**7. References**

[1] Frank L. Lewis, Ayla Gurel, Stjepan Bogdan, Alper Doganalp and Octavian C. Pastravanu (1998) Analysis of deadlock and circular waits using a matrix model for flexible manufacturing systems. Automatica. 34(9): 1083-1100.

**Chapter 0**

**Chapter 20**

**A Forward On-The-Fly Approach in Controller**

Controller synthesis refers to finding a controller which is running in parallel with the system under study and preventing any violation from the given properties. Such a controller guarantees satisfaction of the desired properties; a controller makes an open-loop system to

Controller synthesis can also be explained by game theory as a timed game with two players: environment and the controller. The strategy of the game determines the sequence of actions to be executed. In this context, the objective of controller synthesis is to find a strategy such that no matter what action is executed by the environment, the controller wins absolutely the game. Two main questions arise for the controller: the existence and possibility of implementation. The first question, *Control Problem* says given a system S and a property *ϕ*, does a controller *C* exist for the system S such that *C* running in parallel with S satisfies the property *ϕ* (S||*C* |= *ϕ*). And the second one is the *Controller Synthesis Problem*; if the mentioned controller exists, is there a solution to implement it? First, a system should be modeled and

Among various models used to describe the behavior of S, Timed Automata (*TA* in short) and Time Petri Nets (*TPN* in short) are the well-known. The properties studied in the TPN and

1. Safety properties: Whatever path is traveled, for all situations, a given set of forbidden

2. Reachability properties: Whatever path is traveled, for all situations, a state of a given set

Some research has been done to find algorithms to control these kinds of properties for timed models (TA and TPN), such as [10, 11, 20]. Two known methods in the literature are the backward *fix point* method and the backward/forward *on-the-fly* method. Both methods

> ©2012 Heidari and Boucheneb, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original

©2012 Heidari and Boucheneb, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

**Synthesis of Time Petri Nets**

Additional information is available at the end of the chapter

then, synthesized regarding the desired property.

states (or bad states) are never reached.

TA for control purposes are classified in two main categories:

of states (good states) will eventually be reached.

work is properly cited.

Parisa Heidari and Hanifa Boucheneb

http://dx.doi.org/10.5772/47744

**1. Introduction**

be closed-loop.

