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Neural Networksfor System IdentificationS. Reynold Chu, Rahmat Shoureshi, and Manoel Tenorio

ABSTRACT: This paper presents two ap-proaches for utilization of neural networksin identification of dynamical sy stems. In thefirst approach, a Hopfield network is used toimplement a least-squares estimation fortime-varying and time-invariant systems. Thesecond approach, wh ich is in the frequencydomain, utilizes a set of orthogonal basisfunctions and Fourier analysis to construct adynamic system in terms of its Fourier coef-ficients. Mathematical formulations are pre-sented along with simulation results.

IntroductionArtificial neural networks offer the advan-

tage of performance improvement throughlearning using parallel and distributed pro-cessing. These networks are implementedusing massive connections among process-ing units with variable strengths, and theyare attractive for applications in system iden-tification and control.Recently, Hopfield and Tank [l ] , [2] dem-onstrated that some classes of optimizationproblems can be programmed and solved onneural networks. They have been able toshow the power of neural networks in solv-ing difficult optimization prob lems. H ow-ever, a globally optimal solution is not guar-anteed because the shape of the optimizationsurface can have many local optima. Theobject of this paper is to indicate how toapply a Hopfield network to the problem oflinear system identification. By measuringinputs, state variables, and time derivativesof state variables, a procedure is presentedfor programming a Hopfield network. Thestates of the neurons of this network willconverge to the values of the system param-eters, wh ich are to be identified. Results arepresented from computer simulations for theidentification of time-invariant and time-varying plants.

Tank and Hopfield [2] have also shownthat the network can solve signal decompo-Presented at the 1989 American Control Confer-ence, Pittsburgh, Pennsylvania, June 2 1-23, 1989.Reynold Chu and Rahmat Shoureshi are with theSchool of Mechanical Engineering and ManoelTenorio is with the School of Electrical Engi-neering, Purdue University, West Lafayette, I N47907.

sition problems in which the goal is to cal-culate the optimal fit of an integer-coefficientcombination of basis functions (possibly anonorthogonal set) for an analog signal.However, there are advantages in choosingorthogonal basis functions because of the fi -nality of the coefficients [3]. This paper pre-sents how trigonometric functions can beused so that the Fourier transform of the sig-nal can be generated. A s show n, it is simplerto pose this problem in the form of an adap-tive linear combiner (ALC ), w hich was orig-inally proposed in [4], [5]. The paper de-scribes that by proper selection of learningconstants and the time for updating networkweights, we can transform the continuousversion of the Widrow-Hoff rule into New-tons method in one step, i .e. , weight con-vergence is achieved in only one updatecycle. The convenience provided by this ap-proach is due to the orthogonality of the ba-sis functions. Computer simulations areconducted for two cases: ( I ) signal decom-position of a flat spectrum with nonzero m eanand linear phase delay, and (2) identificationof the frequency response of a mass-spring-damper system subject to periodic inputpulses.

Hopfield Neural ModelThe Hopfield model [6], [7] consists of a

number of mutually interconnected process-ing units called neurons, whose outputs V,are nonlinear functions g of their state U; .The outputs can take discrete or continuousvalues in bounded intervals. Fo r our pur-poses, we consider only the continuous ver-sion of the Hopfield model. In this case, neu-rons change their states, U, according to thefollowing dynamic equation, where Tv ar ethe weights, R, the ith neuron input imped-ance, and 1,the bias input.

N IU i/d t = C jlq- U, R; + Ii ( 1 )It is assumed that all neurons have the sam ecapacitances, thus C is not included in Eq.(1). The dynamics are influenced by thelearning rate X and the nonlinear function g.

J = I

v,=&?(XU;) ( 2 4U, = (l/X)g-(V;) (2b)

The nonlinear function g ( - ) is called the sig-moid function and takes the values of s an d--s as x approaches +ca an d --ca. As Xincreases, g approaches a scaled shifted stepfunction.

Consider the following energy function E.

(3)Hopfield [7] has shown that, if the weightsTa re symmetr ic (T G= T, , ) , then this energyfunction has a negative time gradient. Thismeans that the evolution of dynamic system(1) in state space always seeks the minimaof the energy surface E. Integration of Eqs.(1) and (3) show s that the outputs V , do fol-low gradient descent paths on the E surface.

