Time-Invariant Dynamic Systems IdentificationBased on the Qualitative Features of the Response

Juan J. Flores a, Nelio Pastor a,∗

aDivision de Estudios de Postgrado, Facultad de Ingenierıa Electrica, UniversidadMichoacana de San Nicolas de Hidalgo, Morelia, Mexico

Abstract

The problem of Systems Identification starts with a time-series of observed data and triesto determine the simplest model capable of exhibiting the observed behavior. This opti-mization problem searches the model from a space of possible models. In this paper, wepresent the theory and algorithms to perform Qualitative and Quantitative Systems Identifi-cation for Linear Time Invariant Dynamic Systems. The methods described here are basedon successive elimination of the components of the system’s response. Sinusoidals of highfrequencies are eliminated first, then their carrying waves. We continue with the process un-til we obtain a non-oscillatory carrier. At that point, we determine the order of the carrier.This procedure allows us to determine how many sinusoidal components, and how manyexponential components are found in the impulse response of the system under study. Thenumber of components determines the order of the system. The paper is composed of twoimportant parts, the statement of some mathematical properties of the responses of LinearTime Invariant Dynamic Systems, and the proposal of a set of filters that allows us to im-plement the recognition algorithm. We present the aplication of the proposed methodologyto analysis and modeling of electrical circuits and electrical power systems.

Key words: Dynamic systems, systems identification, differential equations, qualitativereasoning

This research has been supported by CIC-UMSNH grant No. 10.3∗ Correspondig author. Address: Maestro de Avila 90, Fracc. Fray Antonio de San Miguel,Morelia, Michoacan, 58270, Mexico

Email addresses:juanf@zeus.umich.mx (Juan J. Flores), npastor@zeus.umich.mx (Ne-lio Pastor).

Preprint submitted to Engeenering Applications of Artificial Intelligence 10 March 2004

1 Introduction

System identification deals with the problem of building mathematical models ofdynamical systems based on observed data from the system and tries to determinethe simplest model capable of exhibiting the observed behavior. This optimiza-tion problem searches the model from a space of possible models. In traditionalmethods, the process of structural identification has received less attention thanthe parametrization of the model. In most of cases the structural estimation is notgenerally made. Typically, a certain model structure is chosen by the user whichcontains unknown parameters. The choice of an appropiate model structure is mostcrucial for a successful identification application. This choice must be based bothon an understanding of the identification procedure and on insights and knowledgeabout the system to be identified. An alternating way to infer a suitable structure,guided by system knowledge and the collected data set, is presented in this paper.In our approach we limit the search space to Linear Time-Invariant models.

The problem we are assessing in this paper is that of Qualitative Systems Identi-fication for Linear Time Invariant (LTI) Dynamic Systems. This kind of systemscan be represented by Linear ordinary Differential Equations with Constant Coef-ficients. Although this problem may seem over-constrained, there are many impor-tant problems in engineering and physics that can be expressed in mathematicalterms by this kind of differential equations. We could even assess time-varying oreven non-linear systems, if we consider them as piece-wise decomposed by linearapproximations of the original systems. So, this is a limited yet interesting domainto start with.

The Systems Identification process can be decomposed in two steps: the first stepcalled structural (or qualitative) identification, involves determining the qualitativefeatures of the mechanism, i.e. the qualitative form of the system inside the blackbox [Bradley, Easley, & Stolle2001] [Bellazzi, Guglielmann, & Ironi1999][Bradley & Stolle1996] [Kay, Rinner, & Kuipers2000]; once we know the natureof the mechanism, in the second step we proceed to determine the numerical valueof the parameters of the model determined in the first step.

This second part can be done by any optimization process (e.g. minimum square er-ror [Ljung1987] [P.1989], genetic algorithms [Goldberg1998] [Haupt & Haupt1998][Hunt1993] [Pastor2000] [Zhang Zibo1987], etc.). Some algorithms use an opti-mization process to determine both faces at the same time (e.g. genetic algorithms[Downing et al.1996] [Kristinsson & Dumont1992]).

