A Frequency Domain Identification Scheme

Copyright ~ IFAC System Identification. Copenhagen. Denmark. 1994

A FREQUENCY DOMAIN IDENTIFICATION SCHEME Per-Olor Killen Lund Inditute

0/ Technology, Boz 118, S-Ul 00 Lund, SVleden

Ab.tract This work considers on-line estimation of the frequency response of a system. The paper concentrates on analysing the convergence properties of such a scheme. Instead of a _. full order parametric model the estimation is split up into a set of frequency response point estimators. These are shown to converge in the case of linear in the parameters models and narrow band pass filters irrespective of the system order.

Keywords System identification, Frequency response, Frequency domain, Frequency domain estimation, Convergence analysis

invariant. The analysis that follow discrete time.

1. INTRODUCTION

Most controller design techniques require a parametric state space or transfer function model of the plant. Consequently system identification has been primarily concerned with finding full order parametric models, see for instance Ljung (1987). The fit of the process model as well as the resulting controller order and design calculations are in most cases directly related to the model order used. Since modeling is always approximate there will always be trade-offs. By representing the process by its Nyquist curve the process order becomes a secondary issue or differently speaking, it is no longer necessary to fit a specific parametric model over a wide frequency band. A controller design technique that uses this type of process information is given in Kiillen (1992), Kiillen and Wittenmark (1993) and Lilja (1989). In an off-line setting the frequency response can be obtained by different types of spectral methods, see for instance Wellstead (1981). Since the work is originally concerned with adaptive control a different strategy is considered. This work discusses the convergence properties of such a scheme.

IS

made in

Consider discrete time systems described by y

= gT * U + v

(1)

where * denotes the discrete time convolution, u and y are respectively the system input and output and the noise is collected in v. The frequency response to input signals is assumed to satisfy

Here G8(z,8 0 ) is a parametric nominal model and G A (z) is a residual that will depend on the structure of G8(Z, 8) and on 80 , This decomposition is natural in the case of complex processes where simplified models are unavoidable. The under modeling is then naturally captured by GA(z) which is the part of the system response that is not modeled but still depends directly on the input signal. Let the estimation be based on the following filtered least squares criterion.

2. PROBLEM SETUP

1

IN(8)

The problem addressed is that of estimating points on the frequency response of a system. Only single-input single-output systems are considered and the estimation is to be used in an on-line environment. This makes spectral methods less attractive. In order for the frequency response to exist the system is assumed linear and time

N

1

N

= N L 1£,1 2 = N L If * (y k=l

g. * u)1

2

k=l

(3) where f is the pulse response of the regression filter i.e. in the frequency domain (with X(z) denoting the z-transform of x)

EF(Z",) = F(z",)(Y(z",) - G8(z""8)U(z,,,)) (4) 785

with the notation z.., = ei ..," being used to save space. With no further assumptions this is a common and quite general setup. Since the problem is to estimate points on the frequency response of the proceBB, F is chosen as a narrow band pass filter with center frequency at the interesting frequency. Further, for the purpose of convergence analysis, only models that are linear in the parameters will be considered in this paper, i.e. the structure of the estimated model is assumed be

G,(z,9) = L(z)9

where M R = Re(M), NR(Y) = Re(N(Y)) and

M =

J

N(Y)

2

2

IF(z..,)1 IU(z..,)1 L(z..,)HL(z..,)dw (10)

=

J

2

IF(z..,) 1 U"(z..,)Y(z..,)L(z..,)H dw (11)

For a given 80

Y(z..,)

= {L(z..,)8 + G 6 (z..,)} U(z..,) + V(z..,) 0

(12)

(5)

Using this expreBBion the equation to be solved (9) can be written as

Models of this form are for instance FIR-models, first order Taylor expansion models, Goodwin et al. (1991), Laguerre models, Wahlberg (1991b), and Kautz models, Wahlberg (1991a).

where Normally an estimate of the frequency response at a set of frequencies is sought. This is obtained by using a set of frequency response point estimators. For each point a model with few parameters is used. This is pOBBible since the analysis show convergence for a model with at least two parameters irrespective of the true system order. An on-line scheme is obtained by using recursive least squares on band pass filtered data.

(14) Partition 8 into one part, 8., due to proceBB input u and one part, 8.. , due to noise input v defined by

MR8u = NR(G 6 U) MR8.. = NR(V)

Using Parsevals theorem the loss function (3) will be minimized by minimizing, Goodwin et al. (1991)

GT -

G = (L8 + G 6) - L8 0

=G 6 -L8=G.+G..

