Frequency Domain System Identification

6
Frequency Domain System Identification
Ford Motor Company, Dearborn, Michigan, USA

Gang Jin

6.1 6.2 6.3 6.4 6.5

Introduction ....................................................................................... 1069 Frequency Domain Curve-Fitting ........................................................... 1069
6.2.1 M a t r i x Fraction Parameterization • 6.2.2 Polynomial Matrix Parameterization • 6.2.3 Least Squares O p t i m i z a t i o n A l g o r i t h m s

State-Space System Realization ............................................................... 1074
6.3.1 M a r k o v Parameters Generation • 6.3.2 The ERA M e t h o d

Application Studies ..............................................................................
6.4.1 Identification o f a 16-Story Structure • 6.4.2 Identification o f the Seismic-Active Mass Driver B e n c h m a r k Structure

1075 1078 1078

Conclusion ......................................................................................... References ..........................................................................................

6.1 Introduction
A general procedure for the frequency domain identification of multiple inputs/multiple outputs (MIMO) linear time invariant systems is illustrated in Figure 6.1. Typically, one starts with the experimental frequency response function (FRF) of the test system. These FRF data may either be computed from the saved input/output measurement data or measured directly online by a spectrum analyzer. Based on these data, the matrix fraction (MF) or the polynomial matrix (PM) curvefitting technique is applied to find a transfer function matrix (TFM) that closely fits into the FRF data. Detailed algorithms for the curve-fitting are introduced in Section 6.2. Frequently, for the purposes of simulation and control, one needs a statespace realization of the system. This may be achieved by various linear system realization algorithms. In particular, the eigensystem realization algorithm (ERA) is presented in Section 6.3 for this purpose, thanks to its many successes in previous application studies. The Markov parameters, based on the parameters from which the state-space model will be derived, can be easily generated from the identified transfer function matrix. Finally, as a measure of performance, the model FRF is computed and is compared to the experimental FRE This is illustrated in Section 6.4 by means of two experimental application examples.
Copyright© 2005 by AcademicPress. MI rights of reproduction in any form reserved.

6.2 Frequency Domain Curve-Fitting
Frequency domain curve-fitting is a technique to fit a TFM closely into the observed FRF data. Like other system identification techniques, this is a two-step procedure: model structure selection and model parameter optimization. In this context, the first step is to parameterize the TFM in some special forms. Two such forms are introduced in the following: the matrix fraction (MF) parameterization and the polynomial matrix (PM) parameterization. This is always a critical step in the identification because it will generally lead to quite different parameter optimization algorithms and resulting model properties. In particular, this section shows that for the MF form, the parameters can be optimized by means of linear least squares (LLS) solutions. As for the PM parameterization, one has to resort to some nonlinear techniques; specifically, this section introduces the celebrated Gauss-Newton (GN) method. On the other hand, the PM parameterization offers more flexibility in the sense that it allows the designer to specify certain properties of the identified model (e.g., fixed zeros in any input/output channels). This feature may be quite desirable as shown by the application studies in Section 6.4. Before starting the discussion, it is important to make clear the notations that will be used throughout this section. Assume the test system has r input excitation channels and

1069

1070
/ / \ \

Gang fin

/ Experimental \ I-I input-output ~- -- ..~.1

,,measurement/
\ /

I Calculation L__

FRF

MF/PM Curve-fitting

\

\

\

X
Model evaluation Markov parameters generation

@

~

~

ModelFRF
generation

ERA System realization

FIGURE 6.1 GeneralProcedure of Frequency Domain System Identification. Bolded components imply critical steps; dashed components imply steps may be excluded. m output measurement channels. U s e {~((Oi)}i_l,...1 to denote the observed FRF data based on which the TFM G(z 1) will be estimated. To evaluate G(z 1) at discrete frequencies coi, use the map z(coi)= ej%~#w', with ws being the sampling frequency. The curve-fitting error is measured by Frobenius norm ]]-]]F for matrices and by Z-norm ]]. ][2 for vectors. Use I, to denote an identity matrix of dimensions n × n. Substituting the Q(z -1) and R(z 1) polynomials in equations 6.2 and 6.3 and vectorizing the summation, equation 6.4 is changed into the form:

G*(z-1) =
where:
-Z I(OJ1)GT(o)I) ..

arg min

G=Q-IR

llano ÷, IIF,

(6.5)

6.2.1 Matrix Fraction Parameterization
The matrix fraction (MF) parameterization of a TFM takes the following form:
G(z-1) = Q l(z-1)R(z 1),

~=

z P(COl)GT((DI) --Ir --z-l(oJ1)Ir

...

