From fractality to non integer derivation : A fundamental idea for a new process control strategy

Abstract

For many years, it is common to speak of the notion of robustness. But this notion is very wide, even in a same domain such as the automatic control one. In this one, it is frequent to consider the robustness as far as stability is concerned.

Our work concerns a much stricter robustness type, that is to say the robustness as for stability degree or damping versus the plant parameters.

This paper specially tries to find the mathematical origin of this robustness kind in nature. The natural phenomenon considered is the one of the ebb and flow consecutive to the damping of water on a porous dyke.

It appears that the mathematical origin is the non integer derivation. Indeed, taking into account the fractality of porosity and the corresponding recursivity, we show that the differential equation which describes the process is characterized by a non integer order between 1 and 2 : the damping robustness is due to a non integer degree characteristic equation whose roots represented in the operational plane form a constant angle, independently of the motion water mass. The aim is to obtain the same thing in automatic control, that is to say a control which should be characterized by such an equation.

It is indeed interesting to use this fundamental result to try to synthesize a robust control. Besides it is the purpose of the second part of our expansions, since we specify the open loop frequency response of a robust control.