626
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS-11: ANALOG AND DIGITAL SIGNAL PROCESSING, VOL. 40, NO. 10, OCTOBER 1993
Synchronization of Lorenz-B ased Chaotic
Circuits with Applications to Communications
Kevin M. Cuomo, Member, IEEE, Alan V. Oppenheim, Fellow, IEEE, and Steven H. Strogatz
Abstruct- A circuit implementation of the chaotic Lorenz
system is described. The chaotic behavior of the circuit closely
matches the results predicted by numerical experiments. Using
the concept of synchronized chaotic systems (SCS’s), two possible
approaches to secure communications are demonstrated with the
Lorenz circuit implemented in both the transmitter and receiver.
In the first approach, a chaotic masking signal is added at the
transmitter to the message, and at the receiver, the masking
is regenerated and subtracted from the received signal. The
second approach utilizes modulation of the coefficients of the
chaotic system in the transmitter and corresponding detection of
synchronization error in the receiver to transmit binary-valued
bit streams.
The use of SCS’s for communications relies on the robustness
of the synchronization to perturbations in the drive signal. As
a step toward further understanding the inherent robustness,
we establish an analogy between synchronization in chaotic systems, nonlinear observers for deterministic systems, and state
estimation in probabilistic systems. This analogy exists because
SCS’s can be viewed as performing the role of a nonlinear state
space observer. To calibrate the robustness of the Lorenz SCS
as a nonlinear state estimator, we compare the performance
of the Lorenz SCS to an extended Kalman filter for providing
state estimates when the measurement consists of a single noisy
transmitter component.
1. INTRODUCTION
C
HAOTIC systems provide a rich mechanism for signal
design and generation, with potential applications to
communications and signal processing. Because chaotic signals are typically broadband, noiselike, and difficult to predict,
they can be used in various contexts for masking informationbearing waveforms. They can also be used as modulating
waveforms in spread spectrum systems.
A particularly useful class of chaotic systems are those
that possess a self-synchronization property [ 11-[3]. A chaotic
system is self-synchronizing if it can be decomposed into
at least two subsystems: a drive system (transmitter) and a
stable response subsystem (receiver) that synchronize when
coupled with a common signal. For some synchronized chaotic
Manuscript received May 23, 1993. This work was supported by the Air
Force Office of Scientific Research under Grant Number AFOSR-91-0034-A,
the National Science Foundation under Grant DMS-9057433, a subcontract
from Lockheed Sanders, Inc. under ONR Contract N00014-91-C-0125, and
the Office of Naval Research under Grant N00014-93-1-0686. K. M. Cuomo
was supported by the MITLincoln Laboratory Staff Associate Program. This
paper was recommended by Guest Editor L. 0. Chua.
K. M. Cuomo and A. V. Oppenheim are with the Research Laboratory
of Electronics and the Department of Electrical Engineering and Computer
Science, Massachusetts Institute of Technology, Cambridge, MA 02139.
S. H. Strogatz is with the Department of Mathematics, Massachusetts
Institute of Technology, Cambridge, MA 02139.
IEEE Log Number 921 1620.
systems (SCS’s), the ability to synchronize is robust. For
example, in the Lorenz system, the synchronization is highly
robust to perturbations in the drive signal as shown numerically in Section VII. This property leads to some interesting
communications applications. For example, the chaotic signal
masking technique introduced in [4], [5] appears to be a
potentially useful approach to private communications. In a
second approach to private communications, the informationbearing waveform is used to modulate a transmitter coefficient.
The corresponding synchronization error in the receiver can be
used to detect binary-valued bit streams [6], [7].
In a somewhat different context, SCS’s can be viewed
as performing the role of a nonlinear state space observer.
Indeed, the input to a SCS consists of a single component
of the transmitter’s state vector and produces an estimate of
the transmitter’s full-dimensional state vector. In the case of
noise-free measurements, the asymptotic state estimates are
exact, with exponentially fast convergence in the case of the
Lorenz system. Structurally, the SCS can be viewed as an
open-loop nonlinear observer, since information from feedback
signals does not not take part in the synchronization process.
