## You are not fair with LQG!

This point is made by
Carl-Fredrik Lindberg from Uppsala
University in Sweden. He refers to our
Example 9.1 on page 357
where LQG performs rather poorly for the inverse response
process
g(s) = 3(-2s+1)/(5s+1)(10s+1)

Carl-Fredrik has made an
alternative LQG-design which has a good response
similar to that for the loop-shaping design in
Figure 2.17 on page
44 .
Carl-Fredrik penalizes the integral of the error rather than
putting in the integrator explicitly (as we did in Example 9.1)
and he then avoids putting the input weight on du/dt so he can
get a fast response. He tunes the observer by making it
slightly faster than the LQ regulator - this is similar to
the ideas of loop transfer recovery (LTR).

So, Carl-Fredrik is right. LQG may be tuned to work well for this
example - in fact, for a SISO plant a good engineer is probably
able to make * any * design method work, if he/she know how
to use the method.

The question is then how easy
the method is to use (picking weights etc.) and how well it works
on MIMO plants. We still claim that LQG is in often inferior in this
respect to some other methods, such as H-infinity
(weighted sensitivity or loop-shaping)
or H2-optimal control
(which generalizes LQG by putting it in the generalized plant
setting, see Figures 3.13, 9.8 and 9.9, and allows, for example,
for frequency-dependent weights). The problems of tuning LQG controllers
by selecting reasonable (physical) values for the weight Q and R and
the noise V and W, is
the reason why the Loop transfer recovery (LTR) procedure became so
pupular with practicioners during the 1980's. However, we feel that
the methods presented in our book are easier to use and give
better designs.

To illustrate that also other methods work well on this problem, try the file
for an S/KS H-infinity design . Note
that the method and weight selection is * identical * to that in
Table 2.3 on page 60, except that the desired
bandwidth was set to wp = z/2 = 0.25 (which is reasonable for a
plant with a RHP-zero at z=0.5). This also gives a very good response
similar to that in Figure 2.17.
To reduce the input use one may increase Wu from 1 to 2.

## CORRECTION AS OF 12 May 2005

While preparing the second edition of the book, we realized that Carl-Fredrik is not quite correct after all.
He actually designs a two-degrees of freedom controller, and the setpoint response is improved (as shown by the simulation). However,
the feedback part of his controller is identical to the one we have (in the first edition of the book), so the disturbance rejection response (and robustness!) remains poor.