A feedback control in Max-Algebra

Abstract

For timed event graphs, linear models were obtained using Max-Algebra. This paper presents a method to control such systems. After describing the optimal solution of a model tracking problem, we propose a feedback control structure in order to take into account a possible modeling error. We present its construction, its main properties and an algorithm for its practical implementation. An illustrative example is provided.

Stable Model Predictive Control for Constrained Max-Plus-Linear Systems

“…This infeasibility is caused by the fact that the optimal input aims to fulfill the constraint y sys (k) ≤ r sys (k) for all k, which cannot be met using a nondecreasing input sequence. So other residuation-based control design methods that also include this constraint such as (Baccelli et al 1992;Cottenceau et al 2001;Menguy et al 1997) would also yield a control sequence that is not nondecreasing and thus infeasible.…”

“…Let us now compare our MPC method with the other control design methods mentioned in Section 1. The max-plus control approaches proposed in (Baccelli et al 1992;Cottenceau et al 2001;Libeaut and Loiseau 1995;Menguy et al 1997) typically involve an open-loop optimal control problem over a simulation horizon and for a given due date signal r sys such that the output of the system y sys should satisfy y sys (k) ≤ r sys (k) for all k. The solution of this optimal control problem is computed using residuation, resulting in a just-in-time control input. The main disadvantage of this approach is that it cannot cope with tracking problems where the outputs do not occur before the due dates, and that the resulting control input sequence is sometimes decreasing, i.e.…”

“…However, the residuation-based approach does not cope with input and state constraints. Moreover, the methods presented in Menguy et al (1997) and Cottenceau et al (2001) cannot solve tracking problems corresponding to the case when the actual outputs do not necessarily have to occur before the due dates although these situations are often met in many practical applications. Some of these drawbacks are removed in Maia et al (2003), Menguy et al (2000), and Necoara (2006) by using respectively projection, an adaptive approach, or MPC.…”

“…In Baccelli et al (1992) it has been shown that a DES with synchronization but no concurrency can be modeled by a max-plus-linear (MPL) system. Although several authors have already developed methods to compute optimal controllers for MPL systems (Baccelli et al 1992;Cofer and Garg 1996;De Schutter and van den Boom 2001;Kumar and Garg 1994;Libeaut and Loiseau 1995;Menguy et al 1997Menguy et al , 2000Necoara 2006), the literature on stabilizing controllers for this class of systems subject to input and state constraints is relatively sparse. Some of the contributions that partially address this problem include model predictive control (MPC; van den Boom 2001, 2006) and optimal control based on residuation theory (Cottenceau et al 2001;Maia et al 2003;Menguy et al 1997Menguy et al , 2000.…”

Modeling and control of switching max-plus-linear systems with random and deterministic switching

“…Residuation essentially consists in finding the largest solution to a system of max-plus inequalities with the input times as variables and the due dates as upper bounds. Residuationbased approaches for computing optimal input times are presented or used in Boimond and Ferrier (1996), Cottenceau et al (2001), Goto (2008), Lahaye et al (2008), Libeaut and Loiseau (1995), Maia et al (2003) and Menguy et al (1997Menguy et al ( , 2000a. The MPC approach is essentially based on the minimization of the error between the actual output times and the due dates, possibly subject to additional constraints on the inputs and the outputs.…”