Representation from continuous systems to discrete event systems

1. Representation from continuous systems to discrete event systems Dr Hongnian Yu Department of Computing

2. Outline of the presentation • Motivation example • Modelling and control of continuous engineering systems • Petri nets (PN) • PN modelling of manufacturing systems • Performance analysis using PN • Scheduling using PN and AI search • Summary

3. Motivation example Problem Solving = Representation + Reasoning • We can represent numerals in many different ways, • e.g. Arabic, Roman, English, Chinese, etc. • Which one shall we use? It depends on the tasks. • For numerical analysis, we prefer the Arabic representation. E.g. carry out a simple multiplication • (twenty five times thirty five =?) • To write a check, what will we use?

4. Differential Equations A powerful representation tool of continuous engineering systems (1) Mechanical system: Mass-spring-damper, m: mass, k: spring constant, b: friction constant, u(t): external force, y(t): displacement. (2) Electrical system: RLC circuit General form (State space representation)

5. Interesting Issues of Engineering Systems • Controllability • Stability • Observability • Optimality • Adaptation • Robustness disturbance

6. Control Methods • Adaptive control • Adaptive Control of Robot Manipulators Using a Popov Hyperstability Approach, Journal of Systems and Control Engineering, 1995. • Simple adaptive control • Simple adaptive control of processes with uncertain time-delay and Affine linear structured uncertainty, Journal of Control Theory and Application, 2001. • Robust control • Exponentially Stable Robust Control Law For Robot Manipulators, Journal of Control Theory and Applications, 1994. • Combined adaptive and robust control • Robust Combined Adaptive and Variable Structure Adaptive Control of Robot Manipulators, Journal of Robotica, 1998. • Iterative learning control • Model Reference Parametric Adaptive Iterative Learning Control, 15th IFAC World Congress on Automatic Control, 2002.

7. Representation of discrete event systems • Man-made systems • Computer networks • Communication networks • Transportation networks • Power networks • Water networks • Manufacturing systems • Supply chains • Common features: discrete event systems • Representation approaches: various, but not unique • Finite state automata • MAXPLUS algebra • Petri nets • No standard representation (model) like differential equations for continuous dynamic systems

8. Petri nets • A PN is a mathematical formalism and a Graph tool to model and analyze discrete event dynamic systems. It is directed graphs with two types of nodes: places and transitions. Places represent conditions which may be ‘held’ and transitions represent events that may ‘occur’ place: transition: • Enabling rule: A transition t is enabled if and only if all the input places of the transition t have a token. t Initial state • Firing rule: An enabled transition t may fire at marking Mc. Firing a transition t will remove a token from each of its input places and will add a token to each of its output places. Final state t

9. Graphical representation of a Petri net an arc a token 4 a place 2 an arc’s weight a transition

10. Petri nets: Mathematics Model • A Petri net is 5-tuple, PN=(P,T,I,O,M) where • P={p1,p2, ···, pmp} is a finite set of system states; • T={t1,t2,···, tnp} is a finite set of transitions; • I: the input (preincidence) function; • O: output (postincidence) function; • M: the m-component marking vector whose ith component, M(pi) is the number of tokens in the ith place. M0 is an initial marking. • A Petri net from stage k to stage k+1 can be expressed by the following state equation Mk+1 = Mk + CTuk (1) where Mk is the current marking state vector, uk is the control vector and C=O-I is the incident matrix.

11. p1 p3 t1 t2 t3 Example P={p1, p2, p3}; T={t1, t2, t3}; I(t1)={}, I(t2)={p1, p2}, I(t3)={p3}; O(t1)={p1}, O(t2)={p3}, O(t3)={p2} Initial marking: M0=[1, 1, 0]. p2 Using the firing rule , we have M1=M0+etC=[1, 1, 0]+[0, 1, 0]C=[0, 0, 1] where et is the characteristic vector of t: et(x):=1 if x=t, else =0.

