Problem 1
a) For noninteracting capacities with linear resistances subject to a unit-step change in the
input of the first tank, the material balance can be written as follows:
1
1
1
1
1
dy
A R
y
R u(t)
dt
+
=
(S1.1)
2
2
2
2
2
1
1
dy
R
A R
y
y
dt
R
+
=
(S1.2)
subject to the initial conditions
1
1s
1
s
y (0)
y
R u
=
=
2
2s
2
1
1s
2
s
y (0)
y
R
R y
R u
=
=
=
(S1.3)
Defining the deviation variables Y
1
=y
1
-y
1s
, Y
2
=y
2
-y
2s
and Q=u(t)-u
s
, equations (S1.1)-
(S1.2) become:
1
1
1
1
1
dY
A R
Y
R Q
dt
+
=
(S1.4)
2
2
2
2
2
1
1
dY
R
A R
Y
Y
dt
R
+
=
(S1.5)
subject to the initial conditions
1
Y (0)
0
=
2
Y (0)
0
=
(S1.6)
Hence, the transfer functions for equations (S1.4)-(S1.5) are:
1
1
1
1
1
Y (s)
R
G (s)
Q(s)
(A R )s
1
=
=
+
(S1.7)
2
2
1
2
1
2
2
Y (s)
R
R
G (s)
Y (s)
(A R )s
1
=
=
+
(S1.8)
Since the two noninteracting tanks are placed in series, the overall transfer function is the
following:
[
][
]
2
2
1
2
1
1
2
2
Y (s)
R
G(s)
G (s)G (s)
Q(s)
(A R )s
1
(A R )s
1
=
=
=
+
+
(S1.9)
Rewriting (S1.9) in the standard form, gives:
2
2
1
2
2
1
1
2
2
1
1
2
2
Y (s)
R
G(s)
G (s)G (s)
Q(s)
(A R )(A R )s
2(A R
A R )
1
=
=
=
⎡
⎤
+
+
+
⎣
⎦
(S1.10)

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