J(9)

(17)

where

.,,/"

J

(16)

Then 8 = 8u + 8... Now the frequency response estimation error can be partitioned in one part, Gu , due to process input and one part, G.. , due to noise input according to

3. CONVERGENCE ANALYSIS

=

(15)

2

2

IF(z..,)1 IY(z..,) - U(z..,)L(z..,)91 dw(6)

G. = G 6 - L8. G.. = -L8..

-.,,/" where it is assumed that the corresponding transforms exist. The estimated frequency response is then given by

where

8=

,

argminJ(8)

(18) (19)

3.1 Narrow Band Pan Filtering In order to investigate the consequence of using narrow band pass filtering, the error in the estimated transfer function due to noise and under modeling is evaluated through the solutions of (15) and (16).

(8)

Let F be a narrow band pass filter with the effective pass band width Wpb. Introduce the following assumption for the filter

Because of the narrow band filtering this model is only valid in a narrow frequency band. The problem of interest is the following. Will the frequency response estimate converge to a correct value as the band pass filter gets narrower? If so, will it converge irrespective of differences between the order of the model and that of the proceBB?

Assumption A The band pass filter is narrow enough such that the approximation

The solution (8) is obtained by differentiating (6) with respect to 8. Setting the gradient to zero gives

Iw - wol < Wpb/2 elsewhere

(9)

(20) 786

is valid up to some specified degree of accuracy. This means that the filter must be narrow enough, i.e. wp6 < K for some K.

Proof: Follows directly by applying the approximations (21)-(22) to (15)-(16). 0

Then

Remark. Notice that A is singular when L has more than two elements. Further, A has rank two as long as at least two elements of L(e....·") does not have identical phase. Since this will not happen unless L is chosen very poorly and since L will nearly always have more than two elements, (23)-(24) are in all normal cases singular problems with A of rank two.

MR ~ Wpl>

IF(z,.,JI 2 ·IU(z...JI 2 Re {L(z,.,JH L(z,.,J} (21)

NR(X) ~ Wpl> IF(z... JI . Re {U·(z,.,.)X(z....)L(z....)H} 2

(22)

where Z"'o = ei "' oh • The validity of this approximation can be shown as follows. Consider the integral

1=

LEMMA 2 With the approximations (21)-(22) the estimation error at Wo is uniquely given by

J

f(w)g(w)dw

where frepresents the band pass filtering according to

f(w)

G..

J J =

Since b = oRIR + a/I] E 'R.(A) = 'R.( (IR 1/) a solution exists despite the fact that the problem is singular. Further, since A is symmetric

1(w)g(w o + £w)dW

j(w)(g(w o) + £w ~ (wo)

=

= -LA+b(a.)

Proof:

£

with the even function j satisfying f jdw = 1. Then by the variable substitution w = (w - wo )/£ and a Taylor expansion of 9 around Wo the integral becomes I

LA+b(a u ) and G.

(30)

= ~1(w -WO) £

= Gt:. -

Take any vector that belongs to N(A), say 8. Then because of (31)

+ 0(£2»dW

= g(w o) + 0 + 0(£2) since w j(w) is an odd function of w. In this £ represents the filter width and as £ decreases /(w) gets narrower and the integral approaches the used approximation as 0(£2).

Since the non-unique parts of the solutions to (23)-(24) lie in N(A) these do not influence the frequency response estimate at wo' The unique part of the solution to (23)-(24) is given by

The system of equations can now be reformulated according to the following lemma. and

LEMMA

1

Using

the

L(ei ... oh )

(33)

=

approximations

Ik + it{.

(21 )-(22)

and (15)-(16) can be written

(34) where A+ denotes the Moore-Penrose inverse of A. By using the definitions (18)-(19) for G.. and G. the result follows. 0

as

A8.. = b(ou) A8. = b(o.)

(23) (24)

In order to get an explicit solution for the estimation error Lemma 3 below will be helpful

where

A = Re{L(ei... oh)H L(ei ... oh )}

= lRl~

LEMMA

+ I]IT

(25) b(o) = Re{oL(ei... oh)H} = oRIR + 0/1] (26) U(ei... oh)" X(e i ... oh ) . o(X) = . h . h = OR + WI/ (27) U(e''''o )·U(e'...• ) 0 ..

= a(Gt:.U) = Gt:.(ei ...oh )

(28)

0.

U( e i "' oh )" V( ei .... h ) = a(V) = lU(e i .... h ) 12

(29)

3

The solution to

(35) where A and b are given by (25)-(26) and IR and are linearly independent satisfies

lr

L8 787

=0

(36)

Proof:

Let

P= Zl T

(IR lr)

(37)

i)

(38)

= (1

(OR

Z'lT

frequency response error of the estimated model at Wo depends only on the noise to signal ratio. For cases where the filter is not narrow enough for the result to hold it still can be concluded that it is an advantage to use band pass filtering to get small frequency response estimation error at Wo'

(39)

0])

Remark 2. Narrow band paas filters have long response time. This implies that input data to the filter will influence its output for a considerable time period. In an adaptive context this is undesirable since it will then be harder to track process parameter changes. There is thus a trade-off between estimation error and parameter tracking speed.

