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30.4 HH 2 State Feedback Problem

This section shows that the methods presented for output feedback may be readily adopted to permit
2
the design of H optimal constant gain state feedback control laws (control gain matrices G c ) as well.
Generalized Plant Structure for State Feedback

For this case, the generalized plant G (including plant P and weighting functions) takes the following form:

A B 1 B 2
G = G 11 G 12 = C 1 0 n × D 12 = A B (30.241)
z n w
G 21 G 22 CD
I n × n 0 n × n 0 n × n
y w y u
This implies that the measured signals y are the states x of the generalized plant G. As such, all of the modes
of A are observable through C 2 = I n×n .

State Feedback Assumptions

The standard state feedback assumptions are a subset of those required for the output feedback problem
formulation. The state feedback assumptions are as follows.
2
Assumption 30.2 (HH State Feedback Problem)
Throughout this section, it will be assumed that

1. Plant G 22 Assumption. (A, B 2 ) stabilizable.
2. Nonsingular Control Weighting Assumption. R = D 12 D 12 > 0 (D 12 full column rank).
T
3. Regulator Assumption. jwI – A – B 2 has full column rank for all ω.
C 1 D 12
T
It should be noted that if D 12 C 1 = 0, then (3) is equivalent to (A, C 1 ) having no unobservable imaginary
modes. If (A, C 1 ) is detectable, then this is satisfied.
HH 2 Optimal State Feedback Control Law
2
The H optimal controller is given by
K opt = – G c (30.242)

n × n
∈
u
where the control gain matrix G c R is given by
G c = R [ B 2 X + D 12 C 1 ] (30.243)
–
1
T
T
where X ≥ 0 is the unique (at least) positive semi-definite solution of the CARE:

T
1
–
T
(
T
–
1
T
T
T
–
1
1
( AB 2 R D 12 C 1 ) X + X A B 2 R D 12 C 1 ) + C 1 ID 12 R D 12 )C 1 – XB 2 R B 2 X = 0 (30.244)
(
T
–
–
–
–
The closed loop poles that result from the above constant gain state feedback control law are the
eigenvalues of A − B 2 G c . The minimum closed loop norm is given by
(
2 = trace B 1 XB 1 ) (30.245)
T
min T wz
K H
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