hybrid chaos synchronization - peoplechua/papers/barajas-ramirez03.pdf · 1198 j.-g. barajas-ram...

June 11, 2003 13:21 00714

International Journal of Bifurcation and Chaos, Vol. 13, No. 5 (2003) 1197–1216c© World Scientific Publishing Company

HYBRID CHAOS SYNCHRONIZATION*

JUAN-GONZALO BARAJAS-RAMIREZ,GUANRONG CHEN and LEANG S. SHIEH

Department of Electrical and Computer Engineering,

University of Houston, Houston, TX 77204-4005, USA

Received March 19, 2002; Revised March 25, 2002

The problem of hybrid chaos synchronization is investigated, where a digital response subsys-tem is designed to synchronize with an analog drive subsystem. The approach taken is a newprediction-based digital redesign for a continuous-time observer embedded in the response viaan optimal linearization approach of the nonlinear chaotic systems. Three typical but topolog-ically quite different chaotic systems, Chua’s circuit, Duffing oscillator, and Chen’s system, aresimulated thereby validating the novel design proposed in this paper.

Keywords : Chaos synchronization; digital redesign; observer; hybrid system.

1. Introduction

Chaotic systems are characterized by their complexdynamical behaviors, particularly their extremesensibility to initial conditions and parameter vari-ations, which make their behaviors long-term un-predictable. Therefore, the idea that two chaoticsystems can synchronize had been seen as illogi-cal and hence impossible, until the report of Pecoraand Carroll [1990] in which it was shown that chaossynchronization not only is possible but also can bepractical with applications to physical systems andreal-world problems.

Since the work of Pecora and Carroll [1990],the subject of chaos synchronization has been ex-tensively studied from different perspectives by sci-entists and engineers in various fields. Oftentimeschaos synchronization is described by using thedrive–response framework, where the objective is tosynchronize a given drive by a designed responsesubsystem. Usually, the response is a copy of a partor the whole drive, and they are connected uni-directionally by a scalar transmitted signal, such

that both systems produce identical oscillations inan asymptotical sense starting from arbitrary initialconditions.

Notice, however, that almost exclusively thedrive and the response used before are either bothanalog or both digital, while a hybrid analog–digitalsetting (e.g. an analog drive with a digital re-sponse, or vice versa) has not been studied [Chen& Dong, 1998]. This hybrid chaos synchronizationproblem appears to be interesting both theoreticallyand practically, motivating our current investiga-tion reported in this paper.

As is known, a systematic approach to de-sign the response is to reformulate the chaossynchronization as an observer design problem,where synchronization merely is to recover the fullstate trajectory of the drive in the response us-ing the transmitted signal [Grassi & Mascolo, 1997;Huijberts et al., 2000; Jiang, 2002; Morgul & Solak,1997; Nijmeijer, 2001; Nijmeijer & Mareels, 1997].An advantage of the observer-based approach tochaos synchronization is that the solution can befound systematically and the response can be in an

∗Supported by the US Army Research Office under Grant DAAD19-02-1-0321.Barajas-Ramirez is now with CICESE, Depto. De Electronica y Telecomunicaciones, Ensenada, B.C.22860, Mexico.

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arbitrary form, not necessarily the same as that ofthe drive, thus allowing a larger class of systems toperform an intended synchronization using differentmethodologies.

In a communication system, usually the trans-mitter and the receiver are implemented throughdigital computers while the information signals aretransmitted through an analog channel with someanalog devices. Therefore, it is logical to consider acommunication system as a hybrid (partially analogand partially digital) infrastructure, so that whensynchronization is considered (e.g. in coherent com-munications), it is more natural or even necessaryto consider a hybrid setting in the design. In otherwords, it is reasonable and also important to deter-mine some particular aspects of the synchronizationproblem under the hybrid framework for a commu-nication system design. To achieve synchronizationbetween an analog and a digital subsystem, it is es-sential to find a methodology that allows the designof a digital receiver that satisfies the analog featuresand requirements for synchronization, particularlyif the systems are chaotic (used for, e.g. some sortof efficient or secure communications).

It is common practice, in digital implemen-tation of analog controllers, to consider using asufficiently small sampling period so that the per-formance of the analog device can be maintained inthe digital device. Unfortunately, this is not alwaysachievable as desired. It is well known that in thesignal transformation, from analog to digital, thesize of the sampling period has significant effects onthe stability and performance of the entire systemand sometimes they are conflicting in the sense thatif one is improved then the other is degraded. Animportant constraint is the price of implementinga very small sampling period, which can be pro-hibitive due to its CPU-time consumption, or phys-ical limitation such as in biological systems wheresampling (e.g. taking blood or tissues) cannot betoo fast and too frequent. For these reasons, digitalimplementation of analog controllers for nonlinearsystems is an important issue for study, especiallyfor chaotic dynamical systems, which are extremelysensitive to sampling. Some recent advances in thisresearch can be found in [Guo et al., 2000a, 2000b;Shieh et al., 1998; Tsai et al., 1993], where analog–digital conversion was performed from the state-matching digital redesign approach.

State-matching digital redesign is a hybridcontroller design approach, first proposed by Kuo

[1980], where a digital controller is designed foran analog plant by first designing an analog con-troller to satisfy a set of control specifications andthen converting it to an equivalent digital controllersuch that the states of the analog and the dig-ital closed-loop controlled systems can match ateach sampling instant through the entire process.In the last decade, Shieh and his colleagues [Guoet al., 2000a, 2000b; Shieh et al., 1998; Tsai et al.,1993] have thoroughly investigated this techniqueand furthermore developed several types of digi-tal redesign methods. These techniques allows for(sub)optimal control performance when the digi-tal controller is implemented with a relatively largesampling period, and at the same time requiringsignificantly smaller control energy to achieve thecontrol objectives. This is due to the fact that,in general, the amplitude of the piecewise-constantdigital controller is smaller than the maximal am-plitude of the analog controller throughout eachsampling period of the process.

Given the success of the digital redesign tech-nique for hybrid control systems, as describedabove, this paper proposes to apply it to hybridchaos synchronization problems. In this approach, anewly developed prediction-based digital redesign ofobservers will be used as the basic scheme for solv-ing the intended chaos synchronization problem.First, in Sec. 2, a representation of the given chaoticsystem is formulated by using an appropriatelinearized model, using an optimal linearizationmethod. Then, in Sec. 3, using the dual systemprinciple, an analog local observer is optimallydesigned and then converted to a correspondingdigital representation. This digital observer, beingappropriately designed, is used to achieve chaos syn-chronization to the analog drive subsystem. Thedesign procedure, along with the modeling erroranalysis and sampling-time issue in synchronizationperformance, will be discussed in Sec. 4. Finally,in Sec. 5, the design method is applied to threetypical but topologically quite different chaotic sys-tems, namely, Chua’s circuit, Duffing oscillator, andChen’s system, showing its effectiveness and perfor-mance in hybrid chaos synchronization.

2. Observer-Based Chaos

Synchronization

Chaos synchronization may be defined in severalslightly different ways [Nijmeijer et al., 1997;Kolumban et al., 1998; Brown & Kocarev, 2000].

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Hybrid Chaos Synchronization 1199

Here, its definition is taken as follows:Two n-dimensional dynamical systems, the

drive

x(t) = f(x(t)), x(0) = x0, y(t) = h(x(t)) , (1)

and the response,

˙x(t) = f(x(t), y(t)), x(0) = x0, y(t) = h(x(t)) ,

(2)

are said to synchronize if

limt→∞

‖x(t) − x(t)‖ = 0 , (3)

where notation as usual, i.e. f : Rn → Rn is a givennonlinear vector field, initial conditions x0 and x0

are generally different and known, and ‖ · ‖ is theEuclidean norm. The synchronization problem con-sidered is to design the functions h(·) and f(·), ifthey exist, such that the objective (3) is achieved.

