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The operation time T i,j and the meeting time x qc , agv and x agv , asc will be sent by
the supervisory controller to the controller of each individual piece of equipment
at the lower level. The relation above is shown in Fig. 5. Each individual piece of
equipment should then make sure to finish its transport task τ i,j before the deter-
mined operation time T i,j . The controller of each individual piece of equipment
can hereby include additional control objectives, e.g., aimed at energy saving.
T +
T +
1,1
1,2
QC
t
i
x
s +
qc,agv
1,1
T
T
,3
,6
AGV
i
t
x
x
i
x +
1
qc,agv
x +
1
agv,asc
qc,agv
agv,asc
T
T
,4
,5
ASC
t
x
i
x +
1
agv,asc
s
agv,asc
,4
Fig. 5. The relation of time variables
3.2 The Lower-level Controllers
At the lower level, each piece of equipment has a controller that decides on the
continuous-time trajectories of the piece of equipment. In each controller at the
lower level, an optimal control problem is formulated so as to finish the task given
by the higher level within the operation time allowed and the meeting time. The
controller in the lower level can hereby take into account additional objectives,
such as energy consumption minimization. The specific control problem of a
piece of equipment depends on the task that it has to carry out. In particular, in
this case depending on the task and the position from where to start to where to
go, the moving behaviors of each equipment are different. Let z a and z b denote
the origin and destination positions of a piece of equipment. For task τ i,j ,the
controller of a piece of equipment solves the following problem:
min
u j ( t )
J ( z j ( t ) ,u j ( t ))
(27)
subject to
[0 ,T i,j ) , (28)
where J ( z j ( t ) ,u j ( t )) = T i, 0 0 . 5 mz 2 ( t ) 2 is the objective function quantifying the
energy consumption with mass m and velocity z 2 . The piece of equipment starts
its task at t = 0 and has to complete its task before t = T i,j . Here, (28) rep-
resents the continuous-time dynamics of the piece of equipment as explained in
z ( t )= g ( z ( t ) ,u ( t )) , z (0) = z a , z ( T i,j )= z b , t
 
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