Page 123 - Modern Control Systems

Section 2.8 Design Examples 97

is sufficiently = Ai/100), so the water flows
atmospheric pressure, and A 2 small (A 2
is negligible. Thus Bernoulli's equation reduces to
out slowly and the velocity v x
v 2 = VlgH. (2.110)
Substituting Equation (2.110) into Equation (2.109) and solving for H yields
t^
H = - VH + -\-Q h (2.111)

pA x
Using Equation (2.110), we obtain the exit mass flow rate
Qi = P^ 2V2 = (pV2^4 2 )V/7. (2.112)
To keep the equations manageable, define
A 2V2g

1
ft,:-
PA{
k 3 := PV2gA 2.
Then, it follows that
H = h VH + k 2Q h
= k 3VH. (2.113)
Q 2
Equation (2.113) represents our model of the water tank system, where the input is
Q\ and the output is Q 2. Equation (2.113) is a nonlinear, first-order, ordinary differ-
1 2
ential equation model. The nonlinearity comes from the H ^ term. The model in
Equation (2.113) has the functional form

where

/(//, Q x) = k,VH + k 2Q x and h(H, Q x) = fc 3V//.
A set of linearized equations describing the height of the water in the reservoir
is obtained using Taylor series expansions about an equilibrium flow condition.
When the tank system is in equilibrium, we have H = 0. We can define Q* and H*
as the equilibrium input mass flow rate and water level, respectively. The relation-
ship between Q* and H* is given by

Q* = —±\/H* = p\/2gA 2Vm. (2.114)
k 2
This condition occurs when just enough water enters the tank in A\ to make up for
the amount leaving through A 2. We can write the water level and input mass flow
rate as
// = //* + A//, (2.115)
Qx = Q* + AQ lf