Parameterization and identification of multivariable state-space systems: A canonical approach☆
Introduction
The parameterization of linear time-invariant (LTI) dynamical systems and the determination of canonical forms are subjects which widely interested the automatic control community in the 1960s, the 1970s and the 1980s (Denham, 1974, Guidorzi, 1989, Kailath, 1980, Kalman, 1963, Luenberger, 1967, Silverman, 1966). The canonical forms are of importance mainly because they provide a unique parameterization of the studied system in the model set. This property can be exploited in the system identification framework because the canonical forms generally give rise to an identifiable model structure. It can be really useful when the parameters of the model are estimated by optimizing particular cost functions (such as the sum of squared errors (Ljung, 1999)) because an injective parameterization leads to a unique minimum in the parameter space (McKelvey, 1995). They are also a convenient starting point for some control design problems (see, e.g., the pole placement with Ackermann’s formula (Kailath, 1980)). They can be useful for a one-to-one correspondence with particular matrix fraction scriptions of LTI systems. Hence, the parameters of the canonical forms can have a direct interpretation as particular transfer function coefficients. Finally, it is interesting to notice that the increasing need of relevant and accurate model parameterizations for systems described by linear time-varying (Guidorzi & Diversi, 2003), linear parameter-varying (Tóth, 2008) or switched linear structures (Bako et al., 2009, Petreczky, 2006) has triggered a recent surge of interest in linear canonical forms.
Initially developed for single input (SI) systems, most of the methods developed in the 1960s have been extended for multi-inputs multi-outputs (MIMO) systems. The basic idea of the methods developed for MIMO systems mainly consists in selecting particular structural indices, sometimes corresponding to the Kronecker indices (Denham, 1974, Kailath, 1980), so that specific vectors, generally extracted from the observability or controllability matrix of the system, are gathered into a full rank square matrix in order to be used as a similarity transformation (see also, e.g., Ljung, 1999, Appendix 4A).
The parameterization of multivariable systems is still considered as a difficult problem, especially when MIMO minimal forms are required (Ljung, 1999). This challenge is related to the fact that a bijective parameterization does not exist when both the input and the output dimensions are greater than one. The parameterization of MIMO systems is still studied in fields such as system identification (McKelvey, 1995, Petreczky, 2006, Ribarits, 2002). Indeed, the determination of the model structure has a strong impact on the model accuracy and deeply affects the numerical properties of the identification algorithms (McKelvey, 1995), especially when a nonlinear optimization is performed. When LTI state-space systems are considered, in order to get round the problem of the model structure selection, the user can resort to a fully-parameterized state-space model (McKelvey, 1995). The main advantages of this parameterization are its simplicity and the fact that the McMillan degree is the only integer to be estimated. Unfortunately, the introduction of extra degrees of freedom makes this parameterization non-injective. To circumvent these difficulties, several approaches have been developed.
- •
As far as the nonlinear least-squares-based methods are concerned, two main solutions have been suggested. First, particular parameterizations with fewer parameters (such as the tridiagonal matrix form or the output normal form) and dedicated minimization algorithms have been developed (see Verhaegen & Verdult, 2007 for an interesting overview). Then, regularized version of the Gauss Newton algorithm have been considered to deal with fully-parameterized state-space representation (McKelvey, 1995, Ribarits, 2002). However, as nonlinear least-squares-based methods, the related optimization problems are still quite complex because, in many practical cases, local minima are present. Hence, these methods are very dependent on the quality of the initial estimate in order to ensure the global convergence. Interesting developments concerning the initialization for system identification can be found in Tohme (2008) or in Lyzell, Enqvist, and Ljung (2009).
- •
In parallel to these maximum-likelihood methods, the subspace-based algorithms (Katayama, 2005, Van Overschee and De Moor, 1996, Verhaegen and Verdult, 2007) have been introduced in order to face the challenge of fully-parameterized state-space model identification. By storing the input and output data into structured block Hankel matrices, the state-space matrices are estimated, up to a similarity transformation, by extracting particular subspaces related to the system. By using robust linear algebra tools, the model of the system is determined in a non-iterative way without performing any nonlinear optimization. In this particular framework, the degrees of flexibility are translated into the elements of the similarity transformation which parameterizes the equivalence class. Interesting from a numerical point of view, this more flexible model structure and the fact that no explicit cost function is minimized to estimate the model matrices make
- –
the introduction of prior knowledge difficult,
- –
the determination of the identified parameter uncertainty limited to particular invariants such as the poles (Viberg, Ottersten, Wahlberg, & Ljung, 1991) or the frequency responses of the estimated models (Bittanti & Lovera, 2000),
- –
the statistical analysis of the estimates quite tricky (see, e.g., Bauer, 1998, Bauer and Jansson, 2000, Chiuso and Picci, 2004 for the main developments).
- –
The paper is organized as follows. In Section 2, the main notations are introduced and the problem is formulated. Section 3 is dedicated to the description of the state-space canonical form, i.e., the way it is obtained and described as well as its main properties. The use of this parameterization in the identification framework is presented in Section 4. Numerical examples are introduced in Section 5 in order to illustrate the performance of the developed approach. Section 6 concludes this paper.
Section snippets
Problem formulation and notations
Consider the following discrete-time linear time-invariant multivariable state-space form where and are respectively the input, the output and the state of the system. are the state-space matrices of the system relatively to a certain state-space coordinate basis.
