Parametric Identification of Nonlinear Dynamic Systems Based on Correlation Functions

PARAMETRIC IDENTIFICATION OF NONLINEAR DYNAMIC SYSTEMS BASED ON CORRELATION FUNCTIONS R.Haber Department of Automation, Technical University of Budapest, Budapest, Hungary

Abs't raot. The paper introduces the two-etep identification method - least squares parameter estimation based on correlation functions - for nonlinear dynamic systems being linear in the parameters.The identification of the input/output parametric models working in open and closed loops is dealt with. It is shown the relation between the structure of the model, the choice of the multiplicators and the necessary delay time domain. A new method is given for nonlinearity test. The theoretical considerations are demonstrated by simulations and practical applications. Kefh0rdB c Nonlinear systems, identification, correlation met ods, parameter estimation. A correlation method has been applied for nonparametric identification ot nonlinear dynamic systems (Krempl, 1914), but the elaborated new method makes also the parametric identffication possible.

INTRODUCTION Most of the identification algorithms have the common feature that they reduce the disturbances to the output. They differ from each other in the respect that besides the process parameters also estimate the noise model, whose determination is an iterative task. In the case of 001oured noise the simple least squares estimation leads to biased result. But taking the autocorrelation function of the input and the crosscorrelation function of the input and the output signals instead of the measured input and output signals, the one-etep least squares estimation yields unbiased estimation of the process parameters. In the report (Haber, 1977) an extension of the two-step identification method - correlation analysis and least squares parameter estimation - known in linear case (Isermann, 1974) is introduced to nonlinear dynamic systems being linear in the parameters.

NONLINEAR MODELS Modelling of nonlinear dynamic systems is not an easy wo~k. The canonic Volterra series are nonparametric desoriptions and the most gene.al blockoriented model the canonic Schetzen model is parametric but nonlinear in the parameters. Most of the practioal processes can be well approximated by models being similar to those of the linear MISO systems. A linear dynamio MISO model oan be described by the well-known formula: M B. (z-l) yo(k)=l 1. -I ui(k-d i ). (1) i~iA(Z ) On multiplying Eq. (1) by A(z-l) and rewriting it, we get Eq. (2) being linear in the parameters:

This work started during a temnonth scholarship granted by the DAAD /German state Office for Exchange of Academics/ through the Hungarian KKI /Institute for Cultural Relations/. The present research work is based upon work supported by the National Soienoe Foundation and the Institute for CultUral Relations under Grant No NSF-ICR/INT 7B-OB77B.

Yo(k)=gT(u,y)~ •

(2)

A lot of nonlinear S1SO systems oan be desoribed by Eq. (2) if the input signals in Eq. (1) are replaoed by known nonlinear functions of the measured input and output signals of the process /Fig. 1/:

51 5

R. Ha b e r

5 16

u, (k) MI SO

k n ow n u(k)

-

u2 (k)

I ln E' a r

n on lin e ar t ra ns for m

..

d i na m lc

u ~ ( k)

n Y. ( k )

-=-=---

L i=o

u (k )

b , (U(k) , y.(k » z- i

y, (k)

n

S

L

a j (u(k), y.(k»z-I

1=0

Fig. 1 u1.. {k)=f 1.. (u{ k) ,U{ Ic-l), •••• '¥o{ k-l), ¥ 0 ( k-2 ) , ••• ) . (3 ) The class of the nonlinear dynamic s¥stems being linear in the parameters is wideLY spread in the literature because of its simple e8timation and the big information of the description. now three applications of the desoription are presented.(Haber and Keviczky, 1974 and 1976, Haber, 1977 , Haber and Keviczky, 1979). Hammerstein model The model seen in Fig. 2 can be transformed into Fig. 1 if

Fig. 3 and if its time constant has the same character then Ul{ k )=u(k),

u2 {k)=n{le)¥0{k).

(7)

CORRELATIOn AnALYSIS Let us assume that the input signal is measured without aQy error and the disturbanoes are reduced to the output. Let the noise model a general nonlinear one: )-l-c.{z-l)

n{ k )=L

-1

1.

i~ D{

z

Ei (le).

