Structure identification based on steady-state control: Experimental results and applications
Mattia Frasca,1,2,* Dongchuan Yu,3, and Luigi Fortuna1,2,1Dipartimento di Ingegneria Elettrica, Elettronica e dei Sistemi, Universit degli Studi di Catania, viale A. Doria 6, 95125 Catania, Italy
2Laboratorio sui Sistemi Complessi, Scuola Superiore di Catania, Via San Nullo 5/i, 95123 Catania, Italy3School of Automation Engineering, University of Electronic Science and Technology of China, Chengdu 610054, China
Received 3 June 2009; revised manuscript received 8 October 2009; published 24 February 2010
We report experimental results on structure identification of nonlinear systems by a steady-state controlmethod. The idea underlying the method is to drive the nonlinear system to steady state by applying a suitablefeedback control input. It turns out experimentally that this control-based structure identification method can beused for some applications, such as estimation of initial conditions and state variables of nonlinear systems andstructure identification of some special elements. Two attractors of the Chua oscillator are presented to illus-trate the reliability of the suggested techniques under the hypotheses of measurable state variables and physicalaccess to the system for implementing the proportional feedback.
DOI: 10.1103/PhysRevE.81.026212 PACS numbers: 05.45.Xt
Mathematical models can be used to quantitatively under-stand dynamical behavior of natural or artificial systems. Insome cases, the structure of these mathematical models canbe derived from first principles and only some model param-eters have to be determined from experimental data. Thisproblem has been well studied in the literature and recentlyhas regained considerable interest especially for chaotic sys-tems. Several methods have been suggested based on auto-synchronization 112 such methods derive from the basi-lar concept of synchronization in chaos theory 1315,balanced synchronization 16,17, partial synchronization18,19, parametric optimization 2023, nonlinear filters22,24, or special properties of the feedback structure insystems with delayed feedback 25. The problem of param-eter estimation in time-delay systems has been recently ad-dressed with specific techniques 26,27. In particular, in 26to estimate the delay time the idea of disturbing the systemby a short-correlated noisy signal of large amplitude is ex-ploited, and the delay is then identified by analyzing thecorrelation function, while in 27 it is the analysis of thesystem response to regular external impulsive perturbationsthat allows the reconstruction of the time-delay system pa-rameters.
However, many parameter estimation methods are appli-cable under the assumption that the structure of these math-ematical models is known accurately and their performancemay dramatically be deteriorated even in the presence ofsmall structure error. In practice, however, the structure ofthe mathematical models usually is not or only partiallyknown. Therefore, to understand the dynamical behavior ofsystems of interest, one first has to identify the system struc-ture. Such an issue has not been fully investigated especiallyfor complex dynamical systems.
Recently, the use of control-based methods for estimatingthe structure of complex systems has been investigated in
several works 2832. The idea underlying the method is todrive the system to steady states by adding to the systemsuitable feedback control inputs. The method has been suc-cessfully applied to the identification of the system dynamics28, to estimate the topology of a complex network 29,30,and to identify the delays underlying a nonlinear dynamicalsystem 31,32. In this article, we report experimental struc-ture identification of chaotic circuits using the control-basedmethod and show that the control-based structure identifica-tion method can be applied to some applications of physicalinterest, such as system modeling and structure identificationof some special elements.
To illustrate the reliability of the suggested techniques, weconsider a Chua oscillator described by the following dimen-sionless equations 33:
x1 = x2 hx1 ,
x2 = x1 x2 + x3 ,
x3 = x2 x3 , 1
with hx=m1x+0.5m0m1x+1 x1. The Chua oscil-lator is a well-known generalization of the Chuas circuit, inthe sense that it can implement all the dynamics of theChuas circuit 34,35 and, moreover, every dynamics thatcan be generated by any member of the Chuas family can beobtained in the Chua oscillator. So, quoting Chua 36, thiscircuit represents structurally the simplest and dynamicallythe most complex member of the Chuas circuit family.
In this paper, two chaotic attractors generated by the Chuaoscillator are taken into account. The two chaotic attractorscorrespond to two different sets of parameters. In the twoexperiments, the nonlinearity has the same qualitative form,but different parameters. The slopes of the two considerednonlinearities also differ for their sign, this leads to two dif-ferent circuits constituting two interesting case studies.
