Design of ObserversControllers for Discrete Event Systems Using Petri Nets

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Design of Observers/Controllers for Discrete
Event Systems Using Petri Nets

• Alessandro Giua
• DIEE Department of Electrical and Electronic
  Engineering
• University of Cagliari, Italy

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OUTLINE

• 0) Petri nets
• Motivation for discrete event observers
• Relevant literature
• Main idea
• Marking estimation
• Marking estimation with initial macromarking
• Control using observers
• Deadlock recovery and estimate after net time out
• Using timing information to improve the procedure
• Conclusions and future work

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0 PETRI NETS

• A place/transition net is a 4-ple
  N(P,T,Pre,Post)
• P p1, p2, , pm set of places (circles)
• T t1, t2, , tn set of transitions (bars)
• Pre matrix denoting of arcs from places to
  transitions
• Post matrix denoting of arcs from transitions
  to places

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PETRI NETS (contd)
Incidence matrix
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PETRI NETS (contd)

• A marking M assigns to eack place p a nonnegative
  integer number of tokens M(p) (black dots). It
  represents the state of the system.
• A transition t is enabled at a marking M (denoted
  ) if each input place p has a number of
  tokens greater or equal to the number of arcs
  from p to t, i.e., if
• A transition enabled at M may fire taking
  Pre(p,t) tokens from each input place p and
  putting Post(p,t) tokens in each output place,
  yielding the new marking M (denoted
  )

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PETRI NETS (contd)
p1
p2
p4
t2
t3
p3
t1
t6
p8
p5
p7
p5
p6
t4
t5
t2 is enabled and fires
p1
p2
p4
t2
t3
p3
t1
t6
p8
p5
p7
p5
p6
t4
t5
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PETRI NETS (contd)

• Net system (a net N with initial marking M0)
• Set of firable sequences
• Set of reachable markings

Siphon a set of places S such that if a
transition inputs into S then it also outputs
from S (Ex S p1, p2 )
An empty siphon will always remain empty ? all
its output transitions are deadlocked
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1 - MOTIVATION FOR DISCRETE EVENT OBSERVERS

• Supervisory control theory is based on
• language specifications (a set of legal words)

language specification
event-feedback
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MOTIVATION (contd)

• When dealing with Petri nets it is natural to use
  state specifications (a set of legal markings)

state specification
state-feedback
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MOTIVATION (contd)

• A mixed structure is often used
• State specification marking of the net
• Output events transitions firing

• When the net structure and the initial marking
  is known (and the net labeling is deterministic)
  event observation is sufficient to reconstruct
  the net marking.

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MOTIVATION (contd)

• If the initial marking is not completely
  known
• use observers to estimate the marking
  after the word of events w has been observed

• Estimate Bound

C(w) set of w-consistent markings
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MOTIVATION (contd)

• In this talk we present
• Algorithms for computing estimate and consistent
  set
• Algorithms for control using observers
• Properties of the observer

PROBLEM incomplete information due to the
presence of an observer in the control loop may
lead to deadlock.

• We can further extend the approach
• Algorithms for deadlock recovery
• Deadlock analysis of the closed loop system

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2 - RELEVANT LITERATURE

• State-feedback control with partial observability
• - Li Wonham CDC88 T-AC93 (state observ.)
• - Takai, Ushio Kodama T-AC95 (state observ.)
• - Zhang Holloway Allerton95 (event observ.)

Derived nec suff condition for optimality
given a mask.
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LITERATURE (contd)

• DES state estimation for FSM / Predicate
  Transformers
• - Ramadge CDC86 (FMS)
• - Caines, Greiner, Wang CDC88 CDC89 (FMS)
• - Özveren, Willsky T-AC90 (FMS)
• - Kumar, Garg Markus T-AC93 (PT)

Requires to enumerate at each step the set of
consistent states (high complexity). No notion of
estimate error.
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LITERATURE (contd)

• Diagnosis
• - Wang, Schwartz T-net 93 (state estimation)
• - Ushio, Onishi Okuda SMC98 (place
  observation)

• Petri net observability
• - Giua, Seatzu T-AC02, ACC03
• - Giua, Basile, Seatzu ETFA01,CDC01,T-AC
  submitted
• - Meda, Ramirez SMC98 (interpreted nets)
• - Ramirez, Riveda, Lopez ICRA2000

• Partial knowledge of the marking
• - Cardoso, Valette Dubois ICATPN90
• Concept of macromarking and membership
  function.

