On-Line Order Selection and Parameter Estimation-An Experimental ...

On-Line Order Selection and Parameter Estimation-An Experimental Application I

Rosalba Lamanna de R. Universidad Simon Bolivar, Departamentode Procesos y Sistemas, Caracas, Venezuela

Rafael A. Padilla lnstituto de Ingenieria, Centro de Ingenieria de Sistemas, Caracas, Venezuela

Maite Uria de C. Universidad Simon Bolivar, Departamentode Procesos y Sistemas, Caracas, Venezuela Problems arise in selecting the structureof the model (the order in the S .I. S .O . case) and the most suitable identification method. Much workhas been done on the model storageplantispresented.Theexperiorder selection problem (see [ 1.2.3, and 41 ments are carried out on different among others). However. most of the prooperating points around which the plant posed methods are off-line procedures. which can be modeled as a linear, discrete-time, need awholeset of operatingrecords to stochastic and S.I.S.O. system. A simple estimatetheorder.Othermethodsaretoo algorithmforthedetermination of the sophisticated or lengthy and need prior infororder is proposed.Theexperiment is mation about the plant. We develop in this b a s eotdne s t i nrge c u r s i v e lt yh e significance of the reduction in the predic-paper an easy on-line method that determines the order of the system only from the opertion error variancewhenincreasingthe ating observations. The method is based on a modelorder,by means of a modified recursivealgorithm for testingthesignifiF-test. Theresultsofthreewell-known cance of the reduction in the prediction reidentificationprocedures,used to peris increased. form the parameter estimation in different siduals when the model order The validity of the recursive test proposed is conditions, are also reportedfor combased on its off-line version [5,6] and shows parison purposes. that the test can be implemented as part of a Introduction normal on-line identification procedure. We develop an interactive software packWhen studying the dynamic behavior or age ofintegratedalgorithmsthatsimultadesigningtheautomaticcontrol of agiven neously identify the orderand the parameters plant. it is important to have an appropriate of acompletelyunknownmodel.Inorder mathematicalmodel of thesystem.Generally, a linear parametric input-output relation toestimatetheparametervalues,weuse either of threewidelyknownrecursive is proposed, and its unknown coefficients are identification methods: least squares (RLS), estimated by meansof an identification instrumentalvariables (RIV), and modified procedurebasedontheoperatingdata of maximum likelihood (RMML). These have the plant. been discussed extensively by many authors Received October 13, 1982; revised June 8 and [5,7.8]. Severalcomparativestudieson December 13, 1983. Acceptedinrevised form themethodshavealsobeenpublished by Associate Editor R. N. Lobbia. [9. 10. 111. but they have been mainly theoor, at most. based on simulated exretical The work of R. Padilla was supported in part amples. In this case. we analyze the above by CONICIT (National Council for Scientific mentionedalgorithmscomplementedwith and Technological Research), under grant theon-lineorderdetermination.appliedto numbers: S1-1183 and DDCT-ELE-2.

Abstract: The experimental application of

an integrated on-line order determination and a parameter estimation algorithm ato

simulatedexamplesandtotheexperirnental identification of a laboratory model storage plant. Theplant is anopen tank whereoutput flow rate is a function of level, as described in the section that discusses the description of the physical system. It can be modeled as a discrete-time S.I.S.O. system, linear around a given operating point, as follows

A(q-')J(r)= B(q-')u(t)

C(q-')e(t) (1)

where u ( t ) and!(?)

are inputand output signalsat instant r e ( t ) is white noise q-' is the backward shift operator A(q-') = 1

+ a,q

-1

t

- + azq-?

...

+ nnq-n B ( q - ' ) = b'q-' ib2q-'

(2)

+ . . . + b,q-" (3)

C(q-1) = 1

+ clq-' + czq-' + . . . ic&-"

(4)

n is the system order. A model of the storage plant can also be determined on the basis of physical laws for comparison purposes. From a practical point of view, reservoirs are usedin the manufacturing industries as storagetanks.separationdrums,reaction pools, and other systems, and their models for control purposesvary with their operating point,makingthemtypicalobjects of onlineidentificationmethodsasapart of their automation.