Then

A= ppT

(40)

b= PZ'l L(e iwoh ) = lk

(41) (42)

+ ilT

= zi pT

Since b E 'R(A), a solution exists. The solution is given by 9 A+b + 9'l (43)

=

for any 9'l E N(A). Since A and according to Lemma 2

Remark 3. Normally the noise contribution to the estimation error is averaged out as the observation length N grows. In the case of band pass filtering considered here this corresponds to the case when N Wpb -+ 00 as N -+ 00 when the filter width Wpb decreases. From the expressions of Theorem 1 it seems like the noise contribution to the estimation error is not averaged out but instead depends on a noise to signal ratio. This instead corresponds to the case when N Wpb -+ k < 00 as N -+ 00 when Wpb decreases since only the center frequency properties are taken into account in the analysis.

AT (31) is valid

L( eiwoh )9 = L(eiwoh)A + b + (lk92 = L(eiwoh)A+b

+ il;( 2) (44)

and finally

L(eiwoh)A+b = zi pT(ppT)+ PZ'l = Zi:l:2 =OR+Kr]=O

(45)

since when P has full column rank it can be shown

thatpT(ppT)+P=I

0

3.2 ContinuoU6 Time Open Loop Co.6e

In the continuous time case consider the loss function

Now finally the estimation error for narrow band pass filtering is obtained.

r(9) =

1 With assumption A satisfied and with L( eiwoh ) having at least two elements with different phase at Wo i.e. 3 k, I: arg(L(eiwoh)h: :f:. arg(L(e iwoh ))/, it holds THEOREM

1~ If * (y -

g.

* u)1 2 dt

(50)

where * denotes continuous time convolution. Let the frequency response of the model be given by (51)

(46)

Analog to the discrete time case (50) can be written as

(47)

Proof: Applying Lemma 3 to Lemma 1 and using the expressions (18)-(19) for G. and G. gives

L8. = G /1 - 0 .. (48) = G/1 - G/1 = 0 G. = -L8. = - 0 . = _U(eiwoh)"V(eiwoh)/ IU(eiWoh)12 (49)

G.

=G

-00

/1 -

by using the appropriate Parseval relation. The result for the continuous time case is now obtained from the discrete time development by changing z-transform variables X(e iwh ) evaluated along the unit circle into X.: (iw) i.e. Laplace transform variables evaluated along the imaginary axis. Then the following result immediately follows for the continuous time open loop case:

o Remark 1. The theorem states that when a narrow enough band pass filter is is used, the 788

THEOREM 2 With assumption A satisfied and with LC(iw o) having at least two elements with different phase at Wo i.e. 3A:,1: arg(LC(iw o ))1: 1:- arg(LC(iwo))', it holds

(53)

U,(iwotv,(iw o) U,(iwo)"U,(iw o) Proof:

Analog to Theorem 1

(54)

o ........ ,.'L.~_....;......;...-,;.;.

10"

"-

,

..

_ _....;....:...'.:... . .:....'~'.:.;'._....;.._~~.J

10"

10"

3.3 On-line E.timation

The proposed scheme is originally intended for online estimation to be used in an adaptive control context~ The convergence analysis above has been put in an off-line setting. The results are, however, applicable also to the case of using a recursive least squares scheme. It is therefore apparent how to implement an on-line scheme.

Figure 1. The relative eatim.tion error LiGu1 u • function of relative filter width Li"'rd for different number of model parameters: n=2 (aolid), n=3 (duhed), n=4 (dotted) and n=& (duh.dotted).

of parameters in the model but for a given constant data length. The process input u is chosen as a PRBS-signal. This case is depicted in Figure 1. It is seen that the estimation error decreases with filter width. Finite data length effects, however, shows up as loss of convergence for narrow filter widths. This is easy to understand. A narrow enough filter will have a settling time that is larger than the finite length data so that not enough process information has been able to pass through the filter. Not surprisingly it is also seen that a better fit can be obtained by increasing the number of parameters in the model.

3.4 Clo.ed Loop Ca.e

The closed loop case with more general models has also been investigated for the proposed scheme. The noise free case gives similar results to those presented here. In case of noise the analysis is more involved.

4. EXAMPLE In this section an example is given in order to demonstrate the results. Because of finite length data the estimate will not be able to converge exactly to zero. This will be seen in the plots. The estimation error is, however, then normally negligible.

