Ph.D. received on: 16/4/1998

E-mail: : califano@riscdis.ing.uniroma1.it

Tutor: Prof. S. Monaco, Università di Roma ‘La Sapienza’
 
 

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Discrete-Time Systems: A Geometric Approach  __________________________________________________________________________________________________________

Advisor:

Prof. S. Monaco, Università di Roma ‘La Sapienza’

Summary:

In the present work analysis and control laws design problems have been addressed for the class of discrete time nonlinear systems. As well known, this class of systems has a great interest, both theoretical, since discrete time dynamics well describe the behaviour of real systems such as biological and economical ones, and practical, due to the increasing use of digital controllers in control processes.
The study has been developed referring, mainly, to a methodology recently introduced in the literature by Monaco and Normand-Cyrot (1990). This methodology is based on the use of a formal exponential representation in order to describe discrete time dynamics. The interest in such representation lies in the possibility of separating, in the dynamics evolution, the discontinuous contribution of the drift from the continuous action of the control. In particular the dynamics dependence of the control is described by a family of canonical vector fields, which, together with the Lie algebra associated to them, characterize the controllability directions. In this way a suitable geometric framework is defined.
On the basis of this representation the first part of the thesis is devoted to the analysis of discrete time dynamics structural properties. In this context some results regarding the existence of particular canonical forms, the normal form between them, are presented. Moreover starting from the well known conditions of linear feedback equivalence, a method for computing the largest linearizable subsystem is proposed.
Starting from the concept of invariance, which finds in this framework its natural formulation, some local decompositions linked to the reachability and observability concepts are considered. Algorithms for the definition of these decomposition are therefore proposed. Finally, with reference to some geometric structures, properties such as controlled invariance and controllability are analysed.  Also for the last one a constructive algorithm is proposed.
A great deal of efforts  has been devoted to the decoupling problem with stability. In this context the exponential representation allows us to characterize the geometric properties of the class of decoupled systems, in terms of invariant distributions. This allows us, as for continuous time affine systems, to compute an appropriate coordinates change which points out the obstruction encountered to reaching decoupling and stability. It is then possible to give necessary and sufficient conditions for the existence of a regular static state feedback which solves the problem.

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