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:
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Unweighted Principal Component (SSI-UPC)
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Extended Unweighted Principal Component (SSI-UPCX)
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Principal Component (SSI-PC)
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Canonical Variate Analysis (SSI-CVA)
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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 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.
Extracting Modal Parameters from the State Space System
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 solely is given by the row vector of that corresponds to that mode. For this reason this vector is called the initial 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.
Uncertainty quantification of modal parameter estimates
ARTeMIS Modal Pro 5 supports a new approach to compute the variance of the modal parameter estimates, which is available in Stochastic Subspace Identification with Extended Unweighted Principal Component, or in short SSI-UPCX. This technique has been developed in cooperation with Inria/IFSTTAR I4S Team in Rennes, France.
A unique aspect of the SSI-UPCX technique is that uncertainty estimation of the modal parameters is performed in a fast and memory-efficient way. The uncertainty estimation makes SSI-UPCX stand out, compared to today's modal analysis estimation techniques.
It's theoretical premise were developed in:
Theoretical background
The practical value of identifying parameters from data is minor unless the deployed identification procedure provides their confidence bounds. The information about their uncertainty is only useful when the underlying identification algorithm is consistent, that is when the estimated modal parameters converge to their true values as the amount of data tends to infinity. This is the case for the family of stochastic subspace identification methods, which are consistent in identification of parameters of linear-time invariant systems when the noise driving the system is stationary or non-stationary.
Covariance estimates of modal parameters are obtained based on the first-order delta method. This statistical framework allows to characterize the probability distribution of a function of an asymptotically Gaussian variable based on one data set, and without imposing prior information on the distribution of estimated parameters. An illustrative example of the propagation scheme is illustrated on Figure below.
Hereafter the general principles of uncertainty propagation in the context for the subspace methods are recalled. The idea is to relate the Hankel matrix from the data-driven SSI-UPC identification algorithm to the adequate covariance matrices of the underlying data sequences. Those covariance sequences are asymptotically Gaussian variables, which allows to characterize the distribution of the Hankel matrix components also as asymptotically Gaussian. Subsequently, the uncertainty of the Hankel matrix is propagated through all the identification steps to modal parameters. This is an exhaustive technical development which is not enclosed here for brevity. An illustrative example of this procedure is illustrated on Figure below.