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3 - MAIN IDEA

• Initially observed sequence
• Initial marking
• Estimate
• Set of consistent markings

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firing is detected
MAIN IDEA (contd)
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After fires
MAIN IDEA (contd)

• Observed sequence
• Actual marking
• Estimate
• Set of consistent markings

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firing is detected
MAIN IDEA (contd)
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After fires
MAIN IDEA (contd)

• Observed sequence
• Actual marking
• Estimate
• Set of consistent markings

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4 - MARKING ESTIMATION

• Hypothesis
• - The net structure
  is known
• - The transition firing can be observed
• - The initial marking is not known

• Algorithm
• 1 - Initial estimate Let
• 2 - Wait until fires
• 3 - Update previous estimate
• 4 - New estimate
• 5 - goto 2.

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ESTIMATION (contd)
Properties

• Estimate is a lower bound

• Can define place error

• and estimation error

• Error functions are non-increasing

• The set of markings consistent with observation w
  is

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Properties
ESTIMATION (contd)

• An observed word is marking complete if
  

• A net system is

- Marking Observable (MO) if exists a complete
word
- Strongly Marking Observable (SMO) in k steps
if a) all with are complete
b) all with that are not
complete can be continued in a word
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ESTIMATION (contd)

• A net system is

- Uniformly Marking Observable (uMO) if the
system is Marking
Observable for all
- Uniformly Strongly Marking Observable (uSMO)
if the system is SMO for
all
- Structurally Marking Observable (sMO) if the
system is MO for all
- Structurally Strongly Marking Observable
(sSMO) if the system is SMO
for all
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Observer reachability graph
ESTIMATION (contd)

• Each node of the graph is labeled with
• The real marking Mw The estimation error uw Mw
  - mw

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Observer coverability graph
ESTIMATION (contd)

• If the net is unbounded, is it possible to
  construct an observer coverability graph (OCG).
  The error vector u is now only an upper bound.

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Analysis of properties
ESTIMATION (contd)

• Theorem 1
• A net system is

• marking observable iff

• marking observable if there exist a node in the
• OCG with

• strongly marking observable iff in the OCG for
  each dead node and for each node in a cycle

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ESTIMATION (contd)

• Theorem 2
• A net system is

• uniformly MO iff the semi-linear set
• is a home-space for all

This is a finite union of linear sets with the
same period and the home space property is
decidable (Johnen Frutos Escrig 89)

• uniformly SMO only if it is bounded

Similar results hold for structural MO and SMO.
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5 MARKING ESTIMATION WITH INITIAL
MACROMARKING

• We want to exploit additional information on the
  initial marking.

• Example We start from marking (known) and
  the net evolves unobserved until we reach
  at this point we start observations.
• Then we may use the information that

• This characterization in terms of PN reachability
  is hard to use but we can approximate it using a
  matrix of invariants

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MACROMARKING (contd)

• Generalizing, we define an initial macromarking.

• The set of places is written as

• For each , the token content of is
  known to be
• Nothing is known about the marking in

• Let be the char vector of and define

• Macromarking

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MACROMARKING (contd)

• Algorithm (estimation with macromarking)
• 1 Initial estimate with
• 2 - Initial bound
• 3 - Let the current observed word be ww0.
• 4 - Wait until t fires.
• 5 - Update the estimate to
• 6 - New estimate
• 7 - New bound
• 8 - Goto 4.

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Elementary results
MACROMARKING (contd)

• The estimate is a lower bound

• The error functions are non-increasing

• The set of markings consistent with the
  observation w is

• This set can also be characterized as

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6 - CONTROL USING OBSERVERS

• GMEC specifications (linear constraints)

• The set of legal markings is

• Control with known state (actual marking
  is legal)
• Prevent the firing of a transition with
• iff there exists such that

Control pattern
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CONTROL (contd)

• Control with observer
• Prevent the firing of t after w has been
  observed iff
• there exists a legal consistent marking M such
  that the firing of t from M leads to a forbidden
  marking.

Control pattern
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• EXAMPLE

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CONTROL (contd)

• Usually, the control law using observers is not
  optimal since it can disable the firing of
  transitions that do not yield illegal markings.
• Such a control law may easily cause the
  controlled plant to block.
• We want to add to the observer the possibility of
  recovering from deadlocks caused by the
  incomplete information.