To quantify the robustness of the Lorenz SCS to perturbations
in the drive signal, we numerically study the ability of the
Lorenz SCS to provide meaningful state estimates when driven
by a single noisy transmitter component. We then compare
the SCS’s performance to an extended Kalman filter (EKF) to
calibrate the SCS’s performance as a nonlinear state estimator.
This paper represents an expanded version of work reported
earlier by us [8] and is organized as follows. In Section 11,
we summarize the concept of sy;_nchronizedchaotic signals
and systems. In Section 111, we discuss the electronic circuit implementation of the Lorenz system. In Section IV, a
synchronizing receiver circuit that can regenerated the fulldimensional dynamics of the transmitter is discussed and
demonstrated. Sections V and VI provide examples of how
the transmitter and receiver circuits can be used together
for the implementation of new algorithms and approaches
to communications. In Section VII, we emphasize a close
connection between synchronization in chaotic systems and
model-based state estimation using EKFs.
11. BACKGROUND
The Lorenz system [9] is given by
1057-7130/93$03.00 0 1993 IEEE
k = u(y
-
x)
y = T X - y - xz
i = XY - bz
CUOMO et al.: SYNCHRONIZATION OF LORENZ-BASED CHAOTIC CIRCUITS
621
where C J ,r , and b are parameters. As shown by Pecora and
Carroll, the Lorenz system is decomposable into two stable
subsystems. Specifically, a stable ( X I , 2 1 ) response subsystem
can be defined by
and a second stable
(y2, 2 2 )
response subsystem by
y* = T X
-
22
y2
-
x22
= Z Y ~- bZq
(3)
Equation (1) can be interpreted as the drive system, since its
dynamics are independent of the response subsystems. Equations (2) and (3) represent dynamical response systems that
are driven by the drive signals y ( t ) and x ( t ) ,respectively. The
eigenvalues of the Jacobian matrix for the (xi,2 1 ) subsystems
are both negative, thus, 1x1 - X I and 121 - zI -+ 0 as t --t m.
Also, it can be shown numerically that the Lyapunov exponents
of the (yy~,2 2 ) subsystem are both negative; thus, ly2 - yI and
122 - zI --t 0 as t + CO.
As we show below, the two response subsystems can be
used together to regenerate the full-dimensional dynamics that
are evolving at the drive system. Specifically, if the input
signal to the ( ~ 2 2,2 ) subsystem is ~ ( t then
) , the output y*(t)
can be used to drive the ( z l ,z l ) subsystem and subsequently
generate a "new" x ( t ) in addition to having obtained, through
synchronization, y ( t ) and ~ ( t )It. is important to recognize
that the two response subsystems given by (2) and (3) can
be combined into a single system having a three-dimensional
state space. This produces a full-dimensional response system
that is structurally similar to the drive system (1). Further
discussion of this result is given below in the context of the
circuit implementations.
111. THE TRANSMITTER
CIRCUIT
A direct implementation of (1) with an electronic circuit
presents one major difficulty: The state variables in (1) occupy
a wide dynamic range with values that exceed reasonable
power supply limits. However, this difficulty can be eliminated
by a simple transformation of variables. Specifically, for the
coefficients U , r , and b used here, an appropriate transformation is u = 2/10, I I = y/lO, and 111 = 2/20. With this scaling,
the Lorenz equations are transformed to
?
= Ui( W - U )
i, = ru - w
-
20uw
.Lii = 511' - bw.
(4)
This system, which we refer to as the transmitter, can be more
easily implemented with an electronic circuit because the state
variables all have similar dynamic range and circuit voltages
remain well within the range of typical power supply limits.