(B)

)

The measured output y( k )=¥ u (k)

-

co' c, u (k). C2 u 2 (k)

r--

B ( £' ) A ( z" )

-

y. (k)

Fig. 2 Canonic nonlinear d¥namic model being trne ar in the parame ters It can be proved that on extending the modeloomponenats in Eq.(4) to the components of the Volterra series u ( k )=u( k )u( le-l) 4 (5 ) u {k )=u('k)u{ k-2) 5 we get a parametric description being equivale nt to the - second de gree Volterra series. Linear models having signaldependent parameters If the parameters of a continuous or discrete model are 1010wn f unctions of the input and output signals /Fig . 3/, The system can be modelled in the form given i n Fig. 1. For example let the gain of a first order lag term linear dependent on the i nput signal, then ul{k)=U{ k ), U2 {1c)=u2 ( le ) (6)

o

(k)+n{k)

(9)

is disturbed by the noise term. Let us correlate Eq. (9) b¥ x{ k ): ~X¥{T)=$Xy (T)+~xn {L) o

(lO)

and ohoose the multiplioator that it should fulfil Eq. ell): ell) In linear case the assumed noise model is linear, so it is enough that x {k) and £1{le) should not correlate, what is a consequence of their independenoe and the faot that E1 (k) is a noise. The nonlinear noise terms can be eliminated if x{k) has zero mean value. Thus two assumptions are to be met: o let x{ k ) i ndependent of Ei (k)(12) o let E{x{k)r=O • The equation
(13 )

is always valid for open loop s¥stems but it is restricted valid in closed loop ones where the input signal is influenced b¥ the noises through the feedbao k . PARAlILETER ESTIMATION The parameters of the prooess can be determined from the equation

Parametric identification of nonlinear dynamic systems

MB.(z-l) l. -1 cpxu. ('l')


i4A(z)

(14)

l.

being linear in the parameters by the least s~uares /LS/ method. Although Eq. (14) is deterministic after having eliminated the noises, it is practical to estimate the parameters in the domain Tl{"L~"t'2· (15) The difference ('t'2-'t).) has to be larger than the number of the parameters. Of course the correlation functions may not be periodic in the domain (-~1..'·t2). The method described so far, is equivalent to that of applied at linear systems (Isermann, 1974). In nonlinear case it can quite easil.y happen that the chosen multiplicator /x(k)/ does not correlate with every quasiinput signal /u . (k)/. In this case one has to chooSe more multiplicators /x.(k);j=1,2, ••• /. Thus J -1 M B. (z ) x y(l:')=L J. -I j' ?i.A(z)


l.

Let us expound Eq. (16) for several ~ and let denote the output crosscorrelation functions by ~x .y= [x .y('t'l)'·· , <\>xJ.y(T 2)] (17) J

J

and the situationmatrix by ix.uy. Thus Vie get: J (18)

~.y = ~.uy.!: • J

J

Let us arrange the crosscorrelation functions belonging to different multiplicators in the vector [ T

T

Eqs. (19) and (20) are connected by the parameters:

T

] -1

T

o ct>x. u. (t) mus t be line arl.y indeJ l. pendent o

~x;ui(~)

...

must be persistent exciting.

SnIDLATION 1. Example

Let us consider the identification of a Hammerstein model /Fig. 2/. The parameters are: B(z-1)_0.065z-1 +0.048z-2-O.008z-3 z -1 A( z-1 ) - 1-1.5z-1 +0.705z -2 -O.lz -3 (23 )

The input signal was a maximum length PRTS with the periodicity 80 and zero mean value. The multiplicators are chosen from the zero mean value, so the estimation of the constant term is unpossible. Let Xl (k)=u(k)-E{u(k)f

(24)

X2(k)=U2(k)-EtU2(k)} •

(25)

and The crossoorrelation functions are shown in Fig. 4. It can be seen that
Xl

u 2 eT) = <.p

x 2u

( 1:) = 0,

( 26 )

ie. xl(k) does not estimate the quadratic and x 2 (k) the linear terms. Having performed the LS estimation in the domain (-8,40) the true parameters are obtained. 2. Example

~xuy~x.y·

Since xl(k) and x 2 (k) are correlated onl.y by the linear or the quadratic terms, it is practical to use the linear combination of xl(k) and x 2 (k) as a multiplicator. Thus let Xl (k)=u (k)-E {u( le)} +u 2 (k)-E {u 2 (k)}.

