Information Technology Reference
In-Depth Information
Based on the prediction model in (34) and the definition of the control vari-
ables, the constraints of the state
z
in (33) can be translated into:
b
min
≤
Au
≤
b
max
.
A standard formulation of this energy consumption minimization problem can
then be obtained as follows:
u
T
Hu
+
f
T
u
min
u
(35)
subject to
Au
b
min
≤
≤
b
max
(36)
u
max
(37)
where
b
min
and
b
max
are the lower bound and the upper bound of the linear
inequality, respectively,
u
min
and
u
max
are the lower bound and the upper bound
of the control variables. This quadratic programming problem can be solved by
several existing solvers, such as quadprog in Matlab and SCIP in the OPTI
Toolbox [6]. When this problem is solved, the lower-level controller will set the
calculated trajectories as the reference for the piece of equipment.
u
min
≤
u
≤
Determining the Minimal Time Required.
The hierarchical control archi-
tecture as proposed provides a methodology for achieving the minimal makespan
in an energy-ecient way. In this control architecture, the minimal time of each
task is required at the higher level for scheduling tasks. As the interaction be-
tween the higher level and the lower level of the hierarchical control architecture,
thelowerboundofthetimerequiredfor a piece of equipment to carry out a
task needs to be computed. The minimal time required for doing a task de-
pends on the states and continuous-time dynamics of the piece of equipment.
The minimal-time required to complete a certain task
τ
i,j
can be obtained from
the theory of optimal control, as the result of Pontryagin's Minimum Principle.
Application of Pontryagin's Minimum Principle results in the minimization of
two so-called Hamiltonian functions. Details of the theory behind this principle
can be found in [7]. Here we provide the outcome of applying this principle in
our setting.
In our setting, application of the principle yields the control action
u
(
t
)that
minimizes the time for carrying out a task as follows:
⎧
⎨
u
max
for
t
=
t
2
,...,t
b
−
for
t
=
t
1
,...,t
2
u
(
t
)=
0
(38)
⎩
for
t
=0
+
,...,t
1
u
max
where
t
1
and
t
2
are so-called switching points between different control modes,
t
1
≥
t
1
+
,
t
2
≥
t
2
+
,
t
1
≥
and
t
2
≥
t
1
−
t
2
−
(
is a small positive value),
and where
t
1
and
t
b
are calculated as:
⎧
⎨
v
2
max
u
max
v
max
u
max
if
S
≥
S
u
max
t
1
=
(39)
if
S<
v
2
max
u
max
⎩
