36 CHAPTER 2. OVERVIEWOF THE UNDERLYING THEORY
any with a zero real part. In practice we must use a toleranceto determine what is
considered as a zero real part.
Finding a basis for the stable subspace of Hinvolves either an eigenvalue or Schur
decomposition. Numerical errors will be introduced at this stage. In most cases using an
eigenvalue decomposition is faster and less accurate than using a Schur decomposition.
Similarly, forming X=X2X−1
1will also introduce numericalerrors. TheSchur solution
approach, developedby Laub et al. [63, 64, 65], is currently the best numerical approach
to solving the ARE and is used in the software as the default method. An overview of
invariantsubspace metho ds for ARE solution is given by Laub [66]. Accurate solution of
the ARE is still very much an active area of research.
2.3.4 Solving the H∞Design Problem for a Special Case
We will nowlo ok atthe H∞design problem for a simplifying set of assumptions. The
general problem (with assumptions given in Section 2.3.2) can be transformedinto the
simplified one given here via scalings and other transformations. The simplified problem
illustrates the nature of the solution procedure and is actually the problem studied in
DGKF. The formulaefor the general problem are given in Glover and Doyle [59]. The
softwaresolves the general problem.
Consider the following assumptions, with reference to the system in Equation 2.11:
(i)(A,B
1
) stabilizable and (C1,A) detectable;
(ii)(A,B
2
) stabilizable and (C2,A) detectable;
(iii)DT
12[C1D12 ]=[0I];
(iv)B1
D21DT
21 =h0
Ii;
(v)D11 =D22 =0.
Assumption (i) is included in DGKF for technical reasons. The formulae are still correct
if it is violated. Note that, with these assumptions,
e=C1x+D12u,