National Instruments Xmath Interactive Control Design Module ICDM - Manual (Page 105)

Chapter 12 LQG/H-Infinity Synthesis
© National Instruments Corporation 12-11 Xmath Interactive Control Design Module
The system equations of plant, filters, and integrators are as follows:
Standard LQG (All Toggle Buttons “Off”)
In LQG synthesis mode, the controller C minimizes a weighted sum of the
steady-state actuator and output variance:
where E denotes expectation.
Integral Action
When Integral action is enabled, the controller minimizes a variation on the
LQG cost:
where
Plant (P):
Output filter (F
y
):
Input filter (F
u
):
Integrator:
x·
p
A
p
x
p
B
p
u
act
B
p
w
p
w
x
+ + + =
y
p
C
p
x
p
w
s
w
y
+ + =
y˜ C
y˜
x
y˜
w
y˜
+ =
x·
y˜
A
y˜
x
y˜
B
y˜
y
p
W
p
w
F
+ + =
x·
u˜
A
u˜
x
u˜
B
u˜
u
act
+ =
u˜ C
u˜
x
u˜
D
u˜
u
act
+ =
x·
I
y˜ W
I
w
r
+ =
y
I
x
I
w
r
+ =
J E
t ∞ →
lim
x
T
u
T
R
xx
R
xu
R
ux
R
uu
x
T
u
T
ρ
u i ,
u
i
2
,
i 1 =
n
u
∑
+
ρ
y j ,
y
j
2
,
j 1 =
n
y
∑
+
⎝ ⎠
⎜ ⎟
⎜ ⎟
⎛ ⎞
=
J E
t ∞ →
lim
x
T
u
T
R
xx
R
xu
R
ux
R
uu
x
T
u
T
ρ
u i ,
u
i
2
i 1 =
n
u
∑
ρ
y j ,
y
j
2
y
I j ,
2
+
[ ]
j 1 =
n
y
∑
+ +
⎝ ⎠
⎜ ⎟
⎜ ⎟
⎛ ⎞
=
⎝ ⎠
⎜ ⎟
⎜ ⎟
⎛ ⎞
a
y
I j ,
t ( )
1
T
int
-------- y
j
ϒ ( ) ϒ d
0
t
∫
=