40 CHAPTER 2. OVERVIEWOF THE UNDERLYING THEORY
eigenvalueof X∞(and Y∞) is displayed. Theultimate test of the software is to form the
closed loop system and check both its stability and norm. We strongly suggest that the
user always perform this step.
The numerical issues discussed above are very unlikely to arise in low order systems.
Experience has shown that systems with very lightly damped modes are more
susceptible to numerical problems than those with more heavily damped modes.
However, it has been found to be possible, with the software provided, to design
controllers using 60th order interconnection structures with very lightly damped modes.
2.3.6 H2Design Overview
Recall from Section 2.1.2 that the H2norm of a frequency domain transfer function,
G(s), is
kG(s)k2=1
2πZ∞
−∞
Trace[G(ω)∗G(ω )]dω1/2
.
Several characterizationsof this norm are po ssible in terms of input/output signals. For
example, if the unknown signals are of bounded energy, kGk2gives the worstcase
magnitude of the outputs e. Alternatively, if impulses are applied to the inputs of G(s),
kG(s)k2gives the energyof the outputs e.H2synthesis involves finding the controller
which minimizes the H2norm of the closed loop system. This is the same the well
studied Linear Quadratic Gaussian problem.
2.3.7 Details of the H2Design Procedure
The H2design procedure is best explained by contrasting it with the H∞procedure
explained in the previous sections. There are several differences, the most obvious being
that the H2design problem always has a unique, minimizing, solution. The other
difference is that (in addition to the four conditions given in Section 2.3.2) D11 is
required to be zero, even in the general case. If this condition is vio lated no controller
will give a finite H2norm for the closed loop system as it will not roll off as the
frequency goes to ∞.
Wepresent the H2solution in anLFT framework rather than the more well known LQG