Table 1

Description of the Algorithms Parameter Estimation

Completetheoreticalderivations of the U S , RIV, and RMML identification methods can be found in the literature [5,7,8]. These algorithms can be described by using the same recursive formula &t

+ 1) = & t ) + K(t + 1 ) E ( t + 1)

(-y(t - 1):. . . ,

-At u(t - l ) , . . . ,

RLS

u(t - n ) )

(-y(r - l ) , . . . -y(r - R ) , u(r - I), . . . ,

(5a)

where B(t) is thevector ofthe parameter estimates E(t) is the prediction error K ( t + 1) is a gain vector that represents theweightanddirection of thecorrectioncorrespondingtothepresent iteration. We find that P(t)z(r + 1) K(t + 1) = 1 + $(t + l)rP(r)z(r+ 1) (5b)

~

RIV

-4 )

y(t)-[-?;(t - 11, . . . ,-y(t - n ) ,

+

1)

=

P(r)

1 . . . --

U ( t - I), . . . ,

u(t - n),E(t - 1). . . . , ~ ( -t n)]'

where P(t

4,

e(r - 1)

R"L

'

&-I)

1 -y(t - ?I),C(q -? .u(t - 1):. . . ,

- P(t)z(t + I ) $ ( r + l ) T P ( t ) 1 $(t + 1)'P(r)z(r 1)

+

+

(5c) is the prediction error covariance matrix. The contents of the vectors involved in the basicalgorithmoutlinedinequations(5) depend on the method used, as indicated in Table 1 . Structure Determination

The problemof structure determinationfor a linear discrete-time S.I.S.O. system is reduced to finding the orderof the model, that is, the number of parameters that best fits the operating data. However. this is not a trivial matter, because of the prediction errorof the data adjusting process, which normally decreases as the model order increases [ 5 ] ,and thus leads to a model of infinite order. The reduction of the prediction error ensuresthatthehigherordermodel is more accurateonlywhenthisreduction is "significant." Only then is the selection of the higherordermodeljustifiable. An off-line testing procedure based on the Fisher distribution [6] is a simple and well-known method used toestablishthesignificance of the reductionofthepredictionerrorwhen using amodelwith pz parametersinstead of pI ( p z > pl). It can be implemented by usingthenormalizedsums of squares of residuals: V / ( N - p ) oftheidentification process, where r=,v

v = 1E ( t y r= 1

moy 1984

where N is the number of observations p is the number of parameters.

using the F,,.(pz - p I :r The null hypothesis

H , = - -vl(4 -

The hypothesis test contrasts

with

H , :- VI -N -Np -l p z

v 2

VZ

VI

N - pz

A' - PI

Ha : - F d p 2 - PI, t - p z )

using a parameter T , calculated as

It has been demonstrated in the literature [6] that T has a F ( p 2 - p l ,N - p 2 ) distribution based on the statistical independence of V 1 and ( V I - V z ) $if p 2 > p l . We can extend this test to the on-line case by computing T recursively as

(9)

at w% risk level, and therefore the order of the model is increased. However,theoff-lineresultdoesnot guaranteethat T(r) willalsoshowtheassumedFisherdistribution,atleast not for small values o f t , because V l ( t )and Vz(r)are different from Vl and Vz. which correspond to the whole processingof the off-line identification. So as to try to avoid this problem, we compute T ( t ) by usingmodifiedloss functions V i ( [ ) and V;(r), insteadofthe actual residuals V;(t) = A(t)V,(r) = A(r) [Vj(t - 1 ) + ( ~ ~ ( r ) ) ~ ] ! ( j = 1,2)

(10)

where A ( t ) > 0 is a forgettingfactor monotonouslyincreasingto l as t increases.

I

~

~~

Therefore, VI(t) and V;(t) are weighted residuals, which take into account the fact that in the beginning of an on-line identification process, the residuals are far bigger than their true values when the procedure is completed. Actually, this differencehasbeenreported to bethecause of poorresultsinearlier studies [3]. The forgetting factorA ( t ) is used todecreasethecontributionoftheearly iterations results in the sum of residuals. As forspeedingtherate of convergence of an identification algorithm [ 121, analogous factors appear elsewhere in the literature. In order to perform the iterativetesting method outlined above, we suggest carrying outtwosimultaneousidentificationprocesses,oneforadjustingmodel 1 and another for model 2 of increased order. The algorithmwilldeterminewhethermodel2 fits the data better than model 1, depending on the result of the significance test realized at each M iteration. In this case,all the informationof theidentificationprocess 2 is transferred to model 1 and the new model 2 should be initialized (Fig. 1). Of course, the test cannot be repeated until at least ( p +~ 1) steps into the future.

