engineers (chemical industries) by peter englezos [cap 13]

8/22/2019 Engineers (Chemical Industries) by Peter Englezos [Cap 13]

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13Recursive Parameter Estimation

In th is chapter w e present very br ie f ly the basic a lgor i thm for recursive leastsquares estimation and some of its variations for single input - single output sys-tems. These t echn iques are rout ine ly used for on-line parameter estimation in dataacquisi t ion systems. They are presented in this chapter without an y proof for thesake of completeness and with the aim to provide the reader with a quick over-view. For a thorough presentat ion of the material the reader may look at any ofthe f o l low in g references: Soderstrom et al. (1978) , L j u n g an d Soderstrom (1983)Shanmugan and Breipohl, (1988) , Wellstead an d Zarrop (1991). The notat ion thatwil l be used in th i s chapter is different from the one we have used up to now. In -stead we shall fol low the notat ion typically encountered in the analysis an d controlof sampled data systems.

13.1 D ISC R ETE I N P U T - O U T P U T MODELSRecursive est imation methods are rout inely used i n m an y appl icationswhere process measurements become available continuously and we wish to re-estimate or better update on-line th e various process or control ler parameters asth e data b e c o m e available . L et u s consider the l inear discrete-t ime mode l h av i n g

the general structure:A (z 1 ) y n = B(z- ' )u n .k + en (13.1)

where z " 1 is the backward shi f t operator (i.e., y n _ i = z"'yn , y n -2 = z " 2 y n > etc.) an d A(-)an d B(-) are po lyn om ia l s of z " 1 . The i nput variable is un = u(tn) and the output vari-2 18

Copyright 2001 by Taylor & Francis Group, LLC


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Recursive Parameter Estimation 219

able is yn = y(tn). The system has a delay of k s a m p l i n g intervals ( k > l ) . In ex-panded form the system equation becomes(1 + a,z"' + a2z" 2 + . . .+apz" p)y n = (b 0 +blzl +b 2z"2+...+b qz" cl)u,,-k + en (13.2)

orY n = - a,y n .i - a2yn .2 - . . . - apyn .p + b 0un .k + b^,,^., + b 2u n .k .2+...

+ b nu,.k.m + en (13.3)W e shal l present three recursive es t imat ion methods for the es t imat ion o fthe process parameters (a b . . . ,a p , b 0, bh..., b q) that should b e employed accordingto the statistical characteristics of the error term sequence en 's (thestochastic dis-

turbance).

1 3 . 2 RE CU RS I VE L EAS T SQUARES (RLS)In this case we assume that is white noise, ^ n, i.e., th e en 's are identicallyand ind ependen t ly d is tr ibuted n ormal ly with zero mean and a cons tant variance o 2.Thus, t he mo d e l equat ion can be rewri t ten as

Y n = - a , y n _ i -a2yn-2---bOun - k+b lun - k - l +b2un-k-2 + +bqun -k -q +^ nwhich is of the form

y n = M > n - l 9 n - l + ^ n (13.5)where

andn - l . -y n -2 . - -y n -p .un -k 'Un-k-l . - . un-k-q]T (I3-6a)

n _ ! = [ a 1 ) a 2 , . . . , a p , b 0 , b ] , b 2 ) . . . , b q J T (13.6b)Whenever a new measurement , yn, becomes available, the parameter vector

is updated to 9 n by the formula

(13-?)where



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220 Chapter 13

_-

The new estimate of the normalized parameter covariance matrix, P n , is ob-tained from

P n = [ l - K n M > n ] P n _ , (13.9)The updated quanti t ies 9 n an d P n represent our best estimates of the un-known parameters an d their covariance matrix with information up to and inc lud-ing t ime tn . Matrix P n represents an estimate of the parameter covariance matrixsince,

C O K ( 9 n) = o 2 P n . (13.10)The above equat ions are developed us ing the theory of least squares an dmaking use of the matr ix invers ion l e m m a

( A + x x 1 ) - 1 = A - ' + A - ' X ^ A - ' X + I J V A - ' 03. i i )" 1where A " = P n , an d x=i|/n is used.To initialize the algorithm we start with our best initial guess of the pa-rameters, 00 . O ur initial estimate of the covariance matrix P 0 is often set propor-

t iona l to the identi ty matr ix (i.e., P0 = y 2 I). If very l i t t le in format ion i s avai lableabout the parameter values, a large value for y2 should be chosen.O f course, as t ime proceeds and data keeps on accumulating, P n becomessmal ler an d smal ler and hence, the algorithm becomes insensi t ive to process pa-rameter variations.The standard way of el iminat ing this problem is by in troducing a . forgettingfactor A . . Nam ely , i ns tead of m i n i m i z i n g

(13 .12)

we es t imate the parameters b y m i n i m i z i n g the weigh ted leas t squares ob jec t ivefunct ion



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(13.13)

wi th 0< A


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222 Chapter 13

w h i c h is again of the form

where N V i = [ - y n - i . - y n - 2 . - - y n - p .u n _ k , u n _ k _ 1 , . . . , u n _ k _ q ^ n _ i ^ n _ 2 , . . . , ^ n _ r ] T (13.21a)

an den-i = [ a i , a 2 , . . . , a p , b 0 , b I , b 2 ) . . . , b q , c ] , c 2 , . . . , c r ] T (13.21b)

