;
where l is the length of the electrical wire inside the magnetic field Bm. ci and ki (i = 1, 2) being the damping and elasticity coefficient respectively. c2 ¼ c1 þ c02 and k 2 ¼ k 1 þ k 02 , where the quantities c02 and k 02 measure the differences between the damping and elasticity coefficients in the region z > zc and z < zc. q denotes the instantaneous electrical charge of the condenser and z the displacement of the mobile beam. Let us use the dimensionless variables x¼
q ; q0
z y¼ ; l
t ¼ we s;
a0 ¼
w2e q20 ; i20
where q0 is a reference charge of q and w2e ¼
1 ; LC 0
w20 ¼
w0m ; we
l¼
R ; Lwe
k1 ¼
c1 ; mwe
c0 ¼
c02 ; mwe
E0 ¼
c¼
w2m ¼
k1 ; m
w02 m ¼
k 02 ; m
w22 ¼
l2 Bm ; Lq0 we v0 : Lw2e q0
wm ; we
k2 ¼
Bm q0 ; mwe
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Thus the set of differential equations (1) reduces to the following set of Rayleigh oscillator coupled to linear oscillator as €x lð1 a0 x_ 2 Þ_x þ x þ k1 ð1 þ cos 2wtÞ_y ¼ E0 cos wt; €y þ f ðy; y_ Þ k2 ð1 þ cos 2wtÞ_x ¼ 0 with
( f ðy; y_ Þ ¼
cy_ þ w22 y ðc þ c0 Þ_y þ
ð2Þ
for y 6 y c ; ðw22
þ
w20 Þy
w20 y c
for y > y c ;
where we fix yc = 1.0 in the rest of the paper. The dot over a quantity denotes the time derivative. l and a0 are two positive coefficients, c the damping coefficient, w2 the natural frequency, b the nonlinearity coefficient, k1 and k2 the coupling coefficients. w and E0 are respectively the frequency and amplitude of the external excitation. c0 and w0 are the measures of the discontinuities of the dimensionless damping and elasticity coefficients respectively. the amplitude of the parametric coupling with 0 6 < 0.9. For mathematical convenience, we set a0 = 1 in the rest of the paper. The model represents by the Fig. 1 without discontinuity is widely encountered in various branches of electromechanical engineering. In particular, in its linear version, it describes the well-known electrodynamics loudspeaker [9]. In the self-excited nonlinear version, it can serve for various applications such as autonome devices. Moreover, the model can serve as a servo-command mechanisms which can be also used in various applications. Here, one would like to take advantages of nonlinear responses of the model in manufacturing processes. With discontinuity, the model can be seen as a representative of macro or micro electromechanical devices used for cutting, drilling and other machine processes such as loudspeakers [9]. 2.2. Harmonic balance calculations We begin this subsection by determining the amplitudes of the harmonic oscillatory states. For this aim, the harmonic balance method [17] enables us to find the solutions x and y as follows x ¼ a1 cos wt þ a2 sin wt; y ¼ b1 cos wt þ b2 sin wt:
ð3Þ
Inserting the expressions (3) in the generating equations (2) and equating the coefficients of sin wt and cos wt separately to zero (assuming that the terms due to higher frequencies can be neglected), we obtain 3 3 2 2 lw Ar lw a2 þ k1 w 1 þ b2 ¼ E0 ; ð1 w Þa1 þ 4 2 3 3 2 lw Ar lw a1 þ ð1 w2 Þa2 k1 w 1 b1 ¼ 0; 4 2 ð4Þ 2 2 ðX w Þb1 þ Cwb2 k2 w 1 þ a2 ¼ 0; 2 2 2 Cwb1 þ ðX w Þb2 þ k2 w 1 a1 ¼ 0; 2