Description of the Physical System

(1 1) around a certain operating point gives

A simplified representation of the storage plantusedintheexperiments is shownin Fig. 2 . The tank consists of a constant circular transversal section with area y. and the fluid is assumed to be of constant density p. The dynamic behaviorof the system can then be defined as

where

(Fi, x)

p = 0.5 6 7; F. = F . h = ]I* I

:-E

An equivalentdiscrete-timemodelof system (14). using an integration interval small enough. is obtained h(t) = -alh(r - 1)

+ blF,(t -

1)

J

(15)

where V is the liquid holdup h* is the liquid level in the tank F,*,F, arethevolumetricinputand output flows

F, = 6 f i

6

(12) (13)

= CU*

c, is thehydrauliccoefficient,depending on the output orifice diameter (computed by experiments).

Linearization of theinput-outputmodel

Severalidentificationexperimentswere carried out on the plant operating in the different conditions listed in Table 2 . Pseudorandominputperturbationswereproduced onthesystemthroughthemanualglobe valve 3 (Fig. 2), by varying the input flow rate around its operating point Ti. The flow

rQ--

+5

ACQUISITION

!

I!

7

IDENTIFICATIOM MODEL 2

IDENTIFICATION MODEL 1 (pl parameter)

(p2 pzrarneters)

7

1

YES

3

COMPUTATION OF r1t)

4-0 F-TEST

MODEL ORDER 2 (MODEL 1 = MODEL 2 MODEL 2:

Fig. 1.

Integraied algorithm scheme.

8

Centrifugalpump Manometer Globe Valve Rotameter Flow transducer 6. Level transducer 7. Tank 8. Output orifice

1. 2. 3. 4. 5.

Fig. 2.

The storage

plant.

control systems magazine

8

~

~~~~~~

Table 3

Table 2 Method

Operating point

1 2 3 4 5

9

6

. 11.922

8.856

transducer 5 and the level transducer 6 (Fig. 2) were used to gather the input-output data of the plant.

0

0

The present one is an identification experiment, and therefore the on-line data collecting procedure must be performed as such from the plant. The other oneis only an example. In this case, the user has to provide the exact system description, which will be simulated so as to generate the input-output signals. The model order is known, or the order estimationprocedurehastobealso performed.

0

RIV

RMML

Estimates -0.9615 -0.9528 -0.9160 -0.9620 -0.9440

0.2713 0.2688 0.2585 0.2714 0.2664

2,

-0.6022 0.1753 -0.2568 -0.2332 -0.4349 -0.2834 23 0.0985 0.1 129 0.0573 24 -0.0206 0.0452 0.0205 45 -0.0173 -0.0357 -0.0071 0.9996 bl 1.0 0.9985 i2 0.5116 0.9416 0.1627 63 -0.4425 -0.0583 -0.2374 2, 0.5061 22

Theforgettingfactor h(t) if theorder selection procedure is to be done. Thenumber of iterations M between consecutive order tests.

The Identification Package Theidentificationandstructuredetermination algorithms outlined in the section discussingadescriptionofthealgorithms areimplementedas an interactivesoftware package. The program is written entirely in FORTRAN and designed to perform each step from the data acquisition process to the description and plotting of the results. Thepackage is represented by asetof interacting subroutines. The most important one is theidentificationsubroutine,where the algorithm described by equations (5) is implemented. It is used for whichever identification method selected. The identification process is realizedtwiceineachiteration with a different number of parameters if the order testing procedureis taking place. Other supportsubroutinesincludesimulation of input-outputmodels,manipulation of the experimental data, and processing and presentation of the results. The user must specify the following alternative situations:

.

141.93 175.59 324.40 139.60 210.00

42.0 29.1 10.5 48.8 21 .o

RLS

al

RMML, to the estimation of the parameters of system (17), knowing the model order, are presented in Table 3. The evolution of the meansquarepredictionerrorV(t)forthe three processes is shown in Fig. 3.

The number of repetitions of the whole procedure.

In thefollowingsection, we summarize the results of different situations studied.

Case B-Simulated example: Model order selectionand parameter estimationof the following system

Results Case A-Simulatedexample:Identification of the following model y(r) = 0.267y(t - 1)

y(t) = 0.267~(t- 1)

+ 0.2875y(t - 2)

- 0.0646y(t - 3) - 0.018y(t - 4)

+ 0.0042y(t - 5) + u(t - 1) + 0.5u(r - 2) + 0.25u(t - 3) + 0.5e(t - 1) + e(t)

+ 0.2875y(t - 2)

- 0.0646y(t - 3) - 0.018y(r - 4)

+ 0.0042y(t

- 5)

+ u(t - 1) + e(t) (18)

(17)

The results of the application of the three identificationmethods, RLS, RIV,and

Identification method used: RLS. Number of iterations M between two successive order tests: 100. Forgettingfactor: A([) = (t - I)’/?

10

The user must select:

0

0

The parameter estimation method: RLS, RIV, or RMML. The variance of the white noise signal e([) and theamplitudeofthepseudorandom input&), in the case of a simulated example. The initial condition of thevector of estimates kt)and the covariance matrix

5

R IV R LS

Fig. 3.

Meansquareprediction

error (simulatedexample).

PW. moy 1984

9

