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Design considerations for electronic chaos controllers
Chaos, Solitons & Fractat¢, Vol. 9, No. 112, pp. 295-306, 1998 ~) 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0960-0779198 $19.00+ 0.00
Pergamon
Pll: S0960-0779(97100068-4
Design Considerations for Electronic Chaos Controllers MACIEJ
J. O G O R Z A L E K
Department of Electrical Engineering, University of Mining and Metallurgy, al. Mickiewicza 30, 30-059 Krak6w, Poland
Abstract--We analyze general possibilities of electronic implementations of chaos controllers. We consider in particular two classes of chaos controllers: those based on the OGY approach, and occasional proportional feedback (OPF) controllers. Influence of accuracy of digital implementation of the OGY algorithm and its partial hardware implementation is discussed in detail. We describe which parts of the algorithm can be implemented in hardware and how to build a feedback controller with computer-assisted goal determination. As a second task, we consider the OPF controller introduced by Hunt for stabilizing periodic solutions in autonomous chaotic systems. Its original version has a major drawback because it requires an external synchronizing signal. We describe a modified version of Hunt's controller which obviates the need for an external synchronizing signal. Advantages and disadvantages of both methods from the implementation point of view are discussed in detail. © 1998 Elsevier Science Ltd. All rights reserved
1. INTRODUCTION Since the 1990 publication of the seminal paper by Ott, Grebogi and Yorke [1, 2] describing a chaos control method employing specific properties of chaotic motion, chaos control has become a lively studied subject. The widespread interest is basically due to extremely ~interesting and important possible applications. These applications include bio-medical ones (e.g. defibrillation or removal of epilleptic seizures), solid-state physics, lasers, aircraft wing vibrations and even weather control, to name just a few of the daring attempts. Looking at the possible applications alone, it becomes obvious that chaos control techniques and their possible implementations will greatly depend on the nature of the process under considerations. Looking from the control implementation perspective, real systems exhibiting chaotic behavior show many differences. The main ones are: • • • • • •
speed of the phenomena (frequency spectrum of the signals), amplitudes of the signals, existence of corrupting noises, their spectrum and amplitudes, accessibility of the signals for measurements, accessibility of the control (tuning) parameters, acceptable levels of control signals.
Looking for an implementation of a particular chaos controller, we must first look at the above system-induced limitations. How can we measure and process signals from the system? Are there any accessible system variables and parameters which could be used for control task? How do we choose the ones that offer best performance for achieving control? 295
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At what speed do we need to compute and apply the control signals? What is the lowest acceptable precision of computations? Can we achieve control in real-time? A slow system like a bouncing magneto-elastic ribbon (with eigen-frequencies below 1 Hz) is certainly not as demanding as a telecommunication chanel (running possibly at GHz) or a laser when it comes to control it! Considering electronic implementations, one must look at several closely linked areas: sensors (for measurements of signals frown a chaotic process), electronic implementation of the controllers, computer algorithms (if computers are involved in the control process) and actuators (introducing control signals into the system). External to the implementation (but directly involved in the control process and usually fixed based on the measured signals) is the finding of the goal of the control. Many methods have been developed and described in the literature [3-5]. However, most of them remain only of academic interest because of the lack of implementation issues. A control method cannot be accepted as a successful one if computer simulation experiments are not followed by further laboratory tests and physical implementations. Only very few results of such tests are known: among the exceptions are the control of a green light laser [6], the control of a magnetoellastic ribbon [7] and a few other examples. An obvious question arises at this point. Why, despite a wealth of developed methods, are there so few successful implementations and real applications'? In this paper, we try to answer at least partially this question looking at two most appealing methods, namely the O G Y technique and occasional proportional feedback (OPF) control. Among the approaches and methodologies for chaos control described in the literature [3, 8~ 9, 4, 5], these two approaches are of interest because they use the following specific properties of chaotic systems [10]. A chaotic attractor contains an infinite number of unstable periodic orbits embedded within; there exist dense orbits, in the sense that a typical trajectory on the attractor passes arbitrarily close to any point on it (and it also passes arbitrarily close to any of the unstable periodic orbits): very small signals are required to achieve control; and thus these approaches are more realistic for implementation purposes. From the implementation point of view. these two methods are very different. O G Y works on the basis of measured signals and uses a computer to find the goal of the control and make necessary calculations of the control signals: thus all the signals used in calculations are discretized both in time and space. OPF is purely analog: all operations are implemented in hardware. To consider the implementation limitations, let us first look at the principles of operation of both methods.
2. S H O R T
DESCRIPTION
OF THE OGY TECHNIQUE
The O G Y control method [1,2] developed in 1990 uses the two properties mentioned above. The goal of control is to stabilize one of the unstable periodic orbits by perturbing a chosen (accessible) system parameter over a small range about some nominal value. To explain in some detail the action of the O G Y method, let us assume for simplicity that we have a three-dimensional continuous-time system of first-order autonomous ordinary differential equations: dx -
dt
F(x,p),
(1)
where x e N3 is the state and p e l~ is a system parameter that we can change. We also assume that p can be modified within a small interval around its nominal value P0. Thus
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297
p E [Po -- ~Pmax,Po + ~Pmax], where ~Pmax is the maximum admissible change in p. We choose a two-dimensional Poincar6 surface Y. which defines a Poincar6 map P. For ~ ~ y , we denote by P(~) the point at which the trajectory starting from £ intersects X for the first time. Since the vector field F depends on p, the Poincar6 map P also depends on p. Thus, we have P:I~ 2 X R ~ (~,p) ~ P(~,p) E •2.
