New developments for matrix fraction descriptions: A fully-parametrised approach

Abstract

This article aims at giving a new answer for the challenging problem of the parametrisation of multi-input multi-output matrix fraction descriptions. In order to reach this goal, new parametrisations of matrix fraction descriptions, called fully-parametrised left matrix fraction descriptions (F-LMFD), are first introduced. Their structural properties as well as their suitability for multi-input multi-output model description are more precisely analysed. As any over-parametrised model description, the F-LMFD cannot describe a transfer function uniquely. The structure of the space of equivalent F-LMFD is then investigated through the determination of its basis. The study carried out in this article is the prelude to a computational improvement of the identification of matrix fraction descriptions with gradient-based optimisation methods.

Introduction

In this article, the problem of the parametrisation of Multi-Input Multi-Output (MIMO) Linear Time Invariant (LTI) systems of a given order $n_x$ is addressed. More particularly, Matrix Fractions Descriptions (MFD) of MIMO transfers are considered. As underlined in Gevers (2006), methodological research on MIMO system parametrisation has been neglected since the 1990s in aid of subspace-based system identification methods. This can mainly be explained for two reasons. First, the tricky problem of finding appropriate parametrisations of MIMO systems is bypassed when working with subspace-based methods (Gevers, 2006). Second, optimisation algorithms often showed poor convergence properties with the minimal parametrisations used to ensure the model identifiability.

Mostly in the 1980s, several parametrisations of MIMO systems were derived. First, a lot of studies focused on canonical parametrisations (Clark, 1976, Hazewinkel and Kalman, 1976). These studies were motivated by the search for a unique representation of the true rational MIMO system, whose structure depends on a finite set of Kronecker indices. In parallel, in order to bypass numerical problems encountered, e.g., in system identification (Gevers and Wertz, 1987, Glover and Willems, 1974), with canonical parametrisations, pseudo-canonical parametrisations (Gevers & Wertz, 1987), also referred to as overlapping models (Glover and Willems, 1974, Van Overbeek and Ljung, 1982) or multistructural models (Guidorzi & Beghelli, 1982), have been developed. These studies mainly concerned state-space representations (Clark, 1976, Correa and Glover, 1984, Gevers and Wertz, 1987, Glover and Willems, 1974, Van Overbeek and Ljung, 1982). An equivalent work was done about the parametrisation of MIMO transfer functions (Deistler and Hannan, 1981, Guidorzi and Beghelli, 1982) providing a set of parametrisations for both state-space and transfer function representations that have in common three main features: (i) these parametrisations are based on the selection of structural indices, (ii) the set of all rational systems with a fixed order $n_x$ can be covered by a finite number of pseudo-canonical parametrisations (Gevers & Wertz, 1987), (iii) they are minimal parametrisations, i.e., they are made up of the minimal number of descriptive parameters. This latest point comes from the pursued objective of canonical parametrisations in describing every system uniquely. Before the 1990s had indeed predominated the idea that a parametrisation which uniquely represents a system should be chosen so as to get a well-conditioned parameter estimation problem (Gevers, 2006). Another reason can certainly be found in the optimisation algorithm formulation since minimal parametrisations lead to Jacobian matrices of full rank (Wills & Ninness, 2008).

In this article, we introduce new fully-parametrised matrix fraction descriptions of MIMO systems of a given order $n_x$. These parametrisations depend upon a set of structural indices. However, for a given set of indices, these descriptions encompass several canonical descriptions and are therefore less specific. Moreover, we will show that, for a specific and unique choice of indices, the resulting fully-parametrised MFD is related to pseudo-canonical parametrisations. Indeed, for a given order, this full parametrisation is unique and encompasses all the pseudo-canonical descriptions defined in Guidorzi and Beghelli (1982) for MIMO matrix fraction descriptions of order $n_x$. This feature is of main importance when system identification is concerned because the set of model candidates used for model selection must be large enough and well-parametrised to ensure that the algorithm used to optimise the involved cost function gives access to a global minimum.

This article is organised as follows. Section  2 is devoted to the study of the fully-parametrised MFD (F-MFD). In Section  3, the case of F-MFD specified by quasi-constant indices is detailed. Preliminary results about the structure of the equivalence class of MFD are given in this section. In Section  4, the structure of the equivalence class of MFD is further investigated and specified for any choice of structure indices. After comments gathered in Section  5, Section  6 concludes this study and discusses future research perspectives. Before going further, it can eventually be mentioned here that the results and development presented in this article are illustrated with a numerical example which is first introduced in Section  2. The same numerical example is used all along the article with the aim of helping the reader understanding.