Roughly speaking, a control system is ISS if it is globally asymptotically stable in the absence of external inputs and if its trajectories are bounded by a function of the size of the input for all sufficiently large times.
The importance of ISS is due to the fact that the concept has bridged the gap between input–output and state-space methods, widely used within the control systems community.
This made ISS the dominating stability paradigm in nonlinear control theory, with such diverse applications as robotics, mechatronics, systems biology, electrical and aerospace engineering, to name a few.
The notion of ISS was introduced for systems described by ordinary differential equations by Eduardo Sontag in 1989.
[5] Consider a time-invariant system of ordinary differential equations of the form where
This ensures that there exists a unique absolutely continuous solution of the system (1).
To define ISS and related properties, we exploit the following classes of comparison functions.
the following estimate is valid for solutions of (WithoutInputs) System (1) is called input-to-state stable (ISS) if there exist functions
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
it holds that System (1) satisfies the asymptotic gain (AG) property if there exists
it holds that The following statements are equivalent for sufficiently regular right-hand side
(1) is 0-GAS and has the AG property The proof of this result as well as many other characterizations of ISS can be found in the papers [8] and.
[9] Other characterizations of ISS that are valid under very mild restrictions on the regularity of the rhs
[10] An important tool for the verification of ISS are ISS-Lyapunov functions.
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.
[9] Consider a system Define a candidate ISS-Lyapunov function
One of the main features of the ISS framework is the possibility to study stability properties of interconnections of input-to-state stable systems.
-th subsystem of (WholeSys) the definition of an ISS-Lyapunov function can be written as follows.
The interconnection structure of subsystems is characterized by the internal Lyapunov gains
The question, whether the interconnection (WholeSys) is ISS, depends on the properties of the gain operator
System (1) is called integral input-to-state stable (ISS) if there exist functions
Then the estimate (3) takes the form and the right hand side grows to infinity as
As in the ISS framework, Lyapunov methods play a central role in iISS theory.
it holds: An important result due to D. Angeli, E. Sontag and Y. Wang is that system (1) is integral ISS if and only if there exists an iISS-Lyapunov function for it.
defined by An important role are also played by local versions of the ISS property.
A system (1) is called locally ISS (LISS) if there exist a constant
satisfies certain assumptions to guarantee existence and uniqueness of solutions of the system (TDS).
[19] Input-to-state stability of the systems based on time-invariant ordinary differential equations is a quite developed theory, see a recent monograph.
[21][22] In the last time also certain generalizations of ISS concepts to infinite-dimensional systems have been proposed.