(2)
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 example, this could be the case of a chaotic laser intensity which is clearly an unwanted p h e n o m e n o n and the effective power of the laser beam can be enhanced using control to stabilize or eliminate chaotic behavior [6]. Another example of unwanted chaotic behavior is fibrillation, where the heart pumps blood in an inefficient manner. In this application, controlling the heartbeat into a nearly periodic regime is of paramount importance [11].) For simplicity, we assume that this is a period-1 orbit (a fixed point of the, map P). Let us denote by ~v an unstable fixed point of P for p = Po, so thai: P(~F,Po) = ~V. Let the first-order approximation of P in the neighborhood of (~F,P0) be of the form P(~,P) ~ P(~F,Po) + n ' ( ~ - ~F) + w ' ( p --PO),
(3)
where A is a Jacobian matrix of P(',Po) at ~F, and w = OP/Op(~F,Po)is the derivative of P with respect to p. Stabilization of the fixed point is achieved by realizing feedback of the form P(~) =P0 + cT(~ -- ~r:).
(4)
In the original description of the O G Y method [1], the vector c is computed using the expression
c:
Au
G
(5)
where Au is the unstable eigenvalue and fu is the unstable contravariant corresponding left eigenvector of A. Thus the O G Y method relies on a local linearization of the Poincard map in the neighborhood of the chosen unstable fixed point and local linear stabilizing feedback. An advantage of the O G Y method is that all the necessary calculations can be performed off-line on the basis of measurements. These include, for example, finding the unstable periodic orbits, fixing one of them as the goal of the control and computing the variables and parameters necessary for calculation 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 passes close to the fixed point (where the linearization is valid). The assumption about the existence of a dense orbit guarantees that 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 Dressier and Nitsche [12] have proposed a variant of the O G Y method in which only one variable is measured in the system and other variables needed for control are reconstructed using the delayed coordinate method. 3. IMPLEMENTATION PROBLEMS FOR THE OGY M E T H O D
When implementing the O G Y method for a real world application one has to carry out the following series of elementary operations:
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1. acquiring the data, i.e. measuring the (usually scalar) signal from the chaotic system considered. This operation should be performed in such a way as not to disturb the existing dynamics. For further computerised processing, measured signals must be sampled and digitized (A/D conversion): 2. determing a possible control parameter: 3. finding unstable periodic orbits using experimental data (measured time series) and fixing the goal of control: 4. finding parameters and variables necessary for control (as described above); 5. applying the control signal to the system. This step requires continuous measurement of system dynamics in order to determine the moment of application of the control signal (i.e. the moment when the actual trajectory passes in a small vicinity of the chosen periodic orbit) and immediate reaction of the controller (application of the control pulse) in such an event. In computer experiments it has been confirmed that all these steps can be successfully carried out in a great variety of systems thus achieving stabilization of even long-period orbits. There are however general problems when attempting to build an experimental set-up. Despite the fact that the variables and parameters can be calculated off-line, onc has to consider that the signals measured from the system are badly corrupted due to noise, several nonlinear operations associated with the A/D conversion (possibly rounding, truncation, finite wordlength, overflow correction, etc.). Using corrupted signal values and introducing additional errors by the computer algorithms and linearization used for control calculation may result in general failure of the method. Additionally, there arc time delays in the feedback loop, from, for example, a slow reaction of the computer and interrupts generated when sending and receiving data.
3.1 Effec,ts of calculation precision Below we illustrate in a simple example of the calculation of control parameters for stabilization of a fixed point (simplest case) in the Lozi map how A/D conversion accuracy and what follows limited precision calculations affect the possibilities of control. In the tests described below, we judged the quality of computations alone without looking at other problems such as time delays in the control loop. To be able to compare the results of digital manipulations, we first computed the interesting parameters using analytical formulas. We will assume that these are the true values of needed parameters: A. Conditions of calculations: analytical (theoretical results) Coordinates of the fixed point: (0.8879418373,0.8879418373) Control vector g: [0.4038961828,0.4038961828] Jacobian eigenvalues: - 1.913225419, - 0.1553417742 Stable direction: [0.1535007507,09881485105] Unstable direction: [0.8880129457, - 0.459818393] Possibilities of control: successful Further we calculated the same parameters using different wordlength, different arithmetics (overflow rules, rounding or truncation etc.). The results are summarised in B through F below: B. Conditions of calculations: floating point representation, 32 bit wordlength Coordinates of the fixed point: (0.883074402809143,0.881085395812988) Control vector g: [0.420236974954605,0.41124859422476] Jacobian eigenvalues: - 1.913303494453430, -0.0375206992030014
Design considerationsfor electronicchaos controllers
299
Stable direction: [0.111510016024113,0.993763267993927] Unstable direction: [0.882781267166138, - 0.4698422999382] Possibilities of control: successful C. Conditions of calculations: fixed point representation, 12 bit precision, rounding Coordinates of the fixed point: (0.8830,0.8810) Control vector g: [0.4351,0.4656] Jacobian eigenvalues: -1.9220, -0.0314 Stable direction: [0.1155,0.9932] Unstable direction: [0.8828,- 0.4695] Possibilities of control: successful D. Conditions of calculations: fixed point representation, 12 bit precision, truncation Coordinates of the fixed point: (0.8823,0.8803) Control vector g: [0.4562,0.4773] Jacobian eigenvalues: -1.9336, 0.0199 Stable direction: [0.1158,0.9932] Unstable direction: [0.8968, - 0.4422] Possibilities of control: successful E. Conditions of calculations: fixed point representation, 10 bit precision, rounding Coordinates of the fixed point: (0.883,0.881) Control vector g: [0.349,0.361] Jacobian eigenvalues: - 1.899, -0.020 Stable direction: [0.136,0.990] Unstable direction: [0.890, - 0.454] Possibilities of control: often fails F. Conditions of calculations: fixed point representation, 8 bit precision, rounding Coordinates of the fixed point: (0.88,0.88) Control vector g: [0.00,0.00] Jacobian eigenvalues: 0, 0 Stable direction: impossible to determine Unstable direction: impossible to determine Possibilities of control: impossible Comparing the results of computations summarised above, we can easily see that if we are able to achieve an accuracy of two to three decimal digits the calculations are precise enough to ensure proper functioning of the OGY algorithm in the case of the Lozi system. To have some safety margin and robustness of the algorithm, acceptable AID accuracy cannot be lower then 12 bit and probably it would be best to apply 16 bit conversion. This kind of accuracy is nowadays easily available using general purpose A / D converters even at speed of MHz range and higher. Implementing the algorithms, one must consider the cost of implementation: with growing precision and speed requirements, the cost grows exponentially. This issue might be a great limitation when it comes to IC implementations. 3.2 Approximate procedures for finding periodic orbits Another possible source of problems in the control procedure are the errors introduced by algorithms for finding periodic orbits (goals of the control). Using experimental data, we can only find approximations of unstable periodic orbits [13-15]. Commonly used is a simple technique proposed by Lathrop and Kostelich [14] for recovering unstable periodic orbits from an experimental time series. This procedure
