A generalisation of the entropy of a plane curve to R n space is provided and the generalised entropy is used to evaluate the nonlinear behaviour of dynamical systems. The entropy of a curve, first introduced within the theory of thermodynamics of plane curves, has been used to quantify the irregularity of a curve. This paper extends the concept to higher dimensions and provides an algorithmic procedure to evaluate the entropy of a curve evolving in the phase space according to the equations of a dynamical system. The thermodynamic indicator is eventually used to infer some properties about the dynamical system. In particular, according to the proposed indicator, all linear systems evolve at entropy zero, while higher entropies characterise nonlinear systems, leading to an effective criterion for their classification.

In this paper a vision system for the control quality in a factory of corrugated cardboard is described. The problem under study is the automatic detection and monitoring of the sheets of cardboard constituting a unit of sale. A precise measurement of the number of sheets is a difficult task, since, during the process, some sheets must be discarded, because of their inferior quality, or because they have been damaged during the transport. Usually, the number of the sheets is estimated from the ratio between the weight of the unit and the nominal weight of a single sheet. Since the weight of a sheet is uncertain, an estimation based on the weight ratio is poor and its reliability is very low. A different solution is proposed in this paper: it estimates the number of the sheets via a suitable vision system. It offers a very high precision (the middle error is smaller of 2%) and it has the advantages to avoid invasive interactions with the process of production and to be a low cost solution.

Vibrational controI is a non classical technique proposing the utilisation of zero mean parametric vibrations namic system. Several theoretical resuits on vibrational control are available in literature. In the case of nonwhere A0 is a consrant vector andf(t) is an almost periodic vector function with average value equal to zero (MA2 vector). Following [2 J, (1) assumes the form for shaping the response of a hear or nonlinear dy-where F, is a vector function linear with respect tof(t). linear systems computational &culties arise and theoretical results cannot easily be applied. A new averaging technique, based on the Taylor series expan-1 DefiIlitiOnS 1. the vibrations are named as vector additive if (4) sion of the nonlinear system, is proposed in order to overcome such difliculties.FI(k f(0) = L (0 and L(t) is an APAZ vector. If all but the last component of L are zero the vibrations are named as 1. Introduction Vibrational control [l] [2] [3] makes use of zero mean parametric excitation so that on line measurements on AP-forcing 2. the vibrations are named as linear multiplicative if (5) the system are not necessary. If measurements are not and feed forward techniques) fail but vibrational control can be s u m used as an open loop tool for achieving the control objectives. Vibrational control has been proposed as an effeaive control tool in several available, traditional control methods (such as feedback FI(% fW) = B(t) x Following [3], consider the reduced equation x = FJx, f(0) (6) applications<[4], [5], [6], [7], [SI). Nevertheless some difiIculties are present both in the theory and in practi-Assume that its solution is given bycal applicatioi of vibrational control when the controlled plants are nonlinear. Section 2 presents a short description of the traditional procedure to obtain the averaged system Section 3 describes a new automatic procedure in order to produce a suitable expression of the averaged system for every complex equation. Sec-where c is a CoIlstant uniquely dehed for every initial condition xoConsider the transformation of variables:tion 4 descriis two examples of application.) A short description of vibrational control theory for Substituting (8) into (1) 2. Preliminaries nonlinear systems is presented A complete treatment of the problem is reported in [2] and in [3]. y = y (tYl (9)Introducing the averaged equation, (9) becomes Consider the nonlinear system X = F (x, a)x ER", A. Erm (1)with parame&ic vikirations as Z=Z(z) where @7803-1872-2/94/$4.CMlO 1994 IEEE