An analog circuit implementation of the circuit in the
equations in (4) is shown in Fig. 1. The operational amplifiers ( 1)-(8) and associated circuitry perform the operations
of addition, subtraction, and integration. Analog multipliers
implement the nonlinear terms in the circuit equations. By
Fig. 1. Lorenz-based chaotic circuit.
applying standard node analysis techniques to the circuit of
Fig. 1, a set of state equations that govern the dynamical
behavior of the circuit can be obtained. This set of equations
is given by
.]
. RBCl
' [$' &(' $)
U=-
-
Ri 1
1
- -
+
w
R12
Rs
-
-uw
R12
Rg
1
An examination of these equations shows that the circuit time
scale can be easily adjusted by changing the values of the
three capacitors C1, C2, and C3 by a common factor. Thus,
if a factor of R increase in signal bandwidth is desired, it
can be achieved by dividing the three capacitor values by the
same factor. In addition, the coefficients C J ,T , and b can be
independently varied by adjusting the corresponding resistors
Rg, Rll, and Rls. This property allows for a simple way to
independently vary the coefficients of the circuit equations.
For the component values' we have chosen, ( 5 ) is equivalent
to (4) after rescaling time by a factor of 2505. The resulting
coefficients are CJ = 16, T = 45.6, and b = 4.
To illustrate the chaotic behavior of the transmitter circuit,
an analog-to-digital ( A D )data recording system was used to
sample the appropriate circuit outputs at 48-kHz rate and with
16-bit resolution. Fig. 2 shows a sample function and averaged
power spectrum corresponding to the circuit waveform u ( t ) .
The power spectrum is broadband, which is typical of a chaotic
signal. Fig. 2(b) also shows the power spectrum obtained from
a numerical simulation of the Lorenz equations. As we see, the
performance of the circuit and the simulation are consistent.
Fig. 3 shows the circuit's chaotic attractor projected onto the
uru-plane and uw-plane, respectively. These data were obtained
from the circuit using the stereo recording capability of the
A/D system to simultaneously sample the x-axis and y-axis
signals at a 48-kHz rate and with 16-b resolution. A more
detailed analysis of the transmitter circuit is given in [6].
'Resistors (kn): X1. R2. X3. R4. R6. R7. R13. R14. R16. R1T.
R19 = 100 R5. R10 = 49.9 R 8 = 200 X 9 . R12 = 10 E11 = 63.4
R l 5 = 40.2 R18 = 66.5 R20 = 158 Capacitors (pF): C 1 , C2, C 3 = 500
Op-Amps (1)-(8): LF3.53 Multipliers: AD632AD.
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS-11: ANALOG AND DIGITAL SIGNAL PROCESSING, VOL. 40, NO. 10, OCTOBER 1993
628
0
,002
,001
,003
,004
I
,005
Time (sec)
(a)
-- I
I
CO
g -20
4
-40
0
12
6
18
24
This system is obtained from the transmitter equations by
renaming variables from ( U , w , w) to (U,, U,, w,) and then
) the second and third equations.
substituting u ( t )for ~ , ( t in
We refer to this system as the receiver in light of some potential communications applications. We denote the transmitter
state variables collectively by the vector d = ( U , U , w) and
the receiver variable by the vector r = (U,, w,, w,) when
convenient.
By defining the dynamical errors by e = d - r, it is
straightforward to show that synchronization in the Lorenz
system is a result of stable error dynamics between the
transmitter and receiver. Assuming that the transmitter and
receiver coefficients are identical, a set of equations that
governs the error dynamics is given by
61
= a(ez - e l )
(b)
62
= -e2 - 20u(t)e3
Fig. 2. Circuit data: (a) A sample function of u ( t ) . (b) Averaged power
spectrum of u ( t ) .
63
= 5u(t)e2
Frequency (kHz)
- be3
The error dynamics are globally asymptotically stable at
the origin, provided that cr, b >O. This result follows by
considering the 3-D Lyapunov function defined by
The time rate of change of E ( e , t ) along the trajectories is
given by
1 .
