Palle Andersen Rune
Brincker
Structural Vibration Solutions A/S
Niels Jernes Vej 10, Sohngaardsholmsvej
57,
DK-9220 Aalborg East. DK-9000
In the traditional
input-output modal analysis the estimation of modal parameters have been
performed using a somewhat deterministic mathematical framework. One of the
major hurdles for people of this traditional modal community to overcome, when
turning to output-only modal analysis, is the switch of the mathematically
framework. In output-only modal analysis the mathematically framework involves
the use of statistics and introduction of concepts such as optimal prediction,
linear system theory and stochastic processes.
The two general assumptions
made in output-only modal analysis is that the underlying physical system
behaves linearly and time-invariant. The linearity imply that if an input with
a certain amplitude generates an output with a certain amplitude, then an input
with twice the amplitude will generate an output with twice the amplitude as
well. The time-invariance implies that the underlying physical system does not
change in time. One of the typical parametric model structures to use in
output-only modal analysis of linear and time-invariant physical systems is the
stochastic state space system.
(1)
The first part of this model
structure is called the state equation and models the dynamic behavior of the
physical system. The second equation is called the observation or output
equation, since this equation controls which part of the dynamic system that
can be observed in the output of the model. In this model of the physical
system, the measured system response yt
is generated by two stochastic processes wt
and vt. These
are called the process noise and the measurement noise. The process noise is
the input that drives the system dynamics whereas the measurement noise is the
direct disturbance of the system response.
The philosophy is that the
dynamics of the physical system is modeled by the n´n state matrix A. Given an n´1 input vector wt, this matrix transforms the
state of the system, described by the n´1 state vector xt, to a new state xt+1. The dimension n of the state vector xt is called the state space
dimension. The observable part of the system dynamics is extracted from the
state vector by forward multiplication of the p´n observation matrix C. The p´1 system response vector yt is a mixture of
the observable part of the state and some noise modeled by the measurement
noise vt.
The state space model (1) is
only applicable for linear systems that do not have time-varying changes of its
characteristics. However, this is not the only restriction. The only way to
obtain an optimal estimate of a state space model on the basis of measured
system response, is to require that the system response is a realization of a
Gaussian distributed stochastic process that has zero mean.
In other words, in the
applied stochastic framework the system response is modelled by a stochastic
process yt defined
as
(2)
and the principal assumption
is that the measured system response is a realization of this process. It is
seen that this process is completely described by its covariance function
. This means that if we can estimate a state space
model having the correct covariance function this model will completely
describe the statistically properties of the system response. An estimated
model fulfilling this is called covariance equivalent. The estimator that can
produce such model is called an optimal estimator.
Since the system response of
the linear state space model is a Gaussian stochastic process it implies that
,
and
all are Gaussian stochastic processes as well. Since
the input processes
and
are unknown we make the simplest possible assumption
about their statistical properties, which is to assume that they are two
correlated zero-mean Gaussian white noise processes, defined by their
covariance matrices as
(3)
The Gaussian stochastic
process describing the state
is also zero-mean and completely described by its
covariance function
(4)
Using (1) to (4) the
following relations can be established
(5)
The matrix G
is the covariance between system response
and the updated state vector
. The covariance function of
can also be expressed in terms of the system
matrices as
(6)
There are two additional
system matrices turns out to play an important role
(7)
These are the extended
observability matrix
and the reversed extended stochastic controllability
matrix
.
One of the most important parts of all estimation is the ability to predict the measurements optimally. In output only modal analysis this means to be able to predict the measured system response optimally. An optimal predictor is defined as a predictor that results in a minimum error between the predicted and measured system response. If the system response can be predicted optimally it implies that a model can be estimated in an optimal sense.
Recall
that the state vector
completely describes the system dynamics at
time t. In order to predict the
system response
optimally it is necessary to start by defining an optimal predictor
of
. Now assume that we have
measurements
available from some initial time k = 0 to k = t-1. Collect these
measurements in a vector
(8)
In the
Gaussian case the optimal predictor of
is then given by the conditional mean value
(9)
So, the
optimal predictor of
is defined as the mean value of
given all measured system response
from k = 0 to k = t-1. The difference between
and
is called the state prediction error
and is defined as
(10)
This
error is the part of
that cannot be predicted by
.
In order
to predict the system response a similar conditional mean can be formulated for![]()
(11)
The last
part of this equation is obtained by inserting (1) and assuming that
and
from k = 0 to k = t-1 are uncorrelated.
