National Instruments Xmath Interactive Control Design Module ICDM - Manual (Page 108)

Chapter 12 LQG/H-Infinity Synthesis
Xmath Interactive Control Design Module 12-14 ni.com
By clicking the button at the bottom of the Weights window, arbitrary
weight matrices can be loaded from Xmath. The noise variances and
weights selected in this way are simply added to the diagonal weight and
noise matrices determined by the push buttons and sliders of the Weights
window. There are certain limitations and restrictions:
• If R
uu
is zero, none of the weight sliders on the actuators can be
disabled. This is because of the nonsingularity requirement of the input
weight matrix for the regulator problem.
• If Q
yy
is zero, none of the noise variance sliders of the sensors can be
disabled. This is because of the nonsingularity requirement of the
output weight matrix for the estimator problem.
• If a smaller number of actuators have been selected than there are
sensors, setpoint tracking cannot be expected in case the integrator
toggle button has been enabled.
H-Infinity Solution
The H
∞ controller design is done in entirely the same setting as the LQG
controller. Selection of sensors and actuators, and extension with frequency
weighting and integrators is identical to LQG.
The interpretation of weights and noise levels is slightly different. The
objective here is to minimize the maximal singular value of the transfer
function from a normalized version w
n
of w to a normalized version z
n
of z.
The normalization is based on the weights and noise levels as determined
by the Weights window.
More precisely, assume that, in the LQG formulation, E
ww
T = Q
ww
, and that
z is weighted in the quadratic criterion by a positive semi-definite,
symmetric matrix n, Q
zz
.
Then, w is of the form
and z is of the form
where w
n
and z
n
are normalized quantities.
w Q
ww
T
2
---
w
n
=
z R
zz
1
2
---
z
n
=