A simple interpretation of the synchronizationphenomenon is that the drive (1) sends the trans-mitted signal y(t) to the response (2), and theresponse receives the signal and then is forced tosynchronize with (1) in the sense of (3).

This problem may be viewed as an observerdesign problem [Nijmeijer, 2001] and, depending onthe structure of the system, different approachesmay be possible. A typical solution to the problemis to use the so-called output injection structure,which consists of designing the observer as a copyof the drive (but with unknown initial conditions),and then modifying the error signal depending onthe difference between the received transmitted sig-nal and the predicted output signal of the observer.If the observer is appropriately designed, such thatthe error signal (globally and) asymptotically con-verges to zero, then the drive and the response willsynchronize.

For a linear system,

x(t) = Ax(t) + Bu(t), x(0) = x0, y(t) = Cx(t) ,

(4)

with A, B and C as constant matrices of appropri-ate dimensions, an observer can be designed usingthe output injection structure as

˙x(t) = Ax(t) + Bu(t) + KO(y(t) − y(t)) ,

x(0) = x0, y(t) = Cx(t) ,(5)

where KO is the observer gain matrix. Defining theobserver error e(t) as the difference between thedrive state x(t) and the observer state x(t), the errordynamics can be found from (4) and (5), as

e(t) ≡ x(t) − ˙x(t) = (A − KOC)e(t) . (6)

For (A, C), an observable pair, it is sufficientto design KO such that all eigenvalues of thematrix (A − KOC) have negative real parts, whichwill guarantee that the observer asymptoticallyconverges to the drive dynamics.

Systematically designing a working observer fora general nonlinear system still is an open problem,but many results are now available for some specificclasses of nonlinear systems [Nijmeijer et al., 1997;Kolumban et al., 1998; Brown & Kocarev, 2000].In particular, for Lur’e-type systems, namely, thosenonlinear systems that can be expressed in a com-bination of linear dynamics with an additive staticnonlinearity, a generic solution can be found, as anatural extension of the linear case described above.The importance of Lur’e-type systems comes fromthe fact that different benchmark chaotic systems,like Duffing, Chua, Lorenz and Chen systems, areall included in this special yet still quite generalclass.

A situation that further facilities the synchro-nization of some Lur’e-type systems is when thenonlinear parts of the drive-response depend onlyon the output of the system. In this case, the sys-tem takes on the form

x(t) = Ax(t) + Φ(Cx(t)) + Γ(t) ,

x(0) = x0, y(t) = Cx(t) ,(7)

with A and C as constant matrices, Φ : Rn → Rn asmooth function of the state vector, and Γ : R → Rn

a possible external forcing function. In this setting,by simply extending the solution of the linear case,the nonlinear observer is found to be

˙x(t) = Ax(t) + Φ(y(t)) + Γ(t) + KO(y(t) − y(t)) ,

x(0) = x0, y(t) = Cx(t) .(8)

In this case, using the same definition of observererror as defined in (6), the error dynamics are foundto be linear, so that if the pair (A, C) is observable,then it is sufficient to design the observer gain KO

for achieving synchronization.For Lur’e-type systems in general, a desired ob-

server can still be found as an extension of the linearsolution, but in this case the nonlinear observer willhave the form

˙x(t)=Ax(t)+Φ(x(t))+Γ(t)+KO(y(t)−y(t)) ,

x(0)= x0, y(t)=Cx(t) .(9)

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Here, the error dynamics are found, still defined by(6), to be

e(t) = (A − KOC)e(t) + Φ(x(t)) − Φ(x(t)) . (10)

Thus, if the nonlinear function Φ(x(t)) satisfies theLipschitz condition

‖Φ(x1) − Φ(x2)‖ ≤ `‖x1 − x2‖

x1, x2 ∈ Rn ` ∈ R∗ ,(11)

and if (A, C) is an observable pair, then once againit is sufficient to design the observer gain KO suchthat the linear part is asymptotically stable, whichwill guarantee that the observer error will tend tozero asymptotically. Furthermore, if the Lipschitzconstant ` is sufficiently small, then this error willdecay to zero exponentially.

Inspired by the above analysis, it is proposedhere to solve the afore-posted synchronization prob-lem using an observer approach, for a generalnonlinear system of the form

x(t) = F (x(t)) + G(x(t))u(t) ,

x(0) = x0, y(t) = Cx(t) .

The idea is to first represent this nonlinear systemby a set of linear systems of the form

x(t) = Akx(t) + Bku(t) ,

where (Ak, Bk) are constant matrices, found foreach operating point of the nonlinear trajectoryusing an optimal linearization method, as furtherdescribed in the following section. Then, a linearobserver of the form (5) will be designed, such thatthe error dynamics converge to zero in some optimalsense to be specified later.

3. Digital Redesign of Observers

A typical approach to handling nonlinear systemsis to utilize linearization at their operating points,including such as Jacobian analysis for local dynam-ics of autonomous systems. A different linearizationmethod, proposed by Teixeira and Zak [1999], isto generate optimal local models, which provides anew tool for the analysis of nonlinear systems. Thistechnique is particularly useful in the study of dig-ital redesign of chaotic systems [Guo et al., 2000a,2000b]. This optimal linearization method (OLM) isan online linearization technique for finding a localmodel that is linear in both the state and the con-trol terms, and is optimal in some sense [Teixeira &Zak, 1999]. This method is briefly described belowfor completeness of the presentation.

3.1. Optimal linearization

Consider a nonlinear system of the form

x(t) = F (x(t)) + G(x(t))u(t) , (12)

where F : Rn → Rn and G : Rn → Rn×m aresmooth nonlinear vector fields, x(t) ∈ Rn is thestate vector, and u(t) ∈ Rm is the control input. It isdesired to have a local linear model at an operatingpoint of interest, denoted (xop, uop), in the form

x(t) = Aopx(t) + Bopu(t) , (13)

with Aop and Bop being constant matrices ofappropriated dimensions.

A common approach to finding the linear modelis to use the truncated Taylor expansion, but thisresults in an affine rather than a linear model, un-less the operation point is the origin. Even in theideal case of an equilibrium point of the system, theresult is generally not a local model linear in boththe state and the control.

To find a local model, linear in both x and u,which approximates well the dynamical behavior ofthe original nonlinear (12) in the vicinity of the op-erating state xop, it is necessary to find two constantmatrices, Aop and Bop, such that, in a neighborhoodof xop,

F (x(t)) + G(x(t))u(t)

≈ Aopx(t) + Bopu(t) for any admissible u(t)

(14a)

and

F (xop) + G(xop)u(t)

= Aopxop + Bopu(t) for any admissible u(t) .

(14b)

Since the control u is to be designed, and onemust have

G(xop) = Bop , (15)

substituting this into Eqs. (14) yields

F (x(t)) ≈ Aopx(t) (16a)

and .

F (xop) = Aopxop . (16b)

For simplicity, the solution of Eq. (16) will befound by rows. Denote aT

i as the ith row of matrixAop, so that Eq. (16) can be represented as

fi(x(t)) ≈ aTi x(t), i = 1, 2, . . . , n (17a)

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and

fi(xop) = aTi xop, i = 1, 2, . . . , n , (17b)

where fi : Rn → Rn is the ith row of function F (·).Expanding the left-hand side of (17a) about

xop, and then truncating it from the second term,one obtains

fi(xop) + [∇fi(xop)]T (x(t) − xop) ≈ aTi x(t) , (18)

where ∇fi(xop) : Rn → Rn is the gradient column-vector of fi(·) evaluated at xop. With (17b), one canrewrite (18) as

[∇fi(xop)]T (x(t) − xop) ≈ aTi (x(t) − xop) , (19)

in which x(t) is arbitrary but should be close to xop

for a good approximation. The modeling error canbe determined from (19), as

error = ∇fi(xop) − ai .

