Title:  Infinitesimal Modeling of Impulsive Systems: A Non Standard Analysis Approach

Committee: 

Dr. Verriest, Advisor     

Dr. Vela, Chair

Dr. Dieci

Dr. Ames

Abstract:

The objective of this proposal is to introduce a new framework for modeling nonlinear impulsive systems emphasizing the cause versus effect by using the generalized functions defined on the hyperreal space in Nonstandard Analysis (NSA). The discrete jump equation in classical nonlinear impulsive systems can be eliminated by formulating an equivalent generalized ordinary differential equation (GODE) where its generalized solution displays the same jump behavior. The first task is to construct an algebraically structured extended real space in the hyperreals in order to simplify the space of infinitesimals in NSA. The proposed space is denoted as a Krylov space since its construction is similar to the Krylov subspace method in numerical linear algebra. Next, a generalized piecewise continuous function is defined on the Krylov space using two basic operators, scaling and translation. By introducing an extended differentiation on the space of piecewise differentiable functions that satisfies the Leibniz product rule, we derive a singular delta function on the Krylov space. The proposed framework shows that every generalized function can be differentiated, and the multiplication between generalized function is point-wise well defined. Finally, the GODE is now formulated using the extended differentiation, and the piecewise continuous solution is found in the new generalized function space. Now, modeling the cause of the discrete jump in classical impulsive systems is proposed by involving the new singular function as an input to the system. The new framework can be used to model impulsive contact forces acting on several mechanical system.