A Least Squares Approach to the Subspace Identification Problem
L. Bako1,2, G. Mercère3 and S. Lecoeuche 1,2
Abstract—In this paper, we propose a new method for
the identification of linear Multiple Inputs-Multiple Outputs
(MIMO) systems. By introducing a particular user-defined ma-
trix that does not change the rank of the extended observability
matrix when multiplying this latter matrix on the left, the
subspace identification problem is recasted into a simple least
squares problem with all regressors available. Therefore, the
Singular Value Decomposition algorithm which is a customary
tool in subspace identification can be avoided, thus making our
method appealing for recursive implementation. The technique
is such that the state coordinates basis of the estimated
matrices is completely determined by the aforementioned user-
defined matrix, that is, given such a matrix, the state basis
of the identified matrices does not change with respect to the
realization of input-output data.
I. INTRODUCTION
The identification of linear dynamical Multiple Input-
Multiple Output (MIMO) systems achieved in the last two
decades a remarkable development from the so-called sub-
space methods [1], [2], [3]. The main appealing feature
of these methods over the more traditional error prediction
methods [4], is that they directly provide minimal and not
necessarily canonical state space models from input-output
(I/O) data.
However, the application of subspace identification meth-
ods to the estimation of certain types of systems such as
composite systems with linear constituent submodels [5],
switched linear systems [6], [7] or recursive identification
[8], [9], sometimes comes with some technical difficulties.
One issue is related to the multiplicity of possible bases for
representing the system equations in the state space. For
example, in the case of recursive subspace identification,
most of the existing contributions [8], [9] do not guarantee
that the state basis remains fixed during the whole recursive
identification procedure. This is also a crucial problem when
dealing for example with the identification of multi-models
or switched systems. Indeed, in these cases, all the different
submodels of the system must be obtained in the same basis
[6]. Failing to this requirement may have the consequence
of altering the I/O behavior of the system.
A second issue is related to the SVD algorithm that
is generally required in subspace identification. In fact,
subspace methods involve an SVD step in which one decides
arbitrarily on the basis of the range space of the extended
1 Ecole des Mines de Douai, Département Informatique et Automatique,
59508, Douai, France
2 Laboratoire d’Automatique, Génie Informatique et Signal, UMR CNRS
8146, Université des Sciences et Technologies de Lille, 59655, Villeneuve
d’Ascq, France
3 Université de Poitiers, Laboratoire d’Automatique et d’Informatique
Industrielle, 86022, Poitiers, France
observability matrix to be estimated. An important problem
is that the SVD is computationally heavy and technically
hard to update recursively, thus making its use in online
identification a fastidious task [8]. Moreover, updating an
SVD neither solves the problem of coordinate basis deviation
during the estimation. Therefore, an attempt to reconstruct
the I/O behavior from the estimated matrices may result in
a shifted behavior.
In this paper, we focus on developing a new, simple,
efficient and SVD-free identification method for linear dy-
namical state space models, that could overcome the afore-
mentioned problems. Since a state model holds only up to
a similar transformation, one can indeed choose in advance
the basis of the model to be identified. This is carried out
by introducing a user-defined matrix Λf (notation of the
paper), that preserves the rank properties of the extended
observability matrix when multiplying this latter matrix on
the left. We then show that, under the assumption that the
considered system is observable, if one draws Λf randomly
from a uniform distribution for example, we can obtain
a consistent realization of the system. Being based on a
conversion of the subspace estimation problem into a Least
Squares problem, the developed method can be regarded as
an interesting solution to the problem of subspace tracking
frequently encountered in signal array processing, recursive
system identification and many other applications [10].
The outline of the paper is as follows. In Section II we
formulate the subspace identification problem. A relevant
sufficiency of excitation concept is defined and used to
derive conditions that guarantee consistency of the subspace
identification. In Section III, we present a new subspace-
based identification algorithm. Some illustrative simulation
results shown in Section IV demonstrate the applicability of
our method.
II. PROBLEM STATEMENT
We consider a Linear Time-Invariant (LTI) system de-
scribed by a discrete-time model of the form
{
x(t+ 1) = Ax(t) +Bu(t) + w(t)
y(t) = Cx(t) +Du(t) + v(t), (1)
where x(t) ∈ Rn, u(t) ∈ Rnu , y(t) ∈ Rny are respectively
the state, the input and output vector and w(t) ∈ Rn and
v(t) ∈ Rny symbolize the process noise and measurement
noise. Here, these noises are assumed to be zero-mean
white noise processes. A,B,C,D are the system matrices
relatively to a certain basis of the state space.
The identification problem can then be formulated as
follows: given I/O data {u(t), y(t)}Nt=1 generated by a system
Proceedings of the
47th IEEE Conference on Decision and Control
Cancun, Mexico, Dec. 9-11, 2008
WeC04.3
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