This model is of the same form as the one used for RLS and hence the up-dating equations for 0 n an d P n are the same as in the previous section. However,th e disturbance terms, i ; n _ i , ,n.2,..., ^a.n an d hence the updating equations cannot b eimplemented. The usual approach to overcome th i s dif f icul ty is by using the one-step predictor error, n am ely ,H V i = [ - y n - i . - y n - 2 . " . - y n - p .

un-k'un-k-l.-.un-k-q.?n-1.4-2.-.?n-r]T (13.22)

where

J1'' ""' n ~ ' ' n ~ 2 ""' n"2^n-2 =y n -2 - yn-2in-3 =yn-2 - Vn-30n-3 I> (13.23)

where w e have denoted with y n ! n _ i the one-step ahead predictor of yn . Thereforethe recursive extended least squares ( R E L S ) algor i thm is given by the f o l low in gequat ions :

(13.24)



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Recursive Parameter Estimation 2 2 3

Pn =l[l-Kn M/ n r ]pn _, ( 1 3 . 2 5 )

K n =^^ (13 .26)

13.4 R E C U R S I V E G E N E R A L I Z E D LEAST SQUAR ES (R G LS)In this case w e assum e again the d is turbance te rm, en , i s no t w h i t e noise ,rather it is related to ^ n th rough the f o l l o wing transfe r funct ion (noi se fi l ter)

en=n (13.27)C(z-')where

C(z') = 1 + c i z 1 + c2z '2 +...+C . Z 1 (13 .2 8 )The m o d e l equa t io ns n ow b e c o m e

A(z- ' )C(z- ' )yn = B ( z -1)C(z 1)u n .k + ^ (13 .29)or

A ( z - ' ) y n = B ( z - ' ) u n _ k + ^ (13.30)where the t rans formed i npu t and output variables are given b y

y n = C ( z - ' ) y n (13.3 l a )or

y n = y n + C i y n . i + c 2 y n . 2 + . . . + c ry n . r (13.31b)and

u n _ k = C(z') un.k (13.32a)or

u n _ k = u n . k + c 1 u n . k . 1 + c 2 u n . k . 2 + . . . + c ru n .k .r (13.32b)Equation (13.30) can now be rewritten in our usual form as



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224 Chapter 13

(13.33)

where

HVl = [-yn-l.-yn-2.-.-yn-p. Un-k.Un-k-l . - .Un-k-qP (13.34a)

an d9 n _ i = [ a I , a 2 , . . . , a p , b 0 , b 1 , b 2 , . . . , b q ] T (13.34b)

The above equat ions suggest that the u n k n o w n parameters in po lyn om ia l sA(-) an d B(-) can be est imated with R L S with the transformed variables yn an du n _ k . Hav ing po lyn om ia l s A(-) an d B(-) we can go back to Equation 1 3 . 1 an dobtain an est imate of the error term , en , as

en =AV -BVWk (13 .3 5 )where by w e have denoted with A(-) an d B(-) the current estimates of polynomi-al s A(-) an d B(-) . The unknown parameters in polynomia l C(-) can be obtainednext by cons ider ing the noise f i lter transfer funct ion. Namely , Equation 1 3 . 2 7 iswritten as

en = - c le n - i - C 2 e n - 2 - - - c r e n -r+^n (13.36)Th e above equat ion cannot be used directly fo r R LS e s t ima t io n . Ins tead ofthe true error terms ,e n, w e m u s t use the est imated values from Equation 13.35.Therefore, the recursive generalized least squares ( R G L S ) algori thm can be im-

plemented as a two-step est imation procedure:S t e p l . Compute th e transformed variables y n an d u n _ k based o n o u r

knowledge of polynomia l C(-) at t ime t n _ i .Apply RLS on equation A ( z ~ ' ) y n = B ( z " 1 )u n _ |< + ^,, based on in-formation up to t ime tn . Namely , obtain the updated value for theparameter vector ( i . e . the coefficients of the po lyn om ia l s A andB ) , e n

Step 2 . Hav ing 9 n , estimate en as A ( z ~ ' ) y n -B(z~l)un_k .



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Apply R L S t o equat ion : C(z~ )?n = ^ n to get the coefficients o fC ( z ~ l ) w h i c h are used for the computat ion of the t ransformedvariables yn an d un_^ at the nex t sam pling interval.

Essential ly, the idea behind the above R G L S algorithm is to apply the ordi-nary R L S algori thm twice. The method is easy to apply, however, it may havemult ip le convergence points (L jun g an d Soderstrom, 1983).The previously presented recursive algorithms (RSL, R E L S and RGES)utilize the same algorithm (i.e., the same subrout ine can be used) the on ly differ-ence be ing the set up of the state and param eter vectors. That is the primary reasonfor our select ion of the recursive algor ithms presented here. Other recursive algo-r i thms ( i nc lud ing recursive m a x i m u m l ikel ihood) as well as an exhaust ive pres-entation of the subjec t material i n c l ud i n g convergence characterist ics of the abovealgorithms can be fo und in any standard on-l in e iden tif ication book .


engineers (chemical industries) by peter englezos [cap 13]

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