Subharmonic control of chaos with application to a CO2 laser

Chnm.

Pergamon

Solirom

& Fracrok Printed

Vol. 8. No. 9. pp. IU9-1460. 199:’ 0 1997 Elsevier Science Lfd Britain. All rights reserved 0960-0779197 f17.W + 0.00

in Great

PII: SO960-0779(%)00145-2

Subharmonic

Control

of Chaos with Application CO2 Laser

M. BASSO, R. GENESIO, M. STANGHINI

to a

and A. TESI

Dipartimento di Sistemi e Informatica, Universita di Firenze, Via S. Marta 3, 50139 Firenze, Italy

Abstract-This paper proposes a control design method for stabilizing a chaotic signal of a non-linear system to a periodic signal. The underlying idea of the method is the cancellation via feedback control of the subharmonic components of the considered signal. The resulting feedback control scheme possesses a high degree of robustness against system uncertainty and requires a very low control effort. A detailed application to a CO, laser with modulated losses is developed to illustrate the proposed control design technique. 0 1997 Elsevier Science Ltd

1. INTRODUCTION

In the last few years many contributions have addressed the problem of controlling chaos in non-linear systems. The usual paradigm is that chaotic behaviours have to be suppressed by leading the controlled system towards more regular regimes. In particular, several methods for controlling chaos to periodic orbits have been proposed (see Refs (l-51 and references therein). The proposed methods can employ open-loop (see, for example, [6-91) or feedback (see, for example, [lo-131) control schemes, can require low or high control effort and can locally or globally modify the system behaviour. These differences reflect the fact that the proposed ;tpproaches come from various scientific fields and refer to different goals and constraints. When the implementation of the control scheme is the ultimate goal, it appears reasonable :o employ feedback methods that require a small control action and modify the system Jehaviour only locally. Indeed, it is well known [14] that feedback controllers present a righer degree of robustness against system uncertainty, and it is clear that the local nodification requirement becomes quite stringent when the system can work at different operating points, of which only one has to be controlled. Some control design methods Jossessingthese features can be found in [15-181. However, the derived control schemes do lot always possessthe most important feature of any control application, i.e. the simplicity If the controller implementation. This paper proposes a new method to design a control scheme for controlling a chaotic ignal to a periodic one. The underlying idea of the method is the cancellation via feedback :ontrol of the subharmonic components of the considered signal. The resulting feedback ,ontrol scheme possessesa high degree of robustness against system uncertainty and requires I very low control effort. Moreover, it is applicable to quite a general class of non-linear ystems and has the appealing feature that the controller can be implemented via standard inear filters [14]. The reliability of the proposed design technique is verified via a detailed pplication to the control of a CO, laser with modulated losses. The remainder of the paper is organized as follows. Section 2 describes the main features 1449

M. BASSO

ct cd

of the control design method. Sections 3 and 4 deal with the application to the control of a CO? laser. In particular, Section 3 presents the model of the CO? laser to be controlled. while Section 4 contains the control problem formulation and the resulting control scheme. Section 5 reports some concluding remarks.

2. SURHARMONfC

Ct3NTROl.

OF CHAOS

We now describe the basic idea of the method for controlling a chaotic signal of a non-linear system to a periodic signal. Consider the system Y depicted in Fig. 1 where r(t) is an external forcing signal, y(t) is the controlled output signal, 24(t) is the control signal, e(r) is an internal signat and the connection between the last three signals is highlighted. The symbol 3 denotes a summing Mock. Even if it is possible to consider a more general structure for the subsystem ;4p(see the application to a CO2 laser in Sections 3 and 4). it is assumed here for simplicity that 2’ is a linear time-invariant system. Accordingly. the connection can be described in time-domain h! y(r) = l(t) * (e(t) -..If(f)), (1) where 1 denotes the impulse response of 9 and * denotes convolution, and in s-domain by Y(s) - L(s)(E(s) - U(s)),