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A MANUFACTURING EXAMPLE
p1
p3
M(p3)M(p5) ? 3
t1
p4
p12
p7
p10
t4
t7
t3
p11
t6
p6
t2
p8
p5
p9
M(p9) ? 3
t5
p2
Initial macromarking we know the token content
in each cycle
M0(p11)M0 (p12) 1 M0(p1)M0 (p3)M0 (p4)
5 M0(p1)M0 (p5)M0 (p6) 5 M0(p1)M0 (p3)M0
(p6) M0 (p11) 6 M0(p1)M0 (p4)M0 (p5) M0
(p12) 5
M0(p2)M0 (p8) M0 (p9) 6 M0(p2)M0 (p3)M0
(p4)M0 (p7)M0 (p10) 6 M0(p2)M0 (p5)M0
(p6)M0 (p7)M0 (p10) 6 M0(p2)M0 (p3)M0
(p6)M0 (p7)M0 (p10)M0 (p11) 7 M0(p2)M0
(p4)M0 (p5)M0 (p7)M0 (p10)M0 (p12) 6
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( 4 5 1 0 0 1 0 1 0 0 0 1 / 0 0 0 0 0 0 0 0 0 0 0
0 / 1 5 5 6 5 6 6 6 7 6 )
t1
( 4 5 0 1 0 1 0 1 0 0 1 0 / 0 0 0 1 0 0 0 0 0 0 1
0 / 0 4 5 5 4 6 5 6 6 5 )
t4
( 5 5 0 0 0 0 1 1 0 0 1 0 / 1 0 0 0 0 0 1 0 0 0 1
0 / 0 4 4 4 4 6 5 5 5 5 )
t3
( 5 6 0 0 0 0 0 0 0 0 1 0 / 1 1 0 0 0 0 0 0 0 0 1
0 / 0 4 4 4 4 5 5 5 5 5 )
Deadlock
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p1
p3
M(p3)M(p5) ? 3
t1
p4
?
p12
p7
p10
t4
t7
t3
p11
t6
?
p6
t2
p8
p5
p9
M(p9) ? 3
?
t5
p2
Only the green tokens have been detected t6 and
t7 are disabled by the controller
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7 - DEADLOCK RECOVERY AND ESTIMATE UPDATE AFTER
NET TIME-OUT

• IDEA
• use the info that the net is deadlocked to
  improve the estimate (reducing the set of
  consistent markings)

• Theorem In an ordinary net a marking M is dead
  iff
• is a siphon
• for all

Mw
blocking marking
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DEADLOCK RECOVERY (contd)

• Given a structurally bounded net N, a marking M
  is dead iff ? a vector ?0,1m such that

is the characteristic vector of a siphon
contains only empty places
contains all empty places
each transitions has at least a pre arc coming
from
The set of blocking markings of N
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DEADLOCK RECOVERY (contd)
Algorithm (Control pattern updating after net
time-out) Let C C(?,B). Assume f(.,C) has led
the net to a time-out. 1. Let i0 and f0 f(.,
C). 2. Let Tit?T fi(t)1 and let Ni the net
obtained by N removing all
transitions not in Ti. 3. Update the control
pattern to fi1f(., C ? Mb(Ni)) 4. If fi1 fi
THEN exit (the deadlock procedure has failed) 5.
Wait until (a) a transition fires (net has
recovered from deadlock) (b) a new net
time-out occurs let ii1 and go to 2.
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DEADLOCK RECOVERY (contd)
Main advantages of the approach

• A unique linear algebraic formalism for
• state estimation
• control
• deadlock recovery

This procedure is denoted NTO procedure (net
time-out procedure).
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A MANUFACTURING EXAMPLE (contd)
p1
p3
M(p3)M(p5) ? 3
t1
p4
p12
p7
p10
t4
t7
t3
p11
t6
p6
t2
p8
p5
p9
M(p9) ? 3
t5
p2
Initial macromarking we know the token content
in each cycle
M0(p11)M0 (p12) 1 M0(p1)M0 (p3)M0 (p4)
5 M0(p1)M0 (p5)M0 (p6) 5 M0(p1)M0 (p3)M0
(p6) M0 (p11) 6 M0(p1)M0 (p4)M0 (p5) M0
(p12) 5
M0(p2)M0 (p8) M0 (p9) 6 M0(p2)M0 (p3)M0
(p4)M0 (p7)M0 (p10) 6 M0(p2)M0 (p5)M0
(p6)M0 (p7)M0 (p10) 6 M0(p2)M0 (p3)M0
(p6)M0 (p7)M0 (p10)M0 (p11) 7 M0(p2)M0
(p4)M0 (p5)M0 (p7)M0 (p10)M0 (p12) 6
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DEADLOCK RECOVERY (contd)
Proposition if the initial macromarking
is such that
(i.e., each column of V is a P-invariant) then,
for all observed words w,
Theorem 1 if the initial macromarking
is such that and

