**3. Controller synthesis method**

54 Petri Nets – Manufacturing and Computer Science

arc of the node *M* in *G*

 *M*

oriented cycles of *G.*

where is an oriented cycle of *G*,

from happening at *M* by a place *p* iff

**2.2. Theory of regions and synthesis problem13**

The theory of regions is proposed for the synthesis of pure nets given a finite TS15, which can be adopted to synthesize the liveness-enforcing net supervisor (LENS) for a plant model12-13.

First of all, let *T* be a set of transitions and *G* be a finite directed graph whose arcs are labeled by transitions in *T*. Assume that there exists a node *v* in *G* such that there exists a path from it to any node. The objective of the theory of regions is to find a pure PN (*N*, *M*0), having *T* as its set of transitions and characterized by its incidence matrix [*N*](*p*, *t*) and its initial marking *M*0, such that its reachability graph is *G* and the marking of node *v* is *M*0. In

Consider any marking *M* in net (*N*, *M*0). Because (*N*, *M*0) is pure, *M* can be fully

vector of path *<sup>M</sup>* . For any transition *t* that is enabled at *M*, i.e., *t* is the label of an outgoing

Consider now any oriented cycle of a reachability graph. Applying the state equation to a

*N pt t C* 

 

( )*<sup>t</sup>* is a firing vector corresponding to

[ ]( , ) ( ) 0,

result, *<sup>M</sup>* can be arbitrarily chosen. The reachability of any marking *M* in *G* implies that

<sup>0</sup> *Mp M p N p M G* ( ) ( ) [ ]( , ) 0, *<sup>M</sup>*

The above equation is called the reachability condition. Notably, (3) is necessary but not

It is clear that the cycle equations and reachability conditions hold for any place *p*. For each pair (*M*, *t*) such that *M* is a reachable marking of *G* and *t* is a transition not enabled at *M*, *t* should be prevented from happening by some place *p*. Since the net is pure, *t* is prevented

> <sup>0</sup> *M p N p N pt* ( ) [ ]( , ) [ ]( , ) 1 *<sup>M</sup>*

, (*M*, *M*

(2)

where *<sup>M</sup>*

) *G M* [ *t* > *M'* (1)

. There are several paths from *M*0 to *M*.

is the same for all these paths. As a

(3)

*<sup>M</sup>* from *M*0 to *M*. Applying (1)

(4)

is the firing

*,* and *C* is the set of

For convenience, our method follows the interpretation of the theory of regions in13.

the following, *M* denotes both a reachable marking and its corresponding node in *G*.

characterized by its corresponding incidence vector [*N*](*p*, ) *<sup>M</sup>*

(*p*) = *M*(*p*) + [*N*](*p*, ) *MM*

node in and summing them up give the following cycle equation:

*t T*

According to the definition of *G*, there exists an oriented path

sufficient. Hence, spurious markings are beyond this paper.

along the path leads to *M*(*p*) = *M*0(*p*) + [*N*](*p*,) *<sup>M</sup>*

Under the cycle equations, the product [*N*](*p*,) *<sup>M</sup>*

In this section, an efficient controller synthesis method is developed based on the theory of regions. Please note that all transitions of the PN models are regarded as controllable ones.