E ( e , t ) = ;elel
+ e262 + 4 e 3 i 3
Since E is positive definite and E is negative definite, Lyapunov's theorem (Theorem 10.2 in [lo]) implies that e ( t ) 0
as t -+ m. Therefore, synchronization occurs as t
00.
In the Appendix, we show that the rate of synchronization
is exponentially fast. Note also that the transmitter and receiver need not be operating chaotically for synchronization
to occur. In [ 111, a similar Lyapunov argument is given for
the synchronization of the (y, 2 ) subsystem of the Lorenz
equations.
An electronic implementation of the receiver equations (6)
is shown in Fig. 4. Comparison of the receiver circuit with
the transmitter circuit of Fig. 1 shows that they are virtually
0
identical, with the only difference being that the drive signal
-3
0
3
U @ ) replaces the receiver signal u r ( t ) at a key point in the
4t)
circuit. The practical advantage of this similarity is that the
(b)
transmitter and receiver circuits can be built in an identical
Fig. 3 . Circuit data: (a) Chaotic attractor projected onto uv-plane. (b)
way, which helps to achieve perfect synchronization between
Chaotic attractor projected onto uw-plane.
the transmitter and receiver.
To illustrate the synchronization performance of the receiver
Iv. THE RECEIVERCIRCUIT
circuit, the appropriate transmitter and receiver signals were
A full-dimensional response system that will synchronize to simultaneously recorded using the stereo recording capability
the chaotic signals at the transmitter (4) is given by
of the A/D system. In Fig. 5(a), a plot of the actual circuit
outputs ~ ( tversus
)
u r ( t )is shown. Fig. 5(b) shows a similar
21, = cr(u, - U,)
plot for the circuit outputs ~ ( tand
) w,(t). The 45" lines
U, = ru - U,,.
-~OUW,
indicate that nearly perfect synchronization is achieved and
w, = ~ U U ,- bw,
( 6 ) maintained between the transmitter and receiver. The circuit
-+
-+
CUOMO er al.: SYNCHRONIZATION OF LORENZ-RASED CHAOTIC CIRCUITS
629
Chaotic
Transmitter
m(t)
I
t
I I w = r u --v -bw2 0 u w Ll
-
=
wr
~ U V
Act.,
I
U
Fig. 6. Chaotic signal masking system.
f
Fig. 4.
Synchronizing chaotic receiver circuit.
i.e., is not highly sensitive to perturbations in the drive signal
and thus can be done with the masked signal. It is interesting
to note that this idea is not restricted to just the Lorenz circuit,
but has wider potential. For example, Kocarev et al. [ 121 have
also demonstrated our signal masking concept in [4], [5] by
using Chua’s circuit.
Although there are many possible variations, consider, for
example, a transmitted signal of the form s ( t ) = u ( t ) m(t).
It is assumed that for masking, the power level of m ( t ) is
significantly lower than that of u ( t ) .The dynamical system
implemented at the receiver is
+
h, = 1 6 ( ~ , ?L,)
= 4 5 . 6 ~ ( t )- ‘U, - 2 0 s ( t ) ~ ,
tiJ, = 5 S ( t ) ? 1 ,
(b)
Fig. 5. Circuit data: Synchronization of transmitter and receiver signals. (a)
u ( t ) versus u,.(t).(b) r ( t )versus < . r ( t ) .
outputs shown in Fig. 5 reflect a time span of several minutes,
indicating considerable stability of the synchronization.
V. CHAOTICSIGNALMASKING
A potential approach to communications applications is
based on chaotic signal masking and recovery [4]-[8]. In signal
masking, a noiselike masking signal is added at the transmitter
to the information-bearing signal r n ( t ) , and at the receiver the
masking is removed. In our system, the basic idea is to use the
received signal to regenerate the masking signal at the receiver
and subtract it from the received signal to recover m(t).This
can be done with the synchronizing receiver circuit, since the
ability to synchronize is found experimentally to be robust,
-
4211,.
