輸入-狀態穩定性 （Input-to-state stability）簡稱ISS[ 1] [ 2] 非線性 的控制理論 中探討其穩定性的方式。簡單來說，控制系統具有輸入-狀態穩定性也就是指在沒有外在輸入時，系統會漸近穩定，而且在足夠長的時間後，系統軌跡會限制在和輸入大小有關的函數中。
輸入-狀態穩定性之所以重要，是因為此概念連接了輸入-輸出穩定性以及狀態空間法 ，這二個都是控制系統研究者常常使用的工具。輸入-狀態穩定性的標示方式是由Eduardo Sontag [ 3]
考慮非時變常微分方程 ，其形式如下
x
˙
=
f
(
x
,
u
)
,
x
(
t
)
∈
R
n
,
{\displaystyle {\dot {x}}=f(x,u),\ x(t)\in \mathbb {R} ^{n},}
1
其中
u
:
R
+
→
R
m
{\displaystyle u:\mathbb {R} _{+}\to \mathbb {R} ^{m}}
勒貝格測度 有本質確界 的外部輸入，且
f
{\displaystyle f}
利普希茨連續 函數。這可以確保系統(1 絕對連續 的解。
若要定義ISS以及其他相關的性質，需要引入以下的比較函數
K
{\displaystyle {\mathcal {K}}}
K類函數 ）為連續遞增函數
γ
:
R
+
→
R
+
{\displaystyle \gamma :\mathbb {R} _{+}\to \mathbb {R} _{+}}
γ
(
0
)
=
0
{\displaystyle \gamma (0)=0}
K
∞
{\displaystyle {\mathcal {K}}_{\infty }}
γ
∈
K
{\displaystyle \gamma \in {\mathcal {K}}}
β
∈
K
L
{\displaystyle \beta \in {\mathcal {K}}{\mathcal {L}}}
KL類函數 ）為若
β
(
⋅
,
t
)
∈
K
{\displaystyle \beta (\cdot ,t)\in {\mathcal {K}}}
t
≥
0
{\displaystyle t\geq 0}
r
>
0
{\displaystyle r>0}
β
(
r
,
⋅
)
{\displaystyle \beta (r,\cdot )}
系統(1 在原點全域漸近穩定 （0-GAS），若對應的零輸入系統
x
˙
=
f
(
x
,
0
)
,
x
(
t
)
∈
R
n
,
{\displaystyle {\dot {x}}=f(x,0),\ x(t)\in \mathbb {R} ^{n},}
WithoutInputs
是全域李雅普諾夫穩定 ，也就是存在
β
∈
K
L
{\displaystyle \beta \in {\mathcal {K}}{\mathcal {L}}}
x
0
{\displaystyle x_{0}}
t
≥
0
{\displaystyle t\geq 0}
WithoutInputs
|
x
(
t
)
|
≤
β
(
|
x
0
|
,
t
)
.
{\displaystyle |x(t)|\leq \beta (|x_{0}|,t).}
GAS-Estimate
系統(1 輸入-狀態穩定性 （ISS）若存在函數
γ
∈
K
{\displaystyle \gamma \in {\mathcal {K}}}
β
∈
K
L
{\displaystyle \beta \in {\mathcal {K}}{\mathcal {L}}}
x
0
{\displaystyle x_{0}}
u
{\displaystyle u}
t
≥
0
{\displaystyle t\geq 0}
|
x
(
t
)
|
≤
β
(
|
x
0
|
,
t
)
+
γ
(
‖
u
‖
∞
)
.
{\displaystyle |x(t)|\leq \beta (|x_{0}|,t)+\gamma (\|u\|_{\infty }).}
2
上述不等式中的函數
γ
{\displaystyle \gamma }
增益 （gain）。
很明顯的，ISS系統是0-GAS系統，也有有界輸入有界輸出穩定性 （若令輸出等於狀態），不過0-GAS系統不一定是ISS系統。
也可以證明若在
t
→
∞
{\displaystyle t\to \infty }
|
u
(
t
)
|
→
0
{\displaystyle |u(t)|\to 0}
t
→
∞
{\displaystyle t\to \infty }
|
x
(
t
)
|
→
0
{\displaystyle |x(t)|\to 0}
為了要瞭解輸入-狀態穩定性，需要用其他的穩定性術語來重新說明。
系統(1 全域穩定（GS） ，若存在
γ
,
σ
∈
K
{\displaystyle \gamma ,\sigma \in {\mathcal {K}}}
∀
u
{\displaystyle \forall u}
∀
t
≥
0
{\displaystyle \forall t\geq 0}
∀
x
0
{\displaystyle \forall x_{0}}
|
x
(
t
)
|
≤
σ
(
|
x
0
|
)
+
γ
(
‖
u
‖
∞
)
.
