Electronic chaos controller

Chum

Snhrons

& frocra/s. Printed

Vol. 8. No. 9. pp. I-171-14X4. !YYi 0 1997 Elsewer Science Ltd Britain. All rights reserved 0960.0779197 $17.03 + 0.00

m Great

PII: s0960-0779(%)00147-6

Electronic

Chaos Controller

ZBIGNIEW

GALIAS

Department of Electrical Engineering, University of Mining and Metallurgy, al. Mickiewicza 30, 30-059 Krakow, Poland COLM

A. MURPHY

Silicon and Software Systems. Ballymoss Road. Sandyford Industrial Estate, Dublin lg. Ireland MICHAEL

PETER

KENNEDY

Department of Electronic and Electrical Engineering, University College, Dublin 4. Ireland and MACIEJ

J. OGORZALEKt

Chair of Circuits and Systems, Swiss Federal Institute of Technology, CH-1015 Lausanne. Switzerland

Abstract-The occasional proportional feedback (OPF) controller introduced by Hunt [l] and used for stabilizing periodic solutions in a number of chaotic circuits and also used in the stabilization of a chaotic laser has the disadvantage that it requires an external synchronizing signal. In this paper, we describe a modified version of Hunt’s controller which obviates the need for any external signals. We show how the controller has been used successfully to remove chaos in the chaotic Colpitts oscillator. We also describe some theoretical results on the possibilities for controlling a given chaotic system using the OPF method. We prove that the possibility of stabilization depends on the behavior of the system in the neighborhood of the periodic orbit and not on the dimensionality of the attractor on the Poincare surface. 0 1997 Elsevier Science Ltd

1. INTRODUCTION

Many approaches and methodologies have been described in the literature for controlling chaos [2-61. Of the many schemes proposed, two are of particular interest, namely, the OGY (Ott-Grebogi-Yorke) method [7,8] and OPF (Occasional Proportional Feedback) control [l, 91. These two approaches are of interest because they use specific properties of chaotic systems and require very small signals to achieve control; thus, they are more realistic for implementation purposes. There are two fundamental and specific properties of chaotic attractors [lo]: l l

They contain an infinite number of unstable periodic orbits embedded within them. They contain dense orbits in the sense that a typical trajectory on the attractor passes arbitrarily close to any point on it (it also passes arbitrarily close to any of the unstable periodic orbits).

ton leave from Department of Electrical Engineering. University of Mining and Metallurgy, al. Mickiewicza 30, 30-059 Krakow. Poland. 1471

The OGY control method introduced by Ott, Grebogi and Yorke 17.81 in 1990 was designed to take advantage bf these two properties. It atlows stabilization of any of the unstable periodic orbits by perturbing one of the system parameters over a small range about some nominal value. To explain in some detail the action of the OGY method, let us assume that we have a three-dimensional continuous-time system of first-order autonomous ordinary differential equations

where x E R” is the state and p E R is a system parameter which we can change. We also assume that parameter 0 can he modified within a small interval around its nominal value p,, (0 6s[pu - f$J,,ly,~~rl+ h,,J where &J,,,, 15 the maximum admissible change in the parameter p). We choose a two-dimensional Poincarg surface X which defines a PoincarP map F (for 5 E 1. we denote by F(s) the point at which the trajectory starting from 6 intersects I2 for the first time). Since the vector field F depends on p. the PoincarC map P also depends on this parameter p. Thus. we habc PW k’ R E ([.p )-+ P(&p ) E R-.

