Using geometric control and chaotic synchronization to estimate an unknown model parameter

Ubiratan S. Freitas,1,* Elbert E. N. Macau,1,† and Celso Grebogi2

1Laboratório Associado de Computação e Matemática Aplicada, Instituto Nacional de Pesquisas Espaciais,Caixa Postal 515, 12227-010, São José dos Campos, São Paulo, Brazil

2Instituto de Física, Universidade de São Paulo, 05315-970 São Paulo, São Paulo, BrazilsReceived 1 October 2004; published 27 April 2005d

We present a new parameter estimation procedure for nonlinear systems. Such technique is based on thesynchronization between the model and the system whose unknown parameter is wanted. Synchronization isaccomplished by controlling the model to make it follow the system. We use geometric nonlinear controltechniques to design the control system. These techniques allow us to derive sufficient conditions for synchro-nization and hence for proper parameter estimation. As an example, this procedure is used to estimate aparameter of an example serving as a model.

DOI: 10.1103/PhysRevE.71.047203 PACS numberssd: 05.45.Gg, 05.45.Xt

Since 1990f1,2g, chaos synchronization has been a sub-ject of intense study and is now considered to be a funda-mental mechanism behind a variety of behaviors in naturef3g. As a consequence of the intense research effort, a mul-tiplicity of applications was and is being realized in a varietyof areas, e.g., in opticsf4g, communicationf5g, and neuralnetworksf6g. In the majority of applications, we have thescenario where a chaotic system, called the sender or master,generates a signal which is sent over a channel to a receiver.The receiversor slaved uses this signal to synchronize itselfto the master.

Behind this scenario, there is the assumption that the pa-rameters in both sender and receiver systems are identical.However, in a real situation, this assumption is unrealistic.For example, two electronic components are never identical,even if they came from the same assembly line. As a conse-quence, parameters on different systems are unequal withprobability one. This reality has severe consequences in thecontext of chaotic synchronization: progressive parametermismatch may successively imply a large burst of desyn-chronizationsbubbling transitiond, blowout bifurcation, andlag synchronizationf7g, among other phenomena.

In this work we devise a synchronization strategy thatallows us to estimate an unknown parameter of the mastersystem. Other authors have also used synchronization to es-timate unknown parametersf8–10g. Their work, however,does not provide sufficient conditions for a general couplingbetween master and slave to yield synchronization and,hence, a proper parameter estimation. Our approach, on theother hand, provides a straightforward procedure that, givena master system, will check if the strategy is applicable and,if so, will design a coupling that insures synchronization.Such improvement is possible because we base our strategyon geometric controlf11,12g. Presently, our strategy can es-timate a single unknown parameter of the sender, but thiscould be easily extended to many parameters.

The main idea in our approach is to treat synchronizationas a control problemf13g. For a recent application of this

concept to chaotic systems with uncertainties seef14g. In ourapproach, we conceive the receiver as a system to be con-trolled. Its unknown parameter is considered as the controlinput. The controller, which is designed based on the geo-metric control theory, dynamically adjusts this unknown pa-rameter to make the slave follow a desired trajectory. Sincesynchronization is our goal, the desired trajectory is the onegenerated by the master system. The controller then acts asthe coupling between the master and the slave systems. Weshow that the control action, i.e., the unknown parameter,equals the corresponding parameter of the master system,once synchronization is achieved. Actually, using these de-sign techniques we can establish sufficient conditions forcontrollability, and hence for synchronizability and param-eter recovery.

For concreteness, we consider two almost identical

n-dimensional systems, systemA sthe senderd: X=FsX ,pdand systemB sthe receiverd: Y=FsY,ud, whereX ,YPRn arelocal coordinates for a smooth manifoldM, F is a smoothfunction in its variables,p for systemA is a fixed parametervalue, andu for systemB is an unknown and externallyadjustable parameter. We consider that the two systems areinitialized with different initial conditions, and the signalXis transmitted to the receiverssystemBd so that this signal isavailable at all times at the receiver location. Our strategy isdepicted in Fig. 1.

As can be seen from this picture, the controller uses theinformation provided by signalsX andY to instantly adjustthe systemB parameteru until this system is completelysynchronized to the systemA, i.e., Xstd=Ystd for t. ts. Ac-tually, the parameteru is regarded as acontrol input for the

*Electronic address: ubiratan@lac.inpe.br†Electronic address: elbert@lac.inpe.br

FIG. 1. The diagram of the synchronization strategy. The slavesystem is regarded as a system to becontrolled. The controller actson the parameterp in order to make the slave to follow the mastersystem.

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slave system. The controller has to correctly adjust this inputfor the synchronization to occur.

