SYSTEM IDENTIFICATION: A STATISTICAL APPROACH
5.3. APPLIED IDENTIFICATION TECHNIQUES
The modal parameters are identified using three important techniques: NExT-ERA, SSI (data driven) and FSDD. NExT estimates the impulse response function directly from the ambient vibration data, which is based on sound theoretical background. On the other hand, ERA estimates the modal parameters using the impulse response sequence based on the principle of minimal realization theory. Secondly, the system theory, statistics, optimization theory and (numerical) linear algebra culminate in the evolution of 'subspace’ techniques which enables to obtain state sequence directly from input-output data or only output data (ambient). The subspace-methods don't require an explicit model parametrization. Moreover, subspace algorithms have the advantages of being elegant and computationally efficient. SSI as a sub- sect of subspace-methods using only output data (ambient), provide all these advantage in identification of modal parameters. Finally, FSDD, established based on a reasonable theoretical background, doesn’t require any data-fitting as well as parametrization. FSDD also has various advantages over classical pick-picking method which is usually considered as an important methodology. These three techniques are also found to be used frequently for modal identification from ambient vibration data. The major highlights of these techniques are presented in the following sub-sections. Both of the techniques, ERA and SSI identify the state space matrices in discrete time domain and modal parameters are finally evaluated using those identified state space matrices. The state space equations in discrete time domain, considered in this present study, are as follows.
s k1
A s k
B u k
( )
(5.1a)
y k
C s k
E u k
( )
(5.1b) where, [A], [B], [C], [E] represents the discrete time state space matrices, {s} represents the states, k is integer to represent discrete time as t=k(∆t) with ∆t being sampling time, {u} and{y} represent the input and output respectively. Estimation of [A] and [C] are sufficient for identification of modal parameters.
5.3.1. Natural excitation technique with eigensystem realization algorithm (NExT-ERA)
It is a two-step identification where NExT is applied to estimate the impulse responses from ambient data in the first phase. Subsequently, ERA is employed to identify the state space matrices in discrete time domain from the estimated impulse responses. The displacement vector associated to the equation of motion as {x(t)} at time t is related to the modal coordinate as {q(t)} in the form of the following equation.
1
( ) ( ) ( )
n
r r
r
x t q t q t
(5.2)where, n,
and
r
represent the number of DOF, modal matrix and rth mode shape vector respectively. With consideration of proportional damping the rth modal coordinate can be expressed as follows.
2 1
( ) 2 ( ) ( ) T ( )
r r r r r r r
r
q t q t q t u t
m
(5.3)
where, mr represents the rth modal mass. Solution of the Eq. (5.3) can be carried out using the Duhamel integral. Solution can be found out for the displacement vector, {x(t)}, based on the assumption of zero initial condition in the following form.
1
( ) ( ) ( )d
n T
r r r
r
x t u g t
(5.4)where, r( ) 1 exp( r r ) sin( r )
r r
g t t t
m
and r r(1r2)1/ 2 is the rth damped modal frequency. Eq. (5.4) can be transformed for a single output, xik(t) and a single input force, uk(t) at a location k in the following form.
1
( ) exp(- ) sin( )
n
ir kr
ik r r r
r r r
x t t t
m
(5.5)where, ir is the ith component of the rth mode shape vector
r
. The next step is to form the cross-correlation function of two responses, xik and xjk, due to a white-noise input at a particular location k. The cross-correlation function Rijk(T) is defined as the expected value of the product of two responses evaluated at a time separation of T as follows.( ) E[ ( ) ( )]
i jk ik jk
R T x tT x t (5.6)
Based on the assumption of white-noise, the cross-correlation function due to all the input can be found out by summing up contributions from all the input locations as
1
( ) exp( )sin( )
n
ir jr
ij r r r r
r r r
R T A T T
m
(5.7)It can observed from the Eq. (5.7) that the cross-correlation function is expressed as a single sine function with a new phase angle (r) and a new constant multiplier (Ajr ). Hence, the cross-correlation function is a sum of decaying sinusoids of the same form as the impulse response function of the original system as expressed in the Eq. (5.5). Based on this observation, it is well understood that the impulse response functions can be estimated from the ambient vibration data.
In the next step, Markov parameters blocks are formed using the estimated impulse responses to construct finally the Hankel matrix. Subsequently, ERA uses the Hankel matrix to identify the modal parameters. The Markov parameters can be written as
k 1h k C A B
(5.8)
Considering the number of input and output as n1 and n2 respectively, the size of a Markov parameters becomes n2×n1. Now, the Hankel matrix is represented as
( ) ( 1) ( )
( 1) ( 2) ( 1)
– 1
( ) ( 1) ( )
h k h k h k j
h k h k h k j
H k
h k i h k i h k i j
(5.9)
where, i=1, 2, ….. r1–1 and j=1, 2, …. s1–1, with r1 and s1 as integers. Now the size of the Hankel matrix becomes as (n2r×n1s). Hankel matrix for k = 1 i.e. [H(0)] is decomposed using SVD as
0
TH U V
(5.10)
where, the sizes of [U],
and [V]T are (n2r× n2r), (n2r×n1s) and (n1s× n1s) respectively. It is considered that [H(0)] has 2N non-zero singular values (i.e. rank=2N) equivalent to the order of state space system. Therefore, expression for [H(0)] can be rewritten as
0
2N
2N
2N
TH U V
(5.11)
where, the sizes of
U2N
,
2N
and
2
T
VN are (n2r× 2N), (2N×2N) and (2N×n1s) respectively. The estimate of the discrete time state-space are obtained as follows
A 2N
1/ 2 U2N
T H(1)
V2N
2N
1/ 2 (5.12a)
2
1/2 2
2 TN N
B V E (5.12b)
C E1 T H(1)
U2N
2N
1/2 (5.12c)
1E Tand
E2 as appeared in the above equations, are defined as follows,
E1 T
I 0
0 (5.13a) where, each sub-matrices (identity and zero matrices) is of the size (n2×n2).