(21)

(27) The crosscorrelation functions are drawn in Fig. 5. ~x U(L) and ~x u2(~)

(22)

are dependent linearly in the domain (-8,40), but are independent in (-8,79). Having performed the identification in the latter domain, we get the true process parameters.

~x.y = ~xuy.E • The parameter vector can be computed by the LS method "

situationvector. Now these have to be met for the correlation functions, ie

T

~x.y = ~xl y' ••• '~XjY] (19) and the situationmatrices in the matrix thTT ]T ~QY = [ ~xl uy' ••• '~jUY (20)

g = [~XUy~uy

5 17

In linear case the condition of the one-step LS estimation is the persistent excitation and the linear independence of the components in the

1

1

R. Haber

518

3. Example

X1 U(T) 0.1687

40

o

20

10

Aocording to the above, the multiplicator ought to be a signal correlating with all the model oomponents. The output signal fuliils this assumption but it is not noisefree

T

so

30

(28)

- 0 1687

xlu2(T)

o

I 20

30

40

so

T.

'I

o2 X1U (T)

10

20

30

40

so

10

20

30

10

20

30

40

50

0027~

o

(29)

The crossoorrelation functions are presented in Fig. 6. Having performed the estimation in the domain (-8,79) the true parameters are obtained.

'I

o X1Y (1)

Let the multiplicator beoome the output signal in a noise free example I Xl (k)=y(k)-E {y(k)} •

Om371

-OOC035 _

Of oourse the estimation can be performed iteratively similar to the method of the instrumental variables.

-0.02775

30

'I

40

20

so

-0027~ 10~ ...-

-_ _ _ _~1

x l u 2 (T)

50

1}j:=M

0 .000176

Fig. 4

,P

SOT

20

I

00001758

xl Y (1')

40

o

10

20

'I

T

50

30

20

30

50

- 0.168 7 -0019

o

10

20

30

~xl Y (T)

1

Fig. 6 'I I

so

•

4. Example

0,02873

50 20

30

'I

IfOW let the input Signal beoome a PRBS - u=u(-o.25,0.25) - and choose the input signal as the multiplicator

40

Xl (k)=u(k)-E{u(k)l • -0,02873

Fig. 5

(30)

By the means of this multiplicator the linearized model of the Hammerstein model can be identified.

Parametric id entification of nonlinear dynamic s ystems

The crossoorrelation functions are seen in Fig. 7. The obtained linear model is equal to the linear part in Fig. 2 as 2.28125-1.78125 = 1. _AY = ---'--~--";'''':''''----'';;'''' odU 0.25-(-0.25)

iI> Xl U

(31)

The conditions of the estimation are to be met, too. It can be seen in Fig. 4 that ~xlu(T) and $x2u2(~) are dependent in the domain (0,40)/Example 1/ but they are independent in the domain (0,79) /Example 2/. linear~y

4. The multiplioators can also be chosen in view of optimizing a criterion in the estimation theory.

m

NONLINEARITY TEST

000624 8

T -0.0001078

519

10

20

30

40

41X l y(1)

The primer task of the identification is to find the fitting structure. The oardinal point is to decide wether a process is linear or nonlinear. A method, named nonlinearity test is needed, which does not require any long computations.

0.0009432

T -0.00010078

10

20

30

40

Fig. 7 THE CHOICE OF THE ltIDLTIPLICATORS Ori the basis of the above and other simulation examples several consequences oould be drawn for the choice of the multiplicators. 1. Different types of multiplicators can be chosen for the identification of the same structure. In the 1. example one can take the absolute value instead of the square function as both multiplicators are even functions. 2. BY the proper choice of the multiplicatora it oan be aohieved that on~y oertain components of the whole structure would be identified. In the example presented earlier the constant term can be obtained by a constant multiplicator, the linear term by a linear and the quadratic term by a quadratic multiplicator. 3. Instead of more multiplicators an equivalent one can be applied, which is the linear combination of the individual ones. This theorem is illustrated by the examples 1 and 2 and it is proved as follows. When summing the two parts belonging the two multiplicators in Eq. (21), we get an equation system being equivalent to using the multiplicator x(k)~1(k)+x2(k)=

=u(k)+u 2 (k)-E {u(k)+u 2 (k)}. (32)

The method of the multiplicators is suited for the nonlinearity test, to~ Let the process be excited by a test signal of zero mean and let us correlate the output by the multiplicator X(k)=u 2 (k)-E{U 2 (k)} • (33) It can be seen that the crosscorrelation funotion is zero if the process is linear but does not vanish if the prooess has a seoond degree nonlinear term. The nonlinearity test can be applied to the output signal and to the residuals, as well. SIMULATION App~ying the nonlinearity test to the Hammerstein model given in Example 1 the output crosscorrelation function ~x y(~) is not zero and is correlated to 2q,x J "t), as it can be seen in Fig.

