Optimal control synthesis for max-plus linear dynamical systems: performing the open-loop and feedback control policies in Just-in-timecontext

Abstract

Max-Plus Linear Dynamical Systems are systems modeled by Timed Event Graphs (TEG) whose dynamic can be described by Max-Plus Algebra. This thesi deals with control policies applied to the Max-Plus Linear Dynamical Systems. A new multi-objective formulation to control this class of systems is proposed. This formulation is based on optimization problems and it is possible to consider non-convex constraints (in conventional algebra) in the formulation. Two controlpolicies are obtained from the general problem. The first one is the open-loop Just-in-Time Control, which can be developed either in finite horizon or in infinite horizon aiming to saving resources and the optimal control. The necessary and sufficient conditions to solve the problems are presented, as well as the discussion about the computational complexity of the proposed methods in order to solve them. Some concepts on max-plus algebra are used, such as (A,B)-invariantsets, Residuation Theory and the Theory of Semimodules. Due to computational complexity of general method of solution, algebraic properties are used to solve an important class of problems of practical interest. The second control policy is the Feedback control in Just-in-Time context. The conditions for the existence of a feedback matrix are presented. It is also presented a way to find the greatest feedback matrix in order to comply with deadline dates for the system output. Atthe end of each control problem, numerical examples are developed to illustrate the applicability of the proposed methodologies and the relevance of systems here addressed.