Then, it is desired to determine the constantvector, ai, such that it is as close as possible to thegradient, and also satisfies fi(xop) = aT

i xop. Thiscan be considered as the following constrained min-imization problem:

Find the optimal parameters ai that minimizethe following quadratic error function

Ei ≡1

2‖∇fi(xj) − ai‖

22

=1

2([∇fi(xj) − ai])

T [∇fi(xj) − ai] ,

subject to the constraint fi(xj) = aTi xj , i = 1,

2, . . . , n.Since this is a convex constrained optimiza-

tion problem, it can be solved by using theLaGrange multiplier method, which yields theoptimal solution

ai =

∇fi(xop) +fi(xop) − xT

op∇fi(xop)

‖xop‖22

xop for xop 6= 0

∇f1(xop) for xop = 0 ,

(20)

where ‖xop‖22 = xT

opxop is the square magnitudeof the point xop.

An important consideration on the applicationof the resulting optimal linear models in the de-sign of controllers/observers for nonlinear systemsis that it is necessary to check that if the resultinglocal linear model at a particular operating pointturns out to be uncontrollable or unobservable, thenone needs to make sure that the local model is atleast stabilizable or detectable, which is required inthe design of controller/observer.

3.2. Optimal linear observer design

As defined above, synchronization is an asymptoticproperty, but in real world applications it is desir-able to achieve synchronization as quickly as possi-ble. For example, in coherent communications, theefficiency strongly depends on the synchronizationproperties of the system; therefore, achieving syn-chronization in a short period of time is a verydesirable feature. When synchronization is solvedas an observer design problem, one can design theobserver such that the error dynamics have theireigenvalues on the far left of the complex s-plane,so that the convergence to the original dynamics

can be fast. Another desirable characteristic for syn-chronization that can be obtained via the observerdesign approach is to use the minimum error energy,which can be done by applying an optimal designtechnique such that the observer minimizes a pre-assigned performance index.

Optimal design of controllers is a well-developed area in control systems theory, where fora given controllable and observable linear system,

x(t) = Ax(t) + Bu(t), x(0) = x0 ,

y(t) = Cx(t) ,

the optimal state-feedback control law that mini-mizes the performance index

J =

∫

∞

0[x(t)T Qx(t) + u(t)T Ru(t)]dt (21)

with Q ≥ 0 and R > 0, is obtained as

u(t) = −KCx(t) , (22)

where the optimal feedback gain is KC = R−1BTP ,with P being the positive definite and symmetricsolution of the following Ricatti equation [Stefaniet al., 2002]:

AT P + PA − PBR−1BT P + Q = 0 . (23)

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In closed-loop, the optimally control linearsystem has the form

x(t) = (A − BKC)x(t), x(0) = x0 , (24)

and the linear observer is given by

˙x(t) = Ax(t) + Bu(t) + KO(y(t) − y(t)) ,

x(0) = x0 y(t) = Cx(t) ,

along with the observer error dynamics

e(t) ≡ x(t) − ˙x(t) = (A − KOC)e(t) . (25)

Comparing Eqs. (24) and (25), one can see that

(A − KOC)T = AT − CT KTO ,

which has the structure as a state-feedback con-troller. This is the dual property of linear systems,where the observer gain can be determined as thedual of the feedback controller gain. Thus, the op-timal observer gain KO can be found by designingthe optimal control gain KC for the dual system,via A = AT and B = CT , so that KO = KT

C ; orequivalently, KO = PoC

TR−1, where PO is the posi-tive definite and symmetric solution of the followingRicatti equation:

APO + POAT − POCTR−1CPO + Q = 0 .

3.3. A new prediction-based digital

redesign of observers

Digital redesign is a hybrid control design tech-nique, where a pre-design analog controller thatsatisfies a set of control objectives and specifica-tions is converted to an equivalent digital controllersuch that the control performance of the analogand the hybrid control systems matched at least ateach sampling instant throughout the entire controlprocess [Shieh et al., 1998; Tsai et al., 1991].

The advantages of digital redesign are three-fold: First, due to the availability of inexpensivedigital computer devices and systems, a digital con-troller can be implemented at a low cost, regardlessof its notational complexity. Second, the stabilityand performance of a digitally redesign hybrid sys-tem is guaranteed for larger sampling periods, ascompared to the conventional direct digital design.Finally, a controller designed with a theoreticallyhigh gain in the analog setting is converted to alow-gain design in the digital implementation afterthe digital redesign is completed, due to the factthat the control energy is averaged over the wholesampling period [Tsai et al., 1991].

In this section, the existing digital redesignmethodology for controllers is extended to observers

on the basis of their duality, mentioned above, us-ing the recently derived prediction-based digitalredesign technique for feedback controllers [Guoet al., 2000b]. The derivation of this digital redesignmethod for feedback controllers is first reviewedhere for completeness of the presentation. Con-sider a controllable and observable continuous-timesystem,

xC(t) = AxC(t) + BuC(t), xC(0) = x0 ,

yC(t) = CxC(t) ,(26)

where xC(t) ∈ Rn, uC(t) ∈ Rm, yC(t) ∈ RP andA, B, C are constant matrices of appropriate di-mensions. Let the continuous-time state-feedbackcontrol law be

uC(t) = −KCxC(t) + ECr(t) , (27)

where the feedback control gain KC ∈ Rm×n andthe feedforward gain EC ∈ Rm×m have been ob-tained to satisfy a set of control objectives, and r(t)is an m× 1 reference input. As shown in Fig. 1, theoverall controlled system is

xC(t)=(A−BKC)xC(t)+BECr(t), xC(0)=x0,

(28)

Let the state equation of the correspondingdigitally controlled system be described by

xd(t) = Axd(t) + Bud(t), xd(0) = x0 , (29)

where ud(t) ∈ Rm is piecewise-constant, such thatud(t) = ud(kT ) for kT ≤ t < (k + 1)T , and T > 0is the sampling and hold period. Let ud(t) be adiscrete-time state-feedback control law of the form

ud(kT ) = −Kdxd(kT ) + Edr∗(kT ) , (30)

where Kd ∈ Rm×n and Ed ∈ Rm×m are the feed-back and feedforward digital gains, respectably, andr∗(kT ) is a piecewise-constant reference determinedin terms of r(kT ) for tracking purpose. The overalldigitally controlled closed-loop system becomes

xd(t) = Axd(t) + B[−Kdxd(kT ) + Edr∗(kT )] ,

xd(0) = x0 ,(31)

for kT ≤ t < (k + 1)T , where the controller is real-ized using a zero-order-hold device as illustrated inFig. 2.

The digital redesign problem is thus reduced tofinding the digital gains (Kd, Ed) in (30) from thecontinuous-time controller gains (KC , EC) in (27),such that the closed-loop state xd(t) in (31) canclosely match the closed-loop state xC(t) in (28), atall the sampling instants for a given r(t) = r(kT ),k = 0, 1, 2, . . . .

June 11, 2003 13:21 00714


13

)()()( trEtxKtu CCCC +−= , (27)

where the feedback control gain nm

C RK ×∈ and the feedforward gain mm

C RE ×∈ have

been obtained to satisfy a set a of control objectives, and )(tr is an m×1 reference input.

As shown in Figure 1, the overall controlled system is

),()()()( trBEtxBKAtx CCCC +−=! 0)0( xxC = , (28)

Let the state equation of the corresponding digitally controlled system be describe

by

),()()( tButAxtx ddd +=! 0)0( xxd = , (29)

where m

d Rtu ∈)( is piecewise-constant, such that )()( kTutu dd = for TktkT )1( +<≤ ,

and 0>T is the sampling and hold period. Let )(tud be a discrete-time state-feedback

control law of the form

)()()( * kTrEkTxKkTu dddd +−= , (30)

where nm

d RK ×∈ and

mm

d RE ×∈ are the feedback and feedforward digital gains,

respectably, and )(* kTr is a piecewise-constant reference determined in terms of )(kTr

for tracking purpose. The overall digitally controlled closed-loop system becomes.

