ME 132
Solutions # 13
1 The statevariable equations for the system are
˙
x
1
=
x
2
˙
x
2
=
g

c
m
u
2
x
2
1
y
=
x
1
(a) The control
u
needed to maintain
y
=
ζ
is found from evaluating the system at equilib
rium. This gives the equilibrium
parenleftBiggbracketleftBigg
x
1
x
2
bracketrightBigg
,
u,
y
parenrightBigg
=
parenleftBiggbracketleftBigg
ζ
0
bracketrightBigg
, ζ
radicalbigg
mg
c
, ζ
parenrightBigg
The linear system governing small deviations away from the operating point listed above
is
˙
δx
1
˙
δx
2
δy
=
0
1
0
2
g
ζ
0

2
ζ
radicalBig
gc
m
1
0
0
δx
1
δx
2
δu
(b) The observer and controller gains can be found using the standard observer/controller
design methods
K
=
bracketleftBigg

20

100

2
g
ζ
bracketrightBigg
,
L
=
bracketleftBig

4

2
g
ζ

4
bracketrightBig
parenleftbigg

ζ
2
radicalbigg
m
gc
parenrightbigg
For the controller equation given (which is
not
in terms of the deviation variables),
F
can be found by letting
y
be equal to
r
in the steady state
u
=
L
ˆ
x
+
Fζ
ζ
radicalbigg
mg
c
=
L
1
ζ
+
Fζ
⇒
F
=
radicalbigg
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 Spring '08
 Tomizuka
 Thermodynamics, MATLAB Function MATLAB

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