The two predictors (9) and (11) are related through the so-called Kalman filter for linear and time-invariant systems, see e.g. Goodwin et al. [6]
(12)
The
matrix
is called the non-steady state Kalman gain and
is called the innovation and is a
zero-mean Gaussian white noise process. Defining the non-steady-state
covariance matrix of the predicted state vector
as
the Kalman gain
is calculated from
(13)
The last
of these equations is called the Ricatti equation. The Kalman filter predicts
the state
on the basis of the previous
predicted state
and the measurement
. The covariance
of
the innovations
can be determined from the last
equation in (12) as
(14)
Given
that the initial state prediction is
and the initial state prediction
covariance matrix
and assume that we have measurements
available from k = 0 to k = t-1, then this filter
is an optimal predictor for the state space system (1) when the measurements
are Gaussian distributed.
At start up the Kalman
filter (12) will experience a transient phase where the prediction of the state
will be non-steady. However, if the state matrix A is stable the filter
will enter a steady state as time approach infinity. When this steady state is
reached the covariance matrix of the predicted state vector
becomes constant, i.e.
, which imply that the Kalman gain becomes constant
as well, i.e.
. The Kalman filter is now operating in steady state
and is defined as
(15)
The steady state Kalman gain
is now calculated from
(16)
The last equation is now
called an algebraic Ricatti equation. Assuming all matrices but P
is known this equation can be solved using eigenvalue decomposition, see Aoki
[2] and Overschee et al [1].
If the last equation in (15)
is rearranged the following state space system is obtained
(17)
This system is called the
innovation state space system. The major difference between this system and (1)
is that the state vector has been substituted with its prediction, and that the
two input processes of (1) have been converted into one input process – the
innovations. This state space system is widely used as model structure in
output only modal analysis, see e.g. Ljung [3] and Söderström et al. [4].
The Kalman filter defined in
the last section turns out to be the key element in the group of estimation
techniques known as the stochastic subspace techniques.
From (17) it is seen that if
sufficiently many states of (1), let’s say j
states, can be predicted, i.e.
and
, then the A and C matrices can be
estimated from the following least regression problem
(18)
This is a valid approach
since the innovations are assumed to be Gaussian white noise. Since A
and C
are assumed to be time-invariant this regression approach will be valid even
though the predicted state
and
originates
from a non-steady state Kalman filter.
So the fundamental problem
to solve in the stochastic subspace identification technique is to extract the
predicted states from the measured data. To show how this is performed,
consider the state space system in (1) and take the conditional expectation on
both sides of both equations to yield
(19)
Now assume that a recursion
is started at time step q. Inserting
the first equation in (19) recursively into itself i times and finally inserting the result the last of the equations
in (19) leads to the following formulation
(20)
This equation shows the
relation between the initial predicted state
and the prediction of the free (noise free) response
of the system
. By stacking i
equations on top of each other the following set of equations are obtained
(21)
By introducing the vector
as the left-hand side and noticing that the first
part of the right-hand side is equal to the extended observability matrix
we actually obtain the following expression for the
predicted states
(22)
The matrix
is actually the pseudo-inverse of
. This equation shows that if we can estimate
and
for several values of q, we can in fact estimate the predicted states for several values
of q as well.
In this section we will focus on the estimation of the predicted free
response
. We will
estimate a set of vectors
and gather
them column by column in a matrix O.
In order to predict the system response a conditional mean similar to (11) can be formulated.
(23)
This conditional mean is the
prediction of the future system response
given
the past system response from time t =
i+q-1 down to t = q. This
conditional expectation is only an approximation of (11) since the conditioning
vector stops a time t = q and not t = 0. The approximation is only good if
i is sufficiently high. For zero-mean
Gaussian stochastic processes this conditional mean can be calculated by, see
e.g. Melsa et al. [5].
(24)
Since the error
is zero-mean and uncorrelated and is independent of
the conditional mean
and the conditioning vector
the conditional mean (24) is also called the
orthogonal projection of the vector
onto the vector
.
In order to estimate all
elements of
we need to extend (24) to allow estimation of
to
in one operation. This is done by using (8) to
extend the conditional mean
in (24) to the
following
. This results in the following equation for ![]()
(25)
In the last equation a new ip´ip
matrix
is
introduced for simplicity. This matrix is defined as
(26)
Incidentally, the matrix
is also
equal to
(27)
As seen in (18) we need a
bank of predicted state estimates
for q = i
to q = i+j-1 for a sufficiently large
value of j. To
estimate these state in one operation based on the approach in (25) we need to
define the following two matrices
and
as
(28)
(29)
The index p in (29) signifies that the matrix contains system response of the past compared to the system response we are predicting. Since we assume that the system response is stationary, i.e. that