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assumes that we have a series of successive points {xi}, for i - - 0 , 1 ..... N on the system trajectory and, taking one of these points x,,, we search forward for the smallest positive integer k such that tlx,,, +k -x,,, II < e, where e is the specified accuracy. It is further claimed that the orbit detected in this manner lies close to the unstable periodic orbit whose period is approximated by that of the detected sequence. This approach has several drawbacks. Firstly, the results strongly depend on the choice of e and the length of the measured time series. Further, they depend on the choice of norm and n u m b e r of state variables analysed and, moreover, the stopping criterion JIx.... k - x,,,jJ < ~', in the case of discretely sampled continuous-time systems, is not precise enough. This means that one can never be sure how many orbits have been found or whether all orbits of a given period have been recovered. In m a n y applications, however, it is sufficient to find only some of the unstable periodic orbits e m b e d d e d in the attractor and choose one of them. We have developed a set of computer programs [10] for detecting unstable periodic orbits. The L a t h r o p - K o s t e l i c h procedure has been refined by means of a stopping condition based on the distance of the initial point x,,, from the evolving trajectory, not from distinct points belonging to it which are sampled discretely in time ( D / A conversion of measured signals) or computed via numerical integration. (Thus we avoid the problem of not detecting an orbit when x,,, falls between two successive points along the trajectory.) This slightly slows down the computations but the results are more reliable: the problem of not linding some of the orbits due to distance mismatch can be aw)ided. In our experiments [10] we varied the e between 0.000001 and 0.001 and fixed the threshold for distinguishing between the orbits at 0.001. Although with greater ~" more orbits with given period were detected, most of them were later recognised as identical; there was no significant difference in the n u m b e r and shape of different unstable periodic orbits found. As this step is typically carried out off-line, it does not badly affect the whole control procedure. It has been found in experiments that, when the tolerances for detection of unstable orbits were chosen too big, the actual trajectory stabilized during control showed greater variations and the control signal had to be applied every iteration to compensate for inaccuracies. Obviously making the tolerance too big would cause failure of control. Some new methods have been proposed recently [16] which could possibly improve localization procedures for unstable periodic orbits.
~ffects of time delays Several elements in the control loop may introduce time delays that are crucial for functioning of O G Y method. Although all calculations may be done off-line, two steps ~lre of p a r a m o u n t importance: 3.3
• •
detection of the m o m e n t when the trajectory passes the chosen Poincar6 section, determination of the m o m e n t of application of the control signal (close neighborhood of chosen orbit).
When these two steps are carried out by a computer with a lab card, at least a few interrupts (causing time delays) must be generated in order to detect the Poincar6 section and take the decision of being in the right neighborhood and sending the ready control signal. In most experiments with O G Y control of electronic circuits, we were not able to achieve control when the circuits were running in the 10-100 Hz range. We found that, for higher frequency systems, time delays become a crucial point in the whole procedure. The failure of control was mainly due to late arrival of the control pulse: the system was being controlled at a wrong point in space, where the formulas used for calculations were probably no longer
Design considerations for electronic chaos controllers
la
301
d
i
b
_
~x+by+cz+d--O
I n
E O o
I MDAC Fig. 1. Fast Poincar6 section detector for improved OGY implementation.
valid. The trajectory was already far away from the section plane when the control pulse arrived. T o compensate for at least some of the delays, we proposed a hardware solution for a detector of the Poincar6 section and vicinity detector. Block diagrams of these two pieces of equipment are shown in Figs 1 and 2. The Poincar6 section system uses here all (three in our application) state variables to simplify detection. To implement this function using just one variable delay coordinates must be introduced. Realization in hardware would become much more complicated in this case if possible at all (one could think of calculation of suitable time delay by a computer algorithm and storage of necessary time-delayed samples in special-purpose registers).
4. BRIEF RECALL OF THE OPF (HUNT'S) CONTROLLER The occasional proportional feedback (OPF) technique [17, 18, 6] can be considered as a one-dimensional version of the O G Y method. Let us describe the action of the OPF controller for the case of stabilizing a fixed point of the Poincar6 map.
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M.J. OGORZALEK
measured system variables x Y
p - parameter to the system
J t neighborhood detector
trigger signal
~ ~ ' ~ l
__z
1 p - parameter from computer x*
y*
z*
coordinates of the periodic point Fig. 2. ~'-comparalor for detection of the vicinity of a desired periodic orbit.
In the O P F method, the control signal is computed using only one variable, say (t: p(~)
= p , , + c(~. , -
(6)
.~,~).
Adjusting the values of c for which (v is a stable fixed point of the system ~--* P(~,p(()) ensures the proper functioning of the algorithm. In [19,201, we have described some theoretical results concerning the choice of coefficients and possibilities of successful OPF control. The best results are obtained if the unstable eigenvector is parallel to the coordinate which is used for c o m p u t i n g the control signal and the possibility of control using the O P F technique d e p e n d s on the t'orm of the linear a p p r o x i m a t i o n of the system's behavior in the n e i g h b o r h o o d of the periodic orbit. A schematic of H u n t ' s i m p l e m e n t a t i o n of the O P F control m e t h o d is shown in Fig. 3. The
Sample & H o l d Amplifier Input Waveform
I
-1
Analog Gate
~
T Window Comparator
External I Synchronizing Genarator [
J -I
Timing Circuitry
Delay Element
Fig. 3. Hunt's implementation of the OPF method.