From Eq. 1, the classical double scroll attractor shownby the Chuas circuit is obtained for the following param-eters: =1, =9, =14.286, =0, m0=1 /7, and m1=2 /7.The double scroll attractor shown by this circuit is reported
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in Fig. 1. The second circuit is obtained for the following setof parameters: =1, =1.5601, =0.0156, =0.1581,m0=0.7562, and m1=0.9575. The experimental attractor ob-tained with these parameters is shown in Fig. 2.
Let us consider a generic nonlinear system described by
x = fx , 2
where x= x1 ,x2 , . . . ,xnTRn represents the state vector andf= f1 , f2 , . . . , fnT :RnRn is the dynamics of the system.
Let us then add to system 2 a control input of the fol-lowing form:
u = kx , 3
so that the controlled system reads
x = fx kx , 4
where the gain matrix k=diagk1 ,k2 , . . . ,kn is a diagonalmatrix with nonnegative elements ki and = 1 ,2 , . . . ,nTRn is a constant vector to be specified.
It has been shown 28 that if a proper gain matrix k isused, the system 4 can be driven to a steady state x=0satisfying,
fx = kx . 5
To estimate the function fx, one needs to apply Eq. 5 withm different values of the constant vector . In this way mdifferent data pairs xm ,kxmm can be obtained to rep-resent the input-output relation of the function f, from whichthe function f can be estimated. In particular, we will show aneural-network-based approach to estimate the function ffrom the obtained data pairs xm ,kxmm. In fact, theCybenko theorem 37 guarantees that a single hidden-layerfeed-forward neural network is capable of approximating anycontinuous multivariate function to any desired degree ofaccuracy from the input-output data pairs.
Especially, if the system Eq. 4 is driven to a steady stateunder the control signal Eq. 3 with kj =0 for all j i thatis, only the i-th equation is controlled, then one gets
f ix = kixi i , 6
which can similarly be applied to uncover the structural in-formation or property of the function f i.
Above analysis has explicitly shown the basic principle ofthe control-based structure identification method. Here, wefocus on its potential applications of physical interest.
A. System modeling
As a first potential application, the control-based structureidentification method can be applied to system modeling.Actually, after the function f has been estimated with arbi-trary accuracy by using neural network approximation tech-niques 37, the following equation:
y = fy , 7
can thus be considered as a model to represent system Eq.2, where the function f is an estimation of function f.Therefore, one may analyze the dynamical behavior andproperties, the estimation of the initial conditions, and thesynthesis issues of system 2 using the model 7.
1. Analyzing the dynamical behavior of a system from its model
It is usually difficult to analyze the dynamical behavior ofa nonlinear system esp. chaotic system because: i it issensitive to initial conditions and system parameters; and iione, in practice, often cannot change the initial conditions ofa nonlinear system. However, one may change the initialconditions of the model very easily. As a result, one cannumerically analyze the influence of the initial conditions onthe attractor structure or phase trajectories.
2. Estimating initial conditions of a nonlinear system
Let us consider the system 2 with initial conditionsx0=x0, which actually reads
x = fx,x0 , 8
and its model 7 with initial conditions y0=y0, given by
y = fy,y0 . 9
If x0=y0, system 8 and its model 9 may synchronize iden-tically with each other at least in the very beginning stage,
FIG. 1. Experimental results: projection on the x2x3 plane ofthe double scroll attractor. Horizontal axis: 1 V/div; vertical axis200 mV/div.
FIG. 2. Experimental results: projection on the x2x3 plane ofthe chaotic attractor obtained by the Chua oscillator with param-eters =1, =1.5601, =0.0156, =0.1581, m0=0.7562, andm1=0.9575. Horizontal axis: 2 V/div; vertical axis 200 mV/div.
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even when system 8 is chaotic, that is, ytxt forall 0 tTs, with sufficiently small and proper Ts. Other-wise, both systems generally cannot synchronize identicallywith each other and the synchronization error Es, simply de-fined by i=1
N yiTxiT with proper sampling interval T,is a function of the variable y0x0 and x0 can roughly beestimated by searching the minimal value of Es.
