68 CHAPTER 2. OVERVIEWOF THE UNDERLYING THEORY
and Glover [79] independently obtained the following bound on the error induced by
balanced truncation.
Theorem 13
Given a stable, rational, P(s),andP
bal(s), the balanced truncation of orderk<n. Then,
kP(s)−Pbal(s)k∞≤2
n
X
i=k+1
σi
and
kP(s)−Pbal(s)kH≤2
n
X
i=k+1
σi.
Unobservable oruncontrollable modes have a corresponding Hankel singular value of
zero and we can see immediately from the above that their truncation does not affect
the ∞-norm of the system.
2.6.3 Hankel Norm Approximation
Weca nalso consider the problem of finding the kth order co ntrollerwhich gives the
closest fit (in terms of the Hankel norm) to the original system. The results given here
are due to Glover [79]. The first thing to consider is a lower bound on the error, which is
specified in the following lemma.
Lemma 14
Given a stable, rational P(s),andakth order approximation, Pk(s). Then
σk+1 ≤kP(s)−P
k
(s)k
∞
.
This tells us how well we can expect to do in terms of the ∞norm. Actually, there
exists a Pk(s) which achieves this bound. The only problem is that it can have unstable
(or anti-causal)parts to it.