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NonlinearStateSpaceModel
OutputResponse
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Control Systems
Stabilization of Lynx and Hare Populations with LQG Control
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The populations of the hares and lynx are modeled after predator-prey dynamics and use the following variables:
h
a
r
e
p
o
p
u
l
a
t
i
o
n
ℯ
l
y
n
x
p
o
p
u
l
a
t
i
o
n
ℓ
h
a
r
e
p
o
p
u
l
a
t
i
o
n
e
q
l
.
v
a
l
u
e
ℓ
ℯ
l
y
n
x
p
o
p
u
l
a
t
i
o
n
e
q
l
.
v
a
l
u
e
ℊ
h
a
r
e
g
r
o
w
t
h
r
a
t
e
m
a
x
.
p
o
p
u
l
a
t
i
o
n
o
f
h
a
r
e
s
ℯ
h
a
r
e
a
n
d
l
y
n
x
p
o
p
u
l
a
t
i
o
n
s
'
i
n
t
e
r
a
c
t
i
o
n
t
e
r
m
h
a
r
e
c
o
n
s
u
m
p
t
i
o
n
f
a
c
t
o
r
ℓ
ℊ
l
y
n
x
g
r
o
w
t
h
c
o
n
s
t
a
n
t
ℓ
l
y
n
x
m
o
r
t
a
l
i
t
y
r
a
t
e
I
n
[
1
]
:
=
e
q
n
s
=
′
[
t
]
(
ℊ
[
t
]
)
1
-
[
t
]
-
(
(
ℯ
[
t
]
)
ℓ
[
t
]
)
(
+
[
t
]
)
,
′
ℓ
[
t
]
-
ℓ
ℓ
[
t
]
+
(
(
(
ℯ
ℓ
ℊ
)
[
t
]
)
ℓ
[
t
]
)
(
+
[
t
]
)
O
u
t
[
1
]
=
′
[
t
]
ℊ
[
t
]
1
-
[
t
]
-
ℯ
[
t
]
ℓ
[
t
]
+
[
t
]
,
′
ℓ
[
t
]
-
ℓ
ℓ
[
t
]
+
(
ℯ
ℓ
ℊ
[
t
]
ℓ
[
t
]
)
(
+
[
t
]
)
A nonlinear state-space model of the system:
I
n
[
2
]
:
=
n
s
s
m
=
N
o
n
l
i
n
e
a
r
S
t
a
t
e
S
p
a
c
e
M
o
d
e
l
[
e
q
n
s
,
{
{
[
t
]
,
ℯ
}
,
{
ℓ
[
t
]
,
ℓ
ℯ
}
}
,
{
ℊ
[
t
]
,
ℓ
ℊ
[
t
]
}
,
{
[
t
]
,
ℓ
[
t
]
}
,
t
]
/
.
p
a
r
s
O
u
t
[
2
]
=
{
[
t
]
,
2
0
.
5
8
8
}
-
1
1
2
5
2
[
t
]
ℊ
[
t
]
+
[
t
]
(
ℊ
[
t
]
-
(
3
.
2
ℓ
[
t
]
)
/
(
5
0
+
[
t
]
)
)
{
ℓ
[
t
]
,
2
9
.
4
8
1
}
ℓ
[
t
]
(
-
0
.
5
6
+
(
3
.
2
[
t
]
ℓ
ℊ
[
t
]
)
/
(
5
0
+
[
t
]
)
)
[
t
]
ℓ
[
t
]
The lynx population is eliminated with a growth rate of zero:
I
n
[
3
]
:
=
O
u
t
p
u
t
R
e
s
p
o
n
s
e
[
{
n
s
s
m
,
{
2
3
,
2
0
}
}
,
{
0
,
0
}
,
{
t
,
0
,
4
0
}
]
;
P
l
o
t
%
,
{
t
,
0
,
4
0
}
,
p
l
o
t
O
p
t
s
O
u
t
[
3
]
=
h
a
r
e
p
o
p
u
l
a
t
i
o
n
l
y
n
x
p
o
p
u
l
a
t
i
o
n
Specify the system specification and a set noise covariance matrices and control weights:
I
n
[
4
]
:
=
c
v
s
=
{
{
}
}
,
-
2
1
0
D
i
a
g
o
n
a
l
M
a
t
r
i
x
[
{
1
,
1
}
]
O
u
t
[
4
]
=
{
{
}
}
,
1
1
0
0
,
0
,
0
,
1
1
0
0
I
n
[
5
]
:
=
w
t
s
=
{
D
i
a
g
o
n
a
l
M
a
t
r
i
x
[
{
5
,
5
}
]
,
D
i
a
g
o
n
a
l
M
a
t
r
i
x
[
0
.
1
{
1
,
1
}
]
}
O
u
t
[
5
]
=
{
{
{
5
,
0
}
,
{
0
,
5
}
}
,
{
{
0
.
