Input-to-state stability

Input-to-state stability (ISS) is a stability notion widely used to study the 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.

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 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.

The notion of ISS has been introduced by Eduardo Sontag in 1989.