If the receiver has synchronized with s ( t ) as the drive, then
u,(t) rv u ( t ) ,and consequently 7 n ( t ) is recovered as riz(t) =
s ( t ) - u,(t). Fig. 6 illustrates the approach.
Using the transmitter and receiver circuits, we demonstrate
the performance of this system in Fig. 7 with a segment
of speech from the sentence, “He has the bluest eyes.” As
indicated in Fig. 8, the power spectra of the chaotic masking
signal u ( t ) and the speech are highly overlapping, with an
average signal-to-masking ratio of approximately -20 dB. Fig.
7(a) and (b) shows the original speech m(t)and the recovered
speech signal h ( t ) , respectively. For plotting purposes, these
signals were low-pass filtered and downsampled. Clearly, the
speech signal was recovered, and was of reasonable quality in
informal listening tests.
VI. CHAOTICBINARYCOMMUNICATIONS
As a second illustration of the potential use of synchronized
chaotic systems in communications, we describe a system
to transmit and recover binary-valued bit streams [6]. The
basic idea is to modulate a transmitter coefficient with the
information-bearing waveform and to transmit the chaotic
drive signal. At the receiver, the coefficient modulation will
produce a synchronization error between the received drive
signal and the receiver’s regenerated drive signal with an error
signal amplitude that depends on the modulation. Using the
synchronization error, the modulation can be detected.
This modulation/detection process is illustrated in Fig. 9. In
this figure, the coefficient b of the transmitter (4) is modulated
by the information-bearing waveform, m(t ) . The information
is carried over the channel by the chaotic signal u m ( t ) .The
noisy received signal ~ ( t =
) u m ( t ) 7 1 ( t ) serves as the
+
IEEF. TRANSACTIONS ON CIRCUITS AND SYSTEMS-11:
630
,
.
.
.
ANALOG AND DIGITAL SIGNAL PROCESSING, VOL 40, NO. 10, OCTOBER 1993
~.
~
1
’
0.5 1
-0.5
1
1
1
0
0.2
1
0.6
0.4
0.8
1
1.2
1.4
.
l
L
.
.
d
1.6
1.8
0
.01
2
.02
.03
.04
03
.04
.03
.04
(a)
(a)
0.5
I
I
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
01
.02
(b)
2
TIME(sec)
(b)
Fig. 7. Circuit data: Speech waveforms. (a) Original. (b) Recovered.
20 I
I
t
I
.01
0
Time (sec)
Chaotic Masking
Spectrum
-40
.02
(c)
w
Fig. IO.
Circuit data: (a) Modulation waveform. (b) Synchronization error
power. (c) Recovered waveform.
Speech Spectrum
_-
3
Frequency (kHz)
0
6
Fig. 8. Circuit data: Power spectra of chaotic masking and speech signals.
{
= o ( v - U)
y = ru - v - 2 O u w
w = 5uv - b(m(t))w
Fig. 9.
Chaotic communication system.
driving input to the receiver. At the receiver, the modulation
is detected by forming the difference between r ( t ) and the
reconstructed drive signal u r ( t ) .If we assume that the signalto-noise ratio (SNR) of ~ ( tis)large, the error signal e l ( t ) =
r ( t )- u r ( t )will have a small average power if no modulation
is present. However, if, for example, the information waveform
is a binary valued bit stream, with a “1” representing a
coefficient mismatch and a “0” representing no coefficient
mismatch, then e l ( t ) will be relatively large in amplitude
during the time period that a “1” is transmitted and small
in amplitude during a “0’ transmission. The synchronizing
receiver can thus be viewed as a form of matched filter for
the chaotic transmitter signal u(t).