~~~~~

Samplingintervalofthe data collecting process: J = 5 sec. Identification method used: RMML. Number of iterations between two successive order tests: M = 100. Forgetting factor: h(t) = ( t - l)*/tZ. Repetitions of the experiment: 100. Results are presented in Figs. 9 and 10.

and RMML, always assumingas known the model structure(as in equation (15)).

Number of repetitions of the whole process:100.

Results are listedin Table 4. Figures 6 and 7 show the evolution of the estimated values of the parameters when the plant is on the first operating condition. The behaviorof the mean square prediction error is presented in Fig. 8.Finally, the most important aspects of the identification results using each different method are summarized inTable 5 . In this case, we determinequalitativecriteria and reportmeanvaluesbasedon what weobCase C -Experiment: Parameter identiserved during the estimation process in the fication of the storage plant in the five differ- five different operating conditions. ent operating conditions listed in Table 2. Case D-Experiment: On-line order and Sampling interval of the data acquisition parameter estimation of the plant on the first process: J = 5 sec. Identificationmethodsused: RLS. RIV. operating point.

Results are summarized in Figs. 4 and 5. Theplots in Fig. 4 areillustratingthedynamic behavior of the order testing method, summing up the results of the 100 realizations of the whole identification experiment. Everyplotproducestheinformation of how many times the selection of a greater order ( n . instead of R - 1) happens on each M iteration.

n=3

n=2

I

Discussion and Evaluation of the Results Comparison .of the Parameter Estimation Methods The contents of Tables 3 and 4 show that the RLS converges to biased estimates, because of the existence of colored noise in both cases. This behavior was expectedfrom thetheoreticalderivation of the LS and is commonly known[5].On the other hand, the RIV gives poor results in the simulated example of case A, whereas it converges to correct estimates inall the operating pointsof the storage plant that were studied. RMML

'i 90

'I 30 2o

n=4 1

T

t

I T ,

2

3

4

5

6

n

6

n

(5a) ( t = 1000)

:j 60

I 1

1 2 3 4 5 6 7

(4

Fig. 4.

t x 103

3

fJ

3

4

5

(5b) ( t = 3000)

(e)

Number of times the selection of order n occurs on iteration r, after100realizations of the experiment; (a) n = 2, (b) n = 3, (c) n = 4, (d) n = 5 , (e)

n=6.

2

2x10-3

Fig. 5 .

Density of the estimatedordervalue,corresponding to iteration t.

