Biomedical Engineering Reference
In-Depth Information
The eigenvalues of the Jacobian matrix at equilibria are computable, at least
numerically.
Local asymptotic stability
of
x
∗
arises if all eigenvalues have a strictly
negative real part.
Global stability
may be established through a Lyapunov function
(but they are not easy to find). A Lyapunov function for system
x
=
f
(
x
) is a
continuously differentiable function
V
:
R
n
+
→
R
+
satisfying
V
(
x
)
≥
0 for all
n
+
with
V
(
x
)=0if and only if
x
=
x
∗
,and
∂
∂x
x
x
∈
R
≤
0.
2.2.2.3
Different Timescales: Tikhonov's Theorem
Systems in the form of Eq. (
2.2
) whose variables evolve at different timescales can
often be simplified. The main idea is to separate the system into “fast” and “slow”
variables, and assume that the “fast” variables reach a (quasi) “steady state”. This
method allows reducing system in Eq. (
2.2
) to a new system with less variables,
but with essentially the same dynamical behavior. This method can be applied only
under appropriate conditions (briefly stated below) which are known as Tikhonov's
Theorem (see, for instance, [
27
]). Let
x
p
+
q
+
∈
R
,
y
∈
R
,and
ε
1 beasmallreal
number. Consider a system of the form
⎧
⎨
x
=
f
(
x, y, ε
)
,
ε y
=
g
(
x, y, ε
)
,
(
x
(0)
,y
(0)) = (
x
0
,y
0
)
,
(2.3)
⎩
with
f
and
g
sufficiently smooth, under the following hypotheses:
•
H1
(slow manifold): there exists a unique solution,
y
=
g
(
x
), sufficiently
smooth, of
g
(
x, y,
0) = 0; the matrix
∂g/∂y
(
x, g
(
x
)
,
0) has all eigenvalues with
strictly negative real part;
•
H2
(reduced system): the scalar system
x
=
f
(
x, g
(
x
)
,
0)
,x
(0) =
x
0
has a
solution
x
0
(
t
) on an interval [0
,T
] (0
<T<
∞
);
•
H3
:
y
0
is in the basin of attraction of the steady state
g
(
x
0
) of the fast system
ξ
=
g
(
x, ξ,
0).
If hypotheses H1-H3 are satisfied, the system in Eq. (
2.3
) admits a solution
(
x
ε
(
t
)
,y
ε
(
t
)) on [0
,T
]; in addition, lim
ε→
0
+
x
ε
(
t
)=
x
0
(
t
) and lim
ε→
0
+
y
ε
(
t
)=
y
0
(
t
)=
g
(
x
0
(
t
)), uniformly on time on any closed interval contained in (0
,T
].
The variables
y
are “faster”, since
y
evolves very rapidly when compared to
x
.
Hypothesis H1 means that
y
evolves rapidly to a
quasi steady state
value,
y
=
g
(
x
),
depending only on
x
. This quasi steady state evolves on the slow time scale.
2.2.2.4
General Piecewise Affine Systems
The model has the general form
x
i
=
f
i
(
x
)
−
γ
i
x
i
,
1
≤
i
≤
n,
(2.4)
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