--Z P(COI)/r]

:
-Z-I(o~i)~T("~t)

i

i

i

i

z-P(toI){~T(toI)--It --Z-l(tOl)Ir ... --Z P((ol)IrJ

/

(6.6) TT = [G(tOl)..-G(o~l)]. ®T = [Q,... qpRo RI'"Rp]. (6.7) (6.8)

(6.1)

where:
Q(z -1) =Im + QIz 1 + Q z z - 2 + . . . + Qqz-q,

(6.2) (6.3)

R ( z 1) = Ro + R1 z-1 + R2z - 2 + - ' ' + Rpz -p.

Thus, the MF curve-fitting has been reduced to a standard LLS problem, which can be solved by various efficient algorithms (e.g., the QR factorization approach quoted in algorithm of equations 6.29 through 6.31).

The constant matrices Q1 . . . . . Qq and R0, R1. . . . . Rp are referred to as observer Markov parameters in ]uang (1994). From the same reference, the reader may find more detailed material for the MF curve-fitting discussed here and the ERA realization algorithm discussed in a later section. To simplify the notation, without loss of generality, assume p = q. To fit the TFM G(z 1) as in equation 6.1 into the observed FRF data {G(coi)}i=1,..., l, one may solve the following parameter optimization problem:

6.2.2 Polynomial Matrix Parameterization
The polynomial matrix (PM) parameterization of a TFM has the form:
C(z_l) _ B(z-')

~x(z-') '

(6.9)

where:
B(Z - 1 ) =

G*(z

1
1) =

arg min Z

G=Q- R i=1

IlQ(z-l(~°'))G(~°i) - R(z l(co,))ll~.

(6.4)

Bo + B1 z-1 ÷ BzZ -2 ÷ "'" -}- BpZ -p.
a2 z - 2 + . . .

(6.10) (6.11)

ot(z -1) = 1 + alz -1 ÷

+ aqz -q.

6 Frequency Domain System Identification
These equations are the numerator polynomial matrix and the minimal polynomial of the TFM, respectively. Assume that the orders of the numerator and denominator are equal (i.e., p = q). The goal of parameter optimization is to find G*(z -~) with a prespecified order p such that the estimation error is minimized:

1071 Finally, the right-hand side of equation 6.18 is vectorized for standard optimizations:
F(0) = IIW(a)(y Ho)ll~,

(6.19)

G*(z 1) = argmin ~'w2(o3i)llG(toi) _ G(z 1( o ))1l 2 (6.12)

l

G z.....a i=1

Here, an additional term w( • ) is included to allow desirable frequency weighting on the estimation error. In the following, G(z -1) in equation 6.12 will be parametized in a way that allows the inclusion of the fixed zeros. This is done first for the single input/single output (SISO) case; then, it is generalized to the MIMO case. Let the numerator polynomial of the SISO system be the following:
B ( z -1) = B ( z 1). ~ ( z - 1 )

where 0 = [b~ , b~, .... -W aT] T, y is a vector containing bmr, the measured FRF data, and W(a) is a weighting function with variate a. Readers should have no difficulties to derive the detailed expressions of W(a) and H. For the special case when there is no fixed zeros (i.e., t~Yb = 1 in equation 6.14), the results are given in Bayard (1992). It is important to point out that W(a) and H have the following structure:
- W(a)

0

• •"

0
(6.20)

0
W(a) =

W(a)

(6.13) (6.14)

".
0
- XItl 1

0
W(a)
0
:
o

= t~Tb • t~Tb,

o
0
~t12

where b = [b0, bl . . . . . [Jps]w is the numerator parameter vector corresponding to the fixed zeros; b--[b0, bl . . . . . ~p_p,]T is the to-be-estimated numerator parameter vector; and ~ = [1, z 1. . . . . z-Ps] T and t~ = [1, z -1 . . . . . z (P p,)]T are the corresponding z-vectors. Similarly, the denominator polynomial is as follows: cx(z-1) = 1 + ~bTa, (6.15)

(Dll (I)12
i

0
H =

(6.21)

0

.