12. Petri Nets: Time Information • The concept of time is not explicitly given in the original definition of PNs. For performance analysis and scheduling problems, it is necessary and useful to introduce time delays associated with transitions or places in their PN models. • A timed Petri net TPN=(PN,h): • PN is a normal Petri net defined as the before; • h: the time delay associated with the relevant state. Timed place Timed transition

13. Example

14. Batch Plant Flowchart with 1 Reactor and 1 BlenderSynthesising and Analysis of a Batch Processing System Using Petri Nets, 1997. • The Petri net model of the batch plant • Batch plant = charging + reaction + blending + testing & discharging

15. PN Modelling of Solvent Charging Illustration of places and transitions. p1: Reactor available p2: Charging Solvent 1 to the reactor p3. Charging Solvent 2 to the reactor p4: Charging Solvent 3 to the reactor p5: Charging Solvent 4 to the reactor p6: Reaction in progress to the reactor S1: Solvent 1 S2: Solvent 2 S3: Solvent 3 S4: Solvent 4 t1: Start charging solvent 1 t2: Stop charging solvent 1 & start charging solvent 2 t3 : Stop charging solvent 2 & start charging solvent 3 t4: Stop charging solvent 3 & start charging solvent 4 t5: Stop charging solvent 4 & start reaction

16. PN Modelling of Solvent Charging • This is a marked graph since every place has exactly one input and one output transition. • The net is live and reversible since every circuit has at least one token. • It is a safe net since no place has more than one token.

17. Modelling of Reactor and Blender Illustration of places and transitions p7: Charging solvents p8: Reaction in progress to the reactor p9: Discharging reactor & charging blender p10: Blending&testing&discharging R: Reactor available B: Blender available t6: Start charging solvents t7: Stop charging solvents & start reaction t8 : Stop reaction&start charging blender t9: Stop charging&discharging & start blending t10: Stop discharging blender

18. Modelling of Quality Test Illustration of places and transitions p12: Ready for blending p13: Logical place for rejected material p14: Blending p15: Testing p16: Testing fail & require reblending p17: Testing success & discharging blender S5: Blending resource available O1: Operator available for testing O2: Operator available for discharging t12: Pumping to blender finish t13: Start blending t14 : Stop blending & start testing t15: testing finish (fail) t16: testing finish (success) & start discharging blender t17: Start reblending t16: Discharging finish

19. Reachability Graph

20. Final Petri net model for the batch plant

21. Performance Analysis Using Timed Petri Nets • Performance evaluation of a production system provides the ability to perceive clearly the production plan of the system. It is used to identify the bottleneck in the production unit, estimate the raw material required for production and decide operating policies. • Time to charge each solvent is about 30 min. • The total time for charging four solvents into the reactor is about 120 min and this can be reflected as a delay time in place p7. • The time for reaction is 1080 min which represents the delay time in p8. • The time for discharging the reactor/charging the blender is 60 min which represents the delay time in p9. • The time for blending is 360 min. The time for testing and discharging is about 180 min. The delay time in p10 represents the sum of the blending time and the time for testing and discharging.

22. Performance Analysis • When the times are deterministic, we can compute the cycle time of each circuit g: Cg = m(g)/M(g) for g = { 1....q } where q is the number of circuits in the model, m(g) is the sum of place delays in the circuit g, M(g) is the sum of tokens in the circuit g • For a marked graph, the minimum cycle time, Cm is Cm = Max { m(g)/M(g) } • To compute the result, it is important to list all the circuits produced by this model and show the minimum cycle time of each circuit. There are two elementary circuits in our model. • Circuit 1 C1 = 120 + 1080 + 60 =1240 min • Circuit 2 C2 = 60 + (360 + 180)=600 min • Therefore the minimum cycle time is 1240 minutes. The bottle neck machine is in element circuit 1, i.e., the reactor.

23. Scheduling Approaches • The mixed integer linear programming approach: It is similar to the linear programming approach with linear objective function and constraints but some of its variables are integer and others binary. • The critical path scheduling approach (CPA) and the program evaluation and review technique (PERT): Both are network based methods. • The artificial intelligence (AI) based approaches: These include depth-first and breadth-first search approaches, Branch and Bound search approach, best-first search approach, climb hill search approach, beam search approach, A* (heuristic) search approach, etc. These are called the systematic approaches. • The non-systematic approaches: genetic algorithm based approach, simulation annealing approach, etc. • Rule based approaches: Copying the expertise of human schedulers and adopting the tactics that they use. • The simulation based approaches: discrete-event simulation.

