Page 898 - The Mechatronics Handbook

0066_Frame_C30 Page 9 Thursday, January 10, 2002 4:43 PM

Since T wz involves various “sensitivity” transfer function matrices, we say that we have a weighted mixed
sensitivity problem.

Selection of Weighting Functions
Typically, the weighting functions W 1 , W 2 , W 3 are selected to be stable transfer function matrices that
are (at least initially) diagonally structured.
• Sensitivity Weighting. One might select the sensitivity weighting W 1 on the sensitivity S as follows:

W 1 = ---------- I n × (30.34)
k 1
s + e y n y
where k 1 , > 0. The parameter k 1 is typically selected to be large. The parameter is typically
selected to be small. Such selections are made so that S is heavily penalized at low frequencies—
precisely where we want K to make S small.
• Control Weighting. One might select the control weighting W 2 on KS as follows:

W 2 = k 2 I n × (30.35)
u n u

where k 2 > 0 provides a nonsingular penalty on the controls u (i.e., on KS).
• Output Weighting. One might select the output weighting W 3 on T as follows:

k 3 s + z 3 )
(
W 3 = ---------------------- I n × (30.36)
s + p 3 y n y

with k 3 > 0 and z 3 < p 3 . Such a weighting would penalize T more heavily at higher frequencies.
In general, care must be taken in selecting the structure of weighting functions. Inappropriate selections
may result in an ill-posed problem and a very arduous design process. For example, W 1 must be strictly
2 2
proper for H problems—otherwise the H norm of W 1 S makes no sense (since S approaches the identity
at high frequencies). While there exists no precise systematic method for the selection of weighting
functions, the above structures seem to work well (as starting points) in many applications.
Input–Output Representation for Generalized Plant G
To obtain an input–output (transfer function matrix) description for our generalized plant, we must
express the regulated signals z 1 , z 2 , z 3 and the measurements y in terms of the exogenous signals w and
the controls u. Doing so yields

(
z 1 = W 1 z ˆ 1 = W 1 wPu) = W 1 wW 1 Pu (30.37)
–
–
z 2 = W 2 z ˆ 2 = W 2 u (30.38)
z 3 = W 3 z ˆ 3 = W 3 Pu (30.39)

y = w z ˆ 3 = wPu (30.40)
–
–
From this, we obtain the following input–output (transfer function matrix) description for our gener-
alized plant G:

z = G 11 G 12 w (30.41)
y G 21 G 22 u

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