To illustrate the technique, we use a square-wave for m ( t )
as shown in Fig. 10(a). The square-wave produces a variation
in the transmitter coefficient b with the zero-bit and onebit coefficients corresponding to b(0) = 4 and b(1) = 4.4,
respectively. In [6], we show that the averaged power spectrum
of the drive signal with and without the embedded square-wave
present are very similar. Fig. 10(b) shows the synchronization
error power, e : ( t ) , at the output of the receiver circuit. The
coefficient modulation produces significant synchronization
error during a “1” transmission and very little error during a
“0” transmission. Fig. 1O(c) illustrates that the square-wave
modulation can be reliably recovered by low-pass filtering
the synchronization error power waveform and applying a
threshold test. The allowable data rate is, of course, dependent
on the synchronization response time of the receiver system.
Although we have used a low bit rate to demonstrate the
technique, the circuit time scale can be easily adjusted to allow
much faster bit rates.
Note that the ability to communicate digital bit streams
using this method does not depend on the periodic nature
of the square-wave used to demonstrate the technique. The
results apply to aperiodic or random bit streams as well. A
similar approach using Chua’s circuit has also been shown to
work by Parlitz et al. [13].
VII. CHAOTICSYNCHRONIZATION
NONLINEAR
STATE ESTIMATION
AND
Synchronized chaotic systems obtain state estimates of a
nonlinear deterministic system through the stability of the
error dynamics between transmitter and receiver. This seems
analogous to using EKF’s to produce state estimates of nonlinear probabilistic systems. However, SCS’s have an open-loop
63 1
CUOMO et al.: SYNCHRONIZATION OF LORENZ-BASED CHAOTIC CIRCUITS
structure, and it is not obvious that the SCS can provide robust
state estimates when driven by a noisy drive signal. As a
step toward further understanding the inherent robustness of
the Lorenz SCS, we compare the performance of the Lorenz
SCS to an EKF algorithm. The implementation of the EKF is
discussed below.
Chaotic
Transmitter
Lorenz
scs
-
A
dl
x+v
X
A
Probabilistic state estimates of the Lorenz system can be
obtained with an EKF by expressing the Lorenz system
dynamics and measurement model as a dynamical system of
the form
+
+ v(t)
i: = f(z(t)) w(t)
y(t) = Hz(t)
(7)
where w ( t ) and v ( t )represent zero-mean Gaussian white noise
sources having spectral intensities Q ( t ) and R ( t ) ,respectively.
The process noise w ( t ) , and measurement noise, v ( t ) , are
assumed to be uncorrelated, i.e., E ( w ( t ) v ( ~ )=} 0 for all
t and 7 . For our purposes, the observation matrix H will be
constant with only a single nonzero entry. Specifically, since
the observations consist of only the noisy IC component, the
observation matrix H is given by the row vector H = [l 0 01.
Although, in principle, there should be no process noise
term in (7) because the dynamical system that we are studying
is given, numerical experiments have shown that the process
noise is essential in order to avoid rapid divergence of the
Kalman filter state estimates for the Lorenz system. A reason for the divergence becomes clear when considering the
form of the EKF corresponding to (7). Specifically, the EKF
corresponding to (7) is given by
3i: = f(?(t))
-
d =f(d)
7.1. EKF Implementation
+ K(t)[g(t)
-
Hi(t)]
(8)
where i is the current state estimate and K ( t ) is the Kalman
gain. Although (8) captures the exact nonlinear dynamics of
the system, the state estimates will diverge if the Kalman gain,
K ( t ) becomes small. This follows from the fact that a chaotic
system possesses a sensitive dependence on initial conditions,
and any error in the current state estimate will be amplified
if K ( t ) is nearly zero.
Linearizing (7) about the current state estimate, the following incremental system, which is valid for small & ( t ) is
obtained as follows:
6s = F ( 2 ( t ) ) 6 z ( t+
) w(t)
v ( t ) = H S z ( t ) + .(t)
(9)
The time-varying matrix F ( k ( t ) )corresponds to the Jacobian
matrix of the Lorenz system evaluated at the current state
estimate. Since (9) represents a linear time-varying system,
the Kalman filter error covariance, P ( t ) is governed by the
matrix Riccati equation given below:
r = F ( i ( t ) ) P ( t+) P ( t ) F T ( i ( t ) )
+ Q ( t )- P ( t ) H T R - ’ ( t ) H P ( t )
K ( t )= P ( t ) H T R - l ( t ) .
