Information Technology Reference
In-Depth Information
Section 2.3. We consider the continuous-time dynamics model for the behavior
of a piece of equipment as a double integrator:
z
1
(
t
)=
z
2
(
t
)
,
1
(
t
)
∈
[0
,S
]
(29)
u
max
,u
max
]
,
(30)
where
z
1
(
t
)(m) and
z
2
(
t
)(m/s) describe the position and the velocity of equip-
ment, respectively,
u
(m/s
2
) represents the acceleration,
S
is the traveling dis-
tance of equipment, [
z
2
(
t
)=
u
(
t
)
,
2
(
t
)
∈
[
−
v
max
,v
max
]
,
(
t
)
∈
[
−
v
max
,v
max
] is the constraint on
z
2
(
t
), and
u
max
is the
constraint on the acceleration.
−
Solving the Control Problem.
The continuous-time dynamic model can be
discretized for optimization purposes. Due to the linearity of the original system,
the energy consumption problem in that case can be formulated as a standard
quadratic programing problem, which can be solved eciently. The discretized
dynamical model, based on (29) and (30) for a piece of equipment is as follows:
z
(
k
+1)=
1
ΔT
01
z
(
k
)+
0
.
5
ΔT
2
ΔT
u
(
k
)=
Az
(
k
)+
Bu
(
k
)
.
(31)
where
z
(
k
)=
s
(
k
)
v
(
k
)
T
describes the position
s
(
k
)andvelocity
v
(
k
)ofthe
piece of equipment, and
u
(
k
) is the acceleration of the piece of equipment.
The mechanical energy is minimized by means of minimizing the following
objective function:
N
s
J
=
E
(
k
)
,
(32)
k
=1
where
E
(
k
)=0
.
5
mz
2
(
k
)
2
describes the unit kinetic energy of the piece of equip-
ment at time
k
and
N
s
=
T
i,j
+ 1 is the number of discretized steps, with the
T
s
time step size
T
s
.
For the initial condition and final condition,
z
(0) =
s
(0) 0
T
and
z
(
N
s
)=
s
(
N
s
)0
T
(
s
(0) and
s
(
N
s
) are the initial state and the final state of position).
Besides, the constraints of other states can be written in the following form:
z
min
≤
z
(
k
)
≤
z
max
,k
=1
, ..., N
s
−
1
,
(33)
where
z
min
and
z
max
are the constraints of states on
z
(
k
).
The prediction model of this discretized dynamical system (31) shown below
is used to formulate the standard form of optimization problems.
k−
1
z
(
k
)=
A
k−
1
z
(0) +
A
k
B
u
(
k
−
1
−
i
)
,k
=1
, ..., N
s
(34)
i
=0
⎡
⎣
⎤
⎦
u
(0)
u
(1)
.
u
(
N
s
−
The control variables are defined as:
u
=
1)
