Biomedical Engineering Reference
In-Depth Information
Then, once the desired roots of the characteristic equation of the closed-loop dynam-
ics have been prefixed the constants
{
}
c for
j
∈
, , 3
of the time-evolution of I(t)
−
1
are obtained from
KRM
=
since
R
is a non-singular matrix, i.e., an invertible
p
p
matrix. In this sense, note that
Det(R )
=
(r
−
r )(r
−
r )(r
−
r )
≠
0
since
R is a
p
2 13 13 2
{
}
Vandermonde matrix [15] and the roots
(r)
−
for
j
∈
, , 3
have been chosen dif-
j
ferent among them. Namely:
(
)
(
)
F r , r
I(0)
+σ
G r , r
E(0)
+σβ
I(0)S(0)
23
23
1
(r
−
r )(r
−
r )
2
1
3
1
c
(
)
(
)
1
F r , r
I(0)
+σ
G r , r
E(0)
+σβ
I(0)S(0)
13
13
1
=−
c
(25)
2
(r
−
r )(r
−
r )
2
1
3
2
c
3
(
)
(
)
F r , r
I(0)
+σ
G r , r
E(0)
+σβ
I(0)S(0)
12
12
1
(r
−
r )(r
−
r )
3
1
3
2
where
F:
and
2
+
→
G:
are defined as:
2
+
→
( ,
)
w(
− μ + γ
)(v
++μ + γ
)
(
) ;
2
( ,
)
vw(2
+−μ +σ+γ
)
(26)
σ−μ−γ
(r
)E(0)
+σβ
I(0)S(0)
Note that
c
=
3
1
>
0
since I(0)
≥
0
, E(0)
≥
0
, S(0)
≥
0
,
1
(
μ+γ−
r )(r
−
r )
13 1
(
)
23
(
)
23
Fr,r
=
0
,
Gr,r
=−μ − γ >
r
0
,
μ + γ −> and
1
r0
rr0
−> by taking into
3
3
1
account the constraints in (a). On one hand, I(t)
≥
0
∀∈
t
is proved directly from
0
+
(22) as follows. One 'a priori' knows that
> . However, the sign of both c and
c may not be 'a priori' determined from the initial conditions and constraints in (a).
The following four cases may be possible: (i)
1
c0
2
c0
≥
and
3
c0
≥
, (ii)
2
c0
≥
and
3
c0
<
, (iii)
2
c0
<
and
3
c0
≥
, and (iv)
2
c0
<
and
3
c0
<
. For the cases (i) and (ii),
i.e., if
2
c0
≥
, it follows from (22) that:
(
)
(
)
[
]
−
rt
−
r t
−
rt
−
rt
−
rt
−
r t
−
rt
−
rt
I(t)
= + +−−
c e
c e
I(0)
c
c
e
=
c
e
−+
e
c
e
−+
e
I(0)e
≥
0
(27)
1
2
3
1
3
2
3
3
1
2
1
2
1
2
−
rt
−
rt
∀∈
t
where the facts that
I(0)
=+ +≥ and,
c
c
c
0
e
−≥
e
0
and
1
3
0
+
1
2
3
−
rt
−
rt
e
−≥
e
0
∀∈
t
since
rr r
<< have been taken into account. For the case
2
3
0
+
1
2
3
(iii), i.e., if
2
c0
<
and
3
c0
≥
, it follows that:
(
)
[
]
[
]
−
rt
−
r t
−
rt
−
rt
−
r t
−
rt
−
rt
I(t)
=
I(0)
−
c
−
c
e
+
c e
+
c e
=
I(0)
−
c
e
+
c
e
−
e
+
c e
≥
0
(28)
1
2
3
1
2
1
3
2
3
2
3
3
2
3
−
rt
−
−≤
rt
∀∈
t
by taking into account that
I(0)
=++,
c
c
c
e
e
0
∀∈
t
2
1
0
+
1
2
3
0
+
since
1
rr
<
and the fact that:
2
[
]
(r
−
r )(r
−μ−γ −σβ
)
S(0) I(0)
+σ μ+γ−
(
r )E(0)
3
1
3
1
1
I(0)
−=
c
≥
0
(29)
3
(r
−
r )(r
−
− γ
)
3
1
3
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