(10)
v(t) N(O,R)
Continuous
d,
EKF
--+
L,
Fig. 11. Numerical experiment.
Equations (8) and (10) determine the EKF state estimates for
the Lorenz system. In [6], we also discuss the implementation
of a linearized EKF that uses the state estimates from the
Lorenz SCS as the nominal trajectory. This approach improves
the robustness of the linearized EKF to uncertainties in the
transmitter’s initial condition.
7.2. Performance Comparisons
In this subsection, we compare the relative performance of
the Lorenz SCS and EKF in terms of the input and output SNR.
The numerical experiment used to evaluate the output SNR of
these state estimators is illustrated in Fig. 11. As indicated,
the transmitted signal z ( t ) corresponds to the z component of
the Lorenz system. The input signal to both the SCS and EKF
consists of the transmitted chaotic drive signal plus zero-mean
Gaussian white noise. Both estimators receive the identical
input sample values, i.e., the received signal z(2) v ( t ) is
simultaneously observed by the SCS and EKF. An initial rest
condition is imposed on the SCS, and the EKF is initialized by
using the true state of the transmitter at t = 0. State estimates
were computed numerically for 10 s and the first few seconds
of data were discarded to eliminate initial transient effects. The
output SNR was then computed for the state estimates from
the remaining data.
Fig. 12 shows a performance comparison in terms of input/output SNR. The solid curves correspond with state estimates from the EKF, and the dashed curves correspond
with state estimates from the SCS. The performance of the
SCS and EKF are comparable over a wide range of input
SNR’s. A threshold effect is evident for both the SCS and
EKF at low-input SNR’s. Above the threshold, however, the
normalized error in the synchronization of each state variable
is approximately 10 db less than the normalized error in the
drive signal ~ ( t )We
. also point out that the SCS is able to
provide meaningful state estimates when starting from any
initial condition. The EKF, however, requires accurate initial
conditions, or the state estimates may rapidly diverge.
+
7.3. Modeling Errors
As a matter of practical significance, we numerically evaluate the sensitivity of the Lorenz SCS and EKF to coefficient
modeling errors at the receiver. For this experiment, the
transmitter coefficients are held fixed, and all of the receiver coefficients are treated as uniformly distributed random
variables having mean values equal to the corresponding
transmitter coefficient. The variance of the receiver coefficients
632
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS-11:
ANALOG AND DIGITAL SIGNAL PROCESSING, VOL. 40, NO. 10, OCTOBER 1993
Similarly, e g ( t ) 5 O ( ~ C ‘ “ ~Finally,
/ ~ ) . integration of 11 =
c ( e 2 - el), combined with e n ( t ) 5 O(e-’”’),
implies that
e l ( t ) 5 max { O ( e - u t ) , O ( e - k t / 2 ) }Thus,
.
all components of
e ( t ) decay at least exponentially fast.
Lorenz
40
ACKNOWLEDGMENT
The authors thank S. H. Isabelle, Prof. A. S. Willsky, and
Prof. G. W. Womell for many helpful discussions during
various stages of this work.
2
m
2
R5
20
0
REFERENCES
0
-20
0
-10
I0
20
30
40
INPUT SNR
Fig. 12. Performance comparison: Lorenz SCS and EKF
.
A
EKF
(solid)
/
System
(dashed)
0
5
10
15
20
[ I ] L. M. Pecora and T. L. Carroll, “Synchronization in chaotic systems,”
Phys. Rev. Lett., vol. 64, pp. 821-824, Feb. 1990.
