Biomedical Engineering Reference
In-Depth Information
Tabl e 2. 1
Example piecewise affine model and parameter inequalities
x
c
=
κ
c
+
κ
c
s
−
(
x
f
,θ
f
)
s
+
(
x
c
,θ
c
)
s
+
(
x
y
,θ
y
)
s
+
(
u, θ
u
)+
κ
c
s
−
(
x
f
,θ
f
)
− γ
c
x
c
x
y
=
κ
y
+
κ
y
[1
− s
+
(
x
c
,θ
c
)
s
+
(
x
y
,θ
y
)
s
+
(
u, θ
u
)]
− γ
y
x
y
x
f
=
κ
f
[1
− s
+
(
x
c
,θ
c
)
s
+
(
x
y
,θ
y
)
s
+
(
u, θ
u
)]
s
−
(
x
f
,θ
f
)
+
κ
f
s
+
(
x
g
,θ
g
)
s
−
(
x
t
,θ
t
)
s
−
(
x
f
,θ
f
)
×
[1
− s
+
(
x
c
,θ
c
)
s
+
(
x
y
,θ
y
)
s
+
(
u, θ
u
)]
− γ
f
x
f
x
g
=
κ
g
[1
− s
+
(
x
g
,θ
g
)
s
−
(
x
t
,θ
t
)]
s
−
(
x
f
,θ
f
)
− γ
g
x
g
x
t
=
κ
t
s
+
(
x
g
,θ
g
)
s
−
(
x
t
,θ
t
)
s
+
(
x
f
,θ
f
)
− γ
t
x
t
x
r
=
κ
r
s
+
(
x
f
,θ
f
)+
κ
r
− γ
r
x
r
0
<θ
c
<
κ
c
γ
c
<
κ
c
+
κ
c
<θ
c
<θ
c
<
κ
c
+
κ
c
γ
c
γ
c
κ
y
γ
y
<θ
y
<θ
y
<
κ
y
+
κ
y
γ
y
0
<θ
y
<
κ
f
γ
f
<θ
f
<θ
f
<θ
f
<θ
f
<
κ
f
+
κ
f
γ
f
0
<θ
f
<
κ
g
γ
g
0
<θ
t
<θ
t
<
κ
t
γ
t
0
<θ
g
<θ
g
<
Fig. 2.8
Asymptotic behavior of the PWA in the
(
x
f
,x
g
)
plane, for the case
u
=0
.
Thick black
lines
indicate sliding modes [
24
]
3. Damped oscillations around the point
x
g
=
θ
g
and
x
f
=
θ
f
. It is shown that all
trajectories will asymptotically converge to this point, which is an equilibrium in
the sense of Filippov;
4.
x
r
(
t
)
→
κ
r
+
κ
r
γ
r
following the solution
x
f
.
There are also sliding modes along the segments:
x
g
=
θ
g
with
x
f
<θ
f
and
x
g
>θ
g
with
x
f
=
θ
f
(Fig.
2.8
).
The PWA formalism allowed a more rigorous analysis of the complex network of
carbon starvation response in
Escherichia Coli
. Major participants were identified as
well as their roles in the presence or absence of nutritional stress. This PWA network
Search WWH ::
Custom Search