)],()([)()( * kTrEkTxKBtAxtx ddddd +−+=! 0)0( xxd = , (31)

for TktkT )1( +<≤ , where the controller is realized using a zero-order-hold device as

illustrated by Figure 2.

CE )()()( tButAxtx CCC +=! C

CK

)(tr )(tuC )(txC )(tyC+

−

Fig. 1. Continuous-time control system.

Fig. 1. Continuous-time control system.

14

The digital redesign problem is thus reduced to finding the digital gains ),( dd EK

in (30) from the continuous-time controller gains ),( CC EK in (27), such that the closed-

loop state )(txd in (31) can closely match the closed-loop state )(txC in (28), at all the

sampling instants for a given )()( kTrtr = , ,...2,1,0=k .

The solution of the analog system (26), )(txC , at time vTkTtt v +== for

10 ≤≤ v , where v is a tuning parameter, is found to be

∫+

−+− +=vTkT

kT

C

vTkTA

c

kTtA

vC dBuekTxetx v τττ )()()( )()(. (32)

Letting )( vC tu be piecewise-constant, so that the solution reduces to

)()()()()( )()()()(

vC

v

c

v

vC

vTkT

kT

vTkTA

c

kTtA

vC tuHkTxGtuBdekTxetx v +=+= ∫+

−+− ττ , (33)

with vvATAvTv GeeG === )()()( and BAIGBdeBdeH n

v

T

A

t

kT

tAv 1

0

)()( )(: −− −=== ∫∫ν

ττ ττν

ν .

Here, 1)( −− AIG n

v is a shorthand notation, which is always well defined and can

be verified by the cancellation of the formal notation 1−A in the series expansion of the

term integral )( n

v IG − . Therefore, the invertiability of matrix A is not required.

Also, the solution to the digitally control system (31), )(txd , at time

vTkTtt v +== for 10 ≤≤ v , is obtained as

)()()( )()( kTuHkTxGtx d

v

d

v

vd += . (34)

Fig. 2. Digital control system.

dE )()()( tButAxtx ddd +=! C

)(* kTr )(kTud )(txd )(tyd+

−

dK

ZOHT

T

Fig. 2. Digital control system.

The solution of the analog system (26), xC(t),at time t = tv = kT + vT for 0 ≤ v ≤ 1, where v isa tuning parameter, is found to be

xC(tv) = eA(tv−kT )xc(kT )

+

∫ kT+vT

kT

eA(kT+vT−τ)BuC(τ)dτ .

(32)

Let uC(tv) be piecewise-constant, so that thesolution reduces to

xC(tv) = eA(tv−kT )xc(kT )

+

∫ kT+vT

kT

eA(kT+vT−τ)BdτuC(tv)

= G(v)xc(kT ) + H(v)uC(tv) , (33)

with G(v) = (eAvT ) = (eAT )v = Gv and H(v) =∫ tvkT

eA(tv−τ)Bdτ =∫ vT

0 eAτ Bdτ := (Gv − In)A−1B.Here, (Gv − In)A−1 is a shorthand notation,

which is always well defined and can be verifiedby the cancellation of the formal notation A−1 inthe series expansion of the term integral (Gv − In).Therefore, the invertiability of matrix A is notrequired.

Also, the solution to the digitally controlledsystem (31), xd(t), at time t = tv = kT + vT for0 ≤ v ≤ 1, is obtained as

xd(tv) = G(v)xd(kT ) + H(v)ud(kT ) . (34)

Thus, it follows that to obtain the state xC(tv) =xd(tv) under the assumption that xC(kT )=xd(kT ),

it is necessary to have uC(tv) = ud(kT ). This leadsto the prediction-based digital controller:

ud(kT ) = uC(tv) = −KCxC(tv) + ECr(tv)

= −KCxd(tv) + ECr(tv) , (35)

where the future state xd(tv) needs to be predictionbased on the available causal signals xd(kT ) andud(kT ). Substituting (34) into (35) and then solv-ing for ud(kT ), one can find the desired predicteddigital controller, as

ud(kT ) = (Im + KCH(v))−1(−KCG(v)xd(kT )

+ECr(tv))

= −K(v)d xd(kT ) + E

(v)d r∗(kT ) (36)

For practical applications, one can consider vas a tuning parameter for the desired closeness be-tween the predicted digital and continuous states.If one sets v = 1, then the pre-requisite xC(kT ) =xd(kT ) is ensured. Thus, for any k = 0, 1, 2, . . . ,the controller is obtained as

ud(kT ) = −Kdxd(kT ) + Edr∗(kT ) , (37)

where

Kd = (Im + KCH)−1KCG , (38a)

Ed = (Im + KCH)−1EC , (38b)

r∗(kT ) = r(kT + T ) , (38c)

in which G = eAT , H = (G − In)A−1B and r∗(kT )is an alternative form of the original reference r(t)at time t = kT , with the amplitude one-step aheadr(kT+T ) necessary for a good tracking performance

June 11, 2003 13:21 00714


16

with error dynamics in continuous-time, found from (26) and (40), to be

).()()(�)(:)( teCLAtxtxte CCCCC −=−=!!! (41)

The objective here is to find a digital observer of the form

)()()(�)(� kTyLTkTuHTkTxGkTx CddOdOd +−+−= , (42)

where the discrete-time error is defined as

)(�)(:)( kTxkTxkTe ddd −= , (43)

such that the discrete-time error dynamics match the continuous-time error dynamics at

each sampling instant (kTtCd tekTe

=

≈ )()( ), or equivalently, assuming that the

continuous-time observer is asymptotically stable, the original state and the digital state

match (kTtCkTtCd txtxkTx

==

≈≈ )()(�)(� ).

Using the duality, the continuous-time error dynamics described by (41) can be

viewed as a state-feedback control problem, similar to the one shown in (28), where the

input reference is 0)( =tr . Then, applying the prediction-based digital redesign method

derived above, one can find from (39) the discrete-time error dynamics, described in (43),

as

)()()( kTeMNGTkTed −=+ , (44)

where

ATeG = , (45a)

Cn LAIGM 1)( −

−= , (45b)

CGCMIN m

1)( −

+= . (45c)

Further defining 1)( −

+= CMIML md , one can write CGLMN d= and, with the

definition of (43), one has, from (44),

Fig. 3. Continuous-time observer system.

))(�)(()()(�)(� tytyLtButxAtx CcCCCC −++=! C

)(tuC )(� txC )(� tyC

+ )(tyC −

Fig. 3. Continuous-time observer system.

of the digital control system. With this digitally re-designed controller, the discrete-time model of (29)becomes

xd(kT + T ) = (G − HKd)xd(kT ) + HEdr∗(kT ) ,

xd(0) = x0 ,(39)

where the corresponding matrices are as definedabove.

Now, consider a continuous-time observer forthe system in (26), as presented in Fig. 3, which isdescribed by

˙xC(t) = AxC(t) + BuC(t) + LC(yC(t) − yC(t)) ,

xC(0) = x0, yC(t) = CxC(t) ,(40)

with error dynamics in continuous-time, found from(26) and (40), to be

eC(t) ≡ xC(t) − ˙xC(t) = (A − LCC)eC(t) . (41)

The objective here is to find a digital observerof the form

xd(kT ) = GOxd(kT − T ) + HOud(kT − T )

+LdyC(kT ) , (42)

where the discrete-time error is defined as

ed(kT ) ≡ xd(kT ) − xd(kT ) , (43)

such that the discrete-time error dynamics matchthe continuous-time error dynamics at each sam-pling instant (ed(kT ) ≈ eC(t)|t=kT ), or equiva-lently, assuming that the continuous-time observeris asymptotically stable, the original state andthe digital state match (xd(kT ) ≈ xC(t)|t=kT ≈xC(t)|t=kT ).

