Standard Lqg (All Toggle Buttons "Off"); Integral Action - National Instruments Xmath Interactive Control Design Module ICDM User Manual

Standard LQG (All Toggle Buttons "Off")

Integral Action

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