2

2 Linear Time-Invariant Dynamic Systems

Linear ODEs with constant coefficients are the most studied kind of differentialequations; they have complete analytical solutions. So, if we restrict the kind ofsystems we are to analyze to those that can be expressed by an nth-order ordinarydifferential equation with constant coefficients, we know the kind of responses weare to get. We can express the behavior of a system in terms of the exponential andsinusoidal components in the response. We define

En1(t) =∑

1≤i≤n1

aierit (1)

as a summation with at most n exponential terms, and

ESn2(t) =∑

1≤i≤n2

aierit sin ωit (2)

as a summation of exponentially decreasing sinusoidal functions. Note that we arenot interested in giving analytical solutions to the differential equation, but a quali-tative description of its behaviors.

Theorem 1 The response of an nth order system can be expressed as in Equation 3.

X(t) = En1(t) + ESn2(t) (3)

where n1 + 2n2 = n. This result is evident from the definitions of Equations( 1)and (2).

We see that if the second term of Equation 3 does not exist, the response will beacyclic. Otherwise, it is a sinusoidal wave, where En1(t) represents its axis or at-tractor, and ESn2(t) its sinusoidal components.

Note that if we include a forcing function, the system’s response would be de-composed into Natural (the solution to the homogeneous equation) and Forced re-sponses. If we restrict the forcing functions to be of the form eαt sin βt (i.e. con-stant, exponential, or sinusoidal), the forced response always has the same qual-itative form as the forcing function [Boyce & DiPrima1969]. The force responsewould preserve the qualitative form of the response, and only adds one more expo-nential or sinusoidal term to the response.

Let us analyze the qualitative form of the responses, as expressed by Equation (3).This qualitative form can be derived from the qualitative form of its exponentialand sinusoidal components.

3

2.1 Exponential Components

The qualitative behavior of the exponential part of the response is characterized byTheorem 2.

Theorem 2 X(t) = En(t) =∑

1≤i≤n aierit exhibits at mostn extrema (maxima or

minima), including the one whent → ∞.

We will assume, without loss of generality, that r1 > r2 > . . . > rn. Theorem 2 isequivalent to saying that the derivative X ′(t) has at most n different zeroes. Thatis,

X ′(t) =−r1a1e−r1t − . . . − rnane−rnt = 0

= r1a1e−r1t + . . . + rnane−rnt = 0 (4)

Performing the variable change z = e−t, Equation (4) becomes

−X ′(t) = r1a1zr1 + . . . + rnanzrn = 0 (5)

Equation (5) is a polynomial in z of degree r1. Based on Descartes’ theorem [Kurosch1977,Rees, Sparks, & Reethe number of roots of a polynomial are determined by the sign changes in the poly-nomial coefficients. Since we have n terms in polynomial (4), we can have at mostn − 1 changes of sign, and therefore n − 1 roots. There is one more root placedwhere the function becomes zero, and that is when t → ∞.

Given we have at most n roots for Equation (4), we can therefore have at most nextrema for En(t).

2.2 Sinusoidal Components

If the frequencies of two sinusoidal components of Equation (3) are equal, theirshapes are reduced to one [Flores & Farley1995]. If their frequencies are different,they can be seen as the faster sinusoidal mounted on the slower one. If two fre-quencies of the sinusoidal components are very close together, the resulting wavebeats.

The results presented in this section fully characterizes all possible responses of aLTI Dynamic System. In the next section we will describe how to use these resultsto produce a framework for performing Systems Identification at the qualitative andquantitative levels.

4

3 QSI Algorithm

The identification algorithm presented in this section is based on the fact that theresponse of a LTI system can be decomposed in a sumation of exponential terms.If some of those exponential terms are complex, in which case they are conjugatecomplex pairs, each pair forms a sinusoidal. The algorithm that we propose in thispaper takes as input a time-series representing the response of an LTI dynamic sys-tem and is capable of separating the terms of Equation (3) to determine the structureor qualitative form of the system exhibiting the observed behavior. Separating theterms of the system’s response is performed by a filtering process. Figure (1) showsa simplified version of the QSI algorithm. QSI determines the order of the systemby adding the order of all eliminated components. This algorithm analyses the re-sponse and eliminates each component at a time, there are two main functions inQSI algorithm: TPAFilter and ExpFilter. The function TPAFilter (see nextsection) eliminates the sinusoidal components and returns the order correspond-ing to those components, the remainder signal and the parameters. The functionExpfilter eliminates the exponential componentes and returns the order of themodel, the remainder signal and the parameters. At same time that we eliminateeach component, we isolate it to determine its parameters (quatitative or paramet-ric identification).

