Biomedical Engineering Reference
In-Depth Information
(
]
where
. The first step to apply a coordinate trans-
formation based on the Lie derivation is to determine the relative degree of the sys-
tem. For such a purpose, the following definitions are taken into account: (i)
β = β
N
and
−∞
, 0
∩
1
0
−
(
)
(
)
∂
Lhx )
k1
f
−
(
)
(
)
(
)
k
f
is the
k
th-order Lie derivative of
h x(t) along
Lh x(t)
f x(t)
∂
x
(
)
(
)
(
)
0
Lh x(t) h x(t
and (ii) the relative degree r of the system is the
number of times that the output must be differentiated to obtain the input explicitly,
i.e., the number r so that
fx(t) w th
(
)
(
)
k
gf
r1
gf
−
LLh x(t)
=
0
for k
<−
r
1
and
LL h x(t)
≠
0
.
(
)
(
)
(
)
2
gf
From (7),
Lh x(t)
=
LLh x(t)
=
0
while
LLh x(t)
=−μσβ
I(t)
, so the
g
g
f
{
}
[
]
T
3
0
3
0
relative degree of the system is 3 in
DxI
=
ES
∈
I
≠
0
, i.e.,
∀∈
x
+
+
except in the singular surface I
of the state space where the relative degree is not
well-defined. Since the relative degree of the system is exactly equal to the dimension
of the state space for any
xD
=
0
∈
, the nonlinear coordinate change
(
)
I
(t)
=
L h x(t)
0
f
f
2
=
I(t)
(
)
[
]
(
)
E
(t)
=
L h x(t)
=
1 0 0 f x(t)
= − μ+ γ
(
)I(t)
+σ
E(t)
(8)
)
[
]
(
(
)
2
S(t)
=
L h x(t)
= − μ+γ
(
)
σ
0 f x(t)
= μ+γ
(
) I(t)
−σ
(2
μ+σ+γ
)E(t)
+σβ
I(t)S(t)
f
1
allows to represent the model in the called normal form in a neighborhood of any
xD
∈
. Namely:
(
)
(
)
(
)
x(t)
=
f
x(t)
+
g x(t) u(t) ; y(t)
=
h x(t)
(9)
T
where
x(t)
=
I(t)
E(t) S(t)
and:
T
T
(
)
(
)
(
)
(
)
(10)
f
x(t)
=
E(t) S(t)
ϕ
x(t)
; g x(t)
=
0 0
−μσβ
I (t)
; h x(t)
=
I (t)
with:
(
[
]
)
ϕ
x(t)
= μ+ω σβ− μ+σ μ+γ
(
)
(
)(
) I (t)
− μ+ω
(
)(2
μ+σ+γ
)
(t)
[
]
2
.
−
(3
μ+σ+γ+ω
)S(t)
−β ω μ+σ+γ + μ+σ μ+γ
(
)
(
)(
) I (t)
1
(11)
E(t)S(t)
E (t)
2
−β
(2
μ+σ+γ+ω
) I (t)E(t)
−β
I (t)S(t)+
+
(2
μ+σ+γ
)
1
1
I(t)
I(t)
The following result about the system input-output linearization is established.
Theorem 1.
The state feedback control law
(
)
(
)
(
)
(
)
−
Lhx(t)
3
− λ
hx(t)
− λ
Lhx(t)
− λ
Lhx(t)
2
u(t)
=
f
0
1
f
2
f
(12)
(
)
LLh x(t)
2
gf
{
}
where
λ
, for
i , , 2
∈
, are the controller tuning parameters, induces the linear
i
closed-loop dynamics
y(t)
+λ
y(t)
+λ
y(t)
+λ
y(t)
=
0
(13)
2
1
0
around any point x
∈
D
.
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