Supervisory Control of (max,+) Automata: A Behavioral Approach

Abstract

A behavioral framework for control of (max,+) automata is proposed. It is based on behaviors (formal power series) and a generalized version of the Hadamard product, which is the behavior of a generalized tensor product of the plant and controller (max,+) automata in their linear representations. In the tensor product and the Hadamard product, the uncontrollable events that can neither be disabled nor delayed are distinguished. Supervisory control of (max,+) automata is then studied using residuation theory applied to our generalization of the Hadamard product of formal power series. This yields a notion of controllability of formal power series as well as (max,+)-counterparts of supremal controllable languages. Finally, rationality as an equivalent condition to realizability of the resulting controller series is discussed together with hints on future use of this approach.

Determinization of timed Petri nets behaviors

“…This constitutes a serious obstacle for applications. In fact, it turns out that there exist rational controllers only for systems which admit a description in terms of unambiguous (max,+) automata (a slightly larger than deterministic (max,+) automata) (Komenda et al 2009;Badouel et al 2011). For performance evaluation purpose, determinism is also required to compute the optimal and average behaviors of TDES modeled by (max,+) automata (Gaubert 1995).…”

Residuation of tropical series: Rationality issues

“…Then, we focus on the residuation of (max,+) series. After reviewing the results obtained in [9], we examine to what extent they can be applied in the context of supervisory control and underline some drawbacks. We then turn to consider hybrid residuation operations of (max,+) series by (min,+) series and conversely.…”

“…Namely, for any series S 1 , S 2 and for any word w ∈ Σ * , (S 1 • /S 2 )(w) = S 1 (w)/S 2 (w). Based on this fact, it was proposed in [9] to use (max,+) residuation for computing delay controllers. Given a specification series S 1 and a system series S 2 , the residuated (max,+) series gives for each word w ∈ Σ * in supp(S 2 ) the maximum delay (S 1 • /S 2 )(w) that can be added to the worst-case duration S 2 (w) of the sequence of actions w in the plant without exceeding the specified upper bound S 1 (w).…”

“…In this paper, we aim at extending to abstract systems and specifications both given by pairs of lower and upper bound series the supervisory control approach proposed in [9] for (max,+) automata. In [9], the behavior of the closed-loop system is represented by the Hadamard product of the system and controller series, and the controller series is formally computed using residuation theory [3].…”

“…In [9], the behavior of the closed-loop system is represented by the Hadamard product of the system and controller series, and the controller series is formally computed using residuation theory [3]. Namely, when the controller can delay the controllable transitions but it cannot prevent the firing of transitions, the residuation S 1 /S 2 of the (specification) series S 1 by the (system) series S 2 amounts to the Hadamard product of S 1 with the series −S 2 with all coefficients multiplied by −1.…”