[2] -,
“Driving system with chaotic signals,’’ Phys. Rev. A., vol. 44,
pp. 2374-2383, Aug. 1991.
131 T. L. Carroll and L. M. Pecora, “Synchronizing chaotic circuits,” IEEE
Trans. Circuits Sysf., vol. 38, pp. 453456, Apr. 1991.
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25
PERCENT MODELING ERROR
Fig. 13. Sensitivity to modeling errors.
depend on the percentage modeling error being tested, where
modeling error is defined as the ratio of the standard deviation
to the mean value of the receiver coefficients.
Fig. 13 shows a sensitivity comparison between the Lorenz
SCS and EKF state estimators, where the input signal corresponds to the transmitters’s noisy IC component having a 10-dB
input SNR. These curves were generated by selecting receiver
coefficients from ten independent trials having the indicated
error. The slopes of these curves indicate that the sensitivity
of these two state estimators is nearly the same over the range
of modeling errors tested.
APPENDIX
To show that e ( t ) decays at least exponentially fast, let
V = :e:.
2eg. Then, V decays at least exponentially fast
because V = -e: - 4be: 5 -kV for any k < min(2, 2b).
Integration then yields 0 5 V(t) 5 VOe-lit. Next, observe
that +e; 5 V 5 VOe-lit so that e 2 ( t ) 5
+
Kevin M. Cuomo (S’83-M’86) received the B S .
(magna cum laude) and M S . degrees in electrical
engineering from the State University of New York
at Buffalo in 1984 and 1986, respectively. Since
1991, he has been pursuing the Ph.D. degree i n
electrical engineering at the Massachusetts Institute
of Technology.
From 1986 to 1988, he was a research staff
member at Calspan Corporation, Buffalo, NY In
1988, he joined the MIT Lincoln Laboratory, where
he IS currently a staff member specializing in the
development of advanced radar imaging methods He is currently a member
of the MIT Lincoln Laboratory Staff Associate Program
While attending SUNY at Buffalo, he was awarded Departmental Honors
(1984) and was the recipient of a University Fellowship (19841986) and a
Hughes Fellowship (1983-1986) He is also a member of Tan Beta Pi, Eta
Kappa Nu, and Sigma Xi
CUOMO et al.: SYNCHRONIZATION OF LORENZ-BASED CHAOTIC CIRCUITS
Alan V. Oppenheim (S’57-M’65-SM’7 1-F’77) received the S B and S M degrees in 1961 and the
Sc D degree in 1964, all in electrical engineering,
trom the Massachusetts Institute of Technology,
Cambridge, MA
In 1964, he joined the faculty at MIT, where
he is currently distinguished Professor of Electrical
Engineering From 1978 to 1980, he was Aqqociate
Head of the Data Systems Division at MIT Lincoln
Laboratory Since 1977, he has alqo been a Guest
Investigator at the Woods Hole Oceanographic Institution, Woods Hole, MA He 15 the author of several widely used textbooks
on digital signal processing
Dr Oppenheim has been a Guggenheim Fellow, a Sackler Fellow, and has
held the Cecil H Green Distinguished Chair in Electrical Engineering and
Computer Science. He has also received a number of awards for outstanding
research and teaching, including the 1988 IEEE Education Medal, and is an
elected member of the National Academy of Engineering He is also a member
of Tau Beta Pi, Eta Kappa Nu, and Sigma Xi.
633
Steven H. Strogatz is an Associate Professor of applied mathematics at
the Massachusetts Institute of Technology. After receiving the A.B. degree
(summa cum laude) in mathematics from Princeton University in 1980,
he spent two years as a Marshall Scholar at Trinity College, Cambridge
University. He received the Ph.D. degree in applied mathematics from Harvard
University in 1986.
His research deals with applications of dynamical systems in physics and
biology, especially problems involving synchronization of coupled nonlinear
oscillators.
Dr. Strogatz was awarded a Presidential Young Investigator grant from the
National Science Foundation in 1990.