64 CHAPTER 2. OVERVIEWOF THE UNDERLYING THEORY
Several aspects of this procedure are worth noting. For the µanalysis and Dscale
calculation, a frequency grid must be chosen. The range and resolution of this grid is a
matter of engineering judgement. The µanalysis can require a fine g rid in the vicinity of
the lightly damped modes. The order of the initial controller, K0(s), is the same as the
interconnection structure, G(s). The order of K(s) is equal to the sum of the orders of
G(s), ˆ
D(s)andˆ
D
−1
(s). This leads to a trade-off between the accuracy of the fit
between Dand ˆ
D(s) and the order of the resulting controller, K(s).
Another aspect of this to consideris that as the iteration approaches the optimal µ
value, theresulting controllers often have more and more response at high frequencies.
This may not show up in the µcalculation, the Dscale fitting, or a frequency response
of K(s), because the dynamics are occuring outside of the user specified frequency grid.
However these dynamics affect the next H∞design step and mayeven lead to numerical
instability.
The above discussion used an H∞controller to initialize the iteration. Actually any
stabilizing controller can be used. In high order, lightly damped, interconnection
structures, the H∞design of K0(s) may be badly conditioned. In such a case the
software may fail to generate a controller, or may give controller which doesn’t stabilize
the system. A different controller (the H2controller is often a good choice) can be used
to get a stable closed loop system, and thereby obtain Dscales. Application of these D
scales (provided that they do not add significantly many extra states) often results in a
better conditioned H∞design problem and the iteration can proceed.
The robust performance difference between the H∞controller,K0(s), and K(s), can be
dramatic even after a single D-Kiteration. The H∞problem is sensitive to the relative
scalings between vand w(and zand e). The Dscale provides the significantly better
choice of relative scalings for closed loop robust performance. Even the application of a
constant Dscale can havedramatic benefits.
2.6 Model Reduction
High order interconnection structures will result in high order controllers. Often a
controller of significantly lower order will perform almost as well. Approximating a
state-space system by one of lowerorder is referred to as mo del reduction. There are
several techniques availablefor this purp ose in Xµand the background to these
techniques is discussed here.