2

IDENTIFICATION IN CLOSED LOOP V/hen measuring on industrial objeots the opening of the closed loop is general~y not allowed, one has to identify it in closed loop. Fig. 8 presents such a situation. Furthermore only the direct method - based on input/output records - is dealt with. The two-step identification method leads to unbiased results if the multiplicators and the noise are not correlatedo(Kurz and Isermann, 1975). If an external test signal /z(k)/ can be applied, it is practical to

R. Haber

5 20

form the multiplicator from that: Xj{k)=fj{z{k),Z(k-l), ••• ).

(34)

Then ~X .y(!') J'

=
oCU..1:~

(35)

ruLd the regulator is ~ · -1 P ( z-1 )= 3.4 8 5z -1-5.4J3z +2.l5z -2 Q(z-l)

~--------~ ~~-------Fig. 8

(41) The set value is zero and the test signal is a maximum length PRTS /z=z(-O.l,O.l)/ with periodicity 80. The correlation functions used for the estimation are shown in Fig. 9 for the multiplicator x( k)=u( k:)+u 2 ( k)-E{U(1c)+u 2 ( le)} (42) and in Fig. 10 for the multiplicator

Let us consider the case when the multiplicators are functions of the input signal: Xj (le)=f j (u( k) ,u( k-l), ••• ).

x( k) =z (k)+z2 (l<:)-E lz (!c)+z2 (le)}. (43) The true parameters were obtained in both cases in the domain (0,80).

(36)

Assuming the linear noise model

x.y J

J

(T),

't>0 •

IPXl

(37)

0

J'

= x .y. 0 ('1:'),

t7e>0.

(38)

u(Tl

0.0 106

is valid. If the noise model is of Volterra type and the largest difference in the term u(k)u(k-e) is ~ then $x.y('t)

l_z-l

20

-0.010 6

J

hase to be fulfilled (Haber, 1977). The correlation functions cj)X.u. Ct ) J

i=1,2, ••• ,M and

l

~x.y(T) J

must be linearLy independent. In the case of a linear regulator it means that the order of the regulator /n / has to be bigger or e~ual than that of the process /11./

000197

(39 )

In nonlinear closed loops it can be stated, that the regulator should fulfil the linear independence. It does not mean, that a linear or nonlinear regulator would fulfil the condition automaticalLy. SIMULATION The process is of Hammerstein type with the parameters:

· 0.00197

Fig. 9

Parame tric identif ication of nonlin ear dynamic systems

521

the autoc orrela tion funct ion of the multi plica tor is a white one. Furth er relati ons and the detai ls are to be publis hed in a separ ate paper .

4lx1 u (t)

0.0067 5

APPLICATIOlT 20

30

The new metho d has been prove d by a lot of noise free and noisy examp les. An intera ctive ident ificat ion program packa ge named OLID-MISo-NOLI was elabo rated and applie d for mode lling of analog ue model s in hybri d mode succe sfully (Habe r, 1977) . The program system was used for mode lling of an indus trial size steam heate d heat excha nger at the unive rsity of stutt gart. The plant and the linea r ident ifioat ion are desor ibed in (Baur and Iserm ann, 1976) .

-000675 4lx1 u 2 (T)

0.0000658

4l x1y (t)

The outpu t signa l was the differ ence of the water tempe rature /A~/ ard the input Signa l was the chan~e in the mass flow of the steam /AM /. The PQysi cal proce ss is linear Din the case of small excit ation s but the valve chara cteris tic is nonli near. The appra ximat ing model was of Hamm erstei ntype , the test signa l PRTS ./Fig. 11/. The true chara cteristic of the proce ss - measu red by statio nary measu remen ts - and the estim ated one are seen in Fig. 12.

