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Using the Application > Working with Application Tasks > Analysis Tasks > Estimation > Stochastic Subspace Identification >
Stochastic Subspace Identification
In the Stochastic Subspace Identification (SSI) techniques a parametric model is fitted directly to the raw times
series data. A parametric model is a mathematical model with some parameters that can be adjusted to change the
way the model fits to the data. In general, we are looking for a set of parameters that will minimize the deviation
between the predicted system response of the model and measured system response (measurements). This process
is often called model calibration. See the following picture
All known linear and time-invariant time domain modal identification techniques can be formulated in a generalized
form as an innovation state space formulation
where the A-matrix contains the physical information, the C-matrix extracts the information that can be observed in
the system response and the K-matrix contains the statistical information. The statistical information allows for a
covariance equivalent modeling, so that the model can have the correct correlation function and thus also the
correct spectral density function.
The number of parameters in the model is essential. If this number is to small, then the dynamical- and statistical
behavior cannot be modeled correctly. On the other hand, if the number is too high, then the model becomes
over-specified resulting in unnecessary high statistical uncertainties of the model parameters.
So the art of parametric model estimation is to determine a model with a reasonable number of parameters. This
means that what you must do when you are estimating state space models is to choose the model order also known
as the state space dimension, which is the dimension of the A-matrix.
ARTeMIS Modal has five different implementations of the Stochastic Subspace Identification technique. These are:
Unweighted Principal Component (SSI-UPC)
Extended Unweighted Principal Component (SSI-UPCX)
Principal Component (SSI-PC)
Canonical Variate Analysis (SSI-CVA)
Unweighted Principal Component Merged Test Setups (SSI-UPC Merged)
The four first are all working per Test Setup. This means that you extract the modes for each Test Setup and then
use a tool called Select & Link to merge the results to form a global set of modal parameters. The fifth technique do
the merging as a part of the signal processing done in the Prepare Data Task, and the result is that you work with
this tool as if there was only a single Test Setups even though there are multiple Test Setups. This technique is
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dedicated to multiple Test Setups.
See the Technical Paper on the Stochastic Subspace Identification Techniques for a more comprehensive description
about how the Stochastic Subspace Identification techniques work and what the mathematical difference between
the implementations are.
When the stochastic state space system is being estimated using e.g. the Stochastic Subspace Identification
techniques we obtain what is called a realization of the true but unknown system. So the parameters of the state
space system
is only estimates of the true system. You will never be able to estimate the 100% correct parameters but you can
indeed estimate very accurate parameters by not using a too large state space dimension.
The above system is shown in time domain but can of course also be represented in frequency domain by its
transfer function H(z) as below
where z is a frequency dependent complex number. By a complex transformation of this transfer function using the
eigenvectors of A the modal decomposed transfer function appear as
This representation of the transfer function expose all the modal parameters. From the eigenvalues µjdefined as the
diagonal elements of the matrix
the natural frequencies
and damping ratios
are extracted using the following definition
In this equation T is the sampling interval.
The mode shape that are associated with the jth mode is given by the jth column of the matrix
. The last matrix
that completes the modal decomposition contains a set of row vectors. The jth row vector corresponds to the jth
mode. This vector distributes the white noise excitation et in modal domain to all the degrees of freedom. So the
amplitude values of the degrees of freedom depends on this vector as well as the eigenvalue and the mode shape.
At the initial time step the state vector
is zero. This imply that the contribution of e0 to
from a specific mode
that corresponds to that mode. For this reason this vector is called the initial
solely is given by the row vector of
modal amplitude. Since this vector describes how the white noise is distributes in modal domain, this vector
describes the statistical part of the modal decomposition. All the other modal parameters relates to the dynamic
system and are therefore deterministic parameters that should not change if the excitation changes.
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Modal Estimation window - Stochastic Subspace Identification
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