Continuous Systems To Discrete Event Systems

Representation from continuous systems to discrete event systems ,[object Object],[object Object]

Outline of the presentation ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Motivation example ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Problem Solving = Representation + Reasoning

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)

Interesting Issues of Engineering Systems ,[object Object],[object Object],[object Object],[object Object],disturbance ,[object Object],[object Object]

Control Methods ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Representation of discrete event systems ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Petri nets ,[object Object],[object Object],[object Object],Initial state Final state place: transition: ,[object Object],[object Object],t t

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

Petri nets: Mathematics Model ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Example P={p 1 , p 2 , p 3 }; T={t 1 , t 2 , t 3 }; I(t 1 )={}, I(t 2 )={p 1 , p 2 }, I(t 3 )={p 3 }; O(t 1 )={p 1 }, O(t 2 )={p 3 }, O(t 3 )={p 2 } Initial marking: M 0 =[1, 1, 0]. Using the firing rule , we have M 1 =M 0 +e t C=[1, 1, 0]+[0, 1, 0]C=[0, 0, 1] where e t is the characteristic vector of t: e t (x):=1 if x=t, else =0. t 1 t 2 t 3 p 1 p 3 p 2

Petri Nets: Time Information ,[object Object],[object Object],[object Object],[object Object],Timed transition Timed place

Batch Plant Flowchart with 1 Reactor and 1 Blender Synthesising and Analysis of a Batch Processing System Using Petri Nets, 1997. ,[object Object],[object Object]

PN Modelling of Solvent Charging Illustration of places and transitions. p 1 : Reactor available p 2 : Charging Solvent 1 to the reactor p 3 . Charging Solvent 2 to the reactor p 4 : Charging Solvent 3 to the reactor p 5 : Charging Solvent 4 to the reactor p 6 : Reaction in progress to the reactor S 1 : Solvent 1 S 2 : Solvent 2 S 3 : Solvent 3 S 4 : Solvent 4 t 1 : Start charging solvent 1 t 2 : Stop charging solvent 1 & start charging solvent 2 t 3 : Stop charging solvent 2 & start charging solvent 3 t 4 : Stop charging solvent 3 & start charging solvent 4 t 5 : Stop charging solvent 4 & start reaction

PN Modelling of Solvent Charging ,[object Object],[object Object],[object Object]

Modelling of Reactor and Blender Illustration of places and transitions p 7 : Charging solvents p 8 : Reaction in progress to the reactor p 9 : Discharging reactor & charging blender p 10 : Blending&testing&discharging R: Reactor available B: Blender available t 6 : Start charging solvents t 7 : Stop charging solvents & start reaction t 8 : Stop reaction&start charging blender t 9 : Stop charging&discharging & start blending t 10 : Stop discharging blender

Modelling of Quality Test Illustration of places and transitions p 12 : Ready for blending p 13 : Logical place for rejected material p 14 : Blending p 15 : Testing p 16 : Testing fail & require reblending p 17 : Testing success & discharging blender S 5 : Blending resource available O 1 : Operator available for testing O 2 : Operator available for discharging t 12 : Pumping to blender finish t 13 : Start blending t 14 : Stop blending & start testing t 15 : testing finish (fail) t 16 : testing finish (success) & start discharging blender t 17 : Start reblending t 16 : Discharging finish

Final Petri net model for the batch plant

Performance Analysis Using Timed Petri Nets ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Performance Analysis ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Scheduling Approaches ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Petri Net + AI Based Scheduling Methods ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Gantt Chart  Firing sequence: t 1 t 2 t 3 t 4 t 5 t 6 t 7 t 8 leads to c=14 min Firing sequence: t 5 t 6 t 1 t 7 t 2 t 8 t 3 t 4 leads to c=11 min (optimum)

PN Based Intelligent Scheduling Approaches ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Petri Nets Applications ,[object Object],[object Object],[object Object],[object Object],[object Object]

Summary ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Variable structure control ,[object Object],Theorem. For the system (1), if the robust control laws are  (t)=  n (t)+  l (t),  n (t)=W(t)  v (t)+W 0 (t),  l (t)=-(P ll +P cc  -1 P cc )s(t)+P cc E 1 (t) where 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. p is the number of the uncertainty parameters, P cc ,  , P ll  R n  n are symmetric positive definite gain matrices, P 12 =P cc -1   R n  n , P 1 =[P 12 I n  n ]  R n  2n , ,[object Object],Back Dynamic equation: (1)

Adaptive control ,[object Object],Define the control law as  (t)=  n (t)+  l (t) (2) Linear control law: Non-linear adaptive control law: (1) 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. ,[object Object],[object Object],Back

Iterative learning control ,[object Object],Control input: (2) 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 ,[object Object],[object Object],Back

Continuous Systems To Discrete Event Systems

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Continuous Systems To Discrete Event Systems