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152 CHAPTER 4. DEMONSTRATION EXAMPLES
The loopshaping controller had good nominal performance and very poor robust
performance. This was illustrated with a relatively small perturbation. The difference
between the µ and H
∞
controllers was small in both the nominal and perturbed cases.
This may not always be the case for several reasons.
Only a single D-K iteration was performed here. Further iterations would further
improve the performance of the µ controller.
The perturbation chosen for the above simulation was not the worst case one. Note that
here, each of the three closed loop systems will have a different worst case perturbation.
To find these, perform a µ calculation on each closed loop system and use mkpert to
construct the appropriate perturbations.
The theoretical measure of performance is the H
∞
norm of the closed loop transfer
function. When assessing different controllers by simulation we are applying additional,
unformalized performance measures.
In this case, the performance and perturbation channels were about equally scaled. The
resulting D-scales were within an order of magnitude of unity. Choosing a different set
of units for α and/or θ would change the H
∞
norm of the result without changing µ.A
poor choice of engineering units could therefore lead to a larger difference between the
H
∞
controller and the µ controller.