The Plant Model

In order to create the plant model P using NCES, we start by modeling the subsystems of the DES with safe Petri Nets. The composition of these subsystems requires the description of the input/output structure, which we only give here in graphical terms. This input/output structure is formally described, using matrices, in [6], [25]. Two new arcs are defined to interconnect the modules (see Figure 3):

• Event Arcs: these are arcs directed from a transition in one module to a transition in another module. This type of arc describes a ``causal relation'' between the two events: the occurrence of event causes the (simultaneous) occurrence of event .
• Condition Arcs: these are arcs directed from a place p in one module to a transition t in another module. Such an arc describes a situation in which event t can only occur if the condition represented by p is true. However, unlike the case of Petri Nets, the occurrence of event t need not make this condition false.

  
Figure 3: Illustrating Condition and Event Arcs

Thus, if the subsystems of the DES are modeled with the safe Petri Nets, , , ... , then the model obtained by interconnecting the modules with the above condition/event arcs gives the NCES plant model, P. Formally, what we obtain is the following structure:

where:
P is the set of places
T is the set of transitions (events)
is the incidence matrix
is the initial marking
is the input/output structure
is the condition signal matrix
is the event signal matrix.

The input/output structure is described as follows:

where:
is an ordered set of condition input signals
is an ordered set of event input signals
is an ordered set of condition output signals
is an ordered set of event output signals
is the condition input matrix
is the event input matrix
is the condition output matrix
is the event output matrix

The initial marking of the new net is given by the composition of the initial markings of the modules: . Notice, however, that the new structure is not a Petri Net, and so the firing rule needs to be modified. We start by defining three types of enabled transitions:

1. Marking enabled:
   A transition is enabled by marking if, as before, each of its input places is marked with one token.
2. Condition enabled:
   A transition t is condition enabled when each of its input conditions (if any), which are places in other modules, are marked with one token. In addition, if there are any condition inputs from outside the model to the transition, these must also be true.
3. Event enabled:
   Let denote the set of all transitions which are connected to t by an incoming event arc at t. Then, transition t is event enabled if is empty and there are no event inputs to t from outside the model, or if every transition in is marking, condition, and event-enabled and all event inputs to t from outside the model are also true.

Two kinds of transition firings are possible:

• Spontaneous firing: we call a transition spontaneous if it has no incoming event arcs or event signals from outside the model. We denote the set of spontaneous transitions by .
• Forced firing: transitions with incoming event arcs or incoming event signals from outside the model are forced to fire if they are enabled. We denote the set of force transitions by .

All forced transitions occur at the same time instant as the event signal which forces the transition. In the case of spontaneous transitions, the new marking reached is obtained as before (by removing a token from each input place and adding a token to each output place of the firing transition). In the case of forced transitions, the new marking is obtained by removing a token from each input place of each firing transition, and adding a token to each output place of each firing transition. We illustrate the firing rules of NCES with the two-pusher example in Section 6.4. A formal description of the firing rule can be found in [6].

In the same way that Ramadge and Wonham adjoin a means of control to their plant model by identifying the controllable and uncontrollable events, Hanisch and Rausch also distinguish the controllable transitions from the uncontrollable ones. A transition is controllable if it has at least one incoming signal from outside the model. This signal can be a condition signal as well as an event signal. Controllable transitions with incoming condition signals can be disabled or enabled, whereas controllable transitions with an incoming event signal can be forced to fire by means of control. Observability of the system is also important for control purposes. A place is observable if it has a condition output signal, and a transition is observable if it has an event output signal. See Section 6.4 for the plant model of the Two-Pusher Example using NCES.