Feedback Control with Petri net

1
Feedback Control with Petri net

• Supervisory Control Course
• Lecture 5
• Spring 2004

2
Introduction

• Place invariant
• Transition invariant
• Feedback control
• Example 1
• Maximal permissivity property
• Uncontrollable Transitions
• Sensor Failures
• Piston Rod Robotic Assembly Cell

3
Fundamental equations
A firing sequence S is associated with a vector
s, whose jth component is the number of times Tj
is fired in S.
Example Assume that there are 4 transitions in
the PN (T1,T2,T3,T4) then
s1(1,0,0,0)T
S1 T1 is represented by
S2 T1T1T3 is represented by
s2(2,0,1,0)T
Starting from a marking mi, we can reach a new
marking mk by applying afiring sequence S.
In mathematical terms mkmiW.s
Where W is the PNs related incidence matrix.
4
Place Invariant
Definition A vector x is a place invariant (or
P-invariant) if xT.W0
For a given initial marking m0 we obtain
xT.mk xT.m0 For any reachable mk.

• Properties
• P-invariant is a structural property since it
  does not depend on marking.
• corresponds to the weights associated with the
  places.
• Linear combination of two invariants is a
  P-invariant.

5
Transition invariants

• Definition
• A vector y is a transition invariant (or
  T-invariant) if
• W.y 0
• I.e. If there exist a firing sequence S exists
  from a marking mi such that
• sqy
• (where q is a positive number), then S leads back
  to mi.

T-invariants characterizes a set of occurrence
sequences that have no total effect, i.e. Have
the same start and end markings.
6
Feedback control

• Requirements
• The process is modelled by a PN (untimed).
• A set of constraints, which must be satisfied by
  the process, is given.

• Design procedure
• Introduce an slack variable ms. This variable
  represents a place that belongs to the
  Controller net.

There are as many control places as there are
constraints.
7
Feedback control-II
A constraints is defined in the following
form mimj 1
Introduce a slack variable ms 0 into the
constraint to obtain an equality mi mj
ms 0
The slack variable represents a new place Ps
which recieves the excess tokens and making sure
that the sum of tokens in places Pi and Pj is
always less or equal to 1. The place Ps belongs
to the controller net.
8
Feedback control-III
Adding new places in the net results in increase
in size of the incidence Matrix of the overall
controlled system by a row corresponding to
theplace introduced by the slack variable. The
new row belongs