BEHAVIORAL APPROACH OVER A LIPSCHITZ BOUNDED SET

摘要

In this article we discuss on a behavioral approach restricted over a set of functions whose second derivative are bounded to constant C. The condition is equavalent to the Lipschitz condition to the first derivative of the functions. At first characterization of the bounded set restricted by Lipschitz consatant is discussed. Several kinds of examples are introduced and the usefulness of the set is indicated. Secondly Levant VS differentiator is introduced. The principle of the differentiator is a high gain feedback with integrator in the feedback path. By choosing an appropriate parameters and switching speed, variable structure system can realize a high gain feedback system and practical differentiator can be composed. The choice of the parameters are closely related to the Lipschitz constant of the functions. Finally the invariant property of the function is discussed. The Levant differentiator can release the restriction of the properness of the system. Here we examine the conditions which ensure the application of non-proper systems. Discussion in the article is a challenge of the application of the behavioral approach to practical engineering problems. Appropriate restriction of the set of function might lead the behavioral approach from theoretical cite to practical engineering.