34 CHAPTER 2. OVERVIEWOF THE UNDERLYING THEORY
We havepartitioned the matrix into two n×nblocks, X1and X2.IfX
1is invertible,
then
X=X2X−1
1,
is the unique, stabilizing solution to the ARE. The ability to form Xdoesn’t depend on
the particular choice of X1and X2.
Given a Hamiltonian, H,wesaythatH∈dom(Ric) if Hhas no ω axis eigenvaluesa nd
the associated X1matrix is invertible. Therefore, if H∈do m(Ric), we can obtain a
unique stabilizing solution, X. This mapping, from Hto X, is often written as the
function, X= Ric(H).
Tog ivean idea ofthe application of the ARE consider the following lemma (taken from
DGKF).
Lemma 1 Suppose H∈dom(Ric) and X= Ric(H). Then:
a) Xis symmetric;
b) Xsatis���es the ARE,
ATX+XA+XRX −Q=0;
c) A+RX is stable.
This is of course the well know result relating AREs to the solution of stabilizing state
feedback controllers.
AREs can also be used in calculating the H∞-norm of a state-space system. The
approach outlined here is actually that used in the softwarefor thecalculation of
kP(s)k∞. Consider a stable system,
P(s)=AB
C0.