To
interface circuit ~-
Design considerationsfor electronicchaos controllers
303
window comparator, taking the input waveform, gives a logical high when the input waveform is inside the window. This is then ' A N D ' e d 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: they fix the position of the section plane, values of P0 and c. The interface circuit depends explicitly on the chaotic system under control. One of the major advantages of Hunt's controller over O G Y is that the control law depends on one variable only and does not require any complicated calculations (as was the case of O G Y scheme) in o r d e r to generate the required control signal. All the operations can easily be performed by hardware function blocks. 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 O G Y method, let us summarize the main features of the OPF controller: • • • •
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.
4.1 Improved chaos controller for autonomous circuits Recently we have developed an improved chaos controller modifying the essential set-up proposed by Hunt [21, 19, 20]. The modified controller uses Hunt's method without the need for an external synchronising oscillator. Its circuit diagram is shown in Fig. 4. In the modified controller [21], the derivative of the input signal is taken to generate a pulse when it passes through zero. This pulse replaces the driving pulses from the external oscillator as the 'synch' pulse for our Poincar6 map. This obviates the need for the external generator and so makes the controller simpler and cheaper to build. The variable level window comparator is implemented 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 controller 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 monostable flip-flop. The falling edge of this monostable's pulse triggers another monostable, giving a delay. We use the monostable'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 falling 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 modified controller offers, in our opinion, the simplest and most reliable implementation for chaos control. It has been tested successfully on Chua's circuit and Colpitts osciallator working in the kHz range [21, 19, 20] enabling stabilization of unstable periodic orbits up to order 8. Listing its advantages we can summarise that:
304
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OGORZA~EK
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Design considerations for electronic chaos controllers • • • • •
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it uses just one system variable as input; a window around a fixed level sets the region where control is applied; it overcomes the main drawback of using external synchronization; the speed of control is only limited by the frequency limitations of the components used and so one can easily visualise adopting the scheme to controlling even very fast systems; the controller implementation is cheap, simple and easy to build; IC implementation remains one of the possible future issues.
5. CONCLUSIONS There exist several limitations for implementations of electronic chaos controllers. These limitations are imposed by the speed of the considered chaotic system (highest frequency), accuracy of the measurements, errors introduced by approximations and signal processing algorithms and calculations. Taking into account these limitations when looking for an implementation of a controller, we must make a trade-off between high speed (analog implementation without possibilities of pre-specification of the goal, trial-and-error procedure for achieving the desired behavior) and precise knowledge of the orbit which is interesting as a goal of the control. The modified OPF controller works well in many high frequency systems eliminating chaos but without prior knowledge of attainable orbits. The O G Y method is very attractive when precise knowledge of the goal is needed (stabilized orbits offer some optimal type of performance [22]) but it is possible to implement it in very low frequency (slow) systems only. In this study, we did not consider the actuator design problem which, depending on the real application, might pose specific problems. For example, the application of defibrillating signal to the heart might be of paramount importance far above any of the controller and algorithm design problems. The following are interesting areas of further research and developments: • • • •
hardware implementation of goal (unstable orbit) detection for use with the modifications of analog OPF method; development of specific hardware for use with O G Y method; implementation of both methods in high-order (possibly hyper-chaotic) systems; IC implementations of controllers.
Finally, we should mention that there exist other control schemes, such as the delayed feedback method introduced by Pyragas [9], where certainly the controller is the cheapest possible (delay line which in some cases might be even a piece of cable): there is however no tuning possibilities whatsoever when it comes to real implementations. In our opinion, such controllers might also find particular applications but are of little versatility and for this reason we did not look closer into them. Acknowledgement--This research has been supported in part by the Polish Committee of Scientific Research
(KBN), grant 8TllD03109.
REFERENCES
1. Ott, E., Grebogi, C. and Yorke, J. A., Controlling chaos. Phys. Rev. Lett., 1990,6411, 1196-1199. 2. Ott, E., Grebogi, C. and Yorke, J. A., Controlling chaotic dynamical systems. In Chaos: Soviet-American Perspectives on Nonlinear Science, ed. D. K. Campbell. American Institute of Physics, New York, 1990. 3. Chen, G., Control and synchronization of chaotic systems (bibliography). EE Dept, Univ of Houston, TX,
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M. J. O G O R Z A L E K available from ftp: 'uhoop.egr.uh.edu/pub/TeX/chaos.tex" (login name and password: both 'anonymous'), 1996. Ogorzalek. M. J., Chaos control: How to avoid chaos or take advantage of it. J Franklin hTst. B, 1994. 3316, 681-704. Ogorzalek, M. J., Controlling chaos in electronic circuits. Philos. Trans. Roy. Soc, London Ser. A, 1995, 353, 127-136. Roy, R., Murphy, T. W. Jr., Maier, T. D., Gillis, Z, and Hunt, E. R.. Dynamical control of a chaotic laser: Experimental stabilization of a globally coupled system. Phys. Rev. Lett., 1992, 689, 1259--1262. Ditto, W. L., Rauseo, S. N. and Spano, M. I_.., Experimental control of chaos. Pin's. Re~,. Lctg., 199/), 65, 32i 1 3214. ('hen, G. and Dong, X., From chaos to order---Perspectives and methodologies in controlling chaotic nonlinear dynamical systems, h~ternat. ,I. B([hr. Chaos, 1993, 3, 1363-1409. Ogorzalek, M. ,I., Taming chaos: Part 2-control. I E E E Trans. Circuits S wwem.~, b~93, 411111, 7(10 70,'3. Ogorzalek, M. J. and Galias. Z., Characterization of chaos in Chml's oscillator in terms of unstable pcriodic orbits, d. Circuit,s Systems Comput., 1993, 32, 411 429. Garlinkel, A., Spano, M. L.. Ditto, W. L. and Weiss, .I.N., ('ontrolling cardiac chaos. Science, 257, 1992, 1230 1235. l)ressler, U. and Nitsche, G., Controlling chaos using time delay coordinates. Phv,s, Rev. Lett., 1992, 68, 1 4. Auerbach, D., Cvitanovid, P., Eckmann, J.