QSI(X)

degree=0

P=0; Parameter Matrix

(X,k,P)=TPAFilter(X,degree,P)

(X,k,P)=ExpFilter(X,degree,P)

return Model(degree,X,P)

Fig. 1. QSI Algorithm

The function TPAFilter eliminates one sinusoidal component, at a time, startingwith the highest frequency component, the remainder X ∗(t) contains the summa-tion of all the previous components except the eliminated one. After the eliminationof j sinusoidal components, the remainder is:

X∗j (t) = En1(t) + ESn2−j(t) (6)

The elimination of components continues until the rest of the signal is non-oscillatory.

The remainder signal, after extracting the oscillatory components, is a sum of ex-ponential terms and is filtering by ExpFilter function, this filter also performs a

5

successive elimination of components, starting with the component with the slow-est decay rate. Figure 2 illustrates the application of QSI to a sample signal.

0 2 4 6 8 10 12−6

−4

−2

0

2

4

6

time

mag

nitu

de

Fig. 2. Component Elimination Process.

4 Filters

Digital filters have been used mainly for two general purposes: signal separation,for signals that have been previously mixed, and signal restoration, for signals thathave been distorted. In our application, we use filters to separate the different com-ponents that form the impulse response of a LTI system.

4.1 Two-Point AverageFilter

A modification of a two-point difference filter [Smith1999] is proposed as an alter-native to perform the first step in the filtering process. This first step of the filteringprocess detects the points where the derivative of the observed time series changes.The derivative can be computed using Equation 7

x′(t) =x(t + 1) − x(t − 1)

2∆t(7)

where ∆t is the sampling period, x′(t) is the derivative of x at point t, and x(t) isthe input signal at time point t.

Using the approximation given by Equation 7 we determine the time points wherethe sign of the derivative changes, i.e., we are detecting the extrema of x.

Once computed the vectors of maxima and minima, we model the envelope by acubic spline interpolation using the points at the maxima and minima, respectively,

6

to smoothen the upperhigher and lower envelopes. Once determined the envelopes,we compute the carrier by averaging the envelopes at each point of the originalsignal.

This is a simple implementation of a low-pass filter. Using this filter, the midpointvector m is computed.

Vector m represents the remainder X∗(t) defined in Equation 6. A cubic splineinterpolation approximation is used to smoothen the form of the carrier and hasproven to provide better results than the bare filtered data. These ideas were yieldto implement the filter that eliminated the sinusoidal componentsof the response.Figure 3 shows the TPA filtering algorithm. (TPAFilter).

TPAFilter(X, k,P)

Xtrmax=determinemaxima(X)

Xtrmin=determineminima(X)

if oscillatory(X)

k = k + 2

//Compute Envelope

Envelopemax=CubicInterp(Xtrmax)

Envelopemin=CubicInterp(Xtrmin)

//Compute m, middle point vector

X∗=CubicInterp(m)

S = X − X∗

P=P+SinParam(S)

(X, k,P)=TPAFilter(X∗, k,P)

else

X∗ = X

return (X, k,P)

Fig. 3. TPAFilter Algorithm

Figure 4 illustrates one step in the elimination process as performes by the algo-rithm in figure 3. Extrema are marked with plus signs and the computed middlepoints with circles. The signal formed by the middle points constitutes the carrierof the original wave m (i.e. the remainder).

7

1.8 2 2.2 2.4 2.6 2.8 3 3.2

−2

−1.5

−1

−0.5

0

0.5

Maxima envelope

Minima envelope

Middle points

Remainder X*(t)

Fig. 4. One step in the sinusoidal component elim-ination process.

4.2 Exponential Filter

Once we have filtered the oscillatory components of the signal, the remainder willbe a summation of exponentials. In this step we apply a filter to extract each expo-nential component. In this case, to make the extraction of the exponentialcompo-nents, it becomes necessary to compute the parameters of the component that willbe extracted. In the following section a complete description of the parametrizacinprocess is developed. Figure 5 presents the algorithm of this filter, and the Figure 6shows its application.

