7/30/2019 LQG1
2/2
The optimal controller: Theorem 9.1
The optimal control law is ( ) = ( ), where ( ) is obtained
from the corresponding Kalman filter.
The state feedback gain is = 12
.
The matrix = is the unique non-negative definite solution to
the continuous-time Riccati equation (CARE)
0 = +
+ 1 12
N.B. There are two differentCAREs involved, the one above and
the
one for the Kalman filter!
Some technical conditions: (
) stabilizable, (
) and
(
1 ) detectable...
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LQ/LQG: Properties
is always stable
The control law =
( ) (pure state feedback) is optimal,
also for the deterministic case ( 1 = 0 and 2 = 0) LQ =
linear quadratic control
If 1 and 2 have Gaussian distributions the controller is the
optimal controller (Cor. 9.1) LQG = linear quadratic
Gaussian control
Theorem 9.1 = the separation theorem: the optimal observer =
the Kalman filter, combined with the optimal state feedback
(LQ) give the optimal controller! (this is far from
obvious...)
The LQ/LQG controller looks exactly the same for SISO and
MIMO systems
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The servo problem
How can the reference signal ( ) be included?
General solution: Characterize ( ) by its spectrum and model it
in
the same way as a disturbance, i.e. incorporate it in the
model.
Special case, Theorem 9.2: If ( ) is piecewise constant, then
the
criterion
=
2 1
+
( )
2 2
is minimized with the control law ( ) = ( ) +
( ), where
and ( ) are given as in Theorem 9.1, and
is chosen so that
= (
+
)
=0=
(0)
(Then ( ) =
.)
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Example: The effect of 1 and 2
The DC-motor ( ) = 1
(
+1) ( ), 1 = 1 and 2 varied.
0 1 2 3 4 5 6 70
0.2
0.4
0.6
0.8
1
time
y
Q2
= 104
Q2
= 0.01
Q2
= 0.1
Q2
= 1
0 0.5 1 1.5 2
15
10
5
0
5
10
15
20
time
u
Q2
= 104
Q2
= 0.01
Q2
= 0.1
Q2
= 1
Step responses ( = unit step) for the closed loop systems.
The output to the left, the input to the right.
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