(2)

where L(s) is the transfer function of 2, i.e. the Laplace transform of f(t), and Y(s), U(s) and E(s) denote the Laplace transforms of v(t), I and e(r). respectively. Let us assume rhat the forcing signal is sinusoidal. i.e. r(t) -1 R cos(wr,r + 0)

(3)

and consider the behaviour of system 9 without a controller, i.e. u(t) = 0. We assume that

Subharmonic

control

1451

of chaos

the system is in a steady-state condition and displays an output signal containing a subharmonic term, i.e. (4)

Y(l) = YOW+ Y&l where ye(t) :=A + B cos war

(5)

Clearly, since 2’ is linear time-invariant, it turns out that the signal e(f) has the form e(r)=a+bcos(w,t-&)+ccos

yj-f-t) (“”

>

(7)

where the parameters a, b, c, 4 and $ are such that A = L(O)a

B = IWdl~ C=

Ljy I(

)I

c

(8)

0 = arg L(jw,) - 4 Y=argL

(

js

1

-$

We are interested in the problem of dropping the subharmonic term ys from the output signal y via the use of a suitable control signal U. Before discussing our approach for solving the problem, we remark that this problem is strictly related to the problem of controlling a chaotic signal, generated via a period doubling cascade phenomenon, to a periodic one. Indeed, a signal of the form eqn (4) can be viewed as the simplest one (at least for small values of C with respect to (A( + B) for modelling the system motion soon after a period doubling bifurcation has occurred. In this case, the suppression of the subharmonic term will bring back the output signal to a periodic behaviour, cancelling the period doubling bifurcation. Since a sequence of period doubling bifurcations is a typical route to chaos [19], suppression of bifurcations eventually means a change of the system behaviour from a chaotic to a periodic one. Going back to the problem of dropping ys from y, let us assume that the control signal u is generated via a linear system driven by the signal y, i.e.

u(r) = C(f)“Y(f)

(9)

U(s) = C(s)Y(s)

(10)

or equivalently

where c(t) and C(s) are the impulse response and the transfer function, respectively, of the linear system, hereafter called the controller.

M. BASSO PI ui

Our objective is to design the controller in such a way that the resulting controlled system depicted in Fig. 2 displays a periodic output signal J To this purpose, let us consider the following structure for the controller:

where k,. is the gain. 6 :a 0 is the damping factor and A .) I is introduced to bound the frequency response of the controller. From eqns (2) and (IO) it turns out that the relation between the signals e(t) and y(r) becomes )‘(,s) =--..--- Ms!LE.(,s):

I t C(.\)f,( 5)

= [a’(s)E(s)

(12)

Now, if we assume that the gain k, of eqn (1 I) is small enough, it is reasonable to expect that the controlled system still displays a steady-state signal 0 of the form eqn (7), i.e.

whcrr the overbar makes it clear that the parameters can be different from the uncontrolled citsc, As a consequence. taking into account eqns ( 1.3)and ( 12). it follows that the output signal v(f) hecomcs

Subharmonic

control of chaos

14.53

where

A = L’(O)2 E = IL’(jw,)(b C=

L’

I

C

jy

i

(1s)

)I

G = arg L’( jwO) - 4

Now, it can be easily verified that the chosen form of the controller [see eqn (ll)] the following relations:

yields

L’(0) = L(0)

L’tjwd = L(jw0)

(16)

1imL jT =0 1 i+o ( independently of the values of k,. and A. Therefore, for small values of the gain k,. and the damping factor 5 we get ;i+A B-B

(17)

C-90

which, after all, means that the output signal y(t) reduces to yO(t), thus obtaining the desired control goal. Before passing to the control of a CO2 laser, we briefly remark on some of the features of the proposed method. It is clear that we are designing a feedback controller [see eqn (lo)]. The controller turns out to be linear time-invariant and has a structure that makes simple the practical implementation by employing standard linear filters. Moreover, the required control effort is quite low since it is of the order of the amplitude of the subharmonic term to be dropped. The unique requirement for the effective design of the controller is that the signal y(t) is measurable and the summing point in Fig. 1 is accessible. When the system to be controlled satisfies this requirement then the controller can be designed. The free parameters k, ,l,h can be tuned to optimize the control performances.