-P., Gunaratne, G. and Procaccia, I.. Exploring ehaolic motion through periodic orbits. Phys. Ret,. Lett., 1987, 58, 2387-2389. Lalhrop, D. P. and Kostelich, E. J., Characterization of an experimental strange attractor by periodic orbits, Phys, Rev. A, 1989, 411'7, 4028-4032. Schwartz, 1. B., Estimating regions of existence of unstable periodic orbits using computer-based leehniques. S I A M Y. Numer. Anal., 1983, 2t11, 106-120. So, P.. Otl, E., Schiff, S. J., Kaplan, D. T., Sauer, T. and Grebogi. C., Detecting unstable pcliodic orbits in chaotic experimental data. Phys. Rev. Left., 1996, 76, 4705. Hunt, E. R., Stabilizing high-period orbits in a chaotic system: The diode resonator. Phvrs. Roy. Left., 1991, 6715, 1953-1955. Johnson, G. A., Tigner, T. E. and tlunt, E. R., Controlling chaos in Chua's circuit..I. ('ircuiL~ Sv.vtems Cornput., 1993, 31, 109-117. Galias, Z., Murphy, C. A., Kennedy, M. P. and Ogorzalek, M. J., A feedback chaos conlrotler: Theory and implementation. In Proc. 1SCAS'96. Athlnta, Vol. 3, 1996, pp. 121)-124. Galias, Z., Murphy, C, A., Kennedy, M. P. and Ogorzalek, M. J., Electronic chaos controller. Cha()~, Solitons ~¢: FractaZs (in press). Murphy, C, A. and Kennedy, M. P., Chaos controller for autonomous circuits. In Proc. NI)ES'95, Dublin, 28-29 July 1995, pp. 225-228. Hunt, B. R. and Ott. E.. Optimal periodic orbits of chaotic systems, lVtvs. Rel,. Lctt.. 1996.76, 2254-2257.
Pergamon
Pll: S0960-0779(97100068-4
Design Considerations for Electronic Chaos Controllers MACIEJ
J. O G O R Z A L E K
Department of Electrical Engineering, University of Mining and Metallurgy, al. Mickiewicza 30, 30-059 Krak6w, Poland
Abstract--We analyze general possibilities of electronic implementations of chaos controllers. We consider in particular two classes of chaos controllers: those based on the OGY approach, and occasional proportional feedback (OPF) controllers. Influence of accuracy of digital implementation of the OGY algorithm and its partial hardware implementation is discussed in detail. We describe which parts of the algorithm can be implemented in hardware and how to build a feedback controller with computer-assisted goal determination. As a second task, we consider the OPF controller introduced by Hunt for stabilizing periodic solutions in autonomous chaotic systems. Its original version has a major drawback because it requires an external synchronizing signal. We describe a modified version of Hunt's controller which obviates the need for an external synchronizing signal. Advantages and disadvantages of both methods from the implementation point of view are discussed in detail. © 1998 Elsevier Science Ltd. All rights reserved
1. INTRODUCTION Since the 1990 publication of the seminal paper by Ott, Grebogi and Yorke [1, 2] describing a chaos control method employing specific properties of chaotic motion, chaos control has become a lively studied subject. The widespread interest is basically due to extremely ~interesting and important possible applications. These applications include bio-medical ones (e.g. defibrillation or removal of epilleptic seizures), solid-state physics, lasers, aircraft wing vibrations and even weather control, to name just a few of the daring attempts. Looking at the possible applications alone, it becomes obvious that chaos control techniques and their possible implementations will greatly depend on the nature of the process under considerations. Looking from the control implementation perspective, real systems exhibiting chaotic behavior show many differences. The main ones are: • • • • • •
speed of the phenomena (frequency spectrum of the signals), amplitudes of the signals, existence of corrupting noises, their spectrum and amplitudes, accessibility of the signals for measurements, accessibility of the control (tuning) parameters, acceptable levels of control signals.
Looking for an implementation of a particular chaos controller, we must first look at the above system-induced limitations. How can we measure and process signals from the system? Are there any accessible system variables and parameters which could be used for control task? How do we choose the ones that offer best performance for achieving control? 295
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At what speed do we need to compute and apply the control signals? What is the lowest acceptable precision of computations? Can we achieve control in real-time? A slow system like a bouncing magneto-elastic ribbon (with eigen-frequencies below 1 Hz) is certainly not as demanding as a telecommunication chanel (running possibly at GHz) or a laser when it comes to control it! Considering electronic implementations, one must look at several closely linked areas: sensors (for measurements of signals frown a chaotic process), electronic implementation of the controllers, computer algorithms (if computers are involved in the control process) and actuators (introducing control signals into the system). External to the implementation (but directly involved in the control process and usually fixed based on the measured signals) is the finding of the goal of the control. Many methods have been developed and described in the literature [3-5]. However, most of them remain only of academic interest because of the lack of implementation issues. A control method cannot be accepted as a successful one if computer simulation experiments are not followed by further laboratory tests and physical implementations. Only very few results of such tests are known: among the exceptions are the control of a green light laser [6], the control of a magnetoellastic ribbon [7] and a few other examples. An obvious question arises at this point. Why, despite a wealth of developed methods, are there so few successful implementations and real applications'? In this paper, we try to answer at least partially this question looking at two most appealing methods, namely the O G Y technique and occasional proportional feedback (OPF) control. Among the approaches and methodologies for chaos control described in the literature [3, 8~ 9, 4, 5], these two approaches are of interest because they use the following specific properties of chaotic systems [10]. A chaotic attractor contains an infinite number of unstable periodic orbits embedded within; there exist dense orbits, in the sense that a typical trajectory on the attractor passes arbitrarily close to any point on it (and it also passes arbitrarily close to any of the unstable periodic orbits): very small signals are required to achieve control; and thus these approaches are more realistic for implementation purposes. From the implementation point of view. these two methods are very different. O G Y works on the basis of measured signals and uses a computer to find the goal of the control and make necessary calculations of the control signals: thus all the signals used in calculations are discretized both in time and space. OPF is purely analog: all operations are implemented in hardware. To consider the implementation limitations, let us first look at the principles of operation of both methods.
