where Ylqg is the unique positive de nite solution of
AYlqg + YlqgAT YlqgCTV 1CY
;
lqg + W = 0
(12.19)
;
(which can be solved using the methods already described in sections 12.1 and 5.6.3).
The LQG-optimal controller Klqg is thus
Klqg(s) = Ksfb(sI A + BKsfb + LestC) 1L
;
est
(12.20)
;
;
and the optimal LQG cost is
Jlqg = (XlqgW + QYlqg + 2XlqgAYlqg):
(12.21)
T
r
The speci cation H 2
(along with realizability) is therefore achievable if and
k
k
only if
qJlqg, in which case the LQG-optimal controller Klqg satis es the
speci cations.
12.2.1
Multicriterion LQG Problem
The LQG objective (12.6) can be interpreted as a weighted-sum objective for a
related multicriterion optimization problem. We consider the same system as in the
LQG problem, given by (12.4{12.5) the objectives are the variances of the actuator
signals, u1 2rms ... un 2rms
k
k
k
k
u
and some critical variables that are linear combinations of the system state,
c1x 2rms ... cmx 2rms
k
k
k
k
where the process and measurement noises are the same as for the LQG problem
(c1 ... cm are row vectors that determine the critical variables).
We describe this multicriterion optimization problem in our framework as fol-
lows. We use the same plant as for the LQG problem, substituting
2
u1 3
.
6
.. 7
6
7
6
7
z = u
6
n 7
u
6
c 7
6
1x 7
6
.
7
6
.. 7
4
5
cmx
for the regulated output used there. The state-space plant equations for the multi-
criterion LQG problem are therefore given by (12.7{12.15), with
2
0 3
C
c
6
1 7
z = 6 . 7
6
.. 7
4
5
cm
12.2 LINEAR QUADRATIC GAUSSIAN REGULATOR
281
substituted for (12.10) and
Dzu = I0
substituted for (12.13). The objectives are given by the squares of the 2 norms of
H
the rows of the closed-loop transfer matrix:
i(H) = H(i) 22
k
k
where H(i) is the ith row of H and L = nz = nu + m. The hard constraint for this
multicriterion optimization problem is realizability.
For 1 i nu, i(H) represents the variance of the ith actuator signal, and for
nu + 1 i L, i(H) represents the variance of the critical variable ci n x. The
;
u
design speci cation
1(H) a1 ... L(H) aL
(12.22)
therefore, limits the RMS values of the actuator signals and critical variables.
Consider the weighted-sum objective associated with this multicriterion opti-
mization problem:
wt sum(H) = 1 1(H) +
+ L L(H)
(12.23)
where
0. We can express this as
wt sum(H) = Jlqg
if we choose weight matrices
Q = n +1cT1c1 + + LcTmcm
(12.24)
u
R =
( 1 ... m)
(12.25)
diag
(
( ) is the diagonal matrix with diagonal entries given by the argument list).
diag
Hence by solving an LQG problem, we can nd the optimal design for the
weighted-sum objective for the multicriterion optimization problem with functionals
1 ... L. These designs are Pareto optimal for the multicriterion optimization
problem moreover, because the objective functionals and the hard constraint are
convex, every Pareto optimal design arises this way for some choice of the weights
1 ... L (see section 6.5). Roughly speaking, by varying the weights for the LQG
problem, we can \search" the whole tradeo surface.
We note that by solving an LQG problem, we can evaluate the dual function
described in section 3.6.2:
( ) = min 1 1(H) + + L L(H) H is realizable
f
j
g
= Jlqg
given by (12.21), using the weights (12.24{12.25). We will use this fact in sec-
tion 14.5, where we describe an algorithm for solving the feasibility problem (12.22).
282