.

•

0

~mr

qbmr

where a = [al . . . . . at,] w and ~b = [z -1 . . . . . z-P] T. The estimation error to be minimized is then:

These structures enable the design of an efficient optimization algorithm. This will be discussed in the next section.

6.2.3 Least Squares Optimization Algorithms
1 + ~bT(coi)a

/

(6.16)

=/_~l w+ t ° i ) l ( qbr(eoi)a (~(~°i)-[~r(°~i)b'~T(°~i)

__~((oi) . (bT(03i)] i i l )

2,

(6.17)

where for simplicity t~ and ~ are written as functions of coi. For the MIMO case, equation 6.17 is generalized to (recall that m and r denote the numbers of output and input channels respectively):

This section serves two purposes. First, it gives a brief (but general) account on a few of the most important parameter optimization algorithms, namely the linearleast squares (LLS) method, the Newton's method, and the Gauss-Newton's (GN) method. Second, the discussion applies some of the methods to the parameter optimization problems arising from the curvefitting process. In particular, this section presents a fast algorithm based on the GN method to the minimization of equation 6.19. Excellent textbooks in this field are abundant, and this discussion only refers the reader to a few of them: Gill et al., (1981), Dennis and Schnabel (1996), and Stewart (1973).

General Algorithm Development
Let F(O) be the scalar-valued multivariate objective function to be minimized• If the first and second derivatives of F are available, a local quadratic model of the objective function may be obtained by taking the first three terms of the Taylorseries expansion about a point Ok in the parameter vector space:

F=
-

~-~

j=l k=l i=1

~

l 1 W(~i)
• %k(o~i)

+ +~(~oi)a Gik(~o~)
-

(

[*jk(~Oi)bjk

Gjk(O~i) . ~bT(~ol)]

2
(6.18)

1072

Gang ]in F(Ok + p) ~ Fk + g~p + ~pTGkp,
(6.22) where Q has orthonormal columns and where R is upper triangular with positive diagonal elements. The unique solution of the LLS problem: min ]lAx
xERn
-

where p denotes a step in the parameter space and where Fk, gk, and Gk denote the value, gradient, and Hessian of the objective function at Ok. Equation 6.22 indicates that to find a local minimum of the objective function, an iterative searching procedure is required. The celebrated Newton's method is defined by choosing the step p = p k so that Ok+Pk is a stationary point of the quadratic model in 6.22. This amounts to solving the following linear equation:

£1122,

(6.30)

is given by: x* = R 1QTy. (6.31)

Application to the Curve-Fitting Problems
This discussion now returns to the PM curve-fitting problem in equation 6.19. By letting f = W(a)(y-HO), the GaussNewton method in equation 6.28 may be applied. It turns

GkPk=--gk .

(6.23)

In the system identification case, the objective function F(O) is often in the sums of squares form, as in equation 6.19:

1 n F(O)-- ~ Z ) ~ ( 0 ) 2 = i=1

llf(O)ll ,

(6.24)

where 1~ is the ith component of the vector fi To implement the Newton's method, the gradient and Hessian of F are calculated as:

g(O)=l(O)Tf(o) G(O)=J(O)rJ(O)+Q(O),

(6.25) (6.26)

where l(0) is the Jacobian matrix of f and where Q ( 0 ) = }-~=ly~(0)Gi(0), with Gi(O) being the Hessian of)q(0). The Newton's equation 6.23 thus becomes as follows:

(l(Ok))Tj(Ok)+Q(Ok))Pk=--J(Ok)Tf(Ok).

(6.27)

When [[f(0k)[I is small, ]]Q(0k)[] is usually small and is often omitted from equation 6.27. If this is the case, then solving Pk from 6.27 is equivalent to solving the following linear least squares (LLS) problem:

Pk = arg rn~n

IIJ(0k/P + f(Ok/l122•

(6.28)

Equation 6.28 gives the Gauss-Newton method. The LLS problem is often solved by the QR factorization method given in the following algorithm.