(7,

Let us assume that P is differentiable. Suppose that we have selected one of the unstable periodic orbits embedded in the system’s attractor as the goal of our control because. for example, it offers an improvement in system performance over the original chaotic behavior.+ For simplicity. we assume that this is a period-l orbit (a fixed point of the map P),

Let us denote by & an unstable fixed point of P for p = p(, so that P(tr;,pt,) = &. Let the first-order approximation of F in the neighborhood of (4, .p,,) be of the form F(S,p)=P(&,p,,)

‘- A.(t’ -- Cr,) * w.(p -p,,).

f;)

where A is a Jacobian matrix of PC.,,,) at <, _ and w = (dP/8p)(&.,po) is the derivative of P with respect to the parameter p. Stabilization of the fixed point is achieved bv realizing feedback of the form p(() .y I’,, -~c’(: -
(41

In the original description of the OGY method j71, the vector c is computed using the ~-xn-sssimi

where A,, is the unstable eigenvaluc and f,, 1sthe corresponding left eigenvector of A. Thus the OGY method relies on a local linearization of the Poincare map in the neighborhood of the chosen unstable fixed point and locat linear stabilizing feedback. An advantage of the OGY method is that all the necessary calculations can be performed ilff-line 11t1the basis of measurements (e.g. finding the unstable periodic orbits. fixing one of ~hrrn as thr: goal of the control. and catculating the parameters necessary for the computation of the control signal). Once the goal of the control (unstable orbit to be stabilized) has been selected, the control signal is applied only when the observed trajectory passesclose to the fixed point (where the linearization is valid). The assumption about the existence of a dense orbit guarantees that -I his could hc the case. tar example, 01 a chaotic laser intensity which 1s clearly an unwanted phenomenon and the effective power of the taier beam can be enhanced using control to stabilize or eliminate chaotic behavior [L I]. where the heart pumps blood in an inelkient 4nother example ut unwanted chaotic behavior is fibrillation. ~ma!~txr in !hi, application. conlrollinq thr hcan heat UIW a nc.!rh periodic regime is of paramount impurrancr ; !.>i

Electronic

1473

chaos controller

Sample& Hold

Amplifier

circuit

Fig. I. Hunt’s implementation

of the OPF method.

eventually the trajectory will enter the control window. However, the time one has to wait before starting and achieving control might be very long. It should be mentioned here that Dressler and Nitsche [13] have proposed a variant of the OGY method in which only one variable is measured in the system and other variables needed for control are reconstructed using the delayed coordinate method. The occasional proportional feedback (OPF) technique [ 1,9,1 l] can be considered as a one-dimensional version of the OGY method. A schematic of Hunt’s implementation of the OPF control method is shown in Fig. 1. The window comparator, taking the input waveform, gives a logical high when the input waveform is inside the window. This is then ‘AND’ed with the delayed output from the external frequency generator. This logical signal drives the timing block which triggers the sample-and-hold and then the analog gate. The output from the gate, which represents the error signal at the sampling instant, is amplified and applied to the interface circuit which transforms the control pulse into a perturbation of the parameter p. The frequency, delay, control pulse width, window position, width and gain are all adjustable. The interface circuit depends explicitly on the chaotic system under control. One of the major advantages of Hunt’s controller over OGY is that the control law depends on one variable only and does not require any complicated calculations in order to generate the required control signal. The disadvantage of the OPF method is that there is no systematic method for finding the embedded unstable orbits (unlike OGY). For comparison with the method described in this paper, let us summarize the principal features of the OPF controller: l l l l

It uses just one system variable as input. It uses the peaks of this system variable to generate a one-dimensional map. A window around a fixed level sets the region where control is applied. Peaks are located by means of a synchronizing generator, the frequency of which has to be adjusted by either a trial-and-error procedure or by consulting, for example, measured power spectra of the signals.