{\displaystyle |x(t)|\leq \sigma (|x_{0}|)+\gamma (\|u\|_{\infty }).}
GS
系統(1 漸近增益（AG）特性 ，若存在
γ
∈
K
{\displaystyle \gamma \in {\mathcal {K}}}
∀
x
0
{\displaystyle \forall x_{0}}
∀
u
{\displaystyle \forall u}
lim sup
t
→
∞
|
x
(
t
)
|
≤
γ
(
‖
u
‖
∞
)
.
{\displaystyle \limsup _{t\to \infty }|x(t)|\leq \gamma (\|u\|_{\infty }).}
AG
以下的描述都是等效的
[ 4]
(1
(1
(1 在論文中可以找到以上論述的證明，以及許多輸入-狀態穩定性的特性[ 4] [ 5]
ISS-李亞普諾夫函數是驗證輸入-狀態穩定性時的重要工具。
光滑函數
V
:
R
n
→
R
+
{\displaystyle V:\mathbb {R} ^{n}\to \mathbb {R} _{+}}
1
∃
ψ
1
,
ψ
2
∈
K
∞
{\displaystyle \exists \psi _{1},\psi _{2}\in {\mathcal {K}}_{\infty }}
χ
∈
K
{\displaystyle \chi \in {\mathcal {K}}}
α
{\displaystyle \alpha }
ψ
1
(
|
x
|
)
≤
V
(
x
)
≤
ψ
2
(
|
x
|
)
,
∀
x
∈
R
n
{\displaystyle \psi _{1}(|x|)\leq V(x)\leq \psi _{2}(|x|),\quad \forall x\in \mathbb {R} ^{n}}
以及
∀
x
∈
R
n
,
∀
u
∈
R
m
{\displaystyle \forall x\in \mathbb {R} ^{n},\;\forall u\in \mathbb {R} ^{m}}
|
x
|
≥
χ
(
|
u
|
)
⇒
∇
V
⋅
f
(
x
,
u
)
≤
−
α
(
|
x
|
)
,
{\displaystyle |x|\geq \chi (|u|)\ \Rightarrow \ \nabla V\cdot f(x,u)\leq -\alpha (|x|),}
函數
χ
{\displaystyle \chi }
李亞普諾夫增益 （Lyapunov gain）。
若系統(1
u
≡
0
{\displaystyle u\equiv 0}
∇
V
⋅
f
(
x
,
u
)
≤
−
α
(
|
x
|
)
,
∀
x
≠
0
,
{\displaystyle \nabla V\cdot f(x,u)\leq -\alpha (|x|),\ \forall x\neq 0,}
因此
V
{\displaystyle V}
李亞普諾夫函數 。
E. Sontag和Y. Wang得到的重要結論是系統(1 [ 5]
考慮一系統
x
˙
=
−
x
3
+
u
x
2
.
{\displaystyle {\dot {x}}=-x^{3}+ux^{2}.}
定義候選的ISS-李亞普諾夫函數
V
:
R
→
R
+
{\displaystyle V:\mathbb {R} \to \mathbb {R} _{+}}
V
(
x
)
=
1
2
x
2
,
∀
x
∈
R
.
{\displaystyle V(x)={\frac {1}{2}}x^{2},\quad \forall x\in \mathbb {R} .}
V
˙
(
x
)
=
∇
V
⋅
(
−
x
3
+
u
x
2
)
=
−
x
4
+
u
x
3
.
{\displaystyle {\dot {V}}(x)=\nabla V\cdot (-x^{3}+ux^{2})=-x^{4}+ux^{3}.}
選擇李亞普諾夫增益
χ
{\displaystyle \chi }
χ
(
r
)
:=
1
1
−
ϵ
r
{\displaystyle \chi (r):={\frac {1}{1-\epsilon }}r}
可以得到在
x
,
u
:
|
x
|
≥
χ
(
|
u
|
)
{\displaystyle x,u:\ |x|\geq \chi (|u|)}
V
˙
(
x
)
≤
−
|
x
|
4
+
(
1
−
ϵ
)
|
x
|
4
=
−
ϵ
|
x
|
4
.
