考慮一個線性系統，但有非線性的回授，也就是在回授路徑上有非線性元素
φ
(
v
,
t
)
{\displaystyle \varphi (v,t)}
[
μ
1
,
μ
2
]
{\displaystyle [\mu _{1},\mu _{2}]}
μ
1
v
≤
φ
(
v
,
t
)
≤
μ
2
v
,
∀
v
,
t
{\displaystyle \mu _{1}v\leq \varphi (v,t)\leq \mu _{2}v,\ \forall v,t}
[
−
1
/
μ
1
,
−
1
/
μ
2
]
{\displaystyle [-1/\mu _{1},-1/\mu _{2}]}
考慮非線性系統
x
˙
=
A
x
+
B
w
,
{\displaystyle {\dot {\mathbf {x} }}=\mathbf {Ax} +\mathbf {Bw} ,}
v
=
C
x
,
{\displaystyle \mathbf {v} =\mathbf {Cx} ,}
w
=
φ
(
v
,
t
)
.
{\displaystyle \mathbf {w} =\varphi (v,t).}
假設
μ
1
v
≤
φ
(
v
,
t
)
≤
μ
2
v
,
∀
v
,
t
{\displaystyle \mu _{1}v\leq \varphi (v,t)\leq \mu _{2}v,\ \forall v,t}
det
(
i
ω
I
n
−
A
)
≠
0
,
∀
ω
∈
R
−
1
and
∃
μ
0
∈
[
μ
1
,
μ
2
]
:
A
+
μ
0
B
C
{\displaystyle \det(i\omega I_{n}-A)\neq 0,\ \forall \omega \in R^{-1}{\text{ and }}\exists \mu _{0}\in [\mu _{1},\mu _{2}]\,:\,A+\mu _{0}BC}
ℜ
[
(
μ
2
C
(
i
ω
I
n
−
A
)
−
1
B
−
1
)
(
1
−
μ
1
C
(
i
ω
I
n
−
A
)
−
1
B
)
]
<
0
∀
ω
∈
R
−
1
.
{\displaystyle \Re \left[(\mu _{2}C(i\omega I_{n}-A)^{-1}B-1)(1-\mu _{1}C(i\omega I_{n}-A)^{-1}B)\right]<0\ \forall \omega \in R^{-1}.}
則存在
∃
c
>
0
,
δ
>
0
{\displaystyle \exists c>0,\delta >0}
|
x
(
t
)
|
≤
c
e
−
δ
t
|
x
(
0
)
|
,
∀
t
≥
0.
{\displaystyle |x(t)|\leq ce^{-\delta t}|x(0)|,\ \forall t\geq 0.}
條件3稱為「頻率條件」，條件1稱為「區間條件」。
Haddad, Wassim M.; Chellaboina, VijaySekhar. Nonlinear Dynamical Systems and Control: a Lyapunov-Based Approach.. Princeton University Press. 2011. ISBN 9781400841042
![{\displaystyle [\mu _{1},\mu _{2}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/35821c37dd2d1874814353628cd073ad48e13ae8)
![{\displaystyle [-1/\mu _{1},-1/\mu _{2}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1df91131374069c2efde7d1c2c7ffb9731f85677)
![{\displaystyle \det(i\omega I_{n}-A)\neq 0,\ \forall \omega \in R^{-1}{\text{ and }}\exists \mu _{0}\in [\mu _{1},\mu _{2}]\,:\,A+\mu _{0}BC}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c5b5cef13149a8cb4e3bfb51651b5bf3e196f67)
![{\displaystyle \Re \left[(\mu _{2}C(i\omega I_{n}-A)^{-1}B-1)(1-\mu _{1}C(i\omega I_{n}-A)^{-1}B)\right]<0\ \forall \omega \in R^{-1}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b047ca1fcb614242acddef2424c9b6e0840e9e3a)