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where A2r ¼ a21 þ a22 ; B2r ¼ b21 þ b22 ; ( w22 2 X2 ¼ w22 þ w20
for jBr j 6 y c ; for jBr j > y c
and C¼
c
for jBr j 6 y c ;
c þ c0
for jBrj > y c :
We find that the amplitudes Ar and Br are the solutions of the following nonlinear equations 81 4 12 10 27 3 9 9 2 6 9 2 6 2 2 8 l w Ar l w ðM N ÞAr þ l w ðM N Þ þ l w ðF MN Þ A6r 256 32 16 8 3 3 2 9 lw ðF MN ÞðM N Þ l2 w6 E20 A4r 2 16 ð5Þ 3 2 2 3 2 ðF MNÞ þ lMw E0 A2r ðF 2 þ M 2 ÞE20 ¼ 0 2 ( 2 ) 2 2 2 3 3 2 2 2 2 2 2 D DBr ¼ k2 w E0 1 F þ 1þ M lw Ar 2 2 4 with D ¼ ðX22 w2 Þ2 þ C2 w2 ; 3 3 2 3 3 2 2 D ¼ F M lw Ar N þ lw Ar 4 4 2 2 2 2 k1 k2 w ðX2 w Þ 1 4 F ¼ 1 w2 ; D 2 k1 k2 Cw3 1 þ 2 ; N ¼ lw þ D 2 k1 k2 Cw3 1 2 M ¼ lw : D Eqs. (5) are the equations of the amplitudes of harmonic oscillatory states in the general case. We will first analyze the behavior of the self-sustained electromechanical system without discontinuous parameters, before taking into account the effects of discontinuity.
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3. The model without discontinuous parameters 3.1. Harmonic oscillatory states Our aim is to analyze the behavior of the self-sustained electromechanical model without discontinuous parameters. In this case, the amplitudes of harmonic oscillatory states are described by Eqs. (5) but with X2 = w2 and K = c. So that, we provide the amplitudes of the harmonic oscillatory states when the frequency of the external excitation varies and analyze the effects of the amplitude of the parametric modulation on the response curves. By using the Newton–Raphson algorithm, with the following set of physical parameters k1 = 0.08, k2 = 0.4, c = 0.1, l = 0.1, w2 = 1.2, E0 = 1.5, = 0.5, one finds analytically and numerically in Fig. 2 the amplitudes Ar and Br of the harmonic oscillatory states when the frequency w is varied. The curves show the resonance and antiresonance peaks. Analyzing the effects of the parametric modulation on the behavior of the self-sustained electromechanical model, we find that the amplitude affects very shortly the amplitudes Ar(w) and Br(w), and the results are reported in Fig. 4 for several different values of . The variations of Ar() and Br() with w = 1.0 are plotted in Fig. 3 and it appears that Ar() and Br() increase 3.5
(i)
ANALYTICAL RESULTS NUMERICAL RESULTS
3 2.5
Ar 2 1.5 1 0.5 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
7
(ii)
ANALYTICAL RESULTS NUMERICAL RESULTS
6 5
Br
4 3 2 1 0 0
0.5
1
1.5
2
2.5
Fig. 2. Comparison of analytical and numerical frequency-response curves Ar(w) for (i) and Br(w) for (ii) with the parameters l = 0.1, c = 0.1, k1 = 0.08, k2 = 0.4, E0 = 1.5, w2 = 1.2, = 0.5.
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Fig. 3. Amplitudes response-curves Ar() for (i) and Br() for (ii) with the parameters of Fig. 2 and w = 1.0.