The function ExpFilter eliminates the exponential components at a time startingwith the slowest decay rate component. Using the approximation given by equa-tion 7 we determine the extrema of the TPAFilter’s remainder, i.e., the number ofexponential components (see theorem 2). The exponential components are elimi-nated by means of the following process: first we detected the last inflection point(Pi) and the points P1 and P2 are selected (see Figure 6). The points P1 and P2 areused to calculate the exponential component, that will be eliminate. The remainderX∗(t) contains the summation of all the exponentials components except of theeliminated one. In this case, after the elimination of i exponential component, theremainder is

X∗i (t) = En1−i(t) (8)

This process repeats until all the components have been eliminated.

In case that the extrema of remainder be zero, ExpFilter verifies if the residual hasexponential form or not.

For each eliminated exponential component the algorithm adds an unit to the order

8

ExpFilter(X ,k,P)

Xtr=count extrema of X

if Xtr > 0

do

k=k + 1

Pi=InflectionPoint(X)

P=ExpParam(X ,Pi)

X=X − Exponential(P )

Xtr=Xtr − 1

while Xtr > 0

else

k=k + 1

P=ExpParam(X)

X=X − Exponential(P )

return (X, k,P)

Fig. 5. Exponential Filter Algorithm

of the model and at the same time that filter the components their parameters arecalculated (see next section).

5 Quantitative Identification

To extend the capabilities of QSI we can determine the parameters of the responseat the same time we are filtering each component. The next two subsections showhow to perform this task for the exponential and sinusoidal cases.

5.1 Exponentials components

If the remainder signal is non-periodical or non-oscillatory it is a summation ofexponentials. The basis to separate the remainder exponential is theorem 3:

Theorem 3 If x1, x2, ..., xn are components of a sum of exponentialsE(t) withdecay ratesr1, r2...rn respectively, andr1 > r2 > ... > rn, then x1(t

∗) <<x2(t

∗) << ... << xn(t∗), wheret∗ is the time at the last inflection point.

9

Proof: For a summation with only two exponential components let us asume that

x1 (t)= a1e−r1t (9)

x2 (t)= a2e−r2t (10)

and that

r1

r2= α > 1 (11)

It is evident that(

1

α

)2

<< 1 (12)

the hipotesis is:

x1 (t∗) << x2 (t∗) (13)x1 (t∗)x2 (t∗)

<< 1 (14)

if we make

β =x1 (t∗)x2 (t∗)

<< 1 (15)

then

E (t) = −a1e−r1t + a2e

−r2t (16)

is the summation of those exponentials. According to the definition of the secondderivative, we have an inflection point where

E ′′ (t) = 0 (17)

E ′′ (t) = a1r21e

−r1t − a2r22e

−r2t = 0 (18)

solving for t∗

t∗ =ln(

a2

a1

(r2

r1

)2)

(r2 − r1)(19)

10

then we sustitute t∗in the equation 15

β =a1e

−r1

ln

(a2a1

( r2r1

)2)

(r2−r1)

a2e−r2

ln

(a2a1

( r2r1

)2)

(r2−r1)

(20)

=a1e

ln

(a2a1

(r2r1

)2)(

− −r1(r2−r1)

)

a2eln

(a2a1 (

r2r1 )

2)(

− −r2(r2−r1)

) (21)

=a1

a2

(a2

a1

(r2

r1

)2)(− −r1

(r2−r1)

)

(a2

a1

(r2

r1

)2)(− −r2

(r2−r1)

) (22)

=a1

a2

(a2

a1

(r2

r1

)2)( r2−−r1

(r2−r1)

)(23)

=a1

a2

(a2

a1

(r2

r1

)2)

(24)

finally

β =a1

a2

a2

a1

(r2

r1

)2

(25)

β =(

r2

r1

)2

(26)

comparing equation 26 with equation 11, we have that

β =(

1

α

)2

(27)

therefore

β << 1 (28)

In general, since r1 > r2 > r3 > . . . > rn

x1 (t∗) << x2 (t∗) << x3 (t∗) << . . . << xn (t∗) (29)

11

Theorem 3 implies that at and after Pi, the last inflection points of E(t), x1, ..., xn−1

are negligible with respect to xn. So, we can can use the tail of E(t) to approximateand parameterize the slowest decay exponential component of E(t).

