Modeling of the Double-tank process
We will here briefly derive the transfer functions from control signal to measure-
ment signal for the Double-tank process. First, nonlinear differential equations
that describe the system are stated. These are linearized around a chosen oper-
ating point, and Laplace transforms are taken to obtain the transfer functions.
Finally, parameter values are given.
Derivation and linearization of differential equations
Mass balance and Bernoulli’s law give the following differential equation for a
tank with cross-section Aand outflow area a.
Adh
dt −a
p2gh+qin
We now assume that the flow generated by the pump is proportional to the
applied voltage, i.e.
qpump ku
This is motivated by the fact that the time constant of the pump is "small"
as compared to the tank dynamics. In the same way, assume that the level
measurement signal yis proportional to the true level h. Thus we write
ykch
Since the outflow from the upper tank is the inflow to the lower tank we
get the following system of nonlinear differential equations describing the tank
process
dh1
dt −
α
1
p2gh1+
β
u
dh2
dt
α
1p2gh1−
α
2p2gh2(1)
where h1and h2are the water levels in the upper and lower tank respectively.
We also define
α
iai
Aiand
β
k
A1.
Linearizing around an operating point1(h0
1,h0
2)yields
d∆h1
dt −1
T
1
∆h
1+
β
∆u
d
∆h
2
dt 1
T1
∆h1−1
T2
∆h2(2)
where we have introduced
T11
α
1s2h0
1
g,T21
α
2s2h0
2
g
1Note that from (1)we have the condition h0
1
h0
2
(
α
2
α
1)
2
.
1
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