# Mnesors For Automatic Control

Gilles CHAMPENOIS
Collège Saint-André, Saint-Maur, France gilles_champenois@yahoo.fr ABSTRACT. Mnesors are defined as elements of a semimodule over the min-plus integers. This two-sorted structure is able to merge graduation properties of vectors and idempotent properties of boolean numbers, which makes it appropriate for hybrid systems. We apply it to the control of an inverted pendulum and design a full logical controller, that is, without the usual algebra of real numbers.

"Thus the theory of control in engineering, whether human or animal or mechanical, is a chapter in the theory of messages." 
Norbert Wiener (1894-1964)

## I. Introduction

Machines generally must be controlled or constrained to achieve a certain goal by a human pilot or by an automat which is called a controller. When constant action of the pilot is required, we call it continous control or, in the case of the automat, feedback control. For example, the cars can have their speed kept more or less constant by the driver or by an electronic controller. But to those basic tasks, feedback controllers have added new functionalities. They generally take over supervision tasks. For instance, some car electronic systems can alert on a possible danger of collision. That is what one calls discrete event control. Unfortunately, the two cases, continous and discrete control, are based on two different kinds of computing. The former is based on real number computing, the latter on boolean numbers, and each with completely different mathematical operations. Still they have to interact. So it is clearly needed to find a common standard, that is, common computing operations on common numbers. Although software could help by putting both continous and discrete variables in boolean format (coded by 0 or 1), their mixture is clearly artificial. In fact it brings nothing to help the design. Basically, variables are of completely different meaning. So we must say that numerical technology fails to fill the gap here. Control theory alone is made to do the job. For that, we must find some control laws that can combine continous and discrete variables into a single format with common operations. Fuzzy control [1] positively made the way towards a unified control by dropping the regular algebra of 