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Chapter 6 State-Space Design
Xmath Control Design Module 6-22 ni.com
a compensator is shown in Figure 6-6. This figure combines full-state
regulator with gain K
r
and state estimator with gain K
e
.
Combining the plant, or system, equations with those of the regulator and
estimator, you can simplify the system equations for the compensator as
follows:
(6-9)
Figure 6-6. Linear Quadratic Gaussian Compensator (in the Bold Rectangle)
Equation 6-9 describes the state-space equations for both the
continuous-time compensator and the discrete-time compensator if no unit
delay is used between the time at which an input arrives at the system and
the time at which the new output appears. However, if you are working with
a real-time system which enforces a unit delay between the measurement
and the control update, you will need to create a “direct” compensator
in predictor form. With this direct implementation, the system output
equations become the same as the state update equations, multiplied by
a factor of the regulator gain K
r
.
x
ˆ
·
Ax
ˆ
Bu K
e
yCx
ˆ
Du+()–()++=
x
ˆ
·
Ax
ˆ
K
e
Cx
ˆ
– K
e
DB–()u– K
e
y+=
x
ˆ
AK
e
C– BK
e
D–()K
r
–[]x
ˆ
K
e
y+=
uK
r
x
ˆ
– 0()y+=
y
y
x = Ax + Bu
y = Cx + Du
+
–K
r
x = Ax + Bu + K
e
(y – (Cx + Du))
Regulator
Estimator
Controller
x
u
u
+
Plant
n
w