Input-to-state stability (ISS) is a stability notion widely used to study stability of nonlinear control systems with external inputs. 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.
ISS unified the Lyapunov and input-output stability theories and revolutionized our view on stabilization of nonlinear systems, design of robust nonlinear observers, stability of nonlinear interconnected control systems, nonlinear detectability theory, and supervisory adaptive control. 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.
Since that the concept was successfully used for many other classes of control systems including systems governed by partial differential equations, retarded systems, hybrid systems, etc.
Definition
Consider a time-invariant system of ordinary differential equations of the form
where is a Lebesgue measurable essentially bounded external input and 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 ().
To define ISS and related properties, we exploit the following classes of comparison functions. We denote by the set of continuous increasing functions with and the set of continuous strictly decreasing functions with . Then we can denote as functions where for all and for all .
System () is called globally asymptotically stable at zero (0-GAS) if the corresponding system with zero input
is globally asymptotically stable, that is there exist
so that for all initial values
and all times the following estimate is valid for solutions of ()
System () is called input-to-state stable (ISS) if there exist functions
and so that for all initial values , all admissible inputs and all times the following inequality holds
The function in the above inequality is called the gain.
Clearly, an ISS system is 0-GAS as well as BIBO stable (if we put the output equal to the state of the system). The converse implication is in general not true.
It can be also proved that if , then .
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 () is called globally stable (GS) if there exist
such that , and it holds that
System () 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 ISS
2. () is GS and has the AG property
3. () 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 and. Other characterizations of ISS that are valid under very mild restrictions on the regularity of the rhs and are applicable to more general infinite-dimensional systems, have been shown in.
ISS-Lyapunov functions
An important tool for the verification of ISS are ISS-Lyapunov functions.
A smooth function is called an ISS-Lyapunov function for (), if , and positive-definite function , such that:
and
it holds:
The function is called Lyapunov gain.
If a system () is without inputs (i.e. ), then the last implication reduces to the condition
which tells us that is a "classic" Lyapunov function.
An important result due to E. Sontag and Y. Wang is that a system () is ISS if and only if there exists a smooth ISS-Lyapunov function for it.
Examples
Consider a system
Define a candidate ISS-Lyapunov function by
Choose a Lyapunov gain by
.
Then we obtain that for it holds
This shows that is an ISS-Lyapunov function for a considered system with the Lyapunov gain .
Interconnections of ISS systems
One of the main features of the ISS framework is the possibility to study stability properties of interconnections of input-to-state stable systems.