when vary from = 0 to = 0.9. The effects of the l coefficient on the harmonic oscillatory states are also provided in Fig. 5. In this figure, the amplitudes Ar and Br are highly dependent to the l coefficient. Fig. 6(i) and (ii) shows respectively the phase portrait of an electrical Rayleigh oscillator and the linear mechanical oscillator for several different values of . We find through this figure that the limit cycle orbit of the electrical Rayleigh oscillator changes very shortly with the variation of the amplitude while the one of the linear mechanical oscillator shows the multiplicity of the limit cycle orbit with depends to the value of . This means that only the linear mechanical oscillator depends highly to the amplitude . 3.2. Chaotic states We find now some bifurcation structures which appears in the self-sustained electromechanical device as the amplitude E0 of the external excitation evolves. For this purpose, the periodic stroboscopic bifurcation diagram of the coordinates x and y is used to map the transitions (the stroboscopic time period is T = 2p/w). Our investigations show that chaotic behavior appears in our self-sustained electromechanical model with the parameters defined in Fig. 7 with E0 = 4, and the chaotic phase portrait of the model is plotted. Fig. 8 shows a representative bifurcation
R. Yamapi, S. Bowong / Communications in Nonlinear Science and Numerical Simulation 11 (2006) 355–375 363 3.5 (a) (b) (c)
(i) 3 2.5
Ar
2 1.5 1 0.5 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
8 (a) (b) (c)
(ii)
7 6 5
Br
4 3 2 1 0 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Fig. 4. Effects of the amplitude on the frequency-response curves Ar(w) and Br(w) with the parameters of Fig. 2, where w = 1.0 and (a) = 0.3, (b) = 0.6, (c) = 0.9.
diagrams and the variation of the corresponding Lyapunov exponent as the amplitude E0 varies. The curves are obtained by numerically solving Eqs. (2) and the corresponding variational equations, the Lyapunov exponent being defined by Lya ¼ lim
t!1
1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi In dx2 þ dv2x þ dy 2 þ dv2y ; t
ð6Þ
where dx, dvx, dy and dvy are the variations of x, x_ , y and y_ respectively. As E0 increases from zero, the amplitude of the quasi-periodic oscillations exists until E0 = 1.05 where a period-7 orbit takes place. For E0 = 2.05, the self-sustained electromechanical system bifurcates from a period-7 orbit to a period-5 orbit through a small window of chaotic orbit and quasi-periodic orbit. At E0 = 3.4, we have the following transitions: quasi-periodic orbit ) chaotic orbit ) period-4 orbit ) chaotic orbit ) quasi-periodic orbit, leading to periodic oscillations (harmonic oscillations). In summary the chaotic states appear with the set of parameters defined in Fig. 7 and the value of E0 defined as E0 2 ]2.03; 2.16[[]2.29; 2.42[[]3.38; 3.5[[]3.7; 3.77[[]3.79; 3.85[, (see for instance the bifurcation diagram and the variation of the Lyapunov exponent showing in Fig. 8).
364 R. Yamapi, S. Bowong / Communications in Nonlinear Science and Numerical Simulation 11 (2006) 355–375 3.5 (a) (b) (c)
3 2.5
Ar
2 1.5 1
(i) 0.5 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
7 (a) (b) (c)
(ii) 6 5
Br
4 3 2 1 0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Fig. 5. Effects of the l coefficient on the frequency-response curves Ar(w) and Br(w) with the parameters of Fig. 2, where w = 1.0 and (a) l = 0.1, (b) l = 1, (c) l = 3.
3.3. Control of chaos We consider the self-sustained electromechanical system with the parameters of Fig. 7. In this state, the system has a chaotic behavior as it appears in the phase portrait of Fig. 7. Due to the presence of chaos in the system, one would like to suppress it or take advantage of the flexibility and various infinite number of different unstable orbits embedded in the chaotic attractor to turn the system to a desired equilibrium points or the targeting periodic orbit. We following the procedure of an adaptive Lyapunov strategy. Introducing the new variables x1 = x, x2 ¼ x_ , x3 = y and x4 ¼ y_ , Eqs. (2) can then be rewritten as x_ i ¼ hðt; x1 ; x2 ; x3 ; x4 Þ;
ð7Þ
then the controllable forced chaotic self-sustained electromechanical system can be described as 8 x_ 1 ¼ x2 ; > > > < x_ 2 ¼ l 1 x22 x2 x1 k1 ð1 þ cos 2wtÞx4 þ E0 cos wt þ u; ð8Þ > x_ 3 ¼ x4 ; > > : x_ 4 ¼ cx4 w22 x3 þ k2 ð1 þ cos 2wtÞx2 ;
R. Yamapi, S. Bowong / Communications in Nonlinear Science and Numerical Simulation 11 (2006) 355–375 365
Fig. 6. Effects of the amplitude on the phase portrait shown the regular orbit (limit cycle) with the parameters of Fig. 2, where w = 1.0 and (a) = 0.3, (b) = 0.6, (c) = 0.9.