2. CHAOS

CONTROLLER

FOR AUTONOMOUS

CIRCUIT

In this section, we describe a modified controller which uses HuntTs method without the need for an external synchronising oscillator. Hunt uses the peaks of one of the system variables to generate the l-dimensional map. He then uses a window around a fixed level to set the region where control is applied. This

approach means that his controller needs just one of the system variables as input. In order to find the peaks, Hunt’s scheme uses a synchronizing generator. In our modified controller 1141.we simply take the derivative of the input signal and generate a pulse when it passes through zero. We use this pulse instead of Hunt’s external driving oscillator as the ‘synch’ pulse for ours PoincarC map. This obviates the need for the external generator and so makes the controller simpler and cheaper to build. A circuit diagram for our modified controller is given in Fig. 2. The variable level window comparator is impiemented using a window comparator around zero and a variable level shift. Two comparators and three logic gates form the window around zero. The synchronizing generator used in Hunt’s controlier is replaced by an inverting differentiator and a comparator. A rising edge in the comparator’s output corresponds to a peak in the input waveform. We use the rising edge of the comparator‘s output to trigger a monostabie Hip-flop. The falling edge of this monostable’s pulse triggers another monostable. giving a delay. We use the monostabie’s output pulse to indicate that the input waveform peaked a fixed time earlier. If this pulse arrives when the output from the window comparator is high then a monostable is triggered. The output of this monostable triggers a sample-and-hold on its rising edge which samples the error voltage: on its fatling edge, it triggers another monostable. This final- monostable generates a pulse which opens the analog gate for a specific time (the control pulse width). The control pulse is then applied to the interface circuit, which amplifies the control signal and converts it into a perturbation of one of the system parameters, as required. The controller described here shares some of the advantages of Hunt’s original concept: * It uses just one system variable as input. . A window around a fixed level sets the region where control is applied. But it overcomes the main drawback of using external synchronization: . The external synchronization generator is eliminated; the controller has a simpler and potentially cheaper structure. Control pulses are generated when the derivative of the input signal passesthrough zero at a peak: these pulses act as the synchronization signals which generate the Poincare map. l

3. THEORETKAL

RESULTS

ln this section. we address the problem of whether or not it is possible to stabilize a given periodic orbit using the OPF method. Wr describe: our approach for the case of stabilizing a hxed point of the PoincarC map. in the OPF method, the control signal is computed using only one variable. 5, say: p(E) :=p,, 1 c,((,

t, ).

(-6)

We wan1 to find values of (’ for which c, is a stable fixed point of the system [H P(s.p( 6)).

47nF

Level

Shift

Comparntur LM311

I kfl

V cc

Controller

OPF

in Window.

for the modified

Waveform

OPF

Input

Nnnd

diagram

2-Input 74LSOIN

Fig. 2. Circuit

Quad

Modified

controller.

Pulse

Logic

Rails

at required

Supply

phase

Dual

(VC,)

in input

Flip-Flop

Volts

+ s VOIIS

+-I2

waveform

Monostahle 74HCTl23

ample & Hold LF3YX

Analq DC20

Gale IA

Then the trace and determinant QII~ZI _ Lf,$?~. respectively. If

of matrix

A are given by tr A = a,, + 11~~and det A =

1 -+ tr A + det A + C(MZ,+ M’,cI~~- M~~LZ 12)> 0. > 0. I -~det A + C( - M’,LI~:+ r+~~~r,,)

(9)

1 ~ tr A + det A + c( -II*! t ~v,u~?~..M~?N!~).:>0, then (0.0)‘. is a stable fixed point of f(t) k f(O(S)) t’roo,f

(10)

= f(f.c,t,) = A.6 + WC{,

WC have

Now (tt,O)’ is a stable fixed point of f if alt the eigenvalues of W lie within lising the Hurwitz criterion, one can show that. if the inequalities

the unit circle.

i - IT-w i detw 2‘0

(II)

are satisfied. then the eigenvalues of the matrix W lie within the unit circle. For matrix W. the conditions (II) are equivalent to the inequalities (9). H This theorem is formulated for the cast of a linear map f with fixed point 0 = (O,O)-‘. The following theorem extends this result to a nonlinear map P with an arbitrary fised point cr. Ttrtvwern 2. Let P be the map defined in (2) and let E, be a fixed point of P. Let the linear approximation of P he of the form (3). Define

P(t”! --\ P(,$.c.&,). If conditions (9) are satisfied then there exists a neighborhood I/ of er such that P”(c) 3s II -- x. for all < E 1.: (i.e. er is stable for the map (12)). Proof. Apply Theorem 1 to the linear approximation of the map P at <,. n

(12) [,. .