Algorithm 1 (QR Factorization and LLS Solution) Let A E R mxn have full column rank. Then A can be uniquely
factorized into the form: A = QR, (6.29) FIGURE 6.2 Picture of the 16-Story Structure

6

Frequency Domain System Identification
min min

1073

TABLE 6.1 Key Identification Parameters Curve-fitting Test structure 16-Story Benchmark Type MF PM p 16 8 q 16 8 Systemrealization a 32 12 ~ 32 30 n 10 8

x2ERq

IlA.2x2 yl12~.
-

(6.32)

x] c RP

IlAlXl-

(y

-

A=.~)II==.

(6.33)

out that the ]acobian matrix I(0 k) of f has an identical structure as H in equation 6.21. Thus, instead of solving equation 6.28 directly, the following decomposition of LLS problems may be applied.

Algorithm 2 (Decomposition of LLS Problem) Let A E R m×n have full column rank. Decompose A into A = [At, A2], with Al C R mxp and A2 C R mxq. Let the unique QR factorization of A1 be A1 = Q1R~. Then the unique solution x* of the LLS problem of equation 6.30 has the form x* = [x~T, x~ T] r, with xq and x~ being the unique solutions of the following LLS problems:

In these equations, ]~2 = (I - Q1QT)A2. In practice, ](Ok) is divided into ](Ok) = [h(0k), h(0k)], with ll(Ok) corresponding to the block-diagonal terms and 12(0k) the last block matrix column in ](Ok). Solving the LLS problems with 12(0 k) and 11(0 k) thus corresponds to updating the denominator a and the numerators bjk estimations. Moreover, due to its block-diagonal structure, the LLS problem with 11(Ok) should be further decomposed (i.e., the LLS solutions of bjk are independently solved for each j and k). Thus, the computational cost of the optimization algorithm is significantly reduced. To complete the discussion of the algorithm, note that the initial values for the Gauss-Newton iteration may be generated by the classical Sanathanan-Koerner (SK) iteration composed of a sequence of reweighted LLS problems:

f16 -40
-60

f12 • .....
:

f8 -1 -40 • -60
•

:.
:
•

!
:. . . .

-,OOl y: V f ,1 1 -,oo
-120 L . . . . . . 20 " .... " ....... 40 60 ~ -120

8o
-60

40[~

:
:

:
i

I[ -60

z

f4 •

I

- 8 o ~
-100 _120i 20 40 60 : 20

-80
-100

,
40

\f
60

j-120
20 40 60

-60
-80 -100 -120 20 40 60

-60
-80 -100 -120 20 40 60

ook,.... -12orry
-80 .... :
- 1 4 0 F

-60

:

i

:. . . . :

60 ~
-80

~ ....

":
i

i : ....

;

F .......

Y
60

-120I

"

¢" "',

--"

i l~

1

20

40

20

40

60

x ,0
-6o
-'~vt -60[i x

•

.

i

. . . . .

60

:
:

....

:
....
40 .

:

-'°°I, .... V
20 40 t '~

-Y1
60 1

]

-140t

.

20

.

.

60

1 _12ot, ~ , , J
20 40 -60t -80 -100

,oo
-80

-6o

i

i .... i....
i . i. .

-6o
-80

•.•:..

~.
i .

:
i. . . .

, j-,2Ol,
60

i
20

it
40

_-v
60

-60 ' -80 -100
• i

....

•
i

-60" ! -80 . . . . .

i..... i ¸
•

_1oo
-120 20 40 60 F r e q u e n c y (Hz)

_12o|-,.,
-140 ~

~
, ..... ~ • 20 40

~Y1
, ~ 60

-12o i
20 40 60 F r e q u e n c y (Hz)

-12°I
-140t

~. "
--~ " ,

~-:','1
; .1

20

40

60

F r e q u e n c y (Hz)

F r e q u e n c y (Hz)

FIGURE 6.3 Comparison of Experimental and Model FRF for the 16-Story Structure: Magnitude Plot. ~ denotes the input force on the jth floor; xj denotes the output displacement of the jth floor; dotted lines are for measurement data; solid lines are for model output.

1074 0k+a = arg

Gang fin rnoin][W(ak)(y-- H0)[[~,
(6.34) tem realization algorithm (ERA), is selected to construct a model in the state space form. First presented are the formulas to generate the Markov parameters from the TFM, which are the starting point for the ERA method.

with initial condition 00 = 0.