control sys!e/ns magazine

10

~~~

50

I

Fig. 6.

160

Fitting of theparameter 01 (case A, f i s t operatingpoint).

7

> f

gives very good results in all the situations. Consideringonlytheexperimentalidentification processes of case c, the error of the estimation are less than1 Ti, using either RIV or R " L (comparing Tables 2 and 4). The above mentioned results can be explained by the structure of the algorithms. The fact that RMML eliminates the colored noise effects through the polynomial C(4-I) ensures the precision of the estimates in all cases. RIV may behave correctly or not according to the quality of the auxiliary model usedtogeneratetheinstrumentalvariables matrix. The method works satisfactorily on themodel of the tank but not so well on the simulated example, which is a more complex system. However, not only is the precision of an identificationprocedureimportantbutthe convergencespeed is alsoessential. It depends on the numberof operations performed by the algorithm duringan iteration as well as thenumber of iterationsneededforconvergence. Based on the first aspect, the studiedalgorithmsbehaveasexpected.RLS, which needs fewer programming steps and lessstoragecapacity, is thefastestmethod (see Tables 4 and 5). On theotherhand, R " L needs much more time than RIV to completetheidentificationprocess.Nevertheless, R " L givesestimates of theparameters of C ( q - l ) .This is an advantage over the other algorithms, especially for control applications.However, it shouldbenoted that while IUS and RMML need about 72 to 78 iterations to converge in case C (see Table 4 and Figs. 6, 7 , and 8). RIV converges in less than 45 iterations in all operating conditionsof case C. Therefore, RIVis more advantageous because it only needs almost halfof the operating data than those used by RMML to obtain an equally satisfactory fitting. On theotherhand,RMML would certainly show a faster convergence time if initialized closer to the true values. (Remember that the three studied algorithms are initialized without prior knowledge about theplant,which is notgenerallythecase in practice.)

Concluding Remarks 0

Fig. 7.

Fitting of the parameter bl (case A, firstoperatingpoint). 0

RLS is veryfast,stable, and easyto implement.Nevertheless, its estimates arebiased in thepresence of colored noise. Applicability of RLS must therefore be restricted where bias poses no problem. The precision of RIV depends on the behavior of theauxiliarymodel used and therefore varies, depending on the system. RMML always gives good results but is

more difficult to program, needs more storagecapacity,andtakeslonger to Convergence converge. 0 RIV or R " L have to be chosen depending on the specific problem conditions and requirements. These include: speed of convergence;computational effort;informationrequested from the identificationexperiment(particularly about C(q-')); and availability of prior knowledge about the plant.

Table 4 Operating

point

Method

(iter.)

SI

81

61

IUS

1 2 3 4 5

-0.9445 -0.9212 -0.8728 -0.9321 -0.9238

0.2312 0.2053 0.2125 0.2037 0.21 15

74 72 72 73 76

RIV

1 2 3 4 5

-0.9692 -0.9517 -0.9110 -0.9661 -0.9484

0.2614 0.2515 0.2414 0.2310 0.2513

40 44 45 42

1 2 3 4 5

-0.9613 -0.9512 -0.9059 -0.9610 -0.9457

0.2723 68 0.2691 0.2590 69 0.2710 0.2652

42

RMML

0.1883 0.0882 0.1415 0.1982 0.1507

Order Determination The recursive F-test implemented in this papergivesverysatisfactoryresults.The plots in Figs. 4 and 9 give an idea of the speed of detection of a higher order during the process. We can see from Fig. 4 that the selection of order n = 2 and n = 3, in the simu1ated.example (case B), are immediate. More than 90% of the time, the change to n = 2 occurs in the first test (t = 100) and n = 3 occurs in the second test (t = 200). Theswitchingto n = 4 does not occur as early in many cases (see Fig. 4c). The identificationoftherealorderisstillslower; namely,after1000iterations,theresults show 8 4 9 n = 4 (see Fig. 5a).This is due to the very small residuals of the identification eversincethe u5 coefficient is introduced intothemodel,apparentlybecause of the very low value of the real u5. In any event, the estimation of the order is correct in a p proximately 608 of the runs after3000 iterations (34% correspond to n = 4, while only 5% identify n = 6 (see Fig. 5b)). The applicationof the methodto the determination of theorder of thestorageplant model (case D) is a good evidenceof the test reliability. Figure 10 shows that, after 1000 iterations, only 9% of the results give orders greater than n = 1. The modified computation of the residuals brings about a significant improvement of the procedure. By decreasingtheeffect of the prediction errors corresponding to the beginning of the process, we generally obtain less impressive differences between the loss functions of the two models. The F-test is therefore more precise.

