Linear Controller Design: Limits of Performance by Stephen Boyd and Craig Barratt - HTML preview

CHAPTER 12 SOME ANALYTIC SOLUTIONS

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