1.5
1
vx
0.5
0
-0.5
-1
-1.5 -0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
x
Fig. 7. Chaotic phase portrait of the self-sustained electromechanical device with the parameters: l = 4, w = 5, e = 0.5, k1 = 0.01, k2 = 0.06, w2 = 1.2, c = 0.1 and E0 = 4.
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Fig. 8. Bifurcation diagram (i), (ii) and variation of the Lyapunov exponent versus E0 with the parameters of Fig. 7.
where u is the control signal needed to be chosen. The control term u is added to order or guide the chaotic trajectory to meet our specific requirements. Without loss of generality, we assume that ~y ¼ x3 is the output of the chaotic self-sustained electromechanical system (8). Here, we want to design a robust feedback control u that forces the output ~y ¼ x3 to track a smooth (infinitely differentiable) reference trajectory ~y d ðtÞ ¼ x3d ðtÞ. This problem is widely known in the scientific community as the tracking control problem. 3.3.1. Controlling the chaotic trajectory Suppose that we want the mobile beam of the mechanical part of the self-sustained electromechanical system to follow the admissible periodic reference x3d ðtÞ ¼ Br ð1 þ e cos 2xtÞ3 ;
ð9Þ
where Br is the amplitude of the harmonic oscillatory states of the mechanical part defined before. By an admissible periodic reference, we mean that it does not admit a singularity point when it was introduced in the original system. Thus, with the admissible periodic reference trajectory defined above, the desired reference trajectory meeting Eq. (8) is given as
R. Yamapi, S. Bowong / Communications in Nonlinear Science and Numerical Simulation 11 (2006) 355–375 367
8
Br e2 2 e 2 > 2 2 > > x1d ðtÞ ¼ 1þ x2 6x2 sin 2xt þ 3ce cos 2xt x2 þ 6e x t þ > > x k2 2 > > > 2 > 3 e > 2 2 2 > þ ce cos 4xt þ x 36x sin 4xt ; > > > 4 8x 2 >
> > > Br e2 2 < 1þ x2d ðtÞ ¼ x2 þ 6e2 x2 þ 2e x22 6x2 cos 2xt 6cex sin 2xt k2 2 > > > e2 2 2 > 2 > 3ce x sin 4xt þ x 36x cos 4xt ; > > 2 2 > > > > > x3d ðtÞ ¼ Br ð1 þ e cos 2xtÞ3 ; > > > > x ðtÞ ¼ 6Br exð1 þ e cos 2xtÞ2 sin 2xt; > > : 4d ud ðtÞ ¼ x_ 2d lð1 x2 ð2dÞÞx2d þ x1d þ k1 ð1 þ e cos 2xtÞx4d E0 cos xt
ð10Þ
with x_ 2d ðtÞ ¼
2Br ex 2ðx22 6x2 Þ sin 2xt þ 6cx cos 2xt þ 6cex cos 4xt þ eðx22 36x2 Þ sin 4xt : k2
To accomplish the control objective, we introduce the following error states and the change of control law e1 ¼ x1 x1d ; e3 ¼ x3 x3d ;
e2 ¼ x2 x2d ; e4 ¼ x4 x4d ;
v ¼ u ud :
In this way, the tracking error dynamics can be written as follows 8 e_ 1 ¼ e2 ; > > > < e_ ¼ fðe ; e ; e ; x ; tÞ þ v; 2 1 2 4 2d > e_ 3 ¼ e4 ; > > : e_ 4 ¼ ce4 x22 e3 þ k2 ð1 þ e cos 2xtÞe2 ;
ð11Þ
ð12Þ
where
h i 2 fðe1 ; e2 ; e4 ; x2d ; tÞ ¼ l 1 ðe2 þ x2d Þ ðe2 þ x2d Þ lð1 x22d Þx2d e1 k1 ð1 þ e cos 2xtÞe4 :
