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.


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.