1.1. Historical Persp ective
3
from Limeb eer and Hung [1987] and Limeb eer and Halikias [1988] who showed, for
problems transformable to 2
1-block problems, that a subsequent minimal realization
of the controller has state dimension no greater than that of the generalized plant
G
.
This suggested the likely existence of similarly low dimension optimal controllers in the
general 2
2case.
Additional progress on the 2
2-block problems came from Ball and Cohen [1987],
who gaveastate-space solution involving 3 Riccati equations. Jonckheere and Juang
[1987] showed a connection between the 2
1-block problem and previous work by
Jonckheere and Silverman [1978] on linear-quadratic control. Foias and Tannenbaum
[1988] developed an interesting class of operators called skew Toeplitz to study the
2
2-block problem. Other approaches have been derived by Hung [1989] using an
interpolation theory approach, Kwakernaak [1986] using a polynomial approach, and
Kimura [1988] using a method based on conjugation.
The simple state space
H
1
controller formulae to b e presented in this b ook were rst
derived in Glover and Doyle [1988] with the 1984 approach, but using a new 2
2-block
solution, together with a cumbersome back substitution. The very simplicityofthenew
formulae and their similaritywith the
H
2
ones suggested a more direct approach.
Independent encouragement for a simpler approachtothe
H
1
problem came from
papers by Petersen [1987], Khargonekar, Petersen, and Zhou [1990], Zhou and Khar-
gonekar [1988], and Khargonekar, Petersen, and Rotea [1988]. They showed that for
the state-feedback
H
1
problem one can choose a constant gain as a (sub)optimal con-
troller. In addition, a formula for the state-feedback gain matrix was given in terms
of an algebraic Riccati equation. Also, these pap ers established connections between
H
1
-optimal control, quadratic stabilization, and linear-quadratic dierential games.
The landmark breakthrough came in the DGKF pap er (Doyle, Glover, Khargonekar,
and Francis [1989]). In addition to providing controller formulae that are simple and
expressed in terms of plant data as in Glover and Doyle [1988], the metho ds in that
paper are a fundamental departure from the 1984 approach. In particular, the Youla
parameterization and the resulting 2
2-block model-matching problem of the 1984
solution are avoided entirely; replaced by a more purely state-space approachinvolving
observer-based comp ensators, a pair of 2
1block problems, and a separation argument.
The operator theory still plays a central role (as does Redheer's work [Redheer, 1960]
on linear fractional transformations), but its use is more straightforward. The key
to this was a return to simple and familiar state-space to ols, in the style of Willems
[1971], such as completing the square, and the connection between frequency domain
inequalities (e.g
k
G
k
1
<
1), Riccati equations, and sp ectral factorizations. This b ook
in some sense can b e regarded as an expansion of the DGKF paper.
The state-space theory of
H
1
can be carried much further, by generalizing time-
invariant to time-varying, innite horizon to nite horizon, and nite dimensional to
innite dimensional. A ourish of activity has begun on these problems since the publi-
cation of the DGKF paper and numerous results have b een published in the literature,
not surprising, many results in DGKF paper generalize
mutatis mutandis
, to these cases,
which are b eyond the scop e of this bo ok.
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