{\displaystyle {\dot {V}}(x)\leq -|x|^{4}+(1-\epsilon )|x|^{4}=-\epsilon |x|^{4}.}
可得
V
{\displaystyle V}
χ
{\displaystyle \chi }
系統(1
α
,
γ
∈
K
{\displaystyle \alpha ,\gamma \in {\mathcal {K}}}
β
∈
K
L
{\displaystyle \beta \in {\mathcal {K}}{\mathcal {L}}}
x
0
{\displaystyle x_{0}}
u
{\displaystyle u}
t
≥
0
{\displaystyle t\geq 0}
α
(
|
x
(
t
)
|
)
≤
β
(
|
x
0
|
,
t
)
+
∫
0
t
γ
(
|
u
(
s
)
|
)
d
s
.
{\displaystyle \alpha (|x(t)|)\leq \beta (|x_{0}|,t)+\int _{0}^{t}\gamma (|u(s)|)ds.}
3
積分輸入-狀態穩定性（iISS）系統和ISS系統不同，若系統是iISS系統，在有界輸入下其軌跡仍可能會成長到無限大。例如，在所有
r
≥
0
{\displaystyle r\geq 0}
α
(
r
)
=
γ
(
r
)
=
r
{\displaystyle \alpha (r)=\gamma (r)=r}
u
≡
c
=
c
o
n
s
t
{\displaystyle u\equiv c=const}
3
|
x
(
t
)
|
≤
β
(
|
x
0
|
,
t
)
+
∫
0
t
c
d
s
=
β
(
|
x
0
|
,
t
)
+
c
t
,
{\displaystyle |x(t)|\leq \beta (|x_{0}|,t)+\int _{0}^{t}cds=\beta (|x_{0}|,t)+ct,}
隨著
t
→
∞
{\displaystyle t\to \infty }
t
→
∞
{\displaystyle t\to \infty }
局部輸入-狀態穩定性也是一種輸入-狀態穩定性的特性。系統(1 局部輸入-狀態穩定性 （locally ISS、LISS）若存在常數
ρ
>
0
{\displaystyle \rho >0}
γ
∈
K
{\displaystyle \gamma \in {\mathcal {K}}}
β
∈
K
L
{\displaystyle \beta \in {\mathcal {K}}{\mathcal {L}}}
x
0
∈
R
n
:
|
x
0
|
≤
ρ
{\displaystyle x_{0}\in \mathbb {R} ^{n}:\;|x_{0}|\leq \rho }
u
:
‖
u
‖
∞
≤
ρ
{\displaystyle u:\|u\|_{\infty }\leq \rho }
t
≥
0
{\displaystyle t\geq 0}
|
x
(
t
)
|
≤
β
(
|
x
0
|
,
t
)
+
γ
(
‖
u
‖
∞
)
.
{\displaystyle |x(t)|\leq \beta (|x_{0}|,t)+\gamma (\|u\|_{\infty }).}
4
可以觀察到0-GAS系統會有LISS系統的特性[ 6]
也有其他人提出和輸入-狀態穩定性有關的穩定性特性，例如增量輸入-狀態穩定性（incremental ISS）、輸入至輸出動態穩定性（input-to-state dynamical stability、ISDS）[ 7] [ 8]
考慮非時變的時滯微分方程
x
˙
(
t
)
=
f
(
x
t
,
u
(
t
)
)
,
t
>
0.
{\displaystyle {\dot {x}}(t)=f(x^{t},u(t)),\quad t>0.}
TDS
其中
x
t
∈
C
(
[
−
θ
,
0
]
;
R
N
)
{\displaystyle x^{t}\in C([-\theta ,0];\mathbb {R} ^{N})}
TDS
t
{\displaystyle t}
x
t
(
τ
)
=
x
(
t
+
τ
)
,
τ
∈
[
−
θ
,
0
]
{\displaystyle x^{t}(\tau )=x(t+\tau ),\ \tau \in [-\theta ,0]}
f
:
C
(
[
−
θ
,
0
]
;
R
N
)
×
R
m
{\displaystyle f:C([-\theta ,0];\mathbb {R} ^{N})\times \mathbb {R} ^{m}}
TDS
系統(TDS
β
∈
K
L
{\displaystyle \beta \in {\mathcal {KL}}}
γ
∈
K
{\displaystyle \gamma \in {\mathcal {K}}}
ξ
∈
C
(
[
−
θ
,
0
]
,
R
N
)
{\displaystyle \xi \in C(\left[-\theta ,0\right],\mathbb {R} ^{N})}
t
∈
R
+
{\displaystyle t\in \mathbb {R} _{+}}
|
x
(
t
)
|
≤
β
(
‖
ξ
‖
[
−
θ
,
0
]
,
t
)
+
γ
(
‖
u
‖
∞
)
.