It is easily understood that narrow band signals in the estimators will give only a limited amount of excitation. This results in slow parameter convergence for models with many parameters. However, since only the frequency response at Wo = 1 rad/s is of interest, parameter convergence is of small concern in this context.

Consider the process

G ($) _ -,--1---:-::- ($ + 1)5

(55) In order to exploit the effect of finite data length further, in Figure 2, is plotted for n = 4 but with different data lengths. It is clear from this plot that as the data length grows, the estimation error will go to zeros with decreasing filter width.

A FIR-model structure is used in estimating the frequency response at Wo = 1 rad/s for the sampling interval h = 0.5 sec. Let n be the number of model parameters and N the data vector length. For different cases the absolute relative frequency response error llGrel at Wo = 1 rad/s is shown as a function of the relative filter width llWrcl where

Finally in Figure 3 gaussian white noise is added to the process output. In this case n = 4 and N = 10000. The noise contribution G.. shows up as a loss of convergence as the filter gets narrower.

5. CONCLUSIONS In this work the properties of a special type of frequency domain estimator has been investigated. The analysis has been put in an off-line setting but

and [WI, will is the pass band of the filter. First consider the noise free case with different number

789

6. REFERENCES -··........ ... ·.·;.·.··.···;··i··.·;·.·~;··· : ..... ... : .. :.':'.:.:':'; ~

.

;

10

;

..

;:.

.

ASTROM, K. J. and B. WITTENMARK (1989): Adaptive Control. Addison-Wesley, Reading, Massachusetts.

.

~11........•..... ~~! 1~!~11111~1~1~11i~~1~ li ~i:;~ '1~' ~11 i1" ~~~ 1~ ~;~ ...•.. ..•. ;.:.:.: .. ;

;.

10

;

~Hmm~mn~~n~n~1~1i~~~jYTTn~T~n:ir~;j;~::.~:.. ........ :..... ... :.. :.. ~

,·,.~'\04::--

-

~.:.:~:

·

.....

;.~.- - - -

,

!H~j+.WHm!

... :....... : .. :... :':

ASTROM, K. J. and B. WITTENMARK (1990): Computer Controlled Systems-Tbeory and Design. Prentice-Hall, Englewood Cliffs, New Jersey, second edition.

~

GOODWIN, G. C., M. GEVERS, and D. Q. MAYNE (1991): "Bias and variance distribution in transfer function estimation." In 9th IFAC/IFORS Symposium on Identification and System Parameter Estimation, Budapest, Hungary. GOODWIN, G. C. and K. S. SIN (1984): Ada~ tive Filtering, Prediction and Control. PrenticeHall, Englewood Cliffs, New Jersey.

. . , . - - - - - ' 0..

}'llure 2. ACrel Aa a function of Awrcl for dift'en::nt data1ensth.a: N=l000 (aolid), N=4000 (daahed), N=8000 (dotted) and N=l0000 (daah-dotted).

KXLLEN, P .-0. (1992): Frequency Domain Adaptive Control. Lic Tech thesis ISRN LUTFD2/TFRT--3211--SE, Department of Automatic Control, Lund Institute of Technology, Lund, Sweden. KXLLEN, P.-O. and B. WITTENMARK (1993): "A frequency domain adaptive controller." In Preprints 12th IFAC World Congress, Sydney, Australia. LILJA, M. (1989): Controller Design by Frequency Domain Approximation. PhD thesis TFRT1031, Department of Automatic Control, Lund Institute of Technology, Lund, Sweden. WABLBERG, B. (1991a): "Identification of resonant systems using kautz filters." In Proc. of the 30th CDC, Brighton, U.K.

}'ilure S. AC rcl aa a function of Awrcl for n=4 and with output noiae. Noiae to aignal ratio (horisontalline)

WABLBERG, B. (1991b): "System identification using laguerre models." ieeeac, AC-36:5, pp. 551-562.

the results are applicable to on-line estimation. In the noise free case the estimation error is shown to converge to zero when the filter width goes to zero. In a practical situation there will always be a trade-off between filter width and estimation error unless the process is in the model class.

WABLBERG, B. and L. LJUNG (1986): "Design variables for bias distribution in transfer function estimation." ieeeac, AC-31:2. WELLSTEAD, P. E. (1981): "Non-parametric methods of system identification." Automatica, 17:1, pp. 55-69.

Acknowledgement The work has been partly supported by the Swedish Research Council for Engineering Sciences (TFR) under contract TFR91-721.

WITTENMARK, B. and P.-O. KXLLEN (1991): "Identification and design for robust adaptive control." In Preprints European Control Conference ECC '91, pp. 1390-1395, Grenoble, France.

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