The above theorem determines for which values of c successful control is possible. As an cxamplc. tet us consider the cast when the Jacobian matrix is diagonal. Define two special diaynral matrices

\vrth one stable ciyenvalue

and one unstable elgenvaluc each. Note that IA,,/ ‘> I ;, Ih,l.

C‘orollnr:\ .G Assume that rt’, f 0. If the Jacobian A is of the form (13) then there exists C’ ,
Electronic

chaos controller

1477

Proof. First assume that the stable eigenvalue is positive. From Theorem 1, it follows that stabilization is possible if 1 + A, + A,, + A,h,, + c(u’, + w, A,) > 0, 1 - A,YA,,+ c( -rv, A,) > 0. l-A,-A,,+A,A,,+c(-w,+w,A,)>O. These conditions are equivalent to

M,c > _ 1 + A.7+ A,, + AA, = -1 - A,,, I 1 + A, w,c<-

1 - AAt A.\ ’ 1 - A, - A,, + U,,

= 1 - A,,. 1 - A, Since IV, # 0. we can find c for which the above inequalities hold if N’,C<

1 - &A,, -> 4 and

- 1 -A,,,

(l-5)

1 - A,, > - 1 - A,,.

(16) Inequality (16) is equivalent to 2 > 0, which is always true. Inequality (15) is equivalent to 1 + A, > 0. which is also true as IA.,I< 1. Hence it is always possible to choose value c such that the fixed point will be stabilized. Similarly, one can show for the case A, < 0 that existence of c satisfying (9) is equivalent to A,
RESULTS

We tested our controller in two experiments. The first one concerns chaos control in Chua’s circuit [l-5], and the second is control of a chaotic Colpitts oscillator [16,17]. 4.1.

Control qf chaos in Chua’s circuit

During this test, we used the controller set-up as described earlier and the interface circuit supplying the control signals to Chua’s circuit as shown in Fig. 3. The standard op-amp implementation of Chua’s circuit was used [18].

7,(iALIAS CI ol

1. -A!-+

I-y--

--.+

)---

?..

Electronic

Controlled

Stabilised

Period-2

chaos controller

Chua’s

Oscillator:

Orbit.

Vc, 1 IVldiv)

1479

Period-2

VsVcZ

(O.?V/div).

Zero Level for Offset Input Voltagr

Upper Trace:

Offset input Voltage

Lower Trace:

JFET gate Voltage

Time:

?ms/div

lO.SV/div) (O.SV/div)

Fig. 4. Stabilization of a period-2 orbit in Chua’s oscillator: (a) Projection of the orbit in the state-space (1 V/div) vs V,, (0.2 V/div)): (b) Time waveforms of the output signal and the controlling signal (upper offset input voltage (0.5 V/div), lower trace: JFET gate voltage (0.5 V/div). time: 2 ms/div).

(C;, trace:

Figures 4-7 show results of stabilization of four different periodic orbits of periods 2 to 5, respectively. In each figure, we show a projection of the periodic orbit and, below it, the time waveforms of the output signal and the controlling signal. 4.2. Chaos control in the Colpitts oscillator The diagram of the interface circuitry together with the oscillator used in this experiment is shown in Fig. 8. A typical chaotic attractor which can be observed in the Colpitts oscillator

Z. GALIAS Controlled

Chua’s

ef trl

Oscl~lntor:

Period-?

i< shown in Fig. 9(a). By choosing the emitter resistor R,,! as the control parameter. we succeeded m stabilizing several periodic orbits of this system. The orbits shown in (h), (c) and (d) of Fig. 9 were stabilized using a window of 0.4 \’ placed at the bottom lek-hand side of the attractor. As with Hunt’s implementation of OPF control. the underlying unstable periodic orbits were found hy trial and error.