0.00112

30

-0.0011 2

Fig. 10 RELATIOlT TO OTHER METHODS It is Imown, that the one-s tep LS param eter estim ation of the linea r s .y stems is essen tially the solut ion of rul equat ion system consi sting of corre lation funct ions (Astro m and Eykho ff, 1970) . It can be prove d, that the two-s tep LS param eter estimatio n using the model input and output signa ls as multi plica tors can be traced back to the LS metho d, based on the input /outp ut record s by the corres pondi ng reduc tion of the corre lation time domain (Habe r, 1977) . The two-s tep corre lation metho d has common featu res with the Tall.y prinCiple , since both of them are based on the fact that the noise and the model compo nents do not corre late. It can be prove d that the PIeF method (Pete rka and Holou skova , 1970) is equiv alent to a weigh ted two-s tep corre lation metho d "'_ T -l-1 d>

.2- (~xu..y~=XX=XUy ~u.y~::::xx-xy

(44)

and both metho ds are equiv alent if

u(ll[mA)

65 S

1sec =Q.Smrr

yit)[V)

Fig. 11 Furth er appli cation s and the progra m system OLID-MISO-NOLI are descr ibed in (Habe r and Bamb erger, 1979) .

522

R. Haber

this research work. REFEREl'TCES

.1"'30 reI I.

3

:/'~

2

1,...- ~.'-

L

,,A'

.

Mo

1

30 20 Ident.

1,0

50 lk g/hl

Fig. 12 COlWLUSIONS The paper presents a new parametric identification method of nonlinear dynamic systems being linear in the parame te rs • The first part of the paper is a short summary of the nonlinear systems. l'Tonlinear models being linear in the parameters are presented. In addition to the classioal input/ output crosscorrelation functions new correlation funotions are introduoed. Applying nontrivial multiplioators the two-step identification method elaborated for linear systems could be extended to nonlinear dynamic systems, too. The nonlinear correlation functions make not only the parameter estimation but a simple nonlinearity test possible. The identification method is valid both in open and closed loop systems. The new method has been applied in digi tal and hybrid simulations and for the modelling of a heat exchanger. ACKNOWLEDGEllEN"T The author is grateful to Professor Dr.-Ing.R.Isermann, Head of the Institute of Automatio Control at the Teohnioal University of narmstadt for initiating and Dr.L.Keviczky, principal contributor at the Department of Automation, Technical University of Budapest for advising

Astrom, K.J., and P. EYkhoff (1970). System identifioation - A survey. 2nd IFAC Symposium on Identifioation and Parameter Estimation, Prague. Baur, U., and R. Isermann (1976). On~line identification of a steam heated heat exchanger with a process oomputer - Case study. 4th IFAC SympOSium on Identification and System Parameter Estimation, Tbilisi, pp. 252-293. Haber, R. (1977). A method for identifioation of nonlinear dynamic systems by means of process computer - modelbuilding, theory, programpackage, test fin German/. Report, Department of Control and Systemdynamios, university of Stuttgart. Haber, R., and W. Bamberger (1979). OLID-MISO-NOLI - Interactive software paokage for identification and extremum control of nonlinear dynamic prooesses by means of a process oomputer. 2nd IFAC Symposium on Software for Computer control, prague. Haber, R., and L. Keviczky (1974). l'Tonlinear structures for system iaentifioation. Periodioa polytechnica - Eleotrical Engineering vol. 18. pp. 394-404. Haber, R., and L. Keviczky (1976). Identification of nonlinear dynamic systems - Survey paper. 4th IFAC SympOSium On Identification and System Parameter Estimation, Tbilisi, pp. 62-112. Haber, R., and L. Keviczky (1979). Parametric desoription of dynamic systems having signaldependent parameters. 1979 JACC. Denver. Isermann, R. (1974). Processidentifioation fin German/. Springer verlag. pp. 188. Krempl, R. (1974). Application of three-value pseudo random signal for the identification of nonlinear control plants fin German/ Report, Department of Automation, RQhr University, Bochnm. pp. 164. Kurz, H., and R. Isermann (1975). Methods for on-line process identification in closed loop. 6th IFAC World Congress, Boston. Peterka, V., and A. Holouskova (1970) Tally estimate of Astrom model for stochastic systems. 2nd IFAC symposium on Identification and System. parameter Estimation. Prague.