System IdentificationUsing the Hopfield NetworkThe mean-square error typically is used as

a performance criterion in system identifi-cation. Our m otivation is to study whether itis possible to express system identificationproblems in the form of programming aHopfield optimization network. T he systemidentification discussed here is called puru-metric identiJicution, which means estimat-ing the parameters of a mathematical modelof a system or process. Figure 1 shows theproposed structure for system identificationin the time domain. The dynamics of theplant (to be identified) are defined by theusual equations, where Ap an d Bp are un-known m atrices and x an d U are the state andcontrol.

.k = A, x -I- B pu (4)The dyn amic equation of the adjustable sys-tem depends on e, which is the error vectorbetween actual system states x and estimatedvalues y.

y =A,(e , r )x +&(e, t ) u - Ke (5)Therefore, the e m r dynamic equation is afunction of state and control.

e = (A,, - A,T)x+(B , - B,Ju +Ke(6)

310272-170819010400-0031 $01000 990 IEEE

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- 1- - _ _ _ _ _ _ _ -- -I Plant to be identified 1XII i

I [Fig. 1 . Proposed time-domain system identi3cation scheme.

The goal is to minimize simultaneouslysquare-error rates of all states utilizing aHopfield network. T o ensure global conver-gence of the parameters, the energy functionof the network must be quadratic in terms ofthe parameter errors, (A p - A,) an d (B , -B,). However, the error rates ( e ) n Eq. ( 6 )are functions of the parameter errors and thestate errors. The state error depends on y ,which, in turn, is influenced by A, an d B,.Hence, an energy function based on e willhave a recurrent relation with A , and B,. T oavoid this, we use the following energyfunction, where tr defines the trace of a ma-trix, and (.) 'is the transpose of a matrix [ 8 ] .

n 7E = (1/T) 1 ( 1 / 2 ) e , ( t ) T e , (t ) t

= ( l / T ) l T ( 1 / 2 ) ( X- A ,x - B,u)'. (X - A , x - B,u) dt

= trA,[ ( l /T) ~ ' ( 1 / 2 ) x x ' d t ] A :

- r A , [ ( l / T ) io'x x ' d t ]- r B , [ ( l / T ) So u t T t ]+ [ ( l / T ) soT(1 /2)XTXd tj (7 )Equation (7 ) is quadratic in terms of A , an d

B,. Substituting A, x +B,u for x in Eq. (7 )indicates that E is also a quadratic functionof the parameter errors. Based on Eq. ( 7 ) ,

we can program a H opfield network that hasneurons with their states representing differ-ent elements of th e A , an d B, matrices. Fromthe convergence properties of the Hopfieldnetwork, the equilibrium state is achievedwhen the partial derivatives aE/aA, and aE /aB, are zero. This results in the following,where A: an d BT are optimum solutions ofthe estimation problem.

(AT - Ap ) [(1/T) so ' x x Td t ]

(AT - A, ) [ ( 1 / T )lo' u' d t ]+ (B: - B,) [ ( l /T) s o T u u T d t ]= 0

(8b)Derivation of Eqs. (8) assumes that the neu-ron input impedance, Ri,s high enough sothat the second term in Eq. ( 1 ) can be ne-glected. Therefore, AT approaches A, , andB, approaches B, if, and only if, the follow-ing is true.

( U T ) j' [xi [X'lU*] dt # 0 ( 9 )o uIt can be shown [ 9 ] that Eq. ( 9 ) is true if,and only if, x ( t ) an d u ( t ) are linearly inde-pendent in [0, T I . Landau [ l o ] gives a de-tailed discussion on the condition to ensurelinear independence. If state convergence isimportant, all that is needed is an asymptot-ical ly s table Ki n Eq. ( 6 ) . As shown by Eq.( 9 ) , state convergence follows the parameterconvergence.

To show applications of the preceding net-work for system identification, a second-or-

der system with two states and a single inputis considered in two cases: time invariantand time varying. The following shows theresulting T j an d Z, or Eq. (1).[T,I = - (W lo' MI dt

u x 2 0 0 u 2 00 ux , u x 2 0 u 2-

ai,,UX2]' dtIn the case of the time-invariant system,

the following Ap an d B, matrices are to beidentified.