Electronic Controlled

Stabilixd

chaos controller

Chua’s

Period-l

Orbit.

Oscillator:

Vc,

t IVldiv)

tipper

TWX:

Other

Lower

Trace:

JFET gate Voltaes

Time:

Input

Voltage

Period-4

VsVc2

(tJ.?V/div)

(0.5Vldivl

tO.SV/div)

Sm\/dic

Fig. 6. Stabilization of a period-4 orbit in Chua’s oscillator: (a) Projection of the orbit in the state-space (I V/div) vs V,-? (0.2 V/div)): (b) Time waveforms of the output signal and the controlling signal (upper offset input voltage (0.5 V/div), lower trace: JFET gate voltage (0.5 V/div), time: 2 ms/div).

(C; trace

5. CONCLUSIONS

In this paper, we have described a modified controller for stabilizing periodic orbits in chaotic systems using a modified OPF method. We have developed some analytical results concerning the possibilities for stabilizing periodic orbits in chaotic systems using onedimensional methods. We stress that the only essential difference between the OGY and

1%. 7. Stabilization 01 a period-S orbit in Chua’s osciltator: (a) Projection of the orbit in the state-space 1 ii’/div) vs 1; z ((EV/div)): (b) Time waveforms of the output signal and the controlling signal (upper offset input voltapc fll z Vidivl. lower tmcr” JFE’T gate voltage (0.5 \‘/di\ 1. time: 2 ms/div).

(V<-, trace:

UPF methods is the number of variables used to determine the control signal. Furthermore, we have shown that the usual assumption, when using OPF, that the Poincark map is almost one-dimensional is unnecessary. The only assumptions that we need to determine if a given periodic: orbit can be stabilized are associated with the dynamics of the system in the neighborhood of the chosen oFbit.

Electronic

“cc T

L

chaos controller

Colpitts

RC

~

-,%%I,

Oscillator

Il)kQ ii’ f

Fig. 8. Colpitts oscillator

V-

13x3

with Interface Circuit

t,‘lts

:’

;f;“

L

2.?mH

Q,

2N2222.A

V+

1 I? Volt\

\‘-

-I’Volt5

with control interface circuitry.

Fig. 9. (a) Chaotic attractor observed in the Colpitts oscillator: (b) stabilized period-l orbit; (c) stabilized orbit; (d) stabilized period-3 orbit. All plots: k’<-(1 V/div) vs C; (0.5 V/div).

AckrloI~lrcigPnzenr-This Grant 8TllD03109.

research was supported

in part by the Polish Committee

of Scientific

period-2

Research (KBN).

REFERENCES I. Hunt. E. R.. Keeping chaos at bay. IEEE Specrrctnl. 1993,30,32-36. 2. Chen. G.. Control and synchronization of chaotic systems (bibliography).

EE Department,

University

of

Houston. TX: available tram t’fp: uhoop.egr.uh.edu!puhlTcXicha~~~.t~~ (Logan namti and password: bnth anonymous). 1996. 3 C‘hen. Ci. and Dong, X., From chaos to order-perspectives and methodologies in controlling chaotic nonlirwar dynamicat system< fnr. JmBifwcarion rind Chaos. 1993, 3, 1363-1409. 4. Ogorzakk, M. J,, Taming chaos: Part 2--Control. lEEE Tnwts. C’ircuif.~ Svmrnc. 1993. 41, 700-706. 5 Ogorzakk. M. J.. I‘ontrolling chaos in electronic circuit3. fVzllil~~.\.Trans. ??UV SCJC..L~wztlor~ Sw. A. 199.5. 353. IT- I.36. tr Ogorzalek. M. J.. Chaos control: how to av