Now, our aim is to design a robust feedback control law v to make the zero solution of the above system globally asymptotically stable. The control problem will be studied under the following property: System (12) is minimum phase (see [7] for details concerning this property), that is, the subsystem e_ 3 ¼ e4 ; ð13Þ e_ 4 ¼ ce4 x22 e3 þ k2 ð1 þ e cos 2xtÞe2 is asymptotically stable about e1 = 0 and e2 = 0. In other words, the closed-loop system is internally stable [7,5]. From the control point of view, this is a strong property. But this is reasonable for the nearness of a chaotic attractor in the state space and the interaction of all trajectories inside the attractor. So, when (e1, e2) ! (0, 0) (zero dynamics), the subsystem (13) can be rewritten as follows
368 R. Yamapi, S. Bowong / Communications in Nonlinear Science and Numerical Simulation 11 (2006) 355–375
e_ 3 ¼ e4 ; e_ 4 ¼ ce4 x22 e3 :
ð14Þ
We may take V 0 ðe3 ; e4 Þ ¼ 12 ðx22 e23 þ e24 Þ as the Lyapunov function because its derivative is V_ 0 ðe3 ; e4 Þ ¼ ce24 < 0. Then, the LaSalle Theorem [10,22] implies that the trajectories will converge to the invariant set satisfying V_ 0 ¼ 0, equivalently e4 = 0. From Eqs. (14), such invariant set is only e3 = 0. In this way, the zero dynamics subsystem (14) is asymptotically stable. This is why in the sequel, we only need to stabilize the subsystem in e1 and e2 at the origin and the subsystem in e3 and e4 at the origin will be stabilized automatically. So now, we only consider the following subsystem derived from the dynamics system (12): e_ 1 ¼ e2 ; ð15Þ e_ 2 ¼ fðe1 ; e2 ; e4 ; x2d ; tÞ þ v: We choose the linearizing controller v ¼ v0 ¼ fðe1 ; e2 ; e4 ; x2d ; tÞ þ K~e
ð16Þ
with K = (K1, K2) and ~e ¼ ðe1 ; e2 ÞT . Then the closed-loop system can be written as ~e_ ¼ A~e, i.e.,
0 1 e1 e_ 1 ¼ : ð17Þ K 1 K 2 e2 e_ 2 Should K be a Hurwitz vector, that is all the roots of the polynomial p(s) = s2 K2s K1 have negative real parts, then the error is asymptotic stable at the origin. This means that lim ~eðtÞ ¼ 0 () lim xi ðtÞ ¼ xid ðtÞ;
t!þ1
t!þ1
i ¼ 1; 2
and in particular lim ~y ðtÞ ¼ ~y d ðtÞ:
t!þ1
3.3.2. Adaptive controller Although in general, nonlinear controllers show better performance than linear controllers in controlling nonlinear systems which is rather conceivable. One of the main drawbacks of the linearizing controller is the realization of the nonlinear part in addition to a necessary exact knowledge of the system model (i.e., f(e1, e2, e4, x2d, t)). To overcome this problem, we suggest here to construct a linear adaptive controller e ðtÞ~e; v1 ðtÞ ¼ K
ð18Þ
e ðtÞ will be adjusted adaptively such that where K e ðtÞ~e ¼ v0 ðtÞ: v1 ðtÞ ¼ K
ð19Þ
e denotes the optimal gain vector. Substituting v1 in the second equation of sysThe asterisk in K tem (15) yields to e_ 2 ¼ fðe1 ; e2 ; e4 ; x2d ; tÞ þ v1 ¼ fðe1 ; e2 ; e4 ; x2d ; tÞ þ v1 þ v0 v0 ¼ K~eðtÞ þ v1 v0 :
ð20Þ
R. Yamapi, S. Bowong / Communications in Nonlinear Science and Numerical Simulation 11 (2006) 355–375 369
Define B = (0, 1)T, then we can write ~e_ ¼ A~e þ Bðv1 v0 Þ; ~e_ ¼ A~e þ Bðv1 v Þ;
ð21Þ
1
e K e Þ~e: ~e_ ¼ A~e þ Bð K Theorem. Consider the self-sustained electromechanical system described by Eq. (8) and the desired e ðtÞ be the periodic reference defined by Eqs. (10). Assume that the state vector e* is measurable. Let K solution of e_ ðtÞ ¼ h~eT BT P~e; K
