A non-linear set-membership approach for the control of discrete event systems

A variety of problems in non-linear time-evolution systems such as manufacturing plants, operations research, computer networks, etc., can be modelled as min-max-plus systems in which operations of min, max and addition appear simultaneously. It is well know that systems with only maximum (or minimum) constraints can be modelled as max-plus system and handled by max-plus algebra which changes the original non-linear system into linear system in this framework. Several authors have developed methods, in max-plus algebra, to compute control for max-plus systems. In general, these methods use the residuation theory to design a just-intime control such that the output of the controlled system is, on the one hand, less than the desired reference signal but as close as possible to the given reference and, on the other hand, the control is delayed as much as possible. In this paper, we consider min-max-plus systems which are extensions of max-plus systems and non-linear even in the max-plus algebra view. We proposes a new approach to solve the just-in-time control problem for a non-linear min-maxplus system. This problem is cast into the more general framework of constraint satisfaction problems. This makes it possible to propose a new algorithm that contracts the feasible domains for each uncertain variable optimally (i.e., no smaller domain could be obtained) and efficiently.