We assume that both systems can be written in a smoothaffine form as follows:

systemAssenderd:X = GsXd + pHsXd s1d

and

systemBsreceiverd:Y = GsYd + uHsYd, s2d

whereu is the control input. Let us define the synchroniza-tion error Estd=Xstd−Ystd, and then the error dynamics is

E=GsXd−GsYd+pHsXd−uHsYd. Now, suppose that syn-

chronization is achieved. This implies thatEstd=Estd=0, for

t. ts. Thus,Estd at E=0 is given by

E = sp − udHsXd. s3d

With the restrictionE=0, i.e., both systems remain synchro-nized, we conclude that eitherHsXd=0 for t. ts or u=p. Wemay discard the former because otherwise it would implythat our systems are independent of their parameters, whichcontradicts our assumptions. Then, we must conclude that,under these conditions, synchronization implies proper pa-rameter estimation. So, in order to recover the unknown pa-rameter, we must design a control system that synchronizesthe receiver to the sender.

Such control system must act on a nonlinear systemsthechaotic slaved, through a specific inputsthe unknown param-eterd, following a reference signal that is not periodic nor hasnegligible amplitudesthe master’s chaotic trajectoryd. More-over, the tracking error must be as small as possible. Theseare rather difficult requirements for a control system designand traditional linear techniques cannot be used. Any controldesign technique suitable to nonlinear systems could be used.We chose geometric control theory because it allows us tocheck if a nonlinear system is controllable and, if it is, pro-vides a design method that will lead to a successful control-ler. In other words, geometric control theory gives us neces-sary and sufficient conditions for thecontrollability of anonlinear system, and, if the system is controllable, a proce-dure to design a controller.

At this point, let us give some background on geometriccontrol theory. LetM be a smooth manifold of dimensionnandsU ,wd=sU ,x1, . . . ,xnd be a local coordinate chart forM.A smooth vector fieldG on M assigns to everyqPM atangent vectorGqPTqM. Given a smooth vector fieldG anda function f :M→R, the functionGsfdsqd :M→R is calledthe total derivative off along G, or theLie derivativeof falong G and is denoted as LGf. If G is expressed in localcoordinates as the vectorfx1sqd ,x2sqd , . . . ,xnsqdgT, then wehave

LGfsqd = Gsfdsqd = oi=1

n]f

]xifx1,x2, . . . ,xngGifx1,x2, . . . ,xng

= = fGsqd. s4d

For G andH any two smooth vector fields onM, we define

a new vector field, denoted asfG,Hg, or adGH, called theLiebracketof G andH by setting, in local coordinates,

fG,Hg = = HG − = GH, s5d

where=G is the Jacobian matrix ofG.Our strategy can be divided in two steps. In the first one,

we need to find a transformation pair as follows,

Y = FsYd s6d

and

u = CsY,ud, s7d

that transforms Eq.s2d to the following linear system:

yi = yi+1, i = 1, . . . ,n − 1, yn = u. s8d

According to the geometric control theoryf11,15g, thistransformation is always possible wherever there exists anopen regionV,M so that the following conditions hold forall qPV:

• the set of vector fieldshH ,adGH , . . . ,adGn−1Hj is linearly

independent, and• the set of vector fieldshH ,adGH , . . . ,adG

n−2Hj is involu-tive.

A set of vector fields isinvolutive if the Lie bracket ofevery two of its elements can be expressed as a linear com-bination of the vector fields in the set. If these two conditionshold, the transformationss6d ands7d exist and can be calcu-lated as follows:

FsYd = ffsYd LGfsYd ¯ LGn−1fsYdgT, s9d

wherefs·d is a solution of the system of partial differentialequations

LHfsYd = 0,

LHLGfsYd = 0,

A

LHLGn−2fsYd = 0,

LHLGn−1fsYd Þ 0. s10d

Furthermore, the inverse input transformationu=C−1sY,udis given by

C−1sY,ud =u − asYd

bsYd, s11d

where

asYd = LGn fsYd, s12d

and

bsYd = LHLGn−1fsYd. s13d

In the second step, we apply a classical control strategy sothat the system follows a reference in the transformed space.

Suppose this reference,X, is such that

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X˙

i = Xi+1, i = 1, . . . ,n − 1, X˙

n = Xd, s14d

and define the error in the linear space asestd=X1−Y1. Then,

the error dynamics is given byesndstd=Xd− u. A linear statefeedback controller can then be used to stabilize this dynam-ics in the origin. This controller is given by

u = Xd − oi=1

n

kiesi−1d, s15d

where theki are positive constants. This leads to the stableerror dynamics

esnd + oi=1

n

kiesi−1d = 0 s16d

and the controller’s gainski may be chosen to give the de-sired response. Controlled in this way, the slave system isable to follow any reference in the form ofs14d restricted tothe regionV. Since synchronization is desired, we use themaster system as the reference. The transformations6d isapplied to the master systems1d yielding

Xd = FsXd s17d

and Xd=Xn. As this reference signal is a trajectory of themaster system transformed via a diffeomorphism, when theerror estd approaches zero the slave system will be synchro-nized with the master system.