1
,
0
.
}
,
{
0
.
,
0
.
1
}
}
}
Compute an LQR controller:
I
n
[
6
]
:
=
n
s
s
m
=
N
o
n
l
i
n
e
a
r
S
t
a
t
e
S
p
a
c
e
M
o
d
e
l
[
n
s
s
m
,
A
u
t
o
m
a
t
i
c
,
A
u
t
o
m
a
t
i
c
,
A
u
t
o
m
a
t
i
c
,
N
o
n
e
]
;
I
n
[
7
]
:
=
l
q
g
=
L
Q
G
R
e
g
u
l
a
t
o
r
[
n
s
s
m
,
c
v
s
,
w
t
s
,
"
D
a
t
a
"
]
O
u
t
[
7
]
=
S
y
s
t
e
m
s
M
o
d
e
l
C
o
n
t
r
o
l
l
e
r
D
a
t
a
D
e
s
i
g
n
:
l
i
n
e
a
r
q
u
a
d
r
a
t
i
c
g
a
u
s
s
i
a
n
r
e
g
u
l
a
t
o
r
»
M
e
a
s
u
r
e
d
o
u
t
p
u
t
s
c
o
u
n
t
:
2
Obtain the closed-loop system:
I
n
[
8
]
:
=
c
s
y
s
=
l
q
g
[
"
C
l
o
s
e
d
L
o
o
p
S
y
s
t
e
m
"
]
/
/
F
u
l
l
S
i
m
p
l
i
f
y
O
u
t
[
8
]
=
{
,
2
0
.
5
8
8
}
◼
{
ℓ
,
2
9
.
4
8
1
}
◼
{
x
.
1
,
2
0
.
5
8
8
}
◼
{
x
.
2
,
2
9
.
4
8
1
}
◼
◼
◼
Generate a noisy signal to add uncertainty in the population measurement:
I
n
[
9
]
:
=
n
=
1
2
0
;
I
n
[
1
0
]
:
=
T
h
r
e
a
d
R
a
n
g
e
[
0
,
n
]
,
R
a
n
d
o
m
V
a
r
i
a
t
e
N
o
r
m
a
l
D
i
s
t
r
i
b
u
t
i
o
n
0
,
1
.
5
,
{
n
+
1
}
;
P
l
o
t
[
y
v
=
I
n
t
e
r
p
o
l
a
t
i
o
n
[
%
,
t
]
,
{
t
,
0
,
n
}
]
O
u
t
[
1
0
]
=
The closed-loop response results in a stable cyclic response:
I
n
[
1
1
]
:
=
o
r
=
O
u
t
p
u
t
R
e
s
p
o
n
s
e
[
{
c
s
y
s
,
{
2
3
,
2
0
}
}
,
{
y
v
,
0
,
0
}
,
{
t
,
0
,
1
2
0
}
]
;
P
l
o
t
%
,
{
t
,
0
,
1
2
0
}
,
p
l
o
t
O
p
t
s
O
u
t
[
1
1
]
=
h
a
r
e
p
o
p
u
l
a
t
i
o
n
l
y
n
x
p
o
p
u
l
a
t
i
o
n
The parametric plot demonstrates the limit-cycle behavior:
I
n
[
1
2
]
:
=
P
a
r
a
m
e
t
r
i
c
P
l
o
t
[
o
r
,
{
t
,
0
,
6
0
}
,
P
l
o
t
R
a
n
g
e
A
l
l
,
A
x
e
s
O
r
i
g
i
n
{
0
,
0
}
]
O
u
t
[
1
2
]
=
Obtain the controller model:
I
n
[
1
3
]
:
=
c
m
=
l
q
g
[
"
C
o
n
t
r
o
l
l
e
r
M
o
d
e
l
"
]
/
/
S
i
m
p
l
i
f
y
O
u
t
[
1
3
]
=
{
,
2
0
.
5
8
8
}
◼
{
ℓ
,
2
9
.
4
8
1
}
◼
◼
◼
The control effort reveals the equilibrium values for the hare and lynx growth rates:
I
n
[
1
4
]
:
=
O
u
t
p
u
t
R
e
s
p
o
n
s
e
[
c
m
,
J
o
i
n
[
{
0
,
0
}
,
o
r
]
,
{
t
,
0
,
1
2
0
}
]
;
P
l
o
t
[
%
,
{
t
,
0
,
1
}
,
P
l
o
t
R
a
n
g
e
A
l
l
,
P
l
o
t
L
e
g
e
n
d
s
{
,
}
]
O
u
t
[
1
4
]
=
Source Metadata
Citation:
Astrom & Murray 2009, Feedback Systems v2.10b Chapter 3, page 90 3.31
Related Symbols
NonlinearStateSpaceModel
OutputResponse
ParametricPlot
LQRegulatorGains
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