1. Modeling with ordinary Petri Nets Events: Actions that take place in the system The occurrence of these events is controlled by the state of the system. The state of the system is described as a set of conditions. A condition: a predicate or logical description of the state of the system. Events may occur. Occurrence of an event may require some conditions to hold (preconditions). Occurrence of events may cause some preconditions to cease and may cause other conditions (postconditions) to become true.

2. Modeling - machine shop example (1) Conditions aThe machine shop is waiting. bAn order is arrived and is waiting. cThe machine shop is working on the order. dThe order is complete. Events 1An order arrives. 2The machine shop starts on the order. 3The machine shop finishes the order. 4The order is sent for delivery.

3. Modeling - machine shop example (2)

4. Feedback control - Assembly cell (1)

5. Feedback control - Assembly cell (2) Constraints: Each robot shall perform a task at a time: m 2 + m 3  1 m 4 + m 5 + m 6 + m 7  1 M-1 robot does not interrupt S-380 robot: m 1 + m 2 + m 3 + m 5 + m 6 + m 7  1 >>: A piston rod shall be ready at P3 : m 3  1 A pulling tool is required in P4 and P5 : m 4 + m 5  1 A cap is required in P6: m 6  1 Two nuts are required in P7: m 7  1

6. Feedback control - Assembly cell (3) Uncontrollability conditions: Operation of M-1 robot not be interrupted from the point that it pulls the piston rod into the engine block until it has completed fastening the cap on the piston rod.  T 6, T 7, T 8 uncontrollable. 1- Check L.W uc 2- Find R 1 and R 2 so that R 1. W uc + R 2 L. W uc  0 L´= R 1 + R 2 L. R 2 = I, find R 1 by row operation.

7. Feedback control - Assembly cell (4) Uncontrollability conditions

8. Feedback control - Assembly cell (5) Vision system used in M-1 robot has been obscured: Starting and completing the task can be observed, but tracking the robot’s performance in between is not possible  T 5, T 6, T 7 unobservable. 1- Check L´.W uc 2- Find R 1 and R 2 so that R 1. W uo + R 2 L”. W uo  0 L”= R 1 + R 2 L’. R 2 = I, find R 1 by row operation.

9. Feedback control - Assembly cell (6) Uncontrollable and unobservable conditions

10. Concepts in Petri Net (1) Autonomous PN: Neither time nor external synchronization are involved in the model. Boundedness: For every reachable marking, the number of tokens in every place is bounded. Safeness:The marking of every place is either 0 or 1 (Boolean marking). Liveness:Regardless of the evolution, no transition will become unfireable on a permanent basis. Invariants:P-invariants and T-invariants. Concurrency:Firing of transitions are causally independent (I.e. concurrent, they may occur in any order). Synchronization:….

11. Concepts in Petri Net (2) Source transition:without input place. Sink transition:without output place. Deadlock:no transition is enabled. Conflict:between transitions. Conservation:A PN is conservative if it does not lose or gain tokens but merely moves them around.

12. Petri net classes (1)

13. Petri net classes (2) Three main classes: Ordinary PNs:All arcs have weight 1, one kind of tokens, infinite capacity for places, no time involved. Abbreviations:Simplified representations (useful graphical representations), can be mapped to an ordinary PN. Extensions:Some functioning rules are added, to enrich the initial model.

14. Applications Modeling: - Communication protocols in computer systems; (concurrency, synchronization, and resource sharing). - Manufacturing systems; concurrency (two machines working independently) synchronization (a machine is free and a part is ready to be processed by it). Resource sharing (a robot is assigned to handle parts in two machines but not at the same time). - Hybrid Systems (continuous + discrete parts); ex. batch production processes in biotechnological industry, manufacturing systems.

15. Analysis PNs exhibit how a system work. Analysis of a PN consists of seeking properties of the constructed model such as liveness, boundedness, deadlock, …. (they show whether the specifications are fulfilled). Main categories of methods for seeking these properties: Drawing the graph of marking and coverability tree. Using linear algebra. Reducing the PNs.