In the following, assuming that
- (1)
the order of the system is known a priori,
- (2)
is minimal,
- (3)
the matrix is non-derogatory,
Basic idea and construction
Roughly speaking, the basic idea of the developed approach consists in building a similarity transformation such that the corresponding similar state-space representation is expressed into a minimal and identifiable canonical form. To reach this goal, introduce the extended observability matrix defined as follows where is a user-defined index such that . Then, assuming that the system is observable, it holds that has linearly independent rows. The
Application to subspace-based identification
The parameterization problem must be solved when system identification is concerned. The determination of the model structure is “no doubt the most important and, at the same time, the most difficult choice of the system identification procedure” (Chapter 1 Ljung, 1999). When state-space models are considered, according to the method used for the parameter estimation, particular parameterizations must be handled (see Verhaegen & Verdult, 2007 for an interesting description of dedicated
Numerical examples
In this section, numerical properties of the approach presented in this paper are illustrated briefly. For this purpose, a system of the form (1) is considered where and are given by
The example (30) is intentionally chosen such that the modes of the system are not entirely observable (resp. controllable) from a single output (resp.
Conclusion
In this paper, a method has been introduced to build and to estimate a MIMO canonical state-space model directly from data. Contrary to the conventional approaches, the suggested method does not require the so-called observability/controllability indices (or equivalent indices) in order to build canonical forms for multivariable systems. In this article, the similarity transformation matrix depends on a user-defined vector generated such that . It is interesting to point
Guillaume Mercère was born in Cambrai, France, in 1977. He received the M.S. degree in electrical engineering from Caen Engineering School, Caen, France, in 2001, the Ph.D. degree in automatic control from Lille University, Lille, France, in 2004. Since September 2005, he has been an Associate Professor with Poitiers University, Poitiers, France, and a member of the Automatic Control and Electrical Engineering Laboratory of Poitiers. He is currently co-leader of the French Workgroup on System
References (34)
- et al.
Analysis of the asymptotic properties of the MOESP type of subspace algorithms
Automatica
(2000) - et al.
Bootstrap-based estimates of uncertainty in subspace identification methods
Automatica
(2000) - et al.
The asymptotic variance of subspace estimates
Journal of Econometrics
(2004) - et al.
Minimal representations of MIMO time-varying systems and realization of cyclostationary models
Automatica
(2003) - et al.
Propagator-based methods for recursive subspace model identification
Signal Processing
(2008) - et al.
Convergence analysis of instrumental variable recursive subspace identification algorithms
Automatica
(2007) - et al.
Identification of hybrid systems: a tutorial
European Journal of Control
(2007) - et al.
A unifying theorem for three subspace system identification algorithms
Automatica
(1995) - et al.
On-line structured subspace identification with application to switched linear systems
International Journal of Control
(2009) - Bauer, D. (1998). Some asymptotic theory for the estimation of linear systems using maximum likehood methods or...
Canonical forms for the identification of multivariable linear systems
IEEE Transactions on Automatic Control
Parametrizations of linear dynamical systems: canonical forms and identifiability
IEEE Transactions on Automatic Control
Equivalence, invariance and dynamical system canonical modelling. Part I
Kybernetika
Matrix analysis
Linear systems
Mathematical description of linear dynamical systems
SIAM Journal on Control
Cited by (97)
Online state of health estimation of lithium-ion batteries through subspace system identification methods
2024, Journal of Energy StorageData-driven solutions to spacecraft relative attitude-position fault-tolerant control
2023, Advances in Space ResearchAn improved model predictive control approach for fuel efficiency optimization of vessel propulsion systems
2021, Control Engineering PracticeCitation Excerpt :Since the VPSs have strong coupling and complex nonlinearities, which are difficult to capture by the mechanism modeling method, identifying the VPS dynamic model from the historical data is an effective way. At present, the subspace-based technique is a typical data-driven modeling method, which has simple procedures, but it is mainly used in the linear system identification (Mercère & Bako, 2011). Another typical method is based on the maximum likelihood framework, by which the nonlinear state–space model can be obtained by using the expectation maximization algorithm (Schön et al., 2011).
Data-driven digital twin technology for optimized control in process systems
2019, ISA TransactionsThe sensor-actuators stealthy cyber-attacks framework on networked control systems: A data-driven approach
2024, Asian Journal of Control
Guillaume Mercère was born in Cambrai, France, in 1977. He received the M.S. degree in electrical engineering from Caen Engineering School, Caen, France, in 2001, the Ph.D. degree in automatic control from Lille University, Lille, France, in 2004. Since September 2005, he has been an Associate Professor with Poitiers University, Poitiers, France, and a member of the Automatic Control and Electrical Engineering Laboratory of Poitiers. He is currently co-leader of the French Workgroup on System Identification. His main research interests include subspace-based identification and linear parameter-varying systems identification. His current activities focus on heat exchangers, flexible and cable-driven manipulators and aeronautics.
Laurent Bako received a “diplme d’ingénieur” in Electrical Engineering from Ecole Nationale Suprieure d’Ingénieurs de Poitiers and the M.Sc. degree from Universit de Poitiers, both in 2005. He was a visiting researcher in the Center for Imaging Sciences, at the Johns Hopkins University in the SummerFall of 2007. In 2008 he obtained the Ph.D. degree in Automatic Control and Computer Sciences from Universit des Sciences et Technologies de Lille. He has been serving as an assistant professor at Ecole des Mines de Douai, in the Department of Computer Sciences and Automatic Control since December 2008. His research interests are mainly in control theory, system identification, hybrid systems, machine learning.
- ☆
This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Martin Enqvist under the direction of Editor Torsten Söderström.