{\displaystyle \left|x(t)\right|\leq \beta (\left\|\xi \right\|_{\left[-\theta ,0\right]},t)+\gamma (\left\|u\right\|_{\infty }).}
ISS-TDS
在時滯系統的ISS理論中，提出了二個不同的李亞普諾夫型的充份條件：透過ISS Lyapunov-Razumikhin函數[ 9] [ 10] [ 11]
以非時變常微分方程為基礎的輸入-狀態穩定性是已有相當發展的理論。也有研究者將此理論應用在其他的系統中，例如時變系統 [ 12] 混合系統 [ 13] [ 14] [ 15] [ 16] [ 1] [ 17]
^ 1.0 1.1 Iasson Karafyllis and Zhong-Ping Jiang. Stability and stabilization of nonlinear systems. Communications and Control Engineering Series. Springer-Verlag London Ltd., London, 2011.
^ E. D. Sontag. Input to state stability: basic concepts and results. In Nonlinear and optimal control theory, volume 1932 of Lecture Notes in Math., pages 163–220, Berlin, 2008. Springer
^ Eduardo D. Sontag. Smooth stabilization implies coprime factorization. IEEE Trans. Automat. Control, 34(4):435–443, 1989.
^ 4.0 4.1 Eduardo D. Sontag and Yuan Wang. New characterizations of input-to-state stability Archive-It 的存檔 ，存檔日期2011-04-01. IEEE Trans. Automat. Control, 41(9):1283–1294, 1996.^ 5.0 5.1 Eduardo D. Sontag and Yuan Wang. On characterizations of the input-to-state stability property （頁面存檔備份 ，存於網際網路檔案館 ）. Systems Control Lett., 24(5):351–359, 1995.^ Lemma I.1, p.1285 in Eduardo D. Sontag and Yuan Wang. New characterizations of input-to-state stability. IEEE Trans. Automat. Control, 41(9):1283–1294, 1996
^ Lars Grüne. Input-to-state dynamical stability and its Lyapunov function characterization. IEEE Trans. Automat. Control, 47(9):1499–1504, 2002.
^ Z.-P. Jiang, A. R. Teel, and L. Praly. Small-gain theorem for ISS systems and applications. Math. Control Signals Systems, 7(2):95–120, 1994.
^ Andrew R. Teel. Connections between Razumikhin-type theorems and the ISS nonlinear small gain theorem. IEEE Trans. Automat. Control, 43(7):960–964, 1998.
^ P. Pepe and Z.-P. Jiang. A Lyapunov-Krasovskii methodology for ISS and iISS of time-delay systems. Systems Control Lett., 55(12):1006–1014, 2006.
^ Iasson Karafyllis. Lyapunov theorems for systems described by retarded functional differential equations. Nonlinear Analysis: Theory, Methods & Applications, 64(3):590 – 617,2006.
^ Y. Lin, Y. Wang, and D. Cheng. On nonuniform and semi-uniform input-to-state stability for time-varying systems. In IFAC World Congress, Prague, 2005.
^ C. Cai and A.R. Teel. Characterizations of input-to-state stability for hybrid systems. Systems & Control Letters, 58(1):47–53, 2009.
^ D. Nesic and A.R. Teel. A Lyapunov-based small-gain theorem for hybrid ISS systems. In Proceedings of the 47th IEEE Conference on Decision and Control, Cancun, Mexico, Dec. 9-11, 2008, pages 3380–3385, 2008.
^ Bayu Jayawardhana, Hartmut Logemann, and Eugene P. Ryan. Infinite-dimensional feedback systems: the circle criterion and input-to-state stability （頁面存檔備份 ，存於網際網路檔案館 ）. Commun. Inf. Syst., 8(4):413–414, 2008.^ Dashkovskiy, S. and Mironchenko, A. Input-to-state stability of infinite-dimensional control systems. [永久失效連結 ^ F. Mazenc and C. Prieur. Strict Lyapunov functions for semilinear parabolic partial differential equations. Mathematical Control and Related Fields, 1:231–250, June 2011.
![{\displaystyle x^{t}\in C([-\theta ,0];\mathbb {R} ^{N})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e59c78020b95acc309b2f3d04377b19f199c59cc)
![{\displaystyle x^{t}(\tau )=x(t+\tau ),\ \tau \in [-\theta ,0]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8c9d8ffe3ba45e8deaa6d90546569d8bd5ecc78f)
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![{\displaystyle \left|x(t)\right|\leq \beta (\left\|\xi \right\|_{\left[-\theta ,0\right]},t)+\gamma (\left\|u\right\|_{\infty }).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bbb56921a02573c6ea23bf42f8654c4d0b7cabf4)