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5 Application: Predator-Prey Model
In this section, an example is given to illustrate the method given in this chapter for the
design of the FBO for fuzzy bilinear systems with unmeasurable decision variables.
This example concerns the predator-prey model worked out in Keller ( 1987 ).
The dynamics of predator-prey model can be described by the following non-
linear second order system:
8
<
x 1 ¼
_
ax 1
bx 1 x 2
x 2 ¼
_
cx 1 x 2
dx 2
fx 2 u
ð
102
Þ
:
y
¼
0
:
4x 1 þ
x 2
where the state x 1 ð t Þ
represent the prey and predator population,
respectively. The predator population may be decimated by men via the input
variable u
and x 2 ð t Þ
ð
t
Þ
. The coef
cients a, b, c, d are constant birth and death rates, and f is
the extermination rate.
The constants values used in simulation are given by: a = 1.5, b =1,c = 0.3,
d = 1 and f = 0.5.
Assume that the predator-prey model can be affected by one unknown input.
Then the system equation with the term modeling the unknown input is:
8
<
x 1 ¼ ax 1 bx 1 x 2 þ
0
:
1d
x 2 ¼
_
cx 1 x 2
dx 2
fx 2 u
þ
0
:
3d
ð
103
Þ
:
y
¼
0
:
4x 1 þ
x 2
5.1 Fuzzy Bilinear Models Representation
The previous nonlinear model ( 103 ) can be written as:
xt
_
ðÞ ¼
Axt
ððÞ
xt
ðÞþ
Bu t
ðÞþ
Nx t
ðÞ
ut
ðÞþ
Fd
ð
t
Þ
ð
104
Þ
y
ð
t
Þ ¼
Cx
ð
t
Þ
where the matrices A
ð:Þ
, B, N, F and C are respectively given by:
a
bx 2
0
0
0
Ax ð t ðÞ ¼
;
B ¼
cx 2
d
00
0
0
:
1
N
¼
;
F
¼
f
0
:
3
C ¼½
0
:
41
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