0 1 2 3 4 5 6 7 8 9 10−2

−1.5

−1

−0.5

0

0.5

+ *

*

Pi P1

P2

Fig. 6. Important Points in the Procedure of Quan-titative Identification of Exponentials.

To determine the parameters of the slowest decay component of E(t). We start frompoint Pi. Two points, P1and P2, are selected, accordance with following:

P1 = X(i + 1) (30)

P2 = X(i +n

4) (31)

where: n is length of X(t) and i is the time of last inflexion point.

Since we know the component to be eliminated is exponential:

xn = ane−rtn (32)

we substitute p1and p2 and solve for r and a.

r =ln(

x2

x1

)(t1 − t2)

(33)

a =xn

e−rtn(34)

With the values of a and r, we compute the slowest exponential component E1, andsubtract it from the signal, to obtain the remainder X ∗ (t):

E1 (t) = ae−rt (35)

12

X∗ (t) = X∗ (t) − E1 (t) (36)

This procedure repeats until the remainder contains no more exponential compo-nents.

As we can see in this subsection, at the same time that we filter the exponentialcomponents, we compute their respective parameters.

5.2 Sinusoidal components

Similarly to the exponential case, at the same time that we separate the oscillatorycomponents, we can compute their parameters.

Suppose that the subtracted component has the form ae−rt sin (ωt) . Select twopoints where maxima exist, which we will mark them as P1 and P2 (see Figure 7).

0 1 2 3 4 5 6 7 8 9 10−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

time

mag

nitu

de

+

+

P1=(t1,x1)

P2=(t2,x2)

Fig. 7. Quantitative Identification of a SinusoidalComponent

Use P1and P2 to determine the envelope, which represents the exponential part ofthe decaying sinusoidal. a and r, are computed in the same way as the previoussubsection.

To compute ω we first determine the period T

T = t2 − t1 (37)

and substitute it in the next equation:

ω =2π

T(38)

13

The obtained parameters determine the roots of the characteristic polynomial ofthe ODE that models the system. Once we know those roots, the polynomial andtherefore the ODE can be reconstructed from them.

6 Experiment design

We designed an algorithm to generate systems that exhibit a desired qualitativebehavior. The problem is to determine the coefficients and exponents of the ODEso that the desired qualitative features are guaranteed to appear in the response.Since the response is composed of exponentials En (t) and sinusoidal ESn (t), wecan design them separately.

We start with exponential components with decay rates r1, and r2, that is,

En (t) = a1e−r1t + a2e

−r2t (39)

E ′n (t) = −r1a1e

−r1t − r2a2e−r2t = 0 (40)

solving for t

t =ln(−r2a2

r1a1

)− (r1 − r2)

(41)

It is evident that in equation (41) the argument of the natural logarithm must bepositive. That is:

r2a2

r1a1

< 0 (42)

since the values of decay rates r1 and r2 are positive then the sign of the amplitudesmust alternate:

sign (a1) = sign (a2) (43)

To make sure that the value of t is positive the following must hold

(r2 | a2 |) < (r1 | a1 |) (44)

then

r1

r2>

a2

a1(45)

14

assuming the ratio of decay rates is α

α =r1

r2

(46)

a2 < αa1 (47)

which provides the necessary conditions on r ′s and a′s to yield the expected quali-tative shape.

To desing the sinusoidal components, the conditions for the parameters of ESm (t)are the following:

We start with the slowest frequency, ω1, and produce the rest of components ac-cording to the following rules:

ω1 < ω2 < . . . < ωi < . . . < ωn (48)

It would be convenient to have at least four complete periods of the faster sinusoidalmounted on the slower one. That is:

ω2

ω1

≥ 4 (49)

ω2 ≥ 4ω1 (50)

With these characteristics it is possible to produce a time series which exhibits therequired behaviors. In the next section we present a response obtained with thisalgorithm (see Figure 8. That Corresponds to the ODE:

c6d6

dt6x(t) + c5

d5

dt5x(t) + · · ·+ c1

d

dtx(t) + c0x(t) = 0 (51)

With the coefficients

c6 c5 c4 c3 c2 c1 c0

1 3 653.2 1561.8 16751 28450 8783(52)

7 Results

In this section, we present three examples of application of the QSI algorithm. Wewill begin identifying the structure and the parameters of a system whose response

15

has been generated by the module of experiment design (previous section). In thesecond case we presented the step response of a circuit RLC, finally the third caseof study is the simulation of a transmission line of an electric power system. Thesecond and third cases are simulations carried out in MatLab using the simulinktoolbox.