6.3 State-Space System Realization
System realization is a technique to determine an internal
state-space description for a system given with an external description, typically its TFM or impulse response. The name reflects the fact that if a state-space description is available, an electronic circuit can be built in a straightforward manner to realize the system response. There is a great amount of literature on this subject both from a system theoretical point of view (Antsaklis and Michel, 1997) and from a practical system identification point of view (]uang, 1994). In the following, a well-developed method in the second category, the eigensys-

6.3.1 Markov Parameters Generation
To calculate the Markov parameters Y0, Y1, system TFM, first note that:

Y2. . . .

from the

G(Z-1) = ~
i=o

yi Z i.

(6.35)

For the case when G(z-1) is parameterized in the MF form (i.e., G(z 1) = Q - l ( z 1)R(z-1)) ' the system Markov parameters can be determined from:

f16 -50 -100

f12

0 -200 '.L..# -400 .....

f8

0 -200 -400 -600

f4

.
.

.
.

.
.

.
.

.

100 t ........................... 200 -300

-150IU 20 40 60

20

40

60
lOO~

20
-.! ........

40
i ........

60 0
i .... i. ......

20

40

60

-ioo
¢,i

ool= -qN
- U L 40 60 20

-50

......

-1 O0 ........
. . . .

-300[ ..... ~ U

-+OlLjL
20
-200

n
::+..
; .....

-lOOL~
60

o~ i-

i....

-

-200 -400 20 40 60

-2°°vqF 73 iN
-300I .... ~ - - - - U L _ 20 40 60

40

oI
O'

+: :

i.

-4o0I ..................~ - 6 o o
20 40 60
0 ~ ......

+. . . .

i+
60
~ .....

-lOO ......
-150

0i ... : ~ ~ i ....... 0 ........ -50 . . . . . . . i ...i..... -200
i .... i .....

~

-4oo
-600 20 40 60

20

40
~ ......

20

40

60

-200 =g -400 -600 20 40 60 Frequency(Hz)

-2001~,
-400 i -600' -800 : :

:,

-1

O0 -1 O0 -200 20 40 60 Frequency (Hz) 20 40 60 Frequency (Hz)

-200 -300

20 40 60 Frequency (Hz)

FIGURE 6.4

Comparison of Experimental and Model FRF for the 16-Story Structure: Phase Plot. Same notations are used as in Figure 6.3.

6

Frequency Domain System Identification
T A B L E 6.2 Iteration type PM Iteration Record Iteration index 1 2 3 4 1 2 3 4 5 6 FRF residue 651.47 159.46 156.75 156.74 137.03 133.47 128.03 123.81 123.34 123.48 c~ Step norm 100% 1.91% 0.16% 0.24% 10.9% 4.23% 0.56% 0.21% 0.40% 0.15%

1075

13 Step norm 100% 14.8% 1.36% 0.84% 21.8% 8.31% 2.15% 0.50% 1.07% 0.37%

S K

G N

6.3.2 The ERA Method
To solve for a state-space model (A, B, C, D) using the ERA method, first form the generalized Hankel matrices:
Yk Yk+ l

•.

Yk+f3 1
Yk+f3

]
!

H(k-

1) =

Yk+l

Yk+2 • •

] .

Yk+~ ~

Yk+~

••

Yk+c~+13-2 _l

r

(6.39)

Note that in general, c~ and [3 are chosen to be the smallest numbers such that H(k) has as large row and column ranks are possible. Additional suggestions to determine their optimal values are given in Juang (1994). Let the singular value decomposition of H(0) be H(0) = U~,V r, and let n denote the index where the singular values have the largest drop in magnitude. Then, H(0) can be approximated by: FIGURE 6.5 Picture of the Seismic-AMD Benchmark Structure H(0) ~ U ~ V T, (6.40)

Qi=~OQiZ i)(i=~o yi Z i)Pi~_o Ri Z i,

(6.36)

where U, and V, are the first n columns of U and V, respectively, and 1i;~ is the diagonal matrix containing the largest n singular values of H(0). Finally, an nth order state-space realization (A, B, C, D) can be calculated by: a = ~ 1 / 2 U r N ( l ) gn~nl/2,
C = E mU n~n ' r ,;1/2 D = Yo,
B =

by the following iterative calculations starting from Y0 = R0:

X ' I / 2 v Tn E r , "-~n

(6.41)

Rk - ~ - - 1 QiYk i, gk = --~Pi 1 Qi gk-i,

for k = 1. . . . . p. for k = p + 1. . . . . oc.