ð22Þ
e ð0Þ is a Hurwitz vector and h > 0 is a positive constant. And let P > 0 be the positive definite where K matrix solution of AT P þ PA ¼ Q;
Q > 0;
e ðtÞ~eðtÞ, leads to asymptotic tracking. then the control law defined by v1 ðtÞ ¼ K Proof. It is shown that the dynamics of the tracking error are described by system (21). Let e Þ be a Lyapunov function candidate V ð~e; K e K e ÞT ð K e K e Þ: e Þ ¼ ~eT P~e þ 1 ð K V ð~e; K h e Þ along the trajectories of system (21) is given by The time derivative of V ð~e; K e_ e Þ ¼ ~eT ðAT P þ PAÞ~e þ 2~eT ð K e K e ÞT BT P~e 2 ð K e ÞT K e K V_ ð~e; K h 1 e_ T T T T e e ¼ ~e Q~e þ 2ð K K Þ ~e B P~e K : h _e T T If we choose K ¼ h~e B P~e then e Þ ¼ ~eT Q~e 6 kmin ðQÞk~ek2 ; V_ ð~e; K
ð23Þ
ð24Þ
ð25Þ
where kmin(Q) is the smallest eigenvalue of Q. Hence system (21) is Lyapunov stable. Which in turn implies that ~e 2 L1 . Integrating (25) we obtain Z t ~eT~e ds: V ðtÞ 6 V ð0Þ kmin ðQÞ 0
Since V(t) 2 L1 and V(0) is finite, this implies that ~e 2 L2 . Also from system (21), we obviously have ~e_ 2 L1 in addition ~e 2 L1 and ~e 2 L2 . Therefore by BarbalatÕs lemma [8] limt!1~eðtÞ ¼ 0, particularly limt!1~y ðtÞ ¼ ~y d ðtÞ. h The results of the control strategy are implemented in Figs. 9 and 10 and show the efficiency of the adaptive Lyapunov control strategy. We do not return to the original coordinates because it would obscure the understanding of the qualitative properties of the proposed adaptive control scheme. The closed-loop system (21) was simulated with the initial conditions and parameters:
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Fig. 9. Gain vector of the linear controller with the parameters of Fig. 7.
Fig. 10. Tracking error ei(t) = xi(t) xid(t), i = 1, 2, 3, 4.
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(e1(0), e2(0), e3(0), e4(0)) = (3.4375, 4.285, 3.375, 0), ð~k 1 ð0Þ; ~k 2 ð0ÞÞ = (4, 4), Br = 1 and h = 1. In this way, one has
1:5 0:5 P¼ : 0:5 0:5 Fig. 9 delineates the evolution of the gain vector while Fig. 10 shows the simulation results by applying the adaptive controller (18)–(22) to the uncertain error system (15) for tracking the desired signal x3d. Fig. 10(i)–(iv) present the time trajectories of e1(t), e2(t), e3(t) and e4(t), respectively. After a short transient, the tracking errors ei(t), i = 1, 2, 3, 4 converge to the origin and the control objective is attained. An important feature is the following: although the control input is acting only on the subsystem in e1 and e2, (e3, e4) is also stabilized at the origin.
4. The effects of discontinuity on the model The effects of discontinuous parameters are frequently appeared in many engineering system [15,16,12,4,13,14], resulting in abrupt changes of the damping and stiffness coefficients,
Fig. 11. Effects of elasticity on the frequency-response curves yimax of the mechanical oscillator versus w with the parameters of Fig. 2 and w0 = 0.2.