3. CO, LASER

MODEL

We consider a single-mode CO* laser with modulated cavity losses whose behaviour can be described with good accuracy by the so-called four-level model in terms of the following equations [20] i = - ko(l + m sin wot)Z + G(N, - iV,)I & = N, = n;r, = MI =

-

(ZYR + Y$% - GW, - NV + YRMZ+ ~3 (ZYR + YW, + GO’, - NV + ~2% (YR + ~4% + zr& + ZYZP (YR + YIN’, + ZY&

(18)

1454

M. BASSO et ul.

where I represents the field-intensity, N, and Nz denote the populations of the two lasing states and M, and IU, are the global populations of the manifolds of rotational levels. The parameters k0 (the intensity decay rate), G, z, AR?A,. AZ and P have the numerical values: k,,-= 3.18 x 10'

G=K75XlO

h

p = 5.46 x 10" ; = 10

(19)

yK = 7.0 x l(75

y, = X.0 x lo4 y:

= 1.0 x 1O4.

while the parameters m arid 00 represent the amplitude and the frequency of the forcing signai, respectively. It is well known that Exing o0 and increasing m, the system undergoes a cascade of period bifurcations leading to chaos [20]. Recently, in [21] it has been shown that for w0 in the range 1440X 10” rad s ‘. 880 X 10’ rad SC’] system eqn (18) can be reduced (except for a time scaling factor) to the simpler one described by

where the new variables are

(21)

The parameters P’. and r are given by

(22)

while CL,p and p come from the reduction of the linear subsystem of eqn (18) defined by the last two equations and the difference between the second and the third equation from the linear subsystem defined by the third equation of eqn (20). This reduction is obtained by imposing that the transfer functions of the two systems have the same amplitude at o = 0 and w = w0 (see Ref. 1211 for detaiis). In particular, for the working frequency w,) = 62X X 10’ rad s -‘, it turns out that u = 6.767 X 10’. p = 6.626 X lo6 and p = 0.3887.

145s

Subharmonic control of chaos 1.01 0.9 -

(a) .:.y . . : .. . . . . . . .. .. . . . .I.,. : ..>- . ! .-- *

0.8 b ‘2 2 .c 5 Ii B .z z s

0.7 0.6. 0.5 0.4 0.3-

i? 0.2 0.1 OL 0

0.05

0.15

0.10

0.20

0 .2

m

m Fig. 3. Bifurcation diagrams (w,) = 628 X lo-‘rad SC’): (a) simulated: (b) experimental.

Taking the amplitude m as the bifurcation parameter, system eqn complex and chaotic motions represented in the bifurcation diagram diagram appears to approximate rather satisfactorily the qualitative diagram obtained experimentally and reported in Fig. 3(b) (see Ref. [21]

(20) gives rise to of Fig. 3(a). This behaviour of the for more details).

4. LASER CONTROL

The simulated and experimental diagrams of Fig. 3(a) and (b) indicate the presence of a sequence of period doubling bifurcations leading to chaos for m in a certain range. We are interested in designing a feedback controller such that this undesired phenomenon is suppressed and the system is stabilized to a periodic motion. The control constraints are that

M. BASSO CI cd

corrtroi input

measured output

----I-

(4

&( L--..---

i

measured output

1- ---1 -f I I-.-

YR(P’

+

L’)/(zkO)

-_--CT(s)1

1

)

control input

..----.--.I

we can measure the laser field intensity and we must apply the control signal in parallel with the loss modulator. In order to employ the control design method developed in Section 2. we have first to recast system eqn (20) in: the form of Fig. 1. This can be done easily via standard manipulations allowing for the rearrangement of the third order laser model eqn (20) in the block-diagram scheme of Fig. 3(a). It contains two linear subsystems 9 and % of transfer functions (23)

(24) and a non-tinear subsystem X described by the memoryless non-linearity El’(f) F t”’ “1.

