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Chapter 4 Controller Synthesis
© National Instruments Corporation 4-5 MATRIXx Xmath Robust Control Module
here the weighting matrices are transfer matrices, whereas in the LQG
setup they are constants.
A description of the plant in Figure 4-3 is as follows:
• Dynamical system G
dyn
:
• Measured variables y = y
sens
+ n:
• Input weight W
in
:
• Output weight W
out
:
Weight Selection
In the standard LQG formulation, the weighting functions (W
dist
, W
noise
,
W
reg
, and W
act
) are all constant matrices. In fact, if Q
reg
and Q
act
are positive
definite matrices in the LQ cost, for example,
then the following apply:
Similarly, if R
proc
and R
sens
are positive definite matrices corresponding to
the process and measurement noise intensities, then the following apply:
x
·
Ax B
dist
dB
act
u++=
y
reg
C
reg
x=
y
sens
C
sens
x=
d
n
W
dist
0
0 W
noise
w
dist
w
noise
W
in
w
dist
w
noise
==
e
reg
e
act
W
reg
0
0 W
act
y
reg
u
W
in
y
reg
u
==
J
1
2
---
x'Q
reg
xu'Q
act
u+
0
∞
∫
⎝⎠
⎜⎟
⎛⎞
dt=
W
reg
Q
reg
12/
= W
act
Q
act
12/
=
W
dist
Q
proc
12/
= W
noise
Q
sens
12/
=