Biomedical Engineering Reference
In-Depth Information
Proof.
The state equation for the closed-loop system
I(t)
E
(t)
(14)
E
(t)
=
S(t)
(
)
(
)
3
f
ϕ
x(t)
−
L h x(t)
−λ
I(t)
−λ
E(t)
−λ
S(t)
S(t)
0
1
2
is obtained by introducing the contro
l
law (12) in (9) and taking into account the fact
that
(
)
and the coordinate transforma-
tion (8). Moreover, it follows by direct calculations that:
(
L L h x(t)
2
gf
=−μσβ
I(t)
=−μσβ
I (t)
≠
0
xD
∀∈
)
3
3
2
Lh (t)
= σβ μ+ω − μ+γ
(
)
(
)
I(t)
+σ
(
μ+γ
)
+
(2
μ+σ+γ μ+σ
)(
)
(t)
f
(15)
[
]
−σβ ω
I(t) I(t)
+
E(t)
+σ β
2
E(t)S(t)
−σβ
(4
μ+σ+ γ+ω
2
)I(t)S(t)
−σβ
2
I (t)S(t)
2
1
1
1
1
(
)
3
L h x(t) in the state space defined by x(t) via the application of
the reverse coo
rd
inate transformation to that in (8). Then, it follows directly that
(
One may express
)
(
)
3
Lh x(t)
. Thus, the state equation of the closed-loop system in the state
space defined by x(t) can be written as:
=ϕ
x(t)
x(t)
=
Ax(t)
(16)
with
010
A 001
=
.
(17)
−λ
−λ
−λ
0
1
2
Furthermore, the output equation of the closed-loop system is y(t)
=
Cx(t)
with
[
]
C100
=
since y(t)
== . From (16)-(17) and the closed-loop output
I(t)
I(t)
equation, it follows that:
y ) CAex(0)
()
=
At
(18)
{
}
for
denoting the order of the differentiation of y(t) . Finally, the dy-
namics of the closed-loop system (13) is directly obtained from (18).
∈
0, 1, 2, 3
***
{
}
Remark 1
. The controller parameters
λ
, for
i , , 2
∈
, will be adjusted such that
i
the
roots
of
the
closed-loop
system
characteristic
polynomial
(
)
, with
denoting the identity matrix, be
P(s)
=
Det sI
−
A
=
(s
+
r )(s
+
r )(s
+
r )
33
×
I
∈
3
1
2
3
3
{
}
{
}
located at prescribed positions. i.e.,
λ=λ −
(r)
for
i , , 2
∈
and
j
∈
, , 3
,
i
i
j
with
(r)
−
denoting the desired roots of P(s) . If one of the control objectives is to
j
{ }
guarantee the exponential stability of the closed-loop system then
Re r
>
0
for all
j
{
}
j
∈
, , 3
. Then, the values
λ=
rr r
>
0
,
λ=
rr
+
rr
+
r r
>
0
and
0
1 2 3
1
1 2
1 3
2 3
for the controller parameters have to be chosen in order to achieve
such a stability result. It implies that the strictly positivity of the controller parameters
is a necessary condition for the exponential stability of the closed-loop system. ***
λ= + + >
rr r 0
2
1
2
3
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