162 CHAPTER 4. FUZZY CONTROL
The following basic linear relationships are valid for this system, namely,
q =Rh=rate offlow through orifice
qn =(tank input rate offlow)−(tank output rate offlow)
=net tank rate offlow =ADh
Applying the above relationship to tanks 1 and 2 yields, respectively,
qn 1 = A 1 Dh 1 =qi−q 1 =qi−
h 1 −h 2
R 1
qn 2 = A 2 Dh 2 =q 1 −q 2 =
h 1 −h 2
R 1
−
h 2
R 1
These equations can be solved simultaneously to obtain the transfer func-
tions hq^1 i and hq^2 i. The energy stored in each tank represents potential
energy, which is equal toρAh
2
2 ,whereρis thefluid density coefficient.
Since these are two tanks, the system has two energy storage elements,
whose energy storage variables areh 1 andh 2 .Ifweletx 1 =h 1 ,x 2 =h 2 ,
as the state variables, andu=qias the input to tank 1, we can write the
state-variable model for the system as
"
x ̇ 1
x ̇ 2
=
"
−R 11 A 1 R 11 A 1
1
R 1 A 2 −
1
R 1 A 2 −
1
R 2 A 2
#"
x 1
x 2
+
" 1
A 1
0
u
Lettingy 1 =x 1 =h 1 andy 2 =x 2 =h 2 yields the heights of the liquid in
each tank. As such, the set of output equations can be represented as
∑
y 1
y 2
∏
=
∑
10
01
∏∑
x 1
x 2
∏
UsingMatlabSimulink, obtain simulations that show the response of
the liquid level system.
(a) Develop a set of PID control parameters that will effectively control
the input so that desired heights of thefluid can be maintained in
tanks 1 and 2.
(b) Develop a fuzzy control system that can effectively maintain the liq-
uid level heights in tanks 1 and 2.
(c) Discuss the performance of the fuzzy controller and issues concerning
fine tuning.
8.Project: A mathematical model describing the motion of a single-stage
rocket is as follows:
d^2 y(t)
dt^2
=c(t)
μ
m
M−mt
∂
−g
μ
R
R+y(t)
∂
− 0. 5
μ
dy(t)
dt
∂ 2 μ
ρaACd
M−mt
∂