7.1 Example of aplication 1

This example is a time series obtained by module of experiment design (see Fig-ure 8). In this example is illustrated the process that follows QSI to obtain themodel.

0 2 4 6 8 10 12−8

−6

−4

−2

0

2

4

6

time

mag

nitu

de

Fig. 8. Response of a LTI system.

Figure 9 presents the elimination process of the oscillatory components. The thinestline corresponds to the first eliminated component, as we explained previously, itis the highest frequency component. The medium line corresponds to second com-ponent that will be extracted next. Finally the thick line represents the exponentialremainder, after we extracted all oscillatory components.

Figure 10 presents the non-oscillatory carrier signal and its exponential compo-nents, the dotted lines represents the first order exponentials that conform the re-mainder.

The QSI algorithm adds two units to the order value for each eliminated oscillatorycomponent and one unit for each eliminated exponential component. QSI has de-termined that the simplest LTI system capable of exhibiting the observed behavioris of order 6.

c6d6x(t)

dt6+ c5

d5x(t)

dt5+ · · ·+ c1

dx(t)

dt+ c0x(t) = 0 (53)

16

0 2 4 6 8 10 12−6

−4

−2

0

2

4

6

time

mag

nitu

de

Fig. 9. Elimination of oscillatory components.

0 1 2 3 4 5 6 7 8 9 10−4

−3

−2

−1

0

1

2

3

4

time

mag

nitu

de

Fig. 10. The remainder signal and its exponentialcomponents

The solution to (53) has the form shown in equation (3) where the parameters are:

a r ω

2.2021 −0.3210 ±25.0107

3.3375 −0.3211 ±5.0141

−3.2579 −0.3766 0

3.6731 −1.4711 0

(54)

The resulting model has the form shown in Equation 53. The parametric identifica-tion yielded the following ODE parameters.

c6 c5 c4 c3 c2 c1 c0

1 3.1 654.2 1622 16926.7 29413.4 8750.6(55)

17

To validate the obtained model we implemented the Qualitative Similarity Index[Cuberos et al.2002] to compute the Qualitative Correlation Coefficient (QCC).The QCC expresses the measures what so well the qualitative forms of two seriesof time are adjusted.

We also compute the Numeric Correlation Coefficient (NCC). The NCC is thequantitatively measure of adjust between the prospective model response and theestimated model response.

In both cases (QCC and NCC) the result is confined inside the range [0, 1]. WhenCC=1 exists a perfect correlation among the data, when CC=0 is said that there isnot correlation among the data. The intermediate values corresponding to partialcorrelations. The Table 1 shown the approach and it levels of acceptance.

Correlation coeff. Accuracy

1.0 Excelent

0.99 − 0.9 Very Good

0.89 − 0.8 Good

< 0.79 Very BadTable 1Levels of acceptance

The value of the QCC for this particular case was:

QCC = 1.0 (56)

and the Numeric Correlation Coefficient (NCC) was:

NCC = 0.99 (57)

The resulting of this experiement was satisfactory, since it produced values for NCCnear to one.

7.2 Example of aplication 2

In this example we determine the model of the RLC circuit shown in Figure 11.The signal used for this purpose is the step’s response (see 12

where: Rs = 50Ω. L = 0.1Henry C = 1000µF Vs = 100V.

18

Rs

LC

100 v

Fig. 11. Circuit RLC for the example 2

With these values we can obtain the analyticall differential equation that describesthis system (see Eq. 58). That is, the prospective model.

d2i(t)

dt2+ 20

di(t)

dt+ 10000i(t) = 0 (58)

Figure 12 shows the system’s response, the signal used was the current in the in-ductor.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.5

1

1.5

2

2.5

3

3.5

time in seconds

curr

ent i

n am

pers

Fig. 12. Observed response of the RLC circuit ofFigure 11.