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characterized by many unusual and complicated features. Using a formulation which is appropriate for general piecewise system, recent studies on these systems have shown interesting results, particularly for the systems with bi-linear stiffness [12] and for the system with bi-linear damping and stiffness [4]. As we have mentioned before, the discontinuity in our self-sustained electromechanical model appears when the rod T of the mechanical part passes to the two regions delimited by z < zc and z > zc with different damping and elasticity coefficients. When the discontinuity is considered, new interesting phenomena can appear and we analyze the behavior of the self-sustained electromechanical device when the frequency w of the external excitation varies, so that the effects of elasticity and damping on the dynamics of the model are provided. We present in Fig. 11 the frequency-response curves for y > yc (y1max in Fig. 11(i) being the highest values of y above yc) and for y P yc (y2max in Fig. 11(ii) being the highest values of y at the lower side of yc) when the frequency w varies, for different values of the coefficients c0. It appears that the amplitudes of the frequency-responses decrease with the increase of c0. The same structure appears in Fig. 12 when the discontinuity of the elasticity is considered, and the curves shown the jump phenomena. In these two figures, the maximum response amplitude is highly dependent on the discontinuous parameters c0 and w0. In the present case, the response curves bend strongly towards the frequency. This bending results in the family situation in which three solutions are possible over a specific continuous frequency interval, for given frequency and discontinuous parameters c0
Fig. 12. Effects of damping on the frequency-response curves yimax of the mechanical oscillator versus w with the parameters of Fig. 2 and c0 = 0.2.
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Fig. 13. Effects of elasticity on the phase portrait of the mechanical oscillator with the parameters of Fig. 2 and w0 = 0.2; w = 1.0.
Fig. 14. Effects of damping on the phase portrait of the mechanical oscillator with the parameters of Fig. 2 and w0 = 0.2; w = 1.0.
and w0. Obviously, for the cases where the amplitude of the displacement is less than one or equal to one, the solution is the amplitude of the limit cycle solution as we presented in the case of the system without discontinuity. The effects of discontinuity on the phase portrait are provided in Figs. 13 and 14. We find that the self-sustained electromechanical device exhibits a regular behavior only for small values of the discontinuity coefficients. However, the system undergoes bifurcations leading to the appearance of nonperiodic or chaotic responses as it appears in Figs. 13 and 14. Analyzing the effects of the discontinuity on the bifurcation structures, our investigations shown that when the discontinuous parameters vary, the nature of the bifurcation structure
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(for instance, the variation of x versus E0) could not change qualitatively, this is due perhaps to the interaction between the limit cycle and the external excitation, since the self-sustained electromechanical device allows two periodic attractors which are actually in topological equivalence. These results are considered as combined results of the discontinuities, the intrinsic nonlinearity and the external driving.
5. Conclusion This paper deals with the dynamics and chaos control of the self-sustained electromechanical device with and without discontinuity. The harmonic balance method has been used to find the amplitudes of the harmonic oscillatory states. The effects of and l coefficients on the frequency-response curves have been found, using numerical simulations of the equations of motion. The transitions to chaos have been found and it is seen that chaos appears on the self-sustained electromechanical model for some set of parameters. An adaptive controller strategy has been used to drive the model from the chaotic states to a targeting periodic orbit. An improvement of the obtained results is possible. An observer may be added to the closed-loop system to estimate the missing states. The effects of elasticity and damping on the dynamics of the self-sustained electromechanical system have been analyzed.
Acknowledgments Part of this work was done during the visit of R. Yamapi to the Abdus Salam International Centre for Theoretical Physics to attend the Summer School and Conference on Dynamical Systems takes place at Trieste (Italy) from 19 July to 6 August 2004. He would like to thank the organizers of this activity for the invitation, hospitality and financial support. He also thank Professor P. Woafo for enriching contributions.
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