Guillaume Mercère

Review: Propagator-based methods for recursive subspace model identification

The problem of the online identification of multi-input multi-output (MIMO) state-space models in the framework of discrete-time subspace methods is considered in this paper. Several algorithms, based on a recursive formulation of the MIMO Output Error State-Space (MOESP) identification class, are developed. The main goals of the proposed methods are to circumvent the huge complexity of eigenvalues or singular values decomposition techniques used by the offline algorithm and to provide consistent state-space matrices estimates in a noisy framework. The underlying principle consists in using the relationship between array signal processing and subspace identification to adjust the propagator method (originally developed in array signal processing) to track the subspace spanned by the observability matrix. The problem of the (coloured) disturbances acting on the system is solved by introducing an instrumental variable in the minimized cost functions. A particular attention is paid to the algorithmic development and to the computational cost. The benefits of these algorithms in comparison with existing methods are emphasized with a simulation study in time-invariant and time-varying scenarios.

Parameterization and identification of multivariable state-space systems: A canonical approach

In this paper, the problem of determining a canonical state-space representation for multivariable systems is revisited. A method is derived to build a canonical state-space representation directly from data generated by a linear time-invariant system. Contrary to the classic construction methods of canonical parameterizations, the technique developed in this paper does not assume the availability of any observability or controllability indices. However, it requires the A-matrix of any minimal realization of the system to be non-derogatory. A subspace-based identification algorithm is also introduced to estimate such a canonical state-space parameterization directly from input-output data.

Identification for gain-scheduling: a balanced subspace approach

The problem of deriving MIMO parameter- dependent models for gain-scheduling control design from data generated by local identification experiments is considered and a numerically sound approach is proposed, based on subspace identification ideas combined with the use of suitable properties of balanced state space realisations. Simulation examples are used to demonstrate the performance of the proposed approach.

Brief paper: Convergence analysis of instrumental variable recursive subspace identification algorithms

The convergence properties of recently developed recursive subspace identification methods are investigated in this paper. The algorithms operate on the basis of instrumental variable (IV) versions of the propagator method for signal subspace estimation. It is proved that, under suitable conditions on the input signal and the system, the considered recursive subspace identification algorithms converge to a consistent estimate of the propagator and, by extension, to the state-space system matrices.

On-line structured subspace identification with application to switched linear systems

This article is concerned with the identification of switched linear multiple-inputs–multiple-outputs state-space systems in a recursive way. First, a structured subspace identification scheme for linear systems is presented which turns out to have many attractive features. More precisely, it does not require any singular value decomposition but is derived using orthogonal projection techniques; it allows a computationally appealing implementation and it is closely related to input–output models identification. Second, it is shown that this method can be implemented on-line to track both the range space of the extended observability matrix and its dimension and thereby, the system matrices. Third, by making use of an on-line switching times detection strategy, this method is applied to blindly identify switched systems and to label the obtained submodels. Simulation results on noisy data illustrate the abilities and the benefits of the proposed approach.

Identification of a flexible robot manipulator using a linear parameter-varying descriptor state-space structure

This paper presents a new approach for the identification of the dynamical model of flexible manipulators. The structure of the identified model, chosen as a descriptor LPV model, is derived from the original non-linear equations. A set of experiments around different configurations is involved, which is suitable for an accurate measurement of the tip of the manipulator by video camera. The final estimation step is global, allowing the direct identification of the global model based on the collection of local experimental data. In an output error context, a genetic algorithm is used for the minimization of the identification criterion. As a case study, a robotic arm with two flexible segments is considered. Identification results based on simulations including noise show the effectiveness of the approach.

A New Recursive Method for Subspace Identification of Noisy Systems: EIVPM

In this article, a new recursive identification method based on subspace algorithms is proposed. This method is directly inspired by the Propagator Method used in sensor array signal processing to estimate directions of arrival (DOA) of waves impinging an antenna array. Particularly, a new quadratic criterion and a recursive formulation of the estimation of the subspace spanned by the observability matrix are presented. The problem of process and measurement noises is solved by introducing an instrumental variable within the minimized criterion.

Identification of Parameterized Gray-Box State-Space Systems: From a Black-Box Linear Time-Invariant Representation to a Structured One

While determining the order as well as the matrices of a black-box linear state-space model is now an easy problem to solve, it is well-known that the estimated (fully parameterized) state-space matrices are unique modulo a non-singular similarity transformation matrix. This could have serious consequences if the system being identified is a real physical system. Indeed, if the true model contains physical parameters, then the identified system could no longer have the physical parameters in a form that can be extracted easily. By assuming that the system has been identified consistently in a fully parameterized form, the question addressed in this paper then is how to recover the physical parameters from this initially estimated black-box form. Two solutions to solve such a parameterization problem are more precisely introduced. First, a solution based on a null-space-based reformulation of a set of equations arising from the aforementioned similarity transformation problem is considered. Second, an algorithm dedicated to nonsmooth optimization is presented to transform the initial fully parameterized model into the structured state-space parameterization of the system to be identified. A specific constraint on the similarity transformation between both system representations is added to avoid singularity. By assuming that the physical state-space form is identifiable and the initial fully parameterized model is consistent, it is proved that the global solutions of these two optimization problems are unique. The proposed algorithms are presented, along with an example of a physical system.

A Null-Space-based Technique for the Estimation of Linear-Time Invariant Structured State-Space Representations*

Abstract Estimating the order as well as the matrices of a linear state-space model is now an easy problem to solve. However, it is well-known that the state-space matrices are unique modulo a non-singular similarity transformation matrix. This could have serious consequences if the system being identified is a real physical system. Indeed, if the true model contains physical parameters, then the identified system could no longer have the physical parameters in a form that can be extracted easily. The question addressed in this paper then is, how to recover the physical parameters once the system has been identified in a fully-parameterized form. The novelty of our approach is on transforming the bilinear equations arising from the similarity transformation equations as a null-space problem. We show that the null-space of a certain matrix contains the physical parameters. Extracting the physical parameters then requires the solution of a non-convex optimization problem in a reduced dimensional space. By assuming that the physical state-space form is identifiable and the initial fully-parameterized model is consistent, the solution of this optimization problem is unique. The proposed algorithm is presented, along with an example of a physical system.

Identification of switched linear state space models without minimum dwell time

We consider the problem of identifying switched linear state space models from a finite set of input-output data. This is a challenging problem, which requires inferring both the discrete state and the parameter matrices associated with each discrete state. An important contribution of our work is that we do not make the restrictive assumption of minimum dwell time between the switches, as it is customary in methods that deal with such models. We first propose a technique for eliminating the unknown continuous state from the model equations under an appropriate assumption of observability. On a time horizon, this gives us a new switched input-output relation that involves structured intermediary matrices, which depend on the state space representation matrices. To estimate the intermediary matrices, we present a randomly initialized algorithm that alternates between data classification and parameter update via recursive least squares. Given these matrices, the parameters associated to the different discrete states can be computed after a correct estimation of the discrete state.