Using the duality, the continuous-time errordynamics described by (41) can be viewed as astate-feedback control problem, similar to the oneshown in (28), where the input reference is r(t) =0. Then, applying the prediction-based digital re-design method derived above, one can find from (39)

the discrete-time error dynamics, described in (43),as

ed(kT + T ) = (G − MN)e(kT ) , (44)

where

G = eAT , (45a)

M = (G − In)A−1LC , (45b)

N = (Im + CM)−1CG . (45c)

Further defining Ld = M(Im+CM)−1, one canwrite MN = LdCG and, with the definition of (43),one has, from (44),

xd(kT + T ) − xd(kT + T )

= (G − LdCG)[xd(kT ) − xd(kT )] . (46)

To this end, substituting the following identi-ties into (46):

xd(kT + T ) = Gxd(kT ) + Hud(kT ) ,

yC(kT ) = yd(kT ) = Cxd(kT ) ,

CGxd(kT ) = Cxd(kT + T ) − CHud(kT )

= yd(kT + T ) − CHud(kT ) ,

and then solving the resulting equation for xd(kT ),one obtains the new digitally redesigned observerfor system (40), as

xd(kT ) = GOxd(kT − T ) + HOud(kT − T )

+LdyC(kT ) , (47)

where

Ld = (G − In)A−1LC [Im + C(G − In)A−1LC ]−1 ,

(48a)

GO = G − LdCG , (48b)

HO = H − LdCH , (48c)

with G and H as defined above.A realization of system (47) can be obtained

by using a sampler and a unit delay device, as illus-trated in Fig. 4.

June 11, 2003 13:21 00714


17

)](�)()[()(�)( kTxkTxCGLGTkTxTkTx ddddd −−=+−+ . (46)

To this end, substituting the following identities into (46):

)()()( kTHukTGxTkTx ddd +=+ ,

)()()( kTxCkTykTy ddC == ,

)()()()()( kTuHCTkTykTuHCTkTxCkTCGx ddddd −+=−+= ,

and then solving the resulting equation for )(� kTxd , one obtains the new digitally

redesigned observer for system (40), as

)()()(�)(� kTyLTkTuHTkTxGkTx CddOdOd +−+−= , (47)

where

111 ])([)( −−−

−+−= CnmCnd LAIGCILAIGL , (48a)

CGLGG dO −= , (48b)

CHLHH dO −= , (48c)

with G and H as defined above.

A realization of system (47) can be obtained by using a sampler and a unit delay

device, as illustrated by Figure 4.

4. Application of Digital Redesign to Chaos Synchronization

Due to the linear nature of the prediction-based digital redesign methodology

presented above, for chaos synchronization applications it is necessary to apply a

Fig. 4. Digitally redesigned observer system.

))(()()(�)(� kTyLTkTuHTkTxGkTx cddOCOd +−+−=

)(� kTxd

)(tyC )(kTyc

T

)( TkTud −)(tuC )(kTud

1−ZT

Fig. 4. Digitally redesigned observer system.

4. Application of Digital Redesign

to Chaos Synchronization

Due to the linear nature of the prediction-baseddigital redesign methodology presented above, forchaos synchronization applications it is necessary toapply a linearization to the chaotic systems before adiscrete-time observer can be designed to solve thechaos synchronization problem.

There are two cases to consider: (i) the chaoticsystem is, or can be, expressed as a set of linear sub-systems (a piecewise-linear, switching, or discontin-uous case); (ii) a linearization is needed to obtaina linear representation of the nonlinear system (asmooth case). These two cases are further discussedbelow.

4.1. Chaotic piecewise-linear

systems: The switching case

The simplest setting for an application of the digitalredesign methodology, described above, to chaotic

systems is the case where the chaotic systems arepiecewise-linear such as Chua’s circuit family. Inthis case, the system, the observer, and the errordynamics are basically linear, so that the entire sys-tem can be characterized via a simple conditionalchange of models that is convenient for the designof the observer.

In general, a piecewise-linear system can beexpressed as a set of linear systems, continuouslyconnected, in the following form

xC(t) =

A1xC(t) + B1uC(t), for condition 1

A2xC(t) + B2uC(t), for condition 2

......

AqxC(t) + BquC(t), for condition q ,

yC(t) = CxC(t) .(49)

From this representation, the synchronization prob-lem can be solved via constructing a piecewise-linear observer of the form

˙xC(t) =

A1xC(t) + B1uC(t) + LC,1(yC(t) − yC(t)), for condition 1

A2xC(t) + B2uC(t) + LC,2(yC(t) − yC(t)), for condition 2

......

AqxC(t) + BquC(t) + LC,q(yC(t) − yC(t)), for condition q ,

yC(t) = CxC(t) ,

(50)

where the observer gains, LC,i, for i = 1, 2, . . . , q, can be optimally designed so as to obtain fast convergenceand optimal error energy for each linear subsystem of the original dynamics.

The digitally redesigned version of the piecewise-linear observer is obtained as

xd(kT ) =

GO,1xd(kT − T ) + HO,1ud(kT − T ) + Ld,1yC(kT ), for condition 1

GO,2xd(kT − T ) + HO,2ud(kT − T ) + Ld,2yC(kT ), for condition 2

......

GO,qxd(kT − T ) + HO,qud(kT − T ) + Ld,qyC(kT ), for condition q ,

yd(kT ) = Cxd(kT ) ,

(51)

June 11, 2003 13:21 00714


where the digital observer gains, Ld,i, GO,i, andHO,i, are designed for each linear subsystem byusing Eq. (48).

4.2. Chaotic systems of the form

x = F (x) + G(x)u: The

smooth case

Inspired by the system structure, seen from the pre-vious subsection, one knows that in order to designa digital observer that synchronizes to a given gen-eral continuous-time chaotic system it is necessaryto find a set of linear subsystems that can representthe entire dynamical system. To do so, the optimallinearization method (OLM) developed in [Teixeira& Zak, 1999] is applied.

First, the given chaotic system of the form

xC(t) = F (xC(t)) + G(xC(t))uC (t), xC(0) = x0 ,

yC(t) = h(xC(t)) ,(52)

where F (·) is a smooth nonlinear vector field, andG(·) and h(·) are smooth and possibly nonlinearvector fields, is represented by a linear subsystemof the form

xC(t) = AkxC(t) + BkuC(t) ,

yC(t) = CkxC(t) ,(53)

at each operating point xk, k = 0, 1, 2, . . . ,throughout the system trajectory. According to thechaotic dynamics, which will evolve on either astrange attractor, or a periodic orbit, or a fixedpoint, it is not difficult to find an optimal local lin-ear model for each operating point along the chaotictrajectory.

The set of optimal local linear models for thechaotic system, obtained via the OLM, have theadvantage of having the exact model and dynamicsof the original system at the operating points of in-terest, while having the minimum modeling errors

in their neighborhoods, as indicated by (14a) and(14b).

Then, the continuous-time observer for system(52), as illustrated in Fig. 5, is designed, resultingin

˙xC(t)=AkxC(t)+BkuC(t)+LC,k(yC(t)−yC(t)),

xC(0)= x0, yC(t)=CkxC(t),(54)

where the matrices Ak, Bk and Ck, locally repre-senting the chaotic system (53), are similar in struc-ture but not necessarily identical to the matricesAk,Bk and Ck, which were used in the linear observer(54). The observer gains, LC,k, are optimally de-signed such that in the neighborhood of each oper-ating point, the observer converges to the originaldynamics.

From (54), a digital observer can be found byusing the prediction-based digital redesign methodpresented above, yielding

xd(kT ) = GO,kxd(kT − T ) +HO,kud(kT − T )

+Ld,kyC(kT ) , (55)

where

Ld,k = (Gk − In)A−1k LC,k

× [Im + Ck(Gk − In)A−1k LC,k]

−1 , (56a)

GO,k = Gk − Ld,kCkGk , (56b)

HO,k =Hk − Ld,kCkHk , (56c)

with Gk = eAkT and Hk = (Gk − In)A−1k Bk.