After running QSI, the parameters obtained are:

a r ω

20.0092 −9.9932 ±99.5074(59)

The resulting model is shown in Equation 60

d2i(t)

dt2+ 19.9864

di(t)

dt+ 10001i(t) = 0 (60)

19

The qualitative correlation coefficient for this example was:

QCC = 1.0 (61)

And the numeric correlation coefficient was:

NCC = 0.99 (62)

If we compare equation 58 with the equation 60 we can see that coefficients of theobtained model are very close to the coefficients of the prospective model. Addi-tionally, we have two measures of quality that it allows us to evaluate how well themodel reproduces the observed response. The value of QCC indicates that the formof the response of the identified model and the observed response are perfectly cor-related, that is, their qualitative forms are same. The value of NCC allows us toconclude that the obtained model is acceptable, in accordance with the approach ofthe table 1.

7.3 Example of aplication 3

Our next example is about a linear model of a power transmission line. The circuitis shown in Figure 13.

(187.8 KVinst)

150 kmtransm ission line

ScopeRs Ls RL1

RL Load75 M W 20 M var

+i

-

Load current

+-

v

Line voltage

C2C1

Breaker

132.8 kVrm s60 Hz

I_load

V_line

Fig. 13. Linear model of a transmission line

Figure 14 shows the voltage signal in capacitor c2. The transient starts in t = 0.039seconds and finishes in t = 0.1166 seconds. This time window was used for ouridentification process. The values of the elements of the circuit are: Rs = 2.64 Ω.Ls = 70mH. R1 = 5.2 Ω. L1 = 138mH. C1 = C2 = 0.967 µF.

These values lead us to the following analytical differential equation (prospectivemodel):

c4d4

dt4vc(t) + c3

d3

dt3vc(t) + . . . + c1

d

dtvc(t) + c0vc(t) = V s (63)

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With following ODE parameters:

c4 c3 c2 c1 c0

6.76E − 8 5.1E − 6 1 76.94 74.9E5(64)

0.0385 0.1167−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5x 10

5

time in seconds

Vol

tage

in v

olts

Fig. 14. Voltage signal in Capacitor c2

The parameters obtained are:

a r ω

1.3E5 −20.2953 ±2074.86(65)

The resulting model is shown in Equation 60

d2

dt2vc(t) + 40.5907

d

dtvc(t) + 43.05E5vc(t) = 0 (66)

The qualitative correlation coefficient was:

QCC = 0.97 (67)

And the numeric correlation coefficient was:

NCC = 0.94 (68)

The result obtained in this example is interesting, since there is a difference be-tween the order of the expected model and the order of the obtained model, how-ever the QCC and the NCC indicate that this model is acceptable since the twocorrelation measures are greater than 0.90 (see Table 1). On the other hand, if theexpected model is analyzed, we can observe that the coefficients of the fourth and

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third terms are small compared with the rest of the coefficients, It implies that theirscontribution in the response is small, such that it does not influence significantly inthe system’s dynamics. This allows us to conclude that the obtained model is ac-ceptable since it is able to reproduce the dynamics of the system.

8 Conclusions

We have presented the analysis, algorithms, and implementation of a frameworkto perform Qualitative and Quantitative Systems Identification for Linear Time-Invariant Systems.

At the same time that we identified the structure of the system we send each com-ponent to a parametric identification module to complete the identification process.The parametric identification part has been implemented using the fundamentalproperties of the exponentials and trigonometric functions.

The system has been successful in determining the sinusoidal and exponential com-ponents of different test systems. Nonetheless, the filters still need some work onthe detection of noisy components.

The Systems Identification algorithm, including filters have been implemented us-ing Matlab. In our implementation, the input is the observed signal, and the outputis the Qualitative and Quantitative structure of the black-box system.

It is obvious that the limitation of this system identification technique is relatedto the availability of good data and good model structures. Without a reasonabledata record not much can be done, and there are several reasons why such a recordcannot be obtained in certain applications.

An expression for the time complexity is needed. We need to compare how ouralgorithm behaves with respect to traditional approaches.

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