输入-状态稳定性 （Input-to-state stability）简称ISS[ 1] [ 2] 非线性 的控制理论 中探讨其稳定性的方式。简单来说，控制系统具有输入-状态稳定性也就是指在没有外在输入时，系统会渐近稳定，而且在足够长的时间后，系统轨迹会限制在和输入大小有关的函数中。
输入-状态稳定性之所以重要，是因为此概念连接了输入-输出稳定性以及状态空间法 ，这二个都是控制系统研究者常常使用的工具。输入-状态稳定性的标示方式是由Eduardo Sontag [ 3]
考虑非时变常微分方程 ，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}}
勒贝格测度 有本质确界 的外部输入，且
f
{\displaystyle f}
利普希茨连续 函数。这可以确保系统(1 绝对连续 的解。
若要定义ISS以及其他相关的性质，需要引入以下的比较函数
K
{\displaystyle {\mathcal {K}}}
K类函数 ）为连续递增函数
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类函数 ）为若
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 )}
系统(1 在原点全域渐近稳定 （0-GAS），若对应的零输入系统
x
˙
=
f
(
x
,
0
)
,
x
(
t
)
∈
R
n
,
{\displaystyle {\dot {x}}=f(x,0),\ x(t)\in \mathbb {R} ^{n},}
WithoutInputs
是全域李雅普诺夫稳定 ，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
系统(1 输入-状态稳定性 （ISS）若存在函数
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
上述不等式中的函数
γ
{\displaystyle \gamma }
增益 （gain）。
很明显的，ISS系统是0-GAS系统，也有有界输入有界输出稳定性 （若令输出等于状态），不过0-GAS系统不一定是ISS系统。
也可以证明若在
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}
为了要了解输入-状态稳定性，需要用其他的稳定性术语来重新说明。
系统(1 全域稳定（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
系统(1 渐近增益（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 在论文中可以找到以上论述的证明，以及许多输入-状态稳定性的特性[ 4] [ 5]
ISS-李亚普诺夫函数是验证输入-状态稳定性时的重要工具。
光滑函数
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|),}
函数
χ
{\displaystyle \chi }
李亚普诺夫增益 （Lyapunov gain）。
若系统(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}
李亚普诺夫函数 。
E. Sontag和Y. Wang得到的重要结论是系统(1 [ 5]
考虑一系统
x
˙
=
−
x
3
+
u
x
2
.
{\displaystyle {\dot {x}}=-x^{3}+ux^{2}.}
定义候选的ISS-李亚普诺夫函数
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}.}
选择李亚普诺夫增益
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 }
系统(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
积分输入-状态稳定性（iISS）系统和ISS系统不同，若系统是iISS系统，在有界输入下其轨迹仍可能会成长到无限大。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,}
随著
t
→
∞
{\displaystyle t\to \infty }
t
→
∞
{\displaystyle t\to \infty }
局部输入-状态稳定性也是一种输入-状态稳定性的特性。系统(1 局部输入-状态稳定性 （locally ISS、LISS）若存在常数
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
可以观察到0-GAS系统会有LISS系统的特性[ 6]
也有其他人提出和输入-状态稳定性有关的稳定性特性，例如增量输入-状态稳定性（incremental ISS）、输入至输出动态稳定性（input-to-state dynamical stability、ISDS）[ 7] [ 8]
考虑非时变的时滞微分方程
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
系统(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
在时滞系统的ISS理论中，提出了二个不同的李亚普诺夫型的充份条件：透过ISS Lyapunov-Razumikhin函数[ 9] [ 10] [ 11]
以非时变常微分方程为基础的输入-状态稳定性是已有相当发展的理论。也有研究者将此理论应用在其他的系统中，例如时变系统 [ 12] 混合系统 [ 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 的存档 ，存档日期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 （页面存档备份 ，存于互联网档案馆 ）. 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 （页面存档备份 ，存于互联网档案馆 ）. Commun. Inf. Syst., 8(4):413–414, 2008.^ Dashkovskiy, S. and Mironchenko, A. Input-to-state stability of infinite-dimensional control systems. [永久失效链接 ^ F. Mazenc and C. Prieur. Strict Lyapunov functions for semilinear parabolic partial differential equations. Mathematical Control and Related Fields, 1:231–250, June 2011.
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