Feedback Petri Net Control Design In the Presence of Uncontrollable Transitions - Decision and Control, 1995., Proceedings of the 34th IEEE Conference on

This paper describes a computationally efficient method for synthesizing feedback controllers for plants modeled by Petri nets which may contain uncontrollable transitions. The controller, a Petri net itself, enforces a set of linear constraints on the plant. The original set of plant behavioral constraints is transformed to yield a controller which enforces the original constraints without influencing any uncontrollable transitions. 1 Introduct ion A method is proposed in this paper for transforming a set of linear constraints on the behavior of a plant modeled by a Petri net into an equivalent set which accounts for uncontrollable transitions; see [3] for an alternative methodology based on integer programming. The control goal is to realize a set of n, constraints of the form Lpp 5 b where p p is the marking vector of the Petri net modeling the process, L E ZncXm, b E Z n c , m is the number of places in the plant and Z is the set of integers. The inequality is with respect to the individual elements of the two vectors Lpp and b. In [4] it is shown how these constraints can be enforced by a Petri net controller with incidence matrix D, and marking p, placed in a feedback loop with the plant such that the incidence matrix D and marking p of the closed loop system are The controller Petri net is calculated using D, = -LDp Pco = b LPPO (2) where pco and ppo are the initial markings of the controller and plant respectively. The controllers produced are identical to the monitors [Z] of Giua et al. derived independantly via an alternative method. 2 Handling Uncontrollable Transitions Uncontrollable transitions in a plant may not receive any arcs from the places which make up the the external Petri net controller. Let D, E ZmXn" be the incidence matrix of the uncontrollable portion of the process net, where nu is the number of uncontrollable transitions. Assuming no self loops, the Petri net controller given by WM15 2:lO D, = -LOp violates the uncontrollability constraint if LD, contains any elements greater than zero. If we are unable to meet the uncontrollability constraint, then it is necessary to transform the constraint vector L such that the original constraint of Lpp 5 b is still maintained, while obeying the uncontrollability constraint. Lemma I . Let R1 E Zncxm satisfy Ripp _> 0 V pp. Let Rz E ZncXnc p.d. diagonal matrix (3) (4) If L'pp 5 b' where L' = R i + R z L (5) b' = Rz(b+ 1) 1 (6) and 1 is an n, dimensional vector of 1'9, then Lpp 5 b. Proof. The transformed constraint is (RI + Rz L)pp 5 Rz(b + 1) 1. Because all of the elements are integers, the inequality can be transformed into a strict inequality: ( R I + RzL)pp < Rz(b + 1). Because Rz is diagonal and positive definite, RT1R1pP + Ll.rp < b + 1. Assumptions (3) and (4) imply that all elements of the vector 0 Let a plant Petri net with incidence matrix Dp be given with a set of uncontrollable transitions, a set of linear constraints L y 5 b on the net marking. Assume R1 and Rz meet ( 3 ) and (4) and let RylR1pp 2 0 , therefore Lpp 5 b.