2.6. MODEL REDUCTION 69
Consider the problem of finding the stable, order krealization which minimizes the
Hankel norm of the error. Define, Phankel(s) as the minimizing system. Thenwe have,
σk+1 ≤kP(s)−P
hankel(s)kH= inf
Pk(s)stablekP(s)−Pk(s)kH.
This system also satisfies ∞-norm bounds on the error, as illustrated in the following
theorem.
Theorem 15
Given a stable, rational, P(s), and the optimal kth order Hankel norm approximation,
Phankel(s). Then
kP(s)−Phankel(s)k∞≤2
n
X
i=k+1
σi.
Furthermore,there exists a constant matrix, D0, such that
kP(s)−(Phankel(s)+D
0
)k
∞≤
n
X
i=k+1
σi.
Careful examination of the previous section will indicate that the Hankel norm of a
system is independent of the Dterm. The optimal Hankel norm approximation given
above, Phankel(s), is considered to have a zero Dterm. It has the same error bounds as
the balanced truncation. Theorem 15 states that we can find a Dmatrix to addto
Phankel(s) to cut this bound in half.
The most common use of balanced truncations and Hankel norm approximations is to
reduce the order of a controller. Note that this will give a small ∞-normerror with
respect to the open-loop controller. It do es not say anything about the preservation of
closed loop properties. These should always be checked after performing a controller
order reduction.