Representation from continuous systems to discrete event systems

1
Representation from continuous systems to
discrete event systems

• Dr Hongnian Yu
• Department of Computing

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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

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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?

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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)
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Interesting Issues of Engineering Systems

• Stability

• Controllability

• Observability
• Optimality

• Adaptation
• Robustness

disturbance
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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.

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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

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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
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Graphical representation of a Petri net
an arc
a token
4
a place
2
an arcs weight
a transition
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Petri nets Mathematics Model

• A Petri net is 5-tuple, PN(P,T,I,O,M) where
• Pp1,p2, , pmp is a finite set of system
  states
• Tt1,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 k1 can be
  expressed by the following state equation
• Mk1 Mk CTuk (1)
• where Mk is the current marking state vector, uk
  is the control vector and CO-I is the incident
  matrix.

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Example
p1
p3
t1
t2
t3
Pp1, p2, p3 Tt1, t2, t3 I(t1),
I(t2)p1, p2, I(t3)p3 O(t1)p1,
O(t2)p3, O(t3)p2 Initial marking M01, 1,
0.
p2
Using the firing rule , we have M1M0etC1, 1,
00, 1, 0C0, 0, 1 where et is the
characteristic vector of t et(x)1 if xt, else
0.
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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
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Example
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Batch Plant Flowchart with 1 Reactor and 1
Blender Synthesising 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

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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
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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.

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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 Blendingtestingdischarging R
Reactor available B Blender available t6
Start charging solvents t7 Stop charging
solvents start reaction t8 Stop
reactionstart charging blender t9 Stop
chargingdischarging start blending t10 Stop
discharging blender
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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
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Reachability Graph
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Final Petri net model for the batch plant
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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.

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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.

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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.

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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.

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Gantt Chart
Firing sequence t1t2t3t4t5t6t7t8 leads to c14
min
Firing sequence t5t6t1t7t2t8t3t4 leads to c11
min (optimum)
?
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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

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Petri Nets Applications

• Performance analysis
• Optimisation, scheduling, planning
• Simulation
• Control synthesis
• Formal verification and validation

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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

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Variable structure control
Back
Dynamic equation
(1)

• Assumption The bounds of the unknown parameters
  are known, i.e.

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)-(PllPcc?-1Pcc)s(t)PccE1(t)
where
p is the number of the uncertainty parameters,
Pcc, ?, Pll?Rn?n are symmetric positive definite
gain matrices, P12Pcc-1 ? ?Rn?n, P1P12 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.

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Adaptive control
Back

• Dynamic equation

(1)
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.

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Iterative learning control
Back

• Dynamic equation (1)

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 j1,2, 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.