2. S H O R T
DESCRIPTION
OF THE OGY TECHNIQUE
The O G Y control method [1,2] developed in 1990 uses the two properties mentioned above. The goal of control is to stabilize one of the unstable periodic orbits by perturbing a chosen (accessible) system parameter over a small range about some nominal value. To explain in some detail the action of the O G Y method, let us assume for simplicity that we have a three-dimensional continuous-time system of first-order autonomous ordinary differential equations: dx -
dt
F(x,p),
(1)
where x e N3 is the state and p e l~ is a system parameter that we can change. We also assume that p can be modified within a small interval around its nominal value P0. Thus
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p E [Po -- ~Pmax,Po + ~Pmax], where ~Pmax is the maximum admissible change in p. We choose a two-dimensional Poincar6 surface Y. which defines a Poincar6 map P. For ~ ~ y , we denote by P(~) the point at which the trajectory starting from £ intersects X for the first time. Since the vector field F depends on p, the Poincar6 map P also depends on p. Thus, we have P:I~ 2 X R ~ (~,p) ~ P(~,p) E •2.
(2)
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 example, this could be the case of a chaotic laser intensity which is clearly an unwanted p h e n o m e n o n and the effective power of the laser beam can be enhanced using control to stabilize or eliminate chaotic behavior [6]. Another example of unwanted chaotic behavior is fibrillation, where the heart pumps blood in an inefficient manner. In this application, controlling the heartbeat into a nearly periodic regime is of paramount importance [11].) For simplicity, we assume that this is a period-1 orbit (a fixed point of the, map P). Let us denote by ~v an unstable fixed point of P for p = Po, so thai: P(~F,Po) = ~V. Let the first-order approximation of P in the neighborhood of (~F,P0) be of the form P(~,P) ~ P(~F,Po) + n ' ( ~ - ~F) + w ' ( p --PO),
(3)
where A is a Jacobian matrix of P(',Po) at ~F, and w = OP/Op(~F,Po)is the derivative of P with respect to p. Stabilization of the fixed point is achieved by realizing feedback of the form P(~) =P0 + cT(~ -- ~r:).
(4)
In the original description of the O G Y method [1], the vector c is computed using the expression
c:
Au
G
(5)
where Au is the unstable eigenvalue and fu is the unstable contravariant corresponding left eigenvector of A. Thus the O G Y method relies on a local linearization of the Poincard map in the neighborhood of the chosen unstable fixed point and local linear stabilizing feedback. An advantage of the O G Y method is that all the necessary calculations can be performed off-line on the basis of measurements. These include, for example, finding the unstable periodic orbits, fixing one of them as the goal of the control and computing the variables and parameters necessary for calculation 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 passes close to the fixed point (where the linearization is valid). The assumption about the existence of a dense orbit guarantees that 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 Dressier and Nitsche [12] have proposed a variant of the O G Y method in which only one variable is measured in the system and other variables needed for control are reconstructed using the delayed coordinate method. 3. IMPLEMENTATION PROBLEMS FOR THE OGY M E T H O D
When implementing the O G Y method for a real world application one has to carry out the following series of elementary operations:
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1. acquiring the data, i.e. measuring the (usually scalar) signal from the chaotic system considered. This operation should be performed in such a way as not to disturb the existing dynamics. For further computerised processing, measured signals must be sampled and digitized (A/D conversion): 2. determing a possible control parameter: 3. finding unstable periodic orbits using experimental data (measured time series) and fixing the goal of control: 4. finding parameters and variables necessary for control (as described above); 5. applying the control signal to the system. This step requires continuous measurement of system dynamics in order to determine the moment of application of the control signal (i.e. the moment when the actual trajectory passes in a small vicinity of the chosen periodic orbit) and immediate reaction of the controller (application of the control pulse) in such an event. In computer experiments it has been confirmed that all these steps can be successfully carried out in a great variety of systems thus achieving stabilization of even long-period orbits. There are however general problems when attempting to build an experimental set-up. Despite the fact that the variables and parameters can be calculated off-line, onc has to consider that the signals measured from the system are badly corrupted due to noise, several nonlinear operations associated with the A/D conversion (possibly rounding, truncation, finite wordlength, overflow correction, etc.). Using corrupted signal values and introducing additional errors by the computer algorithms and linearization used for control calculation may result in general failure of the method. Additionally, there arc time delays in the feedback loop, from, for example, a slow reaction of the computer and interrupts generated when sending and receiving data.