A , = [ - 0 . 9 4 2 5 1 2 . 5 6 1- 2 .5 6 -0 .9 4 2 5

Figure 2 shows the simulation results of thesystem identification. As shown, it takesabout 0 . 2 sec for the network to learn thesystem parameters, even though their initialestimates may have opposite polarities. Thisfigure represents only A p l l ; imilar resultsmay be obtained for other entries of th e A ,and Bp matrices.

For the case of time-varying systems, theA matrix has a low-frequency harmonic vari-ation represented by[ 0 .9 4 2 5 1 2 .5 6 1- 2 .5 6 -0 .9 4 2 5

. (1 + 0.1 sin 2 r . 0 .0 2 5 t )A, =

OSO! II

-4.01;I-5 .5 0 2 4 6 8 IO

Time, secFig. 2 .plant . Zdent$cation of a time-invariant

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-? -, ,- 1 .

Fig. 3 .plant.

c0 4 8 12 16 20Time, sec

Idenr$cation of U rime-varying

Figure 3shows the identification results. Asshown, very good tracking is achieved. Itshould be noted th at, for the case of the time-invariant system, a rectangular window withan infinite memory is used; whereas, for thecase of the time-varying system, an expo-nentially decaying window using a first-or-der filter is utilized to ensure fast conver-gence in the presence of changing plantparameters. This window has the effect ofemphasizing the most current estimates oferror, rather than the past memory. High-frequency fluctuations observed in Fig. 3may be alleviated by using seco nd- or higher-order filters.

Signal Decompositionand Frequency-ResponseIdentification

Hopfield demonstrated that the networkcan solve signal decomposition problems inwhich a signal is approximated by a linearcombination of a specific set of basis func-tions. The basis functions Hopfield used arenot necessarily orthogo nal. We p refer usingorthogonal functions for the sake of simplic-ity and finality of coefficients of the combi-nation. By finality, we mean that when theapproximation is improved by increasing thenumber of basis functions from n to n + 1 ,the coefficients of the original basis fun ctionsremain unchanged. In particular, we haveused sines and cosin es as the basis functions.The computational network then will be aFourier analysis circuit. To estimate f ( f ) ymeans of a finite-order function s ,rI(t ) , hefollowing energy function is formulated,where x,n(t)can be expressed by a Fourierseries of cos w,,t and sin w, , f , an d U, , =2 d T a n d n = 1 , 2, .. . ,m.E = ( 1 / 2 T ) l ,! +7 [ f ( r ) - x, ,?(r)]'d t (10)

I/

Let us define~ ' ( r )= [ I , co s w , t , sin w , t , . . . .

wT=b o , al, b , , . . , a !, , b,"lco s w,,t, sin w,, , t ] ,

Then, the error signal E ( ? ) is defined as thedifference betweenf(r) and X'W. Expansionof Eq. (10) results in the following quadraticformulation:E =( l / 2 ) W 7 R W- P'W

The function R represents the input correla-tion matrix and the cross-correlation matrixbetween the desired response and the input.

ro +7R = ( l / T ) /P = ( l / T ) S

X ( r ) X r ( r ) dr0

111+T f ( r ) x ( t ) drU

These results can be used to p rogram Hop-field's network. However, since orthogonal-ity of the basis functions produces a diag onalR matrix, it would be sim pler to use Wid-row's ALC than a Hopfield network. Thegradient of the mean-square-error surface,aE /aW , is R W - P. The optimal W * is ob-tained by setting this gradient to zero, re-sulting in W * =R - ' P . The optimal weightswe obtained are indeed the Fourier coeffi-cients of the original signal, f ( t ) ,which canbe rewritten as W * =W - R - ' V , where Vis the gradient of the energy surface, aE/aW,and can be represented by

IU +7v = - (UT) 'j { e ( [ ) , (t )co s W l r ,e( t ) sin w l t , . . , e(r) co s w,t,e ( t ) sin w,, ,rJ7 r (1 2)

The integrand in Eq. (12) is the instanta-neous estimation of the gradient vector in theleast-mean-squares (LMS) algorithm used toestimate the unknown parameters. Based onorthogonality of the basis functions and ap-plications of Eq. (12), W * can be repre-sented byw * =w + ( I i T ) S o +7 {e( t) ,

I

. co s wit, 2 4 4 sin wlr, . . . , 2e ( r )