3. Synchronization between a system and its model
Let us now move to analyze synchronization between asystem and its model through adding proper unidirectionaland bidirectional couplings, and revisit system 2 and itsmodel 7 as an illustrating example. We first treat the unidi-rectional coupling case and add the coupling term yx to the right hand side of Eq. 7, where =diag1 ,2 , . . . ,n denotes the coupling strength vector.In this case, the error system reads,
e = fe + x fx + fx fx e , 10
where e=yx.When the coupling strengths i increase gradually and are
beyond a critical value, the error system Eq. 10 may runinto a small neighborhood around the origin point, sincefx fx is satisfied for any x. In particular, if the identicalsynchronization manifold holds through unidirectional cou-plings of partial state variables i.e., i=0 holds for some i,then the remaining state variables of system 2 can be esti-mated by the model. In this case, the model can actually betaken as a state observer for system 2. The above analysiscan be extended to the bidirectional coupling case. One maysignificantly declare that the model 7 is good if the identi-cal synchronization between system 2 and the model 7holds through adding proper unidirectional and bidirectionalcouplings.
B. Identification of some elements of interest
In some applications, one may know partial structure ofthe whole system as a preknowledge but the structure ofsome functional elements may be unknown. For example, forcoupled systems given by
x1 = g1x1 + h1x1,x2 , 11
x2 = g2x2 + h2x1,x2 , 12
where x1Rn and x2Rn are state vectors of systems 11and 12, respectively; g1 and g2 describe the node dynamics;h1 and h2 are coupling functions. It is reasonable to assumethat the local dynamics of each system may be known ex-actly but the coupling functions are not. It is of interest toestimate functions h1 and h2. By the control-based method,one may obtain estimated f1 and f2, where f1x1 ,x2=g1x1+h1x1 ,x2 and f2x1 ,x2=g2x2+h2x1 ,x2. Thenh1 and h2 can easily be estimated.
Many chaotic systems including Lorenz system, Rsslersystem, and Chuas circuits can be put in the form
x = Ax + hx , 13
where A is a constant matrix and the nonlinearity hx is theessential element that may determine the existence and prop-
erties of chaos. To analyze the dynamical properties of sys-tem 13, one first has to estimate function h.
III. IDENTIFICATION OF THE CHUA OSCILLATORDYNAMICS AND NONLINEARITY
We apply control 3 to the Chua oscillator 1 and pro-pose a method to identify either the whole structure of thecircuit or a part of it i.e., its nonlinearity. In the first case,we only suppose that the order of the system is known andthe state variables are measurable and controllable. In thesecond case, we suppose that only some important structuralinformation needs to be identified in this case the iv char-acteristics of the nonlinear elements.
The equations of the Chua oscillator with control 3 be-come,
x1 = x2 hx1 k1x1 1 ,
x2 = x1 x2 + x3 k2x2 2 ,
x3 = x2 x3 k3x3 3 , 14
where positive ki are control gains and i are constants to bespecified. In the following the two cases are described morein detail.
A. Identification of the Chua oscillator dynamics
Let us first consider the case in which the whole structureof the circuit has to be identified. As it is the case of theimplementation of the Chua oscillator under examination33, all the state variables are assumed measurable.
We start from Eq. 14 and follow this algorithm:1 We apply the control and let = 1 ,2 ,3T vary so
that we obtain m pairs: xi ,i.2 From Eq. 5 we calculate fx=kxi, so that to
obtain m pairs: xi , fxi.3 We can now apply an identification technique to de-
velop a model fitting the data obtained at point 2 i.e., pairsxi , fxi. In particular, we use an approach based on neuralnetworks. In fact, the Cybenko theorem 37 guarantees thata single hidden-layer feed-forward neural network is capableof approximating any continuous multivariate function toany desired degree of accuracy.
The approach illustrated for the Chua oscillator can beeasily adapted to any other continuous nonlinear system andis quite independent of the particular identification techniqueadopted. Other identification techniques can be used in step3. The approach described provides a black box model of thechaotic dynamics and not a reconstruction of the attractor orthe learning of the state variable trends.
B. Identification of the Chua oscillator nonlinearity
Let us now suppose that the only unknown part of theChua oscillator dynamics is the nonlinearity hx. Moreover,let us suppose that ki are such that steady-state control iseffective. Let us focus on the first equation of Eq. 14 andset x1=0 to obtain,
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hx1 = x2 k1x1 1
Fixed 1, 2, and 3, Eq. 15 gives us a value for hx1. Byvarying 1, 2, and 3 we can have a set of measures forhx11 ,2 ,3 and use them to identify the nonlinearityappearing in the circuit.
In our case, it can be demonstrated that varying just oneparameter 1 is enough to have a complete set of mea-sures to identify the nonlinearity hx1, since the steady-statevalue of variable x2 is a function of that of x1. In practice, wekept constant 2...