3.1 Effec,ts of calculation precision Below we illustrate in a simple example of the calculation of control parameters for stabilization of a fixed point (simplest case) in the Lozi map how A/D conversion accuracy and what follows limited precision calculations affect the possibilities of control. In the tests described below, we judged the quality of computations alone without looking at other problems such as time delays in the control loop. To be able to compare the results of digital manipulations, we first computed the interesting parameters using analytical formulas. We will assume that these are the true values of needed parameters: A. Conditions of calculations: analytical (theoretical results) Coordinates of the fixed point: (0.8879418373,0.8879418373) Control vector g: [0.4038961828,0.4038961828] Jacobian eigenvalues: - 1.913225419, - 0.1553417742 Stable direction: [0.1535007507,09881485105] Unstable direction: [0.8880129457, - 0.459818393] Possibilities of control: successful Further we calculated the same parameters using different wordlength, different arithmetics (overflow rules, rounding or truncation etc.). The results are summarised in B through F below: B. Conditions of calculations: floating point representation, 32 bit wordlength Coordinates of the fixed point: (0.883074402809143,0.881085395812988) Control vector g: [0.420236974954605,0.41124859422476] Jacobian eigenvalues: - 1.913303494453430, -0.0375206992030014
Design considerationsfor electronicchaos controllers
299
Stable direction: [0.111510016024113,0.993763267993927] Unstable direction: [0.882781267166138, - 0.4698422999382] Possibilities of control: successful C. Conditions of calculations: fixed point representation, 12 bit precision, rounding Coordinates of the fixed point: (0.8830,0.8810) Control vector g: [0.4351,0.4656] Jacobian eigenvalues: -1.9220, -0.0314 Stable direction: [0.1155,0.9932] Unstable direction: [0.8828,- 0.4695] Possibilities of control: successful D. Conditions of calculations: fixed point representation, 12 bit precision, truncation Coordinates of the fixed point: (0.8823,0.8803) Control vector g: [0.4562,0.4773] Jacobian eigenvalues: -1.9336, 0.0199 Stable direction: [0.1158,0.9932] Unstable direction: [0.8968, - 0.4422] Possibilities of control: successful E. Conditions of calculations: fixed point representation, 10 bit precision, rounding Coordinates of the fixed point: (0.883,0.881) Control vector g: [0.349,0.361] Jacobian eigenvalues: - 1.899, -0.020 Stable direction: [0.136,0.990] Unstable direction: [0.890, - 0.454] Possibilities of control: often fails F. Conditions of calculations: fixed point representation, 8 bit precision, rounding Coordinates of the fixed point: (0.88,0.88) Control vector g: [0.00,0.00] Jacobian eigenvalues: 0, 0 Stable direction: impossible to determine Unstable direction: impossible to determine Possibilities of control: impossible Comparing the results of computations summarised above, we can easily see that if we are able to achieve an accuracy of two to three decimal digits the calculations are precise enough to ensure proper functioning of the OGY algorithm in the case of the Lozi system. To have some safety margin and robustness of the algorithm, acceptable AID accuracy cannot be lower then 12 bit and probably it would be best to apply 16 bit conversion. This kind of accuracy is nowadays easily available using general purpose A / D converters even at speed of MHz range and higher. Implementing the algorithms, one must consider the cost of implementation: with growing precision and speed requirements, the cost grows exponentially. This issue might be a great limitation when it comes to IC implementations. 3.2 Approximate procedures for finding periodic orbits Another possible source of problems in the control procedure are the errors introduced by algorithms for finding periodic orbits (goals of the control). Using experimental data, we can only find approximations of unstable periodic orbits [13-15]. Commonly used is a simple technique proposed by Lathrop and Kostelich [14] for recovering unstable periodic orbits from an experimental time series. This procedure
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assumes that we have a series of successive points {xi}, for i - - 0 , 1 ..... N on the system trajectory and, taking one of these points x,,, we search forward for the smallest positive integer k such that tlx,,, +k -x,,, II < e, where e is the specified accuracy. It is further claimed that the orbit detected in this manner lies close to the unstable periodic orbit whose period is approximated by that of the detected sequence. This approach has several drawbacks. Firstly, the results strongly depend on the choice of e and the length of the measured time series. Further, they depend on the choice of norm and n u m b e r of state variables analysed and, moreover, the stopping criterion JIx.... k - x,,,jJ < ~', in the case of discretely sampled continuous-time systems, is not precise enough. This means that one can never be sure how many orbits have been found or whether all orbits of a given period have been recovered. In m a n y applications, however, it is sufficient to find only some of the unstable periodic orbits e m b e d d e d in the attractor and choose one of them. We have developed a set of computer programs [10] for detecting unstable periodic orbits. The L a t h r o p - K o s t e l i c h procedure has been refined by means of a stopping condition based on the distance of the initial point x,,, from the evolving trajectory, not from distinct points belonging to it which are sampled discretely in time ( D / A conversion of measured signals) or computed via numerical integration. (Thus we avoid the problem of not detecting an orbit when x,,, falls between two successive points along the trajectory.) This slightly slows down the computations but the results are more reliable: the problem of not linding some of the orbits due to distance mismatch can be aw)ided. In our experiments [10] we varied the e between 0.000001 and 0.001 and fixed the threshold for distinguishing between the orbits at 0.001. Although with greater ~" more orbits with given period were detected, most of them were later recognised as identical; there was no significant difference in the n u m b e r and shape of different unstable periodic orbits found. As this step is typically carried out off-line, it does not badly affect the whole control procedure. It has been found in experiments that, when the tolerances for detection of unstable orbits were chosen too big, the actual trajectory stabilized during control showed greater variations and the control signal had to be applied every iteration to compensate for inaccuracies. Obviously making the tolerance too big would cause failure of control. Some new methods have been proposed recently [16] which could possibly improve localization procedures for unstable periodic orbits.
~ffects of time delays Several elements in the control loop may introduce time delays that are crucial for functioning of O G Y method. Although all calculations may be done off-line, two steps ~lre of p a r a m o u n t importance: 3.3
• •
detection of the m o m e n t when the trajectory passes the chosen Poincar6 section, determination of the m o m e n t of application of the control signal (close neighborhood of chosen orbit).
When these two steps are carried out by a computer with a lab card, at least a few interrupts (causing time delays) must be generated in order to detect the Poincar6 section and take the decision of being in the right neighborhood and sending the ready control signal. In most experiments with O G Y control of electronic circuits, we were not able to achieve control when the circuits were running in the 10-100 Hz range. We found that, for higher frequency systems, time delays become a crucial point in the whole procedure. The failure of control was mainly due to late arrival of the control pulse: the system was being controlled at a wrong point in space, where the formulas used for calculations were probably no longer
Design considerations for electronic chaos controllers
la
301
d
i
b
_
~x+by+cz+d--O
I n
E O o
I MDAC Fig. 1. Fast Poincar6 section detector for improved OGY implementation.
valid. The trajectory was already far away from the section plane when the control pulse arrived. T o compensate for at least some of the delays, we proposed a hardware solution for a detector of the Poincar6 section and vicinity detector. Block diagrams of these two pieces of equipment are shown in Figs 1 and 2. The Poincar6 section system uses here all (three in our application) state variables to simplify detection. To implement this function using just one variable delay coordinates must be introduced. Realization in hardware would become much more complicated in this case if possible at all (one could think of calculation of suitable time delay by a computer algorithm and storage of necessary time-delayed samples in special-purpose registers).
