Input-to-state stability


Input-to-state stability 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.
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
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 . The set of unbounded functions we denote by. Also we denote if for all and is continuous and strictly decreasing to zero for all.
System is called globally asymptotically stable at zero 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 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. The converse implication is in general not true.
It can be also proved that if, as, 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 if there exist
such that, and it holds that
System satisfies the asymptotic gain property if there exists
The following statements are equivalent
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

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