GEOMETRIC METHODS IN DYNAMICAL SYSTEMS MODELLING: ELECTRICAL, MECHANICAL AND CONTROL SYSTEMS

摘要

The problems concerning dynamical systems modelling are discussed from a geometric point of view. The constitutive space of a dynamical system is considered as a subset of the tangent bundle to the manifold (the generalized hypersurface) being the available space of the system. The constitutive space of the (differential) dynamical system implicitly encloses all the necessary information which is sufficient to select the solution space and the infinitesimal generator of the system which enables the design of the dynamic behavior of the system. The differential inclusions then appear in a natural way in the modelling of physical systems as implicitly written differential systems, where sets of differential algebraic relations describe the constitution of the system. The constructive definition of a differential dynamical system considered here has been obtained through generalization of the mathematical model of a non-linear electrical network extended next to lumped mechanical systems and control systems. The constructive approach to differential dynamical systems presented in this paper corresponds to the procedure commonly used when a mathematical model of a physical system is set up. The subbundles of the tangent bundles to manifolds being mathematical models of the configuration spaces of dynamical systems are the basic mathematical tools. A general iterative procedure for selecting the solution space of the dynamical system is proposed, which extends the formulation given in the references. Examples illustrating the considerations are given.