12.2
Linear Quadratic Gaussian Regulator
The linear quadratic Gaussian (LQG) problem is a generalization of the LQR prob-
lem to the case in which the state is not sensed directly. For the LQG problem we
consider the system given by
_x = Ax + Bu + wproc
(12.4)
y = Cx + vsensor
(12.5)
where the process noise wproc and measurement noise vsensor are independent and
have constant power spectral density matrices W and V , respectively.
The LQG cost function is the sum of the steady-state mean-square weighted
state x, and the steady-state mean-square weighted actuator signal u:
Jlqg = lim ;
t
x(t)TQx(t) + u(t)TRu(t)
(12.6)
E
!1
where Q and R are positive semide nite weight matrices.
This LQG problem can be cast in our framework as follows. Just as in the
LQR problem, we extract the (weighted) plant state x and actuator signal u as the
regulated output, i.e.,
z = R12u
Q12x :
The exogenous input consists of the process and measurement noises, which we
represent as
wproc
v
= W 12 w
sensor
V 12
with w a white noise signal, i.e., Sw(!) = I. The state-space description of the
plant for the LQG problem is thus
AP = A
(12.7)
Bw = W 12 0
(12.8)
Bu = B
(12.9)
Cz = 0Q1
(12.10)
2
Cy = C
(12.11)
Dzw = 0 0
0 0
(12.12)
Dzu = R120
(12.13)
Dyw = 0 V 12
(12.14)
Dyu = 0:
(12.15)
12.2 LINEAR QUADRATIC GAUSSIAN REGULATOR
279
This is shown in gure 12.2.
P
process
w n
o
1 2
noise
1 2
=
=
W
R
z
q
B
+
r
(sI A) 1 x q
1 2
;
=
;
Q
+
C
u
1 2
measurement noise
+
r
=
V
y
+
K
The LQG cost is
22.
Figure
12.2
kH
k
Since w is a white noise, the LQG cost is simply the variance of z, which is given
by
Jlqg = H 22:
k
k
The speci cations for the LQG problem are therefore the same as for the LQR
problem: realizability and the 2 norm-bound (12.1).
H
Standard assumptions for the LQG problem are that the plant is controllable
from each of u and w, observable from each of z and y, a positive weight is used
for the actuator signal (R > 0), and the sensor noise satis es V > 0. With these
standard assumptions in force, there is a unique controller Klqg that minimizes
the LQG objective. This controller has the form of an estimated-state-feedback
controller (see section 7.4) the optimal state-feedback and estimator gains, Ksfb
and Lest, can be determined by solving two algebraic Riccati equations as follows.
The state-feedback gain is given by
Ksfb = R 1BTX
;
lqg
(12.16)
where Xlqg is the unique positive de nite solution of the Riccati equation
ATXlqg + XlqgA XlqgBR 1BTX
;
lqg + Q = 0
(12.17)
;
which is the same as (12.2). The estimator gain is given by
Lest = YlqgCTV 1
(12.18)
;
280