As an example, we apply these ideas to the Lorenz sys-tem. The master system is taken as the usual equations

x1 = ssy1 − x1d, y1 = rx1 − y1 − x1z1, z1 = − bz1 + x1y1,

s18d

and we suppose the parameterb is unknown. The slave sys-tem is a copy ofs18d but with this parameter treated as aninput

x2 = ssy2 − x2d, y2 = rx2 − y2 − x2z2, z2 = − uz2 + x2y2.

s19d

In order to design the controller, we need to check if thesystems19d is controllable. The system is put in the form ofs2d leading to

GsYd = fssy − xd, rx − y − xz, xygT s20d

and

HsYd = f0, 0, −zgT. s21d

Here the subscripts were dropped for clarity. As the firstcondition, the vector fieldshH ,adGH ,adG

2 Hj must be linearlyindependent. Computing the Lie bracket ofG andH leads to

fG,Hg = f0 − xz − xygT. s22d

Repeating the procedure we obtain

fG,fG,Hgg = 3 sxz

ss − 1dxz− syz− 2x2y

ss + 1dxy+ 2x2z− sy2 − rx24 . s23d

These vector fields are linearly independent ifsxzÞ0. Thesecond required condition is that the sethH ,adGHj must beinvolutive. To check this we must writefH ,adGHg as a linearcombination ofH and adGH. The Lie bracket ofH and adGHis given by

fH,adGHg = f0 xz − xygT. s24d

By defining c1sYd=2xy/z and c2sYd=−1, fH ,adGHg can bewritten as

fH,adGHg = c1sYdH + c2sYdadGH . s25d

Therefore the sethH ,adGHj is involutive in the wholeR3.This shows that the slave system given bys19d is control-lable everywhere inR3 but in the planesxz=0.

The functionfsYd needed to compute the diffeomorphisms9d is a solution to the following system of partial differentialequation

]f

]zz= 0,

]f

]yxz+

]f

]zxy= 0,

−]f

]xsxz+

]f

]yfss − 1dxz− syz− 2x2yg

+]f

]zfss + 1dxy+ 2x2z− sy2 − rx2g Þ 0. s26d

A simple solution to this system isfsYd=fx,0 ,0gT whichleads to the global diffeomorphism

Y = FsYd = 3 x

ssy − xdssr + s2dx − ss + s2dy − sxz

4 . s27d

FIG. 2. The plot ofx in time for both master and slave systems.All quantities are dimensionless.

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This transformation is applied to the slave system giving

the transformed state coordinatesY, and to the master system

giving the referenceX. The value ofXd is obtained differen-tiating the last row ofs27d which leads to

Xd = ssr + s2dssy1 − x1d − ss + s2dsrx1 − y1 − x1z1d

− sfssy1 − x1dzz + x1s− bz1 + x1y1dg. s28d

The controller used is of the form ofs15d, with the gainschosen as to give a fast nonoscillating response. The valueschosen werek0=8000,k1=1200, andk2=60.

The last step in the controller design is the inverse inputtransformationC−1. This transformation relates the controloutput in the linear spaceu to its counterpart in the originalspaceu. The inverse transformation is given bys11d. Tocompute it we need the functionsasYd andbsYd given by

asYd = fssr + s2d − szgssy − xd − ss + s2dsrx − y − xzd

− sx2y s29d

andbsYd=sxz, respectively. Figure 2 shows our results for aLorenz system withsr ,s ,bd= s60,10,83d, with the x statevariable as a function of time for both the master and slave

systems. In this figure and in those that follow, all quantitiesplotted are dimensionless. A plane projection of both attrac-tors is shown in Fig. 3. Synchronization is achieved as ex-pected.

As a byproduct, the control actionu converges to thevalue of the unknown parameterb. This result is shown inFig. 4

In summary, we presented a parameter estimation proce-dure for nonlinear systems that is based on synchronizationand geometric control theory. Such theory allowed us to de-rive sufficient conditions for synchronization and parameterrecovery. We used this technique to successfully estimate anunknown parameter in Lorenz’s system.

The proposed method, however, still needs some im-provements. First, any practical application involves noisydata, and the method must be modified to deal with thiscondition. Such modification is under conclusion and will bepublished in a future paper. Second, the proposed strategyneeds the entire state vectorX as input to the controller. Thisrestricts the technique to applications where master’s statevector can be fully measured. Hopefully this can be relaxedin the future, maybe via the use of geometric control theory.

The authors would like to acknowledge the support ofFAPESP.

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FIG. 3. The projections in the planexz of master’s and slave’sattractors. All quantities are dimensionless.

FIG. 4. The control actionu plotted against time showing con-vergence to the value ofb s 8

3d. All quantities are dimensionless.

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