2
0 0 E
I
(5.13b)
where, each sub-matrices (identity and zero matrices) is of the size (n1×n1).
5.3.2. Stochastic subspace identification (SSI)
Next, SSI-DATA method is considered for evaluation of the modal parameters. The summary of this identification technique are mentioned below. The output block Hankel matrix (consisting of 2i rows and j columns of output block sub-matrices) is represented in two forms as in Eqs. (5.14a) and (5.14b).
0 1 –1
–2 –1 –3
–1 –2 0| –1 1
0|2 –1
|2 –1 2
1 –1
1 2
2 –1 2 2 – 2
j
i i i j
i i i j i
i
i i
i i i j
i i i j
i i i j
y y y
y y y
y y y Y Y
Y y y y Y Y
y y y
y y y
(5.14a)
0 1 –1
– 2 –1 –3
–1 –2
1
1 –1 0|
0|2 –1 –
1|2 –1
1 2 2
2 –1 2 2 –2
j
i i i j
i i i j
i i i j i
i
i i
i i i j
i i i j
y y y
y y y
y y y
Y
y y y Y
Y y y y Y Y
y y y
(5.14b)
One of the key steps in SSI-DATA is projection. The projections are computed as in Eqs.
(5.15a) and (5.15b).
Oi Y2 / Y1 (5.15a)
Oi–1
Y2– / Y1 (5.15b) Three different choices of algorithms are implemented in SSI-DATA based on three choices of weighting matrices for projection matrix: (a) unweighted principal component (UPC) algorithm, (b) principal component (PC) algorithm and (c) canonical variant algorithm (CVA) (Van Overschee and De Moor 1996). Weighting matrices ([W1] and [W2]) are incorporated with projection matrix to form the weighted projection like
W1 Oi W2 . SVD is applied next for the weighted projectionas in Eq. (5.16).
1 2 TW O Wi U V (5.16)
The order is determined by inspecting
. It is here assumed that the order is 2N. Using the significant part of the decomposed matrices as in Eq. (5.16), the extended observability matrix now is expressed as in Eq. (5.17).
2
2
1/ 21 1
i W U N N
(5.17)
i–1
is found out with stripping the last l (number of outputs) rows from
i . Evaluation of the Kalman filter state sequences is carried out using the Eqs. (5.18a) and (5.18b). Here the symbol “†” represents the Moore-Penrose pseudo-inverse of a matrix.
†ˆi i i
X O
(5.18a)
†
1 1 –1
ˆi i Oi
X
(5.18b)
The least-squares solution is carried out using the Eq. (5.19) finally to compute an asymptotically unbiased estimate of [A] and [C].
1 †
|
. ˆ
i ˆ
i i i
A
C Y
X
X
(5.19)
5.3.3. Frequency spatial domain decomposition (FSDD)
Frequency domain decomposition is theoretically established based on the following formula relating the output PSD, Gyy
and stochastic input PSD, Gxx
.
( )
( )
Hyy xx
G G G G
(5.20)
where, [G(ω)] represents the FRF matrix. The input is considered as white noise and hence the stochastic input PSD, Gxx
becomes a constant matrix. This helps in the evolution of FSDD technique. It can be shown that while frequency reaches close to the mth modal frequency as ωm, the output PSD can be approximated as follows.
m
T H
m
yy i m m m
G
(5.21)
where,
m represents the mth mode shape and m is a constant associated to that mode. The above equation is derived taking all the measurement-locations as reference and thus the output PSD becomes as a square matrix. The key step in FSDD is the singular value decomposition of the estimated output PSD as shown below.
ˆ ( )yy ( ) ( ) ( ) H G i U V
(5.22)
It may be mentioned that all the measurements are considered as reference and hence
U( )
becomes similar as
V( )
. While the frequency approaches a modal frequency ωm, the PSD matrix approximates a matrix of rank one. This observation enables the following relation.
1
1
ˆ ( ) 1( ) ( ) ( )
m
H
yy m m m m
G i U U
(5.23)
where, 1( m) and
U1(m)
are the first singular value and first singular vector at the frequency ωm. While the first singular value reaches maximum within the narrow modal frequency range for the mth mode, the corresponding singular vector
U1(m)
becomes an estimate of the mth mode shape. It may be mentioned that the rank of the output PSD is expected to be nearly equal for multiple repeated modes for repeated modal frequencies. Thus the singular values acts as the complex mode indication function (CMIF) (Shih et al. 1988).After the estimation of mode shape, FSDD use enhanced PSD to evaluate natural frequency and damping ratio. Enhanced PSD in the vicinity of the mth mode is evaluated as
1
1
ˆ ( ) ( ) ˆ ( ) ( )
m
H
yy m yy m
eG i U G i U
(5.24)
The enhanced PSD is a scalar function of frequency and can be approximated as an SDOF system in the narrow frequency range around a modal frequency. An SDOF curve fitter can be utilized in the vicinity of a modal frequency for identification of natural frequency and damping ratio of that mode. It may be mentioned that in the case when all the measurements are not considered as reference, similar approach is applicable with FSDD.