(6.37)

If the TFM is parameterized in the PM form, the derivation of the system Markov parameters is almost the same: one starts with Y0 = B0, and continues with the following iterative procedure:

where Er and E~ are the elementary matrices that pick out the first r (the number of system inputs) columns and first m (the number of system outputs) rows of their multiplicands, respectively.

Yk

f Bk -

2~=1aigk

6.4 Application Studies
This section presents two experimental level application studies conducted in the Structural Dynamics and Control/

i,

for k = 1. . . . . p. for k = p + l . . . . . oc.

(6.38)

1076
Um

Gangfin

XgddOt
40 20 0 -20

.~ -o
"(3

20 0

-20
-40, 40 5 .... 10 15 " i ' 20 i .... 25 i .... 30 '. ' 35

5 40 20 0 -20 -40

10

15

20

25

30

35

'

-40
5

iiiiiiiiiiiii
10 15

IIYIIIIIIIIIIIIIIIIII
20 25 30 35

5 20

10

15

20

25

30

35

"0 "(3

"5

0 -40

0 -20 -40 5 10 15 20 25 30 35 20 0 -20 -40 5 10 15 20 25 30 35 5 10 15 20 25 30 35 5 10 15 20 25 30 35

x~ - 2 0

"0 "(3 E X

~5

10
E

X

-50

0
5 10 15 20 25 30 35 F r e q u e n c y (Hz)

-100 5 10 15 20 25 30 35 F r e q u e n c y (Hz)

FIGURE 6.6 Comparison of Experimental and Model FRF for the Seismic-AMD Benchmark Structure: Magnitude Plot. Um denotes the input command to the AMD; xg ddot denotes the input ground acceleration to the structure; xjddot denotes the output acceleration of the jth floor; xmddot denotes the output acceleration of the AMD; Xm denotes the output displacement of the AMD; dotted lines are for measurement data; solid lines are for model output.

Earthquake Engineering Laboratory (SDC/EEL) at University of Notre Dame. This section only presents the results pertinent to identification studies discussed so far in this chapter. For detailed information about these experiments, including experimental setups and/or control developments, the reader may refer to Jin et al. (2000), Jin (2002), and Dyke et aL (1994), respectively.

6.4.1 Identification of a 16-Story Structure
The first identification target is a 16-story steel structure model shown in Figure 6.2. The system is excited by impulse force produced by a PCB hammer and applied individually at the 16th, 12th, 8th, and 4th floors. The accelerations of these floors are selected as the system measurement outputs and are sensed by PCB accelerometers. The goal of the identi-

fication is to capture accurately the first five pairs of the complex poles of the structure. For this purpose, a DSPT Siglab spectrum analyzer is used to measure the FRF data. The sampling rate is set at 256 Hz, and the frequency resolution is set at 0.125 Hz. The experimental FRF is preconditioned to eliminate the second order direct current (dc) zeros from acceleration measurement. The MF parameterization is chosen for the curve-fitting, which is complemented by the ERA method for state-space realization. The key identification parameters are given in Table 6.1. The final discretetime state-space realization has 10 states. The magnitude and phase plots of its transfer functions are compared to the experimental FRF data in Figures 6.3 and 6.4. Excellent agreements are found in all but the very high frequency range. The mismatch there is primarily due to the unmodeled high-frequency dynamics.

6

Frequency D o m a i n System Identification
Um

1077
xgddot 0 -200 -400 ! 5 -100 -200 -300 5 10 15 20 25 30 35 i 10 i 15 .... i 20 ! 25 l 30 35

"0 o x

-1 O0 -200 -300 5 10 15 20 25 30 35

%

5 0 -200 -400 -600 5

10

15

20

25

30

35

t

-100 10 15 20 25 30 35 5 10 15 20 25 30 35

× -lOO
-200 5 10 15 20 25 30 35

_4oot
5 -200 -400 -600 -800

.....
10 15 20 25

...... t
30 35

...... . iiiiiiii iiiiiiiill
-120I . . . . . . . . . 5 10 15 20 25 30 35 L ~ ,; . . . . " : I- U . . . . : . ~'~w~-,,~.,,,~Z.~ t . . . . ~r." ~ : ..... .... ~, 5 10 15 20 25 30 Frequency (Hz) 35

Frequency (Hz)

FIGURE 6.7 Comparison of Experimental and Model FRF for the Seismic-AMD Benchmark Structure: Phase Plot. Same notations are used as in Figure 6.6.

