ERCIM News No.24 - January 1996 - CNR

**by Andrea Gombani**

**The Systems Theory Group has been active at LADSEB - CNR, Padua, since 1973. Strong collaborations have been under way for some time with a number of ERCIM partners: CWI, INRIA and SICS, and more recently with SZTAKI. The group has participated in three European Union projects (two Science projects and one HCM). It is also belongs to the recently formed ERCIM Working Group on Control and Systems Theory (see Joint ERCIM Actions in this issue).
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The Systems Theory group at LADSEB- CNR, Padua, is very active in the areas of Stochastics Systems and Control. All its members have had extended stays (at least three years, as students or visiting professors) at foreign research institutions (mainly in the United States). The members of the group are either permanent staff of the lab (2 scientists) or hold university posts but are also involved in ongoing research activities at LADSEB (5 collaborators). The group has had visits from about 60 scientists working in the field over the past five year.

Some of the most significant research areas include:

Stochastic Realization and Identification of Systems: The stochastic realization problem is to characterise all state space representations of a dynamic system with a given input/output behaviour. The motivation for this problem comes from modelling an observed phenomenon by a dynamic system. Once a representation has been obtained, powerful filtering and control techniques can be used to study the process: hence the relevance of the problem. The Gaussian linear case has been solved by Picci (LADSEB) and Lindquist (SICS). Recent results have been obtained for the nonlinear case. Among the applications we quote the noncausal estimation problem, the backward representation of stochastic processes, the approximation of a given process, the construction of approximate Wiener-Kolmogorov filters admitting an a priori bound on the estimation error. Ideas of stochastic realization have been employed in the construction of factor analysis models, of modelling of multichannel transmission and in the study of stochastic aggregation of linear Hamiltonian systems in statistical mechanics. Stochastic realization tools have been successfully developed to obtain new state space algorithms for identification and new parametrizations of linear systems for identification.

Control: Approximation techniques have been studied for problems of stochastic control with partial observations. In particular, using the measure transformation technique, approximate problems have been obtained whose almost optimal solutions are in fact almost optimal for the original problem as well, with arbitrarily small error. Recently, the connections between piece-wise linear control systems and adaptive controllers have been studied, with the purpose of showing the relevance of adaptive techniques for the control of nonlinear systems. In particular, Fleming's logarithmic transform has been extended to the discrete case.

Conditions have been obtained for the existence of the solution to the matrix Riccati equation for the linear quadratic control problem over a finite horizon, which does not require definiteness of the criterion and the initial condition. Relations between robust control and Hankel operators are being investigated at LADSEB, in addition to approximation problems.

Nonlinear filtering: In this area the focus has been on derivation and approximation of filtering algorithms for nonlinear stochastic systems. The techniques employed were the approximation of the nonlinear filtering equation and measure transformations to construct approximate models. Both techniques yield approximants with an explicit bound on the approximation error, and therefore they allow for a choice of parameters yielding an arbitrary small error. Relations between filters for adaptive linear systems and for nonlinear systems have been studied. Linear filters of minimal dimension for finite state systems have also been investigated. The tools developed so far result particularly useful in a new area, financial mathematics, where some of the research effort is now being concentrated.

Order Estimation: The focus has been on the class of finite state systems For the class of finite Markov chains a complete characterization has been obtained of the consistent estimators of the order based on the penalized maximum likelihood technique. Estimators of the order optimal in a generalized Neyman Pearson sense have also been obtained. Related results have been obtained for the class of Hidden Markov Models.

Andrea Gombani - LADSEB-CNR

Tel: +39-49-8295756

E-mail: gombani@ladseb.pd.cnr.it

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