(25 )

The symbol 8’2denotes a product block. WC observe that this sche-meis not exactly in the form considered in Section 2. Indeed. while the control signal /l(f) correctly enters the summing point before the linear subsystem .Y, the output .i ,(rj of this Mock is not directly measurable. Fortunately, we can recover this signal rather easily, since it is proportional to the logarithm of the laser field intensity [see eqn (21)]. which is directly avaitable (apart from a scalar factor) at the output w of the block SC’.

Subharmonic

control

1457

of chaos

I.“” I

I 0 Fig.

5. Trajectories

(4

0.002

0.004

0.006

0.008

0.010

0.012

0.014

orbit (solid in the (w = e.“, .r Z) plane: (a) stabilized slowly varying limit cycle.

0.016

line).

I

0 ‘18

unstable

orbit

(dashed

line);

(11)

After all, this leads to consider a non-linear dynamic feedback controller consisting of a linear system (a linear filter) of transfer function eqn (ll), that is C(s) = k,

(26)

driven by a non-linear memoryless system implementing the logarithm (a logarithmic amplifier), as reported in Fig. 4(b).

M. BASSO er ul.

I--------‘-

----r-v

ig !_

-__.---~-._ 9 2

(8~) apn@?ew urnnaads Jarnod p?zqxu~o~

Subharmonic control of chaos

1459

From a conceptual point of view, we have simply modified the transfer function eqn (23) into the transfer function L’(s) =

ko s + k&(s)

’

(27)

In accordance with the theory developed in Section 2, this ensures in a simple way that the subharmonic frequency wo/2 cannot be present in x, and therefore in the measured signal w (proportional to the laser intensity), which is then stabilized to a periodic solution. Numerical simulations confirm the validity of the proposed control scheme, also showing that the required control energy is rather small: the amplitude of the control signal results in about 3% of the external forcing signal amplitude. As an indicative example of the control performances, Fig. 5(a) reports a stabilized limit cycle in comparison with the corresponding unstable limit cycle existing in the original system after the first period doubling bifurcation (m = 0.08, w0 = 628 X lo3 rad s-l). The controller parameters are k, = 5 x lo-’ l= 0.5

(28)

A = 1.5. Moreover, the control scheme exhibits robustness against system uncertainty. As an example, Fig. 5(b) shows that the control goal is still achieved even when the parameter k,, is modulated in the form k. 1 +O.lsinst

. >

The control scheme has been tested experimentally. The experiments largely confirm the simulation results. As an example, Fig. 6(a) and (b) shows the time-domain behaviour and the power spectrum magnitude of the laser intensity without control for m = 0.18, w0 = 690 x 10” rad s-’ ( ra th er inside the chaotic region), while Fig. 6(c) and (d) show the corresponding diagrams when the control is active. Evidently, the subharmonic term wo/2 is strongly dropped. A more detailed discussion on the experimental results can be found in

PI5. CONCLUSIONS

A control design method for controlling the chaotic dynamics of a non-linear system to a periodic motion is presented. The method relies on the suppression via feedback control of the subharmonic components of the considered signal. The resulting feedback control scheme shows a high degree of robustness against system uncertainty and requires a limited control effort. The scheme is applied to the control of a CO2 laser with modulated losses exhibiting a period doubling route to chaos. Numerical simulations as well as experimental tests confirm the achievement of the control goal. Ack,lowledgemenrs-The authors wish to thank F. T. Arecchi. R. Meucci and M. Ciotini of the Istituto Nazionale di Ottica. Firenze. for relevant discussions on the method and for making possible its real application.

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19. Moon. F. C., Chaotic and Fractal Dvnamio.