You are here : Control System Design  Index  Book Contents  Appendix D  Section D.3 D. Properties of ContinuousTime Riccati EquationsD.3 The stabilizing solution of the CTARE
We see from Section D.2 that we have as many
solutions to the CTARE as there are ways of partitioning the
eigevalues of H into the groups
and
.
Provided that
is stabilizable and that
has no unobservable modes in the
imaginary axis, then
has no eigenvalues in the imaginary
axis. In this case, there exists a unique way of partitioning the
eigenvalues so that
contains only the stable
eigenvalues of
.
We call the corresponding (unique) solution
of the CTARE the stabilizing solution and denote it by
.
Lemma D.4
ProofFor part (a), we argue as follows:
Consider (D.1.11) and (D.1.14). Then
which implies that
If we consider only the first row in (D.3.5), then, using (D.1.8), we have
Hence, the closedloop poles are the eigenvalues of and, by construction, these are stable. We leave the reader to pursue parts (b), (c), and (d) by studying the references given at the end of Chapter 24.
