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Chapter 4 Controller Synthesis
© National Instruments Corporation 4-9 MATRIXx Xmath Robust Control Module
If no error message occurs, then is guaranteed. However,
this does not preclude the possibility that either or that
.
For the former case, there are two checks:
•Use the
linfnorm( ) function to compute .
• Compute the graph versus ω.
If by about 6 dB or more, then you can decrease
gamma and try
again.
When
gamma is very large, the specification (Equation 4-1) is easily
met. In this case, the
hinfcontr( ) function returns a controller that
approximately minimizes the H
2
norm of H
ew
while satisfying
Equation 4-2.
Gamma can be interpreted as a “knob” that smoothly
transforms the H
2
optimal (LQG) controller, (with gamma large), to a
H
∞
optimal controller (with ).
Similarly, for a large
gamma, the controller has good RMS performance
with the noise spectra determined by the weights W
dist
and W
noise
. For a
small
gamma, the controller has good worst-case performance for noise
spectra that lay below the weights W
dist
and W
noise
.
Example 4-1 Example of hinfcontr( )
Referring to Figure 4-2, suppose G has the state space description,
where:
1. The extended system matrix for G is:
A = 1;
B1 = [1,0]; B2 = 1; B = [B1,B2];
C1 = [1;0]; C2 = 1; C = [C1;C2];
D11 = zeros(2,2); D12 = [0;1]; D21 = [0,1]; D22 = 0;
D = [D11,D12; D21,D22];
G = system(A,B,C,D);
nw = 2; nz = 2;
H
ew
∞
γ≤
H
ew
∞
γ«
γ
opt
H
ew
∞«
H
ew
∞
σ
max
H
ew
jω()[]
H
ew
∞
γ«
gamma γ
opt
≈
x
·
xd+= u+
yxn+=
z
x
u
= v
d
n
=