70 79 77

Table 5 V(t) Convergence storage Algorithm (iter.) t = 50 t = 100

RLS RIV R 1.50 "L

73 43 73

5.00 0.25

4.80 0.23 0.48

Relative Implementation requirement Bias Yes Sometimes No

60% 100% 140%

Easy Less easy Difficult

Conclusions RRlL R IU I

50

Fig. 8.

Meansquarepredictionerror(case

>

I

lo0

A, firstoperatingpoint).

t

In this paper,we have presented the results of an experimental application of order determination and parameter identification methods to a storage plant. We have used both simulated examples. and real operating data of the plant in order to study the behavior of theon-lineorderestimation test proposed andthreewell-knownparameteridentification algorithms.

[SI

[9] [IO]

Fig. 9.

Number of times the selection of order n occurs on iteration t, after 100 realizations of the experiment; (a) n = 2, (b) n = 3, (c) n = 4.

On iteration t , after 100 realizations of the experiment: (a) n = 2, (b) n = 3, (c) n = 4

[I 11

[I21

[I31

lineardiscrete-timesystemsusingtheinstrumental variable method,” IEEE Trans. Automat.Contr., AC12, pp. 707-718, 1967. J. Gertler and C. BinyBsz. “A recursive (online)maximumlikelihoodidentification method,” IEEE Trans. Auromat. Contr., AC19,pp.816-820:1974. G. N. Saridis,”Comparison ofsix on-line identification algorithms,” Automarica, VOI.10,pp.69-79,1974. D. Graupe, V. Jain, and J. Salahi, “A comparativeanalysis of variousleast-squares identification algorithms,“ Azdtomatica, V O ~ . 16,pp.663-681,1980. P. James, P. Souter,and D. Dixon, “Acomparison of parameter estimation algorithms fordiscretesystems,” Chem.Eng. Sci., VOI. 29, pp. 539-547, 1974. T. Soderstrom,L.Ljung, and I. Gustavsson, “Acomparativestudyofrecursive identification methods,” Report 7427, Lund Institute of Technology, Sweden, 1974. P.C. Young, “Aninstrumentalvariable method forreal-timeidentification of a noisy process,” Automatica, vol. 6, 1970.

Rosa,ba

Lamanna

de R. was born in

I

Salerno, Italy, on August 2, 1956.She received a degree in chemical engineering (19778) and an M.Sc. degree in chemical engineering simulation and process control (1982), both 7 : .~~. from the Universi. = ~ . dad Simbn Bolivar, Caracas, Venezuela.Since1979,she has been with t h e Processes and Systems Department of the UniversidadSimbnBolivarwhere she is now Assistant Professor. Her research interests are currently focused on parameter and structure identification and adaptive control of linear multivariable systems. ~

.E=_~ ~ = = i

~~

~

0

i

1

Fig. 10.

2

3

4

n

Density of the estimated order value, corresponding to iteration

t = 1OOo. Wehavefound thattherecursiveF-test implemented to select the model order is correct in 80% of the runs. We have also observed its parsimony, since in less than 10% of the runs, the results show orders greater than therealones.Theseconclusions,and the simplicity of the method, make its inclusionin an on-lineidentification procedure highly recommendable.Theon-lineorder selection method is also suitable in the case of multivariable systemsif the systemcan be described by linearinput-outputequations. The test would be performed so as to find the structure of every output equation. However, this isachievedwith a correspondingincrease in the computationaleffortofthe experiment.

References [l] C. M . Woodside,“Estimation oftheorder may 1984

of linearsystems.” Automatica, vol. 7, _. UP. 727-733,1971. [2] J. C. Chow, “On estimating the orders ofan autoregressive moving average process with uncertain observations,”IEEE Trans. Automar. Contr., October1972. [3] A. Van Den‘Boom and A. Van Den Enden, ‘The determination of the orders of process and noise dynamics,” Automarica, vol. 10, pp.245-256,1974. [4]P.Young,A.Jakeman,andR.McMurtrie, “An instrumental variable method of model orderidentification,” Automatica. vol.16, pp.281-294,1980. [j] K. J. AstromandP.Eykhoff,”System identification-a survey,“ Automatica, VOI. 7, pp. 123-162,1971. [6] K. J. Astrom,“Lectures on theidentification problem-the least squares method,’’ Report 6806, Lund Institute of Technology, Sweden.1968. [7] K. J. Wong and E. Polak. “Identification of

~ =~

~

~

~

~~~

~

~

~~~

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~ ~~

Maite Una de C . was @-:~ born in Caracas, Ve-

nezuela, on March re18, 1952. She ceived a degreein chemical engineering in 1976 from the Universidad Sim6n Bolivar and an M.Sc. degree in theory and practiceof automatic control from the University of Manchester Institute of Science and Technology in 1978. She has been with the Processes and SystemsDepartmentof the Universidad Sim6n Bolivarsince1979,whereshe isnow Assistant Professor. Her research interests include modeling and identification of multivariable systems and application to electric arc furnaces. 13

=

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