then the closed loop system will never time out
if the following constraint set does not admit
feasible solutions
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DEADLOCK RECOVERY (contd)
Definition the maximal control pattern for a set
C is where and
Theorem 2 if the initial macromarking
is such that
and
then the closed loop system will
always recover from a time-out if the following
constraint set does not admit feasible solutions
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A MANUFACTURING EXAMPLE (contd)
Initial macromarking the net is a marked graph
each cycle corresponds to a
P-invariant the initial macromarking
is such that
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A MANUFACTURING EXAMPLE (contd)
Theorem 1 does not apply the following
constraint set admits feasible solutions
the net might time out (it
actually does)
Theorem 2 does apply the following constraint
set does not admit feasible solutions
the closed loop system with
net time-out recovery is
deadlock-free
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8 - USING TIMING INFORMATION TO IMPROVE THE
PROCEDURE

• We extend the previous approach to exploit
  available information on the timing
  structure so as to obtain a better estimate of
  the set of consistent markings.
• A known delay time ?(t) is associated to each
  transition.
• We say that a transition t has timed-out at
  time now if it has been control enabled without
  firing during now- ?(t), now.
• We can be sure that at time now the actual
  marking Mw is such that ? Mw t?.

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USING TIMING INFORMATION (contd)
IDEA If Tto is the set of transitions that have
timed out at time now, we know for sure that the
actual marking is such that We compute a
(possibly) less restrictive control pattern using
as set of consistent markings i.e., for all
we compute
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USING TIMING INFORMATION (contd)
The new approach is denoted TTO procedure
(transition time-out procedure).

• Main Advantages
• Accelerates the state estimation
• Accelerates the deadlock recovery procedure
• Enables to recover from partial deadlocks

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A MANUFACTURING EXAMPLE (contd)
p1
p3
M(p3)M(p5) ? 3
t1
p4
p12
p7
p10
t4
t7
t3
p11
t6
p6
t2
p8
p5
p9
M(p9) ? 3
t5
p2
Delays

• d(t1) 2 d(t2) 5
• d(t3) 3 d(t4) 1
• d(t5) 2 d(t6) 6
• d(t7) 3

Transition time-out Transition firing
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9 CONCLUSIONS

• We provided a unique linear algebraic formalism
  for state estimation, control, deadlock
  recovery.
• We showed how timing information can be used to
  accelerate the state estimation and to detect the
  observer induced deadlock.
• Some sufficient conditions for deadlock recovery
  have been derived.

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FUTURE WORK

• Language completeness
• A word is language complete if
  This may allow to use observers
  in event feedback.

• Partial event observability
• assume some events are unobservable or
  undistinguishable. This may destroy the linear
  algebraic formalism in the general case. Look for
  restricted cases.

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FUTURE WORK (contd)

• Associate a probabilistic structure to the
  transition firing and define
  , where is the probability of having a
  complete word after k firings. Under which
  conditions ?

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REFERENCES

• A. Giua, C. Seatzu, Observability of
  place/transition nets, IEEE Trans. on Automatic
  Control, Vol. 47, No. 9, pp. 1424-1437,
  September, 2002.
• F. Basile, A. Giua, C. Seatzu, Petri net control
  using event observers and timing information,
  IEEE 2002 Conference on Decision and Control (Las
  Vegas, NV, USA), pp. 787-792, Dec. 2002.
• A. Giua, C. Seatzu, Deadlock characterization
  for Petri nets controlled using GMEC's and
  observers, 2003 American Control Conference
  (Denver, CO, USA) , pp. 320-325, June, 2003.
• A. Giua, J. Júlvez, C. Seatzu, 2003 American
  Control Conference (Denver, CO, USA) , pp.
  326-331, June, 2003.
• A. Giua, F. Basile, C. Seatzu, Observer based
  state-feedback control of timed Petri nets with
  deadlock recovery, IEEE Trans. on Automatic
  Control, submitted.