Max-plus algebra for discrete event systems-some links to structural controllability and structural observability

One can consider a system of production to be a dynamic system. Moreover, it is interesting to describe the system by equations in which state vector, input and output vectors appear. This can help when carrying out synthesis of the system. The Petri net representation of a system allows one to define three vectors mentioned above. If the corresponding Petri net is an event graph (i.e. each place has exactly one input and one output transition), it is possible to apply a mathematical model for this system. The model is known in literature as max algebra or max-plus algebra R_max=(R∪{ε},⊕,⊗) where R∪{ε} is the set of real numbers extended with the “minus infinity” element ε=-∞. This set is endowed with two binary combining operations x⊕y, x⊗y defined to be max {x,y}, (x+y) respectively. In using this algebra the problem takes on the character of a problem of linear algebra. From the definitions of structural controllability and structural observability, the authors aim to show necessary and sufficient conditions for the system to be structurally controllable, resp. structurally observable. These conditions are similar to the conditions of controllability and observability in the conventional algebra for dynamic systems