User`s guide

LQG Design
7-9
Optimal State-Feedback Gain
In LQG control, the regulation performance is measured by a quadratic
performance criterion of the form
The weighting matrices are user specified and define the trade-off
between regulation performance (how fast goes to zero) and control effort.
The first design step seeks a state-feedback law that minimizes the
cost function . The minimizing gain matrix is obtained by solving an
algebraic Riccati equation. This gain is called the LQ-optimal gain.
Kalman State Estimator
As for pole placement, the LQ-optimal state feedback is not
implementable without full state measurement. However, we can derive a
state estimate such that remains optimal for the output-feedback
problem. This state estimate is generated by the Kalman filter.
with inputs (controls) and (measurements). The noise covariance data
determines the Kalman gain through an algebraic Riccati equation.
The Kalman filter is an optimal estimator when dealing with Gaussian white
noise. Specifically, it minimizes the asymptotic covariance
of the estimation error .
Ju() x
T
Qx 2x
T
Nu u
T
Ru++{}td
0
∞
∫
=
QNR
,,
xt
()
uKx
–=
Ju
()
K
uKx
–=
x
ˆ
uKx
ˆ
–=
x
ˆ
·
Ax
ˆ
Bu L y
v
Cx
ˆ
– Du–()++=
u
y
v
Eww
T
()Q
n
,= Evv
T
()R
n
,= Ewv
T
()N
n
=
L
Exx
ˆ
–()xx
ˆ
–()
T
()
t∞→
lim
xx
ˆ
–
x
ˆ
u
y
v
Kalman
estimator