4. BRIEF RECALL OF THE OPF (HUNT'S) CONTROLLER The occasional proportional feedback (OPF) technique [17, 18, 6] can be considered as a one-dimensional version of the O G Y method. Let us describe the action of the OPF controller for the case of stabilizing a fixed point of the Poincar6 map.
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M.J. OGORZALEK
measured system variables x Y
p - parameter to the system
J t neighborhood detector
trigger signal
~ ~ ' ~ l
__z
1 p - parameter from computer x*
y*
z*
coordinates of the periodic point Fig. 2. ~'-comparalor for detection of the vicinity of a desired periodic orbit.
In the O P F method, the control signal is computed using only one variable, say (t: p(~)
= p , , + c(~. , -
(6)
.~,~).
Adjusting the values of c for which (v is a stable fixed point of the system ~--* P(~,p(()) ensures the proper functioning of the algorithm. In [19,201, we have described some theoretical results concerning the choice of coefficients and possibilities of successful OPF control. The best results are obtained if the unstable eigenvector is parallel to the coordinate which is used for c o m p u t i n g the control signal and the possibility of control using the O P F technique d e p e n d s on the t'orm of the linear a p p r o x i m a t i o n of the system's behavior in the n e i g h b o r h o o d of the periodic orbit. A schematic of H u n t ' s i m p l e m e n t a t i o n of the O P F control m e t h o d is shown in Fig. 3. The
Sample & H o l d Amplifier Input Waveform
I
-1
Analog Gate
~
T Window Comparator
External I Synchronizing Genarator [
J -I
Timing Circuitry
Delay Element
Fig. 3. Hunt's implementation of the OPF method.
To
interface circuit ~-
Design considerationsfor electronicchaos controllers
303
window comparator, taking the input waveform, gives a logical high when the input waveform is inside the window. This is then ' A N D ' e d 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: they fix the position of the section plane, values of P0 and c. The interface circuit depends explicitly on the chaotic system under control. One of the major advantages of Hunt's controller over O G Y is that the control law depends on one variable only and does not require any complicated calculations (as was the case of O G Y scheme) in o r d e r to generate the required control signal. All the operations can easily be performed by hardware function blocks. 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 O G Y method, let us summarize the main features of the OPF controller: • • • •
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.
4.1 Improved chaos controller for autonomous circuits Recently we have developed an improved chaos controller modifying the essential set-up proposed by Hunt [21, 19, 20]. The modified controller uses Hunt's method without the need for an external synchronising oscillator. Its circuit diagram is shown in Fig. 4. In the modified controller [21], the derivative of the input signal is taken to generate a pulse when it passes through zero. This pulse replaces the driving pulses from the external oscillator as the 'synch' pulse for our Poincar6 map. This obviates the need for the external generator and so makes the controller simpler and cheaper to build. The variable level window comparator is implemented 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 controller 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 monostable flip-flop. The falling edge of this monostable's pulse triggers another monostable, giving a delay. We use the monostable'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 falling 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 modified controller offers, in our opinion, the simplest and most reliable implementation for chaos control. It has been tested successfully on Chua's circuit and Colpitts osciallator working in the kHz range [21, 19, 20] enabling stabilization of unstable periodic orbits up to order 8. Listing its advantages we can summarise that:
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Design considerations for electronic chaos controllers • • • • •
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it uses just one system variable as input; a window around a fixed level sets the region where control is applied; it overcomes the main drawback of using external synchronization; the speed of control is only limited by the frequency limitations of the components used and so one can easily visualise adopting the scheme to controlling even very fast systems; the controller implementation is cheap, simple and easy to build; IC implementation remains one of the possible future issues.
5. CONCLUSIONS There exist several limitations for implementations of electronic chaos controllers. These limitations are imposed by the speed of the considered chaotic system (highest frequency), accuracy of the measurements, errors introduced by approximations and signal processing algorithms and calculations. Taking into account these limitations when looking for an implementation of a controller, we must make a trade-off between high speed (analog implementation without possibilities of pre-specification of the goal, trial-and-error procedure for achieving the desired behavior) and precise knowledge of the orbit which is interesting as a goal of the control. The modified OPF controller works well in many high frequency systems eliminating chaos but without prior knowledge of attainable orbits. The O G Y method is very attractive when precise knowledge of the goal is needed (stabilized orbits offer some optimal type of performance [22]) but it is possible to implement it in very low frequency (slow) systems only. In this study, we did not consider the actuator design problem which, depending on the real application, might pose specific problems. For example, the application of defibrillating signal to the heart might be of paramount importance far above any of the controller and algorithm design problems. The following are interesting areas of further research and developments: • • • •
hardware implementation of goal (unstable orbit) detection for use with the modifications of analog OPF method; development of specific hardware for use with O G Y method; implementation of both methods in high-order (possibly hyper-chaotic) systems; IC implementations of controllers.
Finally, we should mention that there exist other control schemes, such as the delayed feedback method introduced by Pyragas [9], where certainly the controller is the cheapest possible (delay line which in some cases might be even a piece of cable): there is however no tuning possibilities whatsoever when it comes to real implementations. In our opinion, such controllers might also find particular applications but are of little versatility and for this reason we did not look closer into them. Acknowledgement--This research has been supported in part by the Polish Committee of Scientific Research
(KBN), grant 8TllD03109.
REFERENCES
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