24. Petri Net + AI Based Scheduling Methods • Scheduling: • Based on reachability tree analysis (for simple Petri nets) • Uses reduced reachability space for more complex Petri nets • Example: A 2 product & 2 processor system is used to illustrate the method. • Problem statement: • A complete description of the problem discussed is as follows: • The objective function to be minimised is the time makespan required to complete all the jobs. • The given constraints are: • precedence relationships among the jobs; • fixed number of resources and prescribed job-resource assignment. • The goal is to find a sequential order of jobs that satisfies the above conditions.

26. Gantt Chart Firing sequence: t1t2t3t4t5t6t7t8 leads to c=14 min Firing sequence: t5t6t1t7t2t8t3t4 leads to c=11 min (optimum) 

27. PN Based Intelligent Scheduling Approaches • A scheduling approach using Petri net modelling and a Branch & Bound search, Proc. IEEE International Symposium on Assembly and Task Planning, 1995. • Planning through Petri Nets, Proc. of the Sixteenth Workshop of the UK Planning and Scheduling Special Interest Group, 1997. • Petri Net-Based Closed-Loop Control and On-line Scheduling of the Batch Process Plant, Proc. of CONTROL 98, 1998. • Rule-Based Petri Net Modelling and Scheduling of Flexible Manufacturing Systems, Proc. of 14th NCMR Conference, 1998. • Generic Net Modelling Framework for Petri Nets,IASTED International Conference on Intelligent Systems and Control, 1999 • Integrating Petri Net Modelling and AI Based Heuristic Hybrid Search for Scheduling of FMS,Journal of Computer in Industry, 2002. • Advanced Scheduling Methodologies for FMS using Petri Nets and Artificial Intelligence,IEEE Trans on Robotics and Automation, 2002. • Petri Nets, Heuristic Search and Natural Evolution: Promising Scheduling Algorithm for Job Shop Systems, Proc. of The Third International ICSC Symposia on INTELLIGENT INDUSTRIAL AUTOMATION, 1999 • Petri net Modelling and Witness Simulation of Manufacturing Systems, Proc. of Third World Manufacturing Congress, 2001

28. Petri Nets Applications • Performance analysis • Optimisation, scheduling, planning • Simulation • Control synthesis • Formal verification and validation

29. Summary • Two types of systems • Natural (continuous) engineering systems • A powerful representation tool, differential equation, is available • Many analysis approaches have been developed • Man-made (discrete event) systems • Many representation approaches are proposed, but none of them is as powerful as the differential equation • Complexity • Uncertainty

30. Back Variable structure control Dynamic equation: (1) Theorem. For the system (1), if the robust control laws are (t)=n(t)+l(t), n(t)=W(t)v(t)+W0(t), l(t)=-(Pll+Pcc-1Pcc)s(t)+PccE1(t) Assumption: The bounds of the unknown parameters are known, i.e. where p is the number of the uncertainty parameters, Pcc, , PllRnn are symmetric positive definite gain matrices, P12=Pcc-1 Rnn, P1=[P12 Inn] Rn2n, then for a reasonably small positive constant , all the signals in the system are bounded and E(t) tends to zero with at least an exponential rate that is independent of the excitation. • Exponentially Stable Robust Control Law For Robot Manipulators, Journal of Control Theory and Applications, 1994.

31. Back Adaptive control (1) Dynamic equation: Define the control law as (t)=n(t)+l(t) (2) Linear control law: Non-linear adaptive control law: Theorem. The control system (1) with the control law (2) is globally convergent, that is E(t) asymptotically converges to zero and all internal signals are bounded. • Adaptive Control of Robot Manipulators, Proc. IEEE/RSJ International Conference on Intelligent Robots and Systems, 1992. • Adaptive Control of Robot Manipulators Using a Popov Hyperstability Approach, Journal of Systems and Control Engineering, 1995.

32. Back Iterative learning control Control input: (2) Dynamic equation: (1) Parameter ILC law: (3) Theorem: For the robot system described by (1), if the control law (2) and the parameter iterative learning law (3) are used, the desired joint trajectories and their up to 2nd order derivatives are bounded, and the initial tracking errors (0)=0 and (0)=0 for j=1,2…, then the following properties hold: i ii iii • Parametric Iterative Learning Control of Robot Manipulators, Proc. of the Chinese Automation Conference, 1999. • Model Reference Parametric Adaptive Iterative Learning Control, 15th IFAC World Congress on Automatic Control, BARCELONA, Spain, 2002.