. co s w,r, 2e(r) sin ~ , r } ~ d r1 3 )Therefore, the famous Widrow-Hoff LMSrule for this problem , in a continuous form ,

would be as shown in the following, whereq l , . . . , qZ,n I are the gain constants thatregulate the speed and stability of the ad-aptation process.d W l d r = { q l e ( t ) , ze(r)co s w , f , q 3 e ( t )

* sin w,r , . . . , ~ ~ , ~ e ( r )os w,t,~ ~ ~ + ~ e ( r )in w,,,tJ7 (14)

Because Eq. (14) modifies the weights basedon instantaneous estimates of the gradient,V , he adaptive process does not follow thettue line of steepest descent on the mean-square-error surface. Therefore, th e follow-ing scheme for changing the algorithm isproposed. First, let

'71 = (1/T)'72 ='73 = . . . 72m =7 2 m + I = 2 / T

Then apply Eq. (14) to accumulate theweight changes while keeping all the w eightconstant until the end of the T seconds. Theweight corrections are then based on theaverage of gradient estimates over a periodT. Because of the diagonal R matrix, ourmethod is equivalent to Newton's method ofsearching the performance surface. By suc-cessive increases of T in each cycle (and,hence, the decrease of learning gain), w e areable to eliminate the variations of weightsdue to low-frequency components inf(t). Asuggested sequence of periods for averagingis T, 2T, 3T, . . . . Each time the period isincreased, we need to add more weights andneurons. Hence, the frequency resolution isdetermined by the available resources and isimproved as T increases. Moreover, if theinitial selection o f T happens to be the periodoff @ ) or its integer multiples, then w e reachthe correct weights in a sing le search period.This can be seen by integrating Eq. (13) overa [to, to + TI time interval with the initialweight being W , then we will get Eq. (13).

Figures 4 an d 5 show the magnitude andphase results of decomposing a signal havingfrequency com ponents from direct current to10 Hz with 0.5-Hz increments. We use anetwork capa ble of 1-H z resolution. Th e ini-tial guess of Tis 1 sec, which does not pro-vide good results. After extending Tt o 2 sec ,we can identify quite accurately all the com-ponents within our frequency resolution.Figures 6 an d 7 are the simulation results ofidentifying the frequency response of an un-known plant subject to periodic input pulseswith a period of 2 sec. The pulse train isformed by a series of cosine waves with fre-quency components from direct current to 10Hz with increments of 0.5 Hz and 0.05 mag-nitude for all comp onents. The output of theplant is analyzed by the Fourier network.

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Kahmat Shoureshi I( anAssociate Professor an dChairman o f Manufactur-ing and Matenuls Pro-cessing area in the Schoolof Mcchanical Engineer-ing at Purdue University.He completed his gradu-ate dudieb ut MIT in1981. His research inter-ests include. intelligentcontrol an d diagnosticsystems using an alyticalisymb olic processors andneural network s; active and sem iactive control ofdistributed parameter systems, including flexiblestructures and acoustic plants; and manufacturingautomation, including autonomous machines androbotic manipulators. He was the recipient of the

American Automatic Control Council EckmanAward in 1987 for his outstandin g contribu tionsto the field of au tomatic control. He is currentlyinvolved in several indus trial and government re-search studies. He is a Senior Member of IEEE.Manoel F. Tenorio re-ceived the B.Sc.E.E. de-gree from the National In-stitute of Telecommuni-cation, Brazil, in 1979;the M.Sc.E.E. degreefrom Colorado State Uni-versity in 1983; and the'h.D. degree in computerengineerin g from the Uni-versity of Southern Cali-fornia in 1 987. In 198 9,I

he led a product design group as the Director ofResearch and Development at C.S. Componentsand Electronic Systems in Brazil; from 1982 to1985 , he was a Research Fellow for the NationalResearch Council of Brazil. Wh ile completing hisdoctoral studies, he taught graduate level coursesin artificial intelligence at USC, UCLA, andRockwell International n Los Angeles. Currently,he is an Assistant Professorat the School of Elec-trical Engineering, Purdue University, where hisprimary research inte rests are parallel and distrib-uted system s, artificial intelligence, and neuralnetworks. He is the organizer of the interdiscipli-nary faculty group at Purdue called the SpecialInterest Group in Neurocomputing (SIGN) andheads the Parallel Distributed Structures Labora-tory (PDS L) in the School of Electrical Engineer-ing.

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