6.4.2 Identification of the Seismic-Active Mass Driver Benchmark Structure
The second identification target of this discussion is a threestory steel structure model, with an active mass driver (AMD) installed on the third floor to reduce the vibration of the structure due to simulated earthquakes. A picture of the system is given in Figure 6.5. This system has been used recently as the ASCE first-generation seismic-AMD benchmark study. The benchmark structure has two input excitations: the voltage command sent to the AMD by the control computer and the ground acceleration generated by a seismic shaker. The system responses are measured by four accelerometers for the three floors and the AMD and one linear variable differential transformer (LVDT) for the displacement of the AMD. The sampling rate is 256 Hz, and the frequency resolution is 0.0625Hz. Due to noise and nonlinearity, only the frequency range of 3 to 35Hz of the FRF data is considered to be accurate and, thus, this range is used for the identification. A preliminary curve-fitting is carried out using the MF parameterization. The identified model matches the experi-

mental data accurately in all but the low-frequency range of channels corresponding to the AMD command input and the acceleration outputs. A detailed analytical modeling of the system reveals that there are four (respectively two) fixed dc zeros from AMD command input to the structure (respectively AMD) acceleration outputs: lira Gx~um(S) -- ki,
s-e0 S4

i = 1, 2, 3.

(6.42) (6.43)

lim G~um ( S) - kin.
s-*O S2

The G~,~ and G~,,u,~ are the transfer functions from AMD command input u m to structure and AMD acceleration outputs, respectively. The ki and km are the static gains of these transfer functions with the fixed dc zeroes removed. These fixed zeros dictate the use of the PM curvefitting technique to explicitly include such a priori information. Again, key identification parameters are presented in Table 6.1. The outputs of the parameter optimization iterations are

1078 documented in Table 6.2. The final discrete-time state-space realization has eight states as predicted by the analytical modeling. The magnitude and phase plots of its transfer functions are compared to the experimental FRF data in Figures 6.6 and 6.7. All the input output channels are identified accurately except for the (5,2) element, which corresponds to the ground acceleration input and the displacement output of the AMD. The poor fitting there is caused by the extremely low signal-tonoise ratio.

Gang Jin

References
Antsaklis, P., and Michel, A. (1997). Linear systems. New York: McGraw-Hill. Bayard, D. (1992). Multivariable frequency domain identification via two-norm minimization. Proceedings of the American Control Conference, 1253-1257. Dennis, Jr., J.E., and Schnabel, R.B. (1996). Numerical methods for unconstrained optimization and nonlinear equations. Philadelphia: SIAM Press. Dyke, S., Spencer, Jr., B., Belknap, A., Ferrell, K., Quast, E, and Sain, M. (1994). Absolute acceleration feedback control strategies for the active mass driver. Proceedings of the World Conference on Structural Control 2, TPh51-TPI:60. Gill, EE., Murray, W., and Wright, M.H. (1981). Practical optimization. New York: Academic Press. Jin, G., Sain, M.K., and Spencer, Jr., B.E (2000). Frequency domain identification with fixed zeros: First generation seismic-AMD benchmark. Proceedings of the American Control Conference, 981-985. Jin, G. (2002). System identification for controlled structures in civil engineering application: Algorithm development and experimental verification. Ph.D. Dissertation, University of Notre Dame. Juang, J.N. (1994). Applied system identification. Englewood Cliffs, NJ: Prentice Hall. Stewart, G.W. (1973). Introduction to matrix computations. New York: Academic Press.

6.5 Conclusion
This chapter discusses the identification of linear dynamic systems using frequency domain measurement data. After outlining a general modeling procedure, the two major computation steps, frequency domain curve-fitting and state-space system realization, are illustration with detailed numerical routines. The algorithms employ TFM models in the form of matrix fraction or polynomial matrix and require respectively linear or nonlinear parameter optimizations. Finally, the proposed identification schemes are validated through the modeling of two experimental test structures.