The digital observers obtained in (51) and (55)can be realized, as illustrated in Fig. 6, where thetransmitted signal yC(kT ) is obtained via a sam-pler, and the control input uC(kT − T ), if it existsfor the system, is obtained via a sampler and a unitdelay device from the corresponding continuous-time signals.

20

at the operating points of interest, while having the minimum modeling errors in their

neighborhoods, as indicated by (14a) and (14b).

Then, the continuous-time observer for system (52), as illustrated by Figure 5, is

designed, resulting in

)),(�)(()()(�)(�, tytyLtuBtxAtx CCkCCkCkC −++=

! 0�)0(� xxC = , (54)

)(�)(� txCty CkC = ,

where the matrices kA , kB and kC , locally representing the chaotic system (53), are

similar in structure but not necessarily identical to the matrices kA , kB and kC , which

were used in the linear observer (54). The observer gains, kCL , , are optimally designed

such that in the neighborhood of each operating point, the observer converges to the

original dynamics.

From (54), a digital observer can be found by using the prediction-based digital

redesign method presented above, yielding

)()()(�)(�,,, kTyLTkTuHTkTxGkTx CkddkOdkOd +−+−= , (55)

where

1

,

1

,

1

, ])([)( −−−

−+−= kCknkkmkCknkkd LAIGCILAIGL , (56a)

kkkdkkO GCLGG ,, −= , (56b)

kkkdkkO HCLHH ,, −= , (56c)

with TA

kkeG = and kknkk BAIGH 1)( −

−= .

Fig. 5. Continuous-time synchronization system.

)(tuC

kC

)())(())(()( tutxGtxFtx CCCC +=! ))(( txh C

)(txC )(tyC

))(�)(()()(�)(�, tytyLtuBtxAtx CckCCkCkC −++=

!

)(� txC )(� tyC

+−

Fig. 5. Continuous-time synchronization system.

June 11, 2003 13:21 00714


21

The digital observers obtained in (51) and (55) can be realized, as illustrated by

Figure 6, where the transmitted signal )(kTyC is obtained via a sampler, and the control

input )( TkTuC − , if it exists for the system, is obtained via a sampler and a unit delay

device from the corresponding continuous-time signals.

The difference between the matrices of (53) and (54) is due to the different ways

the local linear models were derived by using the OLM. In the next section, the effects of

such modeling errors and the effects of sample-hold time will be both studied in more

details.

4.3 Modeling errors and sample-hold time effects on synchronization performance

The OLM is an on-line linearization method, where a local model is found in

terms of the current operating point of interest along the trajectory of the nonlinear

system. For clarity, one can re-express the local model in (53) as

)()()()()( tuxBtxxAtx CkkCkkC +=! ,

)()()( txxCty CkkC = ,

where kx is the point of interest in the trajectory of (52) at time ,...2,1,0, =kk . It is clear

that one does not have access to the state kx in the observer. So, the best one can do is to

use the structure of kA , obtained through (20), and to use the observer states, to generate

)(kTud

Fig. 6. Digitally redesigned synchronization system.

))(( txh C)())(())(()( tutxGtxFtx CCCC +=!

)(tuC )(txC )(tyC

)(� kTxd

)(kTyc

T

)()()(�)(�,,, kTyLTkTuHTkTxGkTx CkddkOdkOd +−+−=

)( TkTud −

1−ZT

Fig. 6. Digitally redesigned synchronization system.

The difference between the matrices of (53) and(54) is due to the different ways the local linearmodels were derived by using the OLM. In the nextsection, the effects of such modeling errors and theeffects of sample-hold time will be both studied inmore detail.

4.3. Modeling errors and

sample-hold time effects on

synchronization performance

The OLM is an online linearization method, wherea local model is found in terms of the current op-erating point of interest along the trajectory of thenonlinear system. For clarity, one can re-express thelocal model in (53) as

xC(t) = Ak(xk)xC(t) + Bk(xk)uC(t) ,

yC(t) = Ck(xk)xC(t) ,

where xk is the point of interest in the trajectory of(52) at time k, k = 0, 1, 2, . . . . It is clear that onedoes not have access to the state xk in the observer.So, the best one can do is to use the structure ofAk, obtained through (20), and to use the observerstates, to generate the current model (to be used inthe observer). Then, one can rewrite the proposedobserver (54) as

˙xC(t) = Ak(xk)xC(t) + Bk(xk)uC(t) + LC,k(yC(t)

− yC(t)) ,

yC(t) = Ck(xk)xC(t) .

To simplify the analysis, assume that G(·) andh(·) in (52) are constant matrices. Then, Bk andCk in (53) are equal to Bk and Ck in (54), respec-tively. Thus, the modeling error between the real lo-cal model and the one in the observer can be definedas

ξk ≡ Ak(xk) − Ak(xk) = Ak −Ak . (57)

The error dynamics at each operating point aregiven by

eC(t) = (Ak − LC,kCk)eC(t) + ξkxC(t) . (58)

The solution of (58) in the vicinity of the currentoperating point will have the form ,

eC(t) = exp(Ak−LC,kCk)t eC(0)

+ exp(Ak−LC,kCk)t ⊗ξkxC(t) ,

where ⊗ represents the convolution operation.Then, by designing Lc,k to have a high gain ob-server, one expects that the proposed observer willconverge to the original dynamics, from neighbor-hood to neighborhood along the system trajectory.Of course, an extremely high gain is not recom-mended since it may not be practical. Also, thenonlinear effects will make the convergence slow.Moreover, due to the use of local approximations,this can prevent the synchronization from happen-ing. All these reflect some design tradeoff betweenthe speed of convergence and the size of the mod-eling neighborhoods for approximation accuracy.Nevertheless, the resulting digital counterpart is alow-gain design, suitable for implementation main-taining the convergence characteristics of the analogdesign.

In a digital realization of the proposed observer(55), the sample-hold time T has a direct effect onthe performance of the observer. In the implementa-tion, the actualization of the local models is done foroperating points that are apart from one another.This implies that the linear models that representa larger part of the nonlinear trajectory will pro-duce larger modeling errors in general. In choosingthe sampling period, it is necessary to consider thenature of the system, so that if the original trajec-tory changes at a fast rate, then a smaller samplingtime should be used. This basically means that the

June 11, 2003 13:21 00714


sampling period needs to be chosen carefully, underthe implicit assumption that the sampling theoremalways remains to be satisfied.

5. Numerical Simulations

Three typical yet topologically quite differentchaotic systems are discussed in this section forhybrid synchronization: the piecewise-linear Chua’scircuit, the smooth but nonautonomous Duffing os-cillator, and the smooth autonomous 3D Chen’ssystem.

5.1. Piecewise-linear systems:

Chua’s circuit

The chaotic Chua’s circuit is a very popular bench-mark in the study of chaotic phenomena, for whichdifferent expressions are available in the litera-ture. A particularly interesting one for the presentdiscussion is its piecewise-linear representation in

the dimensionless form [Chen & Dong, 1998]:

x1(t) = α(x2(t) − x1(t) − f(x1(t))

x2(t) = x1(t) − x2(t) + x3(t)

x3(t) = −βx2(t) ,

(59)

where f(x1(t)) is a piecewise-linear function of x1(t)described by

f(x1(t)) = m0x1(t) + 0.5(m1 − m0)(|x1(t) + 1|

−|x1(t) − 1|)

with m0, m1 < 0. This circuit can also be expressedas a set of switching linear systems:

x(t) =

A1x(t) + B1, for x1(t) > 1.0

A2x(t) + B2, for |x1(t)| ≤ 1.0

A3x(t) + B3, for x1(t) < −1.0 ,

y(t) = Cx(t) ,

(60)

with

A1 = A3 =

−α(1 + m0) α 0

1 −1 1

0 −β 0

, A2 =

−α(1 + m0) α 0

1 −1 1

0 −β 0

,

B1 =

−α(m1 − m0)

0

0

, B2 =

0

0

0

, B3 =

α(m1 − m0)

0

0

, C = [1 0 0] .

