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N4SID-VAR Method for Multivariable Discrete Linear Time-variant
System Identification
Alexander E. Robles and Mateus Giesbrecht
School of Electrical and Computer Engineering, University of Campinas, Av. Albert Einstein 400, Campinas - SP, Brazil
Keywords:
System Identification, Subspace Methods, Time-variant System Identification.
Abstract:
In this paper, a method for multivariable discrete linear time-variant system identification is presented. This
work is focused on slowly multivariable time-variant systems, so that it is possible to define time intervals,
defined as windows, in which the system can be approximated by time-invariant models. In each window,
a variation of N4SID that uses Markov parameters is applied and a state space model is estimated. For that
reason the proposed method is defined as N4SID-VAR. After obtaining the models for all windows, the error
between system model outputs are calculated and compared to the system outputs. The N4SID-VAR was
tested with a time-variant multivariable benchmark and the results were accurate. The proposed method was
also compared to the MOESP-VAR method and, for the tested benchmark, the N4SID-VAR was faster and
more accurate than the MOESP-VAR algorithm.
1 INTRODUCTION
System identification consists in the search for a
mathematical model that can describe the behavior of
a dynamical system, from the observed input-output
signals (Ljung, 1999), (Katayama, 2005). A signifi-
cant part of activities and researches in system identi-
fication focuses on time-invariant dynamical systems.
However, there are innumerable systems in nature that
are multivariable with nonlinear and time-variant be-
havior. To deal with last problem, the time-variant
systems can be approximated by linear time-invariant
systems, as long as these systems vary slowly (Tama-
riz et al., 2005).
During the last two decades, subspace-based
methods have been extensively studied to address the
problem of identifying multivariable discrete linear
time-invariant systems (Katayama, 2005). From that
methods, the most popular are the MOESP (Verhae-
gen and Dewilde, 1992) and the N4SID (Overschee
and Moor., 1994). Both methods have a mathematical
support in linear matrix algebra.
The MOESP method is based on LQ decomposi-
tion of a matrix formed by input-output data, where:
L is a lower triangular matrix and Q is an orthogo-
nal matrix. From a block of the matrix L a singular
value decomposition (SVD) is performed, from that
it is possible to find out the system order and its ob-
servability matrix. With this last matrix it is possible
to obtain the matrices C and A corresponding to the
model in state space. The final step is to form a linear
equation and apply the least squares method and esti-
mate the matrices B and D of the model. The method
is detailed in the section 4 of this paper.
Another subspace method called N4SID (Numer-
ical Algorithms for Subspace State Space System
Identification), the same way as the MOESP, is based
on a LQ decomposition of data matrices. However,
in this case this decomposition is interpreted as the
oblique projection of future outputs in the subspace
of the past inputs and outputs, towards the future in-
puts. From these projections the system states are
estimated. With the states, inputs and outputs, the
matrices A, B,C and D can be determined using a
simple least squares method. This method and a
variant proposed in (Clavijo, 2008) are presented in
the section 5. A method inspired by MOESP and
called MOESP-VAR, was introduced and developed
in (Tamariz et al., 2005). The method is initialized
by splitting the total input and output data into data
groups that are associated with time intervals in which
the system exhibits a slow change and can be ap-
proximated by a time-invariant system. The MOESP
method is applied to the data of each interval, result-
ing in a linear time-invariant for the system in each of
the time intervals. Following the same concept, the
N4SID-VAR method is proposed in this article. The
first step of the proposed method is to split the data
502
Robles, A. and Giesbrecht, M.
N4SID-VAR Method for Multivariable Discrete Linear Time-variant System Identification.
DOI: 10.5220/0006907505020509
In Proceedings of the 15th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2018) - Volume 1, pages 502-509
ISBN: 978-989-758-321-6
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