VIRTUAL CELLULAR MANUFACTURING
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A Definition of Hybrid Systems The last several decades have witnessed an explosive development in th e theory of dynamical systems, much of which is oriented toward equations of the form
= F(x,p)
(1)
where F : V C R ~ • R k --+ R n is a smooth map defined on an open, connected subset of Euclidean space and possibly dependent upon a vector of parameters, p E R k. However, many areas of application frequently involve hybrid systems: dynamical systems which require a mixture of discrete and continuously evolving events. Natural examples of such situations are mechanical systems which involve impacts, control systems that switch between a variety of feedback strategies, and vector fields defined on manifolds described by several charts. In this note we present a uniform approach for representing hybrid systems which applies to a wide variety of problems and, in addition, is particularly well suited for computer implementation. Simulation tools which exploit the approach that we describe are being developed in a Computer program called d s t o o l [1]. Our theoretical objective may be viewed as the natural extension of Equation (1) in two fundamental ways: First, we wish to generalize the underlying domain of F, denoted V above, to include open sets composed of many components where the vector field may vary discontinuously from component to component. Second, we want to accomodate the situation that discrete events depending upon the phase space coordinates or time may occur in the flow of the dynamical system. It is worth pointing out that these goals represent fundamental changes in the theoretical character of the dynamics. By satisfying the first requirement, we greatly increase the class of systems under study, but
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must generalize the underlying theory that supports the simulation and analysis of such systems. The second extension demands a formulation that treats the discrete event components, or some representation of them, as equal members of the underlying phase space with the continuous parts. Assume the problem domain may be decomposed into the form:
v = Uv, oLE!
where I is a finite index set and Vs is an open, connected subset of R ~. We shall refer to each element in this union as a chart. Each chart has associated with it a (possibly time-dependent) vector field, f~ : Vs x R -+ R'L Notice that the charts are not required to be disjoint. Moreover, for a,/3 9 I we do not require continuity, or even agreement, of the vector fields on the intersection set V~ N VZ. We introduce further structure by requiring that for each a 9 I, the chart Vs must enclose a patch, an open subset Us satisfying U~ C Vs. The boundary of Us is assumed piecewise smooth and is referred to as the patch boundary. Together, the collection of charts and patches is called an atlas. To implement this model, a concrete representation of the patch boundaries must be selected. For each a 9 [ we define a finite set of boundary functions, hs,i : Va ~ R, i 9 J~Y, and real numbers called target values, C~#, for i 9 jbf that satisfy t
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