USE PROPERTIES OF DISTRIBUTIONS TO MODEL THE SYSTEMS WITH SEVERE NONLINEARITIES

Many times, in science and technology, we need to model and simulate some phenomena and with this we encounter lots of difficulties caused by the presence of certain nonlinear elements, which cannot be linearly approximated, called severe nonlinearities. This kind of nonlinearities contains discontinuities and non-derivable points. In other words, many nonlinearities are defined by some relations and not by the functions. This situation appears most frequently in the study of the systems when we have a mathematical model to simulate them. A solution proposed in this paper is to use a distributions theory that extends the class of functions. In order to refer to the class of nonlinear systems for which the theory of distributions will be applied, a brief presentation of nonlinear systems versus linear systems will be made. In the following paper the distribution concept, the elementary distributions were defined and several distributions and properties were illustrated by a simulation. Using properties of distributions, complex nonlinear problems, like control systems design, can be treated much easy. In this paper we can see models and simulations results for several discrete control applications used in control engineering and power electronics. The distributions are suitable for software implementation and by this offers very good support for embedded systems with software-oriented solution. Many distributions models were known for a long time, but were not used in applications. In this paper the distributions are modeled, simulated and many applications which can be used as guides in the future, was developed. As a conclusion, the distribution properties are very suitable for the development of nonlinear mathematical models especially for ones with severe nonlinearities, and through this to design controllers or other nonlinear devices.