2.3. H∞AND H2DESIGN METHODOLOGIES 35
Choose γ>0 and form the following Hamiltonian matrix,
H=Aγ
−2
BBT
−CTC−AT.
The following lemma gives a means of checking whether or not kP(s)k∞<γ.Aproofof
this lemma is given in DGKF although it is based on the work of Anderson [60],
Willems [61] and Boyd et al.[62].
Lemma 2 The following conditions are equivalent:
a) kP(s)k∞<γ;
b) Hhas no eigenvalues on the ω axis;
c) H∈dom(Ric);
d) H∈dom(Ric) and Ric(H)≥0(if (C,A) is observable then Ric(H)>0).
As the above illustrates, AREs play a role in both stabilization and H∞-norm
calculations for state-space systems. Before giving more detail on the H∞design
problem (Section 2.3.4), we will discuss some of the issues that arise in the practical
calculation of ARE solutions.
We can summarize an ARE solution method as follows:
(i) Form the Hamiltonian, H.
(ii)CheckthatHhasno ω axis eigenvalues.
(iii) Find a basis for the stable subspace of H.
(iv)CheckthatX
1is invertible.
(v) Form X=X2X−1
1.
The first issue to note is that it is difficult to numerically determine whether or not H
has ω axis eigenvalues. A numerical calculation of the eigenvalues is unlikely to give