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SUBSPACE IDENTIFICATION

There is a demand from industry for reliable identification tools   suited for the identification of large systems with tens of inputs and outputs. Traditional parameter-optimization-based identification methods (such as PEM) are difficult to apply   to  multivariable system. They may lead to  unreliable results due to many local minima and bad conditioning.  On the other hand,  for the identification of multivariable state-space models, modern subspace identification methods have been shown to compare very favorably with traditional PEM methods. Contributions to subspace identification have been given by the unit of Padova in the past years, both by developing new algorithms  and by providing statistical analysis of    several existing  algorithms.
 The main open questions in subspace  identification  regard closed-loop identification, the statistical properties of the estimates   and reliable recursive identification.
 Subspace Identification of closed loop was an open problem until a couple of years ago when   new algorithms were presented  with essential contributions from the Padova group. The  statistical properties of subspace methods are still largely unknown. New transparent expressions for the asymptotic variance of subspace estimates have recently been found both for open loop and “closed-loop” algorithms but there is much work to be done in the area.
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RECIPROCAL PROCESSES:  IDENTIFICATION AND COVARIANCE EXTENSION

Modeling and identification of finite support signals (for example signals depending on one or two spatial coordinates like images)  has traditionally been approached by  describing these signals as   truncations of stationary signals defined on the whole time axis. This approach leads to well-known problems and drawbacks related to truncations, border effects etc.
  A class of statistical models called reciprocal processes which intrinsecally live on a finite support set  has   recently been popularized in a series of papers by  A. Krener, B. Levy   and co-workers.
We believe that stationary reciprocal processes are a very natural class of models for treating estimation and   processing   of signals defined on a finite set.  In particular the identification of these processes is an important open problem which we have just started to look at [I18],[I 19]. Its solution would surely lead to a significant advance in several areas of signal processing and control.
 
MODELING AND IDENTIFICATION OF LINEAR MECHANICAL SYSTEMS

 The identification of linear models for large mechanical systems is a very important  research area with diverse  applications such as the analysis,  design and non-destructive testing of large compliant structures. The parameters of interest (e.g. mass and stiffness matrices) are defined  in terms of   continuous-time   models but identification algorithms are invariably based on discrete data and produce discrete-time models. Quite often the recovery of a continuous-time model from a discretization   is a very ill-conditioned problem and    therefore one needs special discretization techniques  which permit an accurate  recovery of  the continuous-time parameters from the identified discrete-time model. One would also like to    preserve  the physical meaning of basic mechanical quantities such as total energy etc.  The theory of variational integrators, a new discretization  technique   of the Lagrange equation of motion of a mechanical system,  leads to simple and well-conditioned transformation formulas for the recovery of the continuous time parameters from the discretized model. We plan to use this theory in identification. Preliminary results [I 22] seem encouraging.