For the parameter set α = 9, β = 14 + 2/7,m0 = −5/7 and m1 = −8/7, the system (59) or(60) is in the chaotic regime and produces the well-known double-scroll attractor.

From the representation in (60), a digital ob-server was obtained as the digitally redesignedversion (51) of the continuous-time observer (50).Numerical simulation of the continuous-time ob-server was carried out using MatLab and a fourth-

order Runge–Kutta integration algorithm, witha fixed integration step tf = 0.005 s and asample- hold period of T = 0.1 s (equivalentto 20 times of the integration step). The initialconditions were xc(0) = [0.1, 0.1, 0.1]T andxd(0) = [−0.21, 0.78, 0.38]T . The results are pre-sented in Fig. 7. The observer gains in each lin-ear subsystem, obtained for the weighting matricesQ = 10000 I3 and R = 1 are presented in Table 1.

Table 1. Gains for the continuous-time and digitally redesignedobserver for Chua’s circuit (T = 0.1 s).

Subsystem LC Ld

1 [109.3368; 92.9664; 16.1054] [0.9414; 0.5507; −0.2979]2 [105.5021; 92.9601; 16.1031] [0.9294; 0.6571; −0.3566]3 [109.3368; 92.9664; 16.1054] [0.9414; 0.5507; −0.2979]

June 11, 2003 13:21 00714


5.2. Smooth systems

5.2.1. Duffing oscillator

The Duffing oscillator with a periodic driving termcan be expressed in the state–space form as [Chen& Dong, 1998]

x1 = x2(t)

x2(t) = −p1x2(t) − p2x1(t) − p3x31 + q cos(wt) .

(61)

The solution of (61) for the parameter set p1 =−1.1, p2 = 0.4, p3 = 1, q = 1.8 and w = 1.8 isknown to be chaotic.

To design a digital observer such that adiscrete-time response system and the continuous-

time Duffing oscillator synchronize, one has to de-termine the optimal linear model for each operatingpoint along the chaotic trajectory of the system.Following the procedure described above, a set oflinear models in the form of (53) for the Duffingoscillator was obtained, where the external forcingterm q cos(wt) can be easily regenerated in the re-sponse. Then, system (61) is represented at eachsampling point by

x(t) = [x1(t), x2(t)]T

= Akx(t) + Bk(u(t) + q cos(wt)) ,

y(t) = Ckx(t) ,

where

Ak =

[

0 1

−p2 − 3p3x2k1 −p1

]

for ‖xk‖22 = 0 ,

Ak =

0 1

−p2 − 3p3x2k1 +

2p3x4k1

‖xk‖22

−p1 +2p3x

3k1xk2

‖xk‖22

for ‖xk‖

22 6= 0 ,

Bk = [0 1]T and Ck = [1 0] ,

(62)

where xk = [xk1, xk2]T and ‖xk‖

22 = [xk1, xk2]

[xk1, xk2]T is the square magnitude of the operat-

ing point.The designed digital observer obtained for the

Duffing oscillator with Eq. (55) was simulated usingMatLab for a fixed integration step tf = 0.005 s anda sample-hold period of T = 0.05 s (equivalent to 10times of the integration step). The initial conditionswere xc(0) = [0.1, 0.1]T and xd(0) = [−0.20, 8.8]T .The results are presented in Fig. 8. The observergains at each operating point xk were obtained withthe weighting matrices Q = 10000 I2 and R = 1.The evolution is presented in Fig. 9.

5.2.2. Chen’s system

The chaotic Chen’s system is recently coined [Chen& Ueta, 1999; Ueta & Chen, 2000], which isdescribed by

x1(t) = a(x2(t) − x1(t))

x2(t) = (c − a)x1(t) − x1(t)x3(t) + cx2(t)

x3(t) = x1(t)x2(t) − bx3(t) ,

(63)

where a, b, c are real parameters. This system hasa chaotic attractor for the parameter set a = 35,b = 3 and c = 28. This chaotic attractor is not topo-logically equivalent to that of the familiar Lorenzsystem and is found to be the dual system to theLorenz system [Lu et al., 2002].

Following the procedure again, a set of locallinear models for Chen’s system were found, in theform

x(t) = [x1(t), x2(t), x3(t)]T = Akx(t) + Bku(t) ,

y(t) = Ckx(t) ,

where

Ak =

−a a 0

c − a − xk3 c −xk1

xk2 xk1 −b

for ‖xk‖22 = 0 ,

June 11, 2003 13:21 00714


24

For the parameter set 7/5,7/214,9 0 −=+== mβα and 7/81 −=m , the system

(59) or (60) is in the chaotic regime and produces the well-known double-scroll attractor.

From the representation in (60), a digital observer was obtained as the digitally

redesigned version (51) of the continuous-time observer (50). Numerical simulation of

the continuous-time observer was carried out using MatLab and a fourth-order Runge-

Kutta integration algorithm, with a fixed integration step 005.0=ft s and a sample-hold

period of 1.0=T s (equivalent to 20 times of the integration step). The initial conditions

were T

cx ]1.0,1.0,1.0[)0( = and T

dx ]38.0,78.0,21.0[)0(� −= . The results are presented in

Figure 7. The observer gains in each linear subsystem, obtained for the weighting

matrices 310000 IQ = and 1=R , are presented in Table 1.

0 0.5 1 1.5 2 2.5 3

0

1

2

X1

Chua Circuit convergence at T=0.1S

0 0.5 1 1.5 2 2.5 3

0

0.5

1

X2

0 0.5 1 1.5 2 2.5 3

-2

-1

0

X3

Time

(a) (a)

25

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

1

2

3

Chua Circuit observer error at T=0.1S

Err

or 1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

-2

-1

0

Err

or 2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

-2

0

2

Err

or 3

Time

Fig. 7. Digitally redesigned observer for Chua�s circuit (T=0.1s):

(a) Convergence of the digital observer,

(b) Error signals (continuous and digital).

Subsystem CL dL

1 [109.3368; 92.9664; 16.1054] [0.9414; 0.5507; -0.2979]

2 [105.5021; 92.9601; 16.1031] [0.9294; 0.6571; -0.3566]

3 [109.3368; 92.9664; 16.1054] [0.9414; 0.5507; -0.2979]

Table 1. Gains for the continuous-time and digitally redesigned observer

for Chua�s circuit (T=0.1s).

5.2 Smooth systems

5.2.1 Duffing oscillator

The Duffing oscillator with a periodic driving term can be expressed in the state-

space form as [Chen & Dong, 1998]

).cos()()()(

)()(

3

1312212

21

twqxptxptxptx

txtx

+−−−=

=

!

!

(61)

(b) (b)

Fig. 7. Digitally redesigned observer for Chua’s circuit (T = 0.1 s): (a) convergence of the digital observer, (b) error signals(continuous and digital).

June 11, 2003 13:21 00714


27

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0

0.5

1X

1

Duffing system convergence at T=0.05S

0 0.5 1 1.5 2 2.5 3 3.5

-2

0

2

4

6

8

X2

Time

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0

0.5

1

Err

or 1

Duffing system observer error at T=0.05S

0 0.5 1 1.5 2 2.5 3 3.5 4

-8

-6

-4

-2

0

Err

or 2

Time

Fig. 8. Digitally redesigned observer for the Duffing oscillator (T=0.05s):



(a)

(b)

(a)

27

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0

0.5

1

X1

Duffing system convergence at T=0.05S

0 0.5 1 1.5 2 2.5 3 3.5

-2

0

2

4

6

8

X2

Time

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0

0.5

1

Err

or 1

Duffing system observer error at T=0.05S

0 0.5 1 1.5 2 2.5 3 3.5 4

-8

-6

-4

-2

0

Err

or 2

Time

Fig. 8. Digitally redesigned observer for the Duffing oscillator (T=0.05s):



(a)

(b) (b)

Fig. 8. Digitally redesigned observer for the Duffing oscillator (T = 0.05 s): (a) convergence of the digital observer,(b) error signals (continuous and digital).

June 11, 2003 13:21 00714


28

0 0.5 1 1.5 2 2.5 3 3.5 4100

100.5

101

101.5

102

102.5L

c1

Observer gain values for Duffing system with T=0.05S

0 0.5 1 1.5 2 2.5 3 3.5 40.82

0.83

0.84

0.85

0.86

Ld

1

Time

0 0.5 1 1.5 2 2.5 3 3.5 4

50

100

150

200

Lc

2


0 0.5 1 1.5 2 2.5 3 3.5 4

0.5

1

1.5

2

Ld

2

Time

Fig. 9. Gains for the continuous-time and digitally redesigned observer

for the Duffing oscillator (T=0.05s):

(a) Observer gain for state 1x (continuous and digital)

(b) Observer gain for state 2x (continuous and digital).

(b)

(a) (a)

28

0 0.5 1 1.5 2 2.5 3 3.5 4100

100.5

101

101.5

102

102.5

Lc

1Observer gain values for Duffing system with T=0.05S

0 0.5 1 1.5 2 2.5 3 3.5 40.82

0.83

0.84

0.85

0.86

Ld

1

Time

0 0.5 1 1.5 2 2.5 3 3.5 4

50

100

150

200

Lc

2


0 0.5 1 1.5 2 2.5 3 3.5 4

0.5

1

1.5

2

Ld

2

Time


for the Duffing oscillator (T=0.05s):


(b) Observer gain for state 2x (continuous and digital).

(b)

(a)

(b)

Fig. 9. Gains for the continuous-time and digitally redesigned observer for the Duffing oscillator (T = 0.05 s): (a) observergain for state x1 (continuous and digital), (b) observer gain for state x2 (continuous and digital).

June 11, 2003 13:21 00714


30

equivalent digital version using equation (55). The chaotic Chen�s system was then

simulated using MatLab, with a Runge-Kutta fourth-order algorithm with a fixed

integration step 001.0=ft s and a sample-hold period of 025.0=T s (equivalent to 25

times of the integration step). The initial conditions were T

cx ]37,0,10[)0( −= and

T

dx ]12.3,70.32,71.4[)0(� −−= . The results are presented in Figure 10. The observer gains

at each operating point kx were obtained with the weighting matrices 310000 IQ = and

1=R . The evolution is presented in Figure 11.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-10

0

10

X1

Chen system convergence at T=0.025S

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-20

0

20

X2

0 0.5 1 1.5

10

20

30

X3

Time

(a) (a)

31

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-5

0

5

Err

or 1

Chen system observer error at T=0.025S

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

0

20

40

Err

or 2

0 0.5 1 1.5

0

10

20

30

Err

or 3

Time

Fig. 10. Digitally redesigned observer for Chen�s system (T=0.025s):



0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2125

130

135

140

145

Observer gain values for Chen system with T=0.025S

Lc

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0.8

0.82

0.84

0.86

Ld

1

Time

(b)

(a)

(b)

Fig. 10. Digitally redesigned observer for Chua’s system (T = 0.025 s): (a) convergence of the digital observer, (b) errorsignals (continuous and digital).

June 11, 2003 13:21 00714


Ak =

−a a 0

c − a − xk3 +x2

k1xk3

‖xk‖22

c +xk1xk2xk3

‖xk‖22

−xk1 +xk1x

2k3

‖xk‖22

xk2 −x2

k1xk2

‖xk‖22

xk1 −xk1x

2k2

‖xk‖22

−b −xk1xk2xk3

‖xk‖22

for ‖xk‖22 6= 0 ,

Bk = [0 0 1]T and Ck = [1 0 0] ,

with xk = [xk1, xk2, xk3]T being the current oper-

ating point in the chaotic trajectory, and ‖xk‖22 =

[xk1, xk2, xk3][xk1, xk2, kk3]T .

Using this representation, a digitally re-designed observer was obtained by first designinga continuous-time observer in the form (54) andthen converting it into the equivalent digital versionusing Eq. (55). The chaotic Chen’s system wasthen simulated using MatLab, with a Runge–Kutta fourth-order algorithm with a fixed in-tegration step tf = 0.001 s and a sample-holdperiod of T = 0.025 s (equivalent to 25 timesof the integration step). The initial conditionswere xc(0) = [−10, 0, 37]T and xd(0) = [−4.71,−32.70, 3.12]T . The results are presented in Fig. 10.The observer gains at each operating point xk

were obtained with the weighting matrices Q =10000 I3 and R = 1. The evolution is presentedin Fig. 11.

6. Conclusions

In this paper, a solution to the hybrid chaos syn-chronization problem was derived from the pointof view of observer design, where the drive isa continuous-time system and the response is adiscrete-time system. In this approach, a set of locallinear systems that together describe the entire non-linear system are first modeled, using either theswitching structure of the system or the optimallinearization method. Then, a high-gain optimalobserver is designed for each local model, wherethe set of optimal observers are converted to theirequivalent low-gain digital counterparts by usingthe newly developed prediction-based digital re-design method. Important observations about theproposed method include:

The approach works for hybrid systemssynchronization, with an analog drive and a digi-tal response, or vice versa.

31

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-5

0

5E

rro

r 1

Chen system observer error at T=0.025S

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

0

20

40

Err

or 2

0 0.5 1 1.5

0

10

20

30

Err

or 3

Time

Fig. 10. Digitally redesigned observer for Chen�s system (T=0.025s):



0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2125

130

135

140

145


Lc

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0.8

0.82

0.84

0.86

Ld

1

Time

(b)

(a) (a)

Fig. 11. Gains for the continuous-time and digitally redesigned observer for the Chen system (T = 0.025 s): (a) observergain for state x1 (continuous and digital), (b) observer gain for state x2 (continuous and digital), (c) observer gain for statex3 (continuous and digital).

June 11, 2003 13:21 00714


32

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

220

240

260

280


Lc

2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

1.2

1.3

1.4

1.5

Ld

2

Time

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

-50

0

50


Lc

3

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

-0.5

0

0.5

Ld

3

Time


for the Chen system (T=0.025s):


(b) Observer gain for state 2x (continuous and digital)

(c) Observer gain for state 3x (continuous and digital).

(b)

(c)

(b)

32

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

220

240

260

280


Lc

2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

1.2

1.3

1.4

1.5

Ld

2

Time

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

-50

0

50


Lc

3

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

-0.5

0

0.5

Ld

3

Time


for the Chen system (T=0.025s):


(b) Observer gain for state 2x (continuous and digital)

(c) Observer gain for state 3x (continuous and digital).

(b)

(c) (c)

Fig. 11. (Continued )

The methodology can be applied to a generalnonlinear system without the demand of satisfyinga Lipschitz condition for synchronization.

The maximum sample-hold time used has tosatisfy the sampling theorem, but it can be rela-tively large — even so large that other digital designmethods cannot accept.

Since local linearization is used, it is necessaryto chose operating points of interest to be as close toeach other as possible, so as to minimize the mod-eling errors.

It has been observed that there sometimes willbe a small steady-state error in the digital imple-mentation of the hybrid synchronization systems, inparticular for a general smooth system. The mainreason may be due to the linear representation ofthe nonlinear system, since this occurs more oftenfor complex trajectories such as Chen’s attractor.

A simple communication scheme based on theapproach studied in this paper has been designedand tested, from the chaos masking approach, whichworks very well except when Chen’s system is used

June 11, 2003 13:21 00714


due to the topological complexity of its attractor. Incomparison, if Chua’s circuit is used then the com-munication system works very well even for largesampling times.

Therefore, more research efforts are neededto carry out for the hybrid chaos synchronizationdesign and analysis.

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