Input-to-state stability

Input-to-state stability (ISS) is a stability notion widely used to study stability of nonlinear control systems with external inputs. Roughly speaking, the system is ISS if it is globally asymptotically stable in the absence of external inputs and if its trajectory stays for large enough times below certain bound, which depends on the norm of the input. Importance of ISS concept is due to the fact, that it has bridged the gap between the input–output and the state-space methods, widely used within the control systems community. The notion of ISS has been introduced by Eduardo Sontag in 1989.

Definition

Consider a time-invariant system of ordinary differential equations of the form

${\dot {x}}=f(x,u),\ x(t)\in \mathbb {R} ^{n},$ 1

where $u:\mathbb {R} _{+}\to \mathbb {R} ^{m}$ is a Lebesgue measurable essentially bounded external input and $f$ is a Lipschitz continuous function w.r.t. the first argument uniformly w.r.t. the second one. This ensures that there exists a unique absolutely continuous solution of the system (1).

We denote by ${\mathcal {K}}$ the set of continuous increasing functions $\gamma :\mathbb {R} _{+}\to \mathbb {R} _{+}$ with $\gamma (0)=0$. The set of unbounded functions $\gamma \in {\mathcal {K}}$ we denote by ${\mathcal {K}}_{\infty }$. Also we denote $\beta \in {\mathcal {K}}{\mathcal {L}}$ if $\beta (\cdot ,t)\in {\mathcal {K}}$ for all $t\geq 0$ and $\beta (r,\cdot )$ is continuous and strictly decreasing to zero for all $r>0$.

System (1) is called globally asymptotically stable at zero (0-GAS) if the corresponding system with zero input

${\dot {x}}=f(x,0),\ x(t)\in \mathbb {R} ^{n},$ WithoutInputs

is globally asymptotically stable, that is there exist $\beta \in {\mathcal {K}}{\mathcal {L}}$ so that for all initial values $x_{0}$ and all times $t\geq 0$ the following estimate is valid for solutions of (WithoutInputs)

$|x(t)|\leq \beta (|x_{0}|,t).$ GAS-Estimate

System (1) is called input-to-state stable (ISS) if there exist functions $\gamma \in {\mathcal {K}}$ and $\beta \in {\mathcal {K}}{\mathcal {L}}$ so that for all initial values $x_{0}$, all admissible inputs $u$ and all times $t\geq 0$ the following inequality holds

$|x(t)|\leq \beta (|x_{0}|,t)+\gamma (\|u\|_{\infty }).$ 2

The function $\gamma$ in the above inequality is called the gain.

Clearly, ISS system is 0-GAS as well as BIBO stable (if we put output equal to the state of the system). The converse implication is in general not true.

It can be also proved that if $|u(t)|\to 0$, when $t\to \infty$, then $|x(t)|\to 0$, $t\to \infty$.

Characterizations of input-to-state stability property

For an understanding of ISS, its restatements in terms of other stability properties are of great importance.

System (1) is called globally stable (GS) if there exist $\gamma ,\sigma \in {\mathcal {K}}$ such that $\forall x_{0}$, $\forall u$ and $\forall t\geq 0$ it holds that

$|x(t)|\leq \sigma (|x_{0}|)+\gamma (\|u\|_{\infty }).$ GS

System (1) satisfies asymptotic gain (AG) property if there exists $\gamma \in {\mathcal {K}}$: $\forall x_{0}$, $\forall u$ it holds that

$\limsup _{t\to \infty }|x(t)|\leq \gamma (\|u\|_{\infty }).$ AG

The following statements are equivalent : 1. (1) is ISS

2. (1) is GS and possesses AG property

3. (1) is 0-GAS and possesses AG property

ISS-Lyapunov functions

Main article: ISS-Lyapunov function

An important tool for verification of ISS property is ISS-Lyapunov functions.

A smooth function $V:\mathbb {R} ^{n}\to \mathbb {R} _{+}$ is called an ISS-Lyapunov function for (1), if $\exists \psi _{1},\psi _{2}\in {\mathcal {K}}_{\infty }$, $\chi \in {\mathcal {K}}$ and positive definite function $\alpha$, such that:

$\psi _{1}(|x|)\leq V(x)\leq \psi _{2}(|x|),\quad \forall x\in \mathbb {R} ^{n}$

and $\forall x\in \mathbb {R} ^{n},\;\forall u\in \mathbb {R} ^{m}$ it holds:

$|x|\geq \chi (|u|)\ \Rightarrow \ \nabla V\cdot f(x,u)\leq -\alpha (|x|),$

The function $\chi$ is called Lyapunov gain.

If a system (1) is without inputs (i.e. $u\equiv 0$), then the last implication reduces to the condition

$\nabla V\cdot f(x,u)\leq -\alpha (|x|),\ \forall x\neq 0,$

which tells us that $V$ is a "classical" Lyapunov function.

An important result due to E. Sontag and Y. Wang is that a system (1) is ISS if and only if there exists a smooth ISS-Lyapunov function for it.

Examples

Consider a system

${\dot {x}}=-x^{3}+ux^{2}.$

Define a candidate ISS-Lyapunov function $V:\mathbb {R} \to \mathbb {R} _{+}$ by $V(x)={\frac {1}{2}}x^{2},\quad \forall x\in \mathbb {R} .$

${\dot {V}}(x)=\nabla V\cdot (-x^{3}+ux^{2})=-x^{4}+ux^{3}.$

Choose Lyapunov gain $\chi$ by

$\chi (r):={\frac {1}{1-\epsilon }}r$.

Then we obtain that for $x,u:\ |x|\geq \chi (|u|)$ it holds

${\dot {V}}(x)\leq -|x|^{4}+(1-\epsilon )|x|^{4}=-\epsilon |x|^{4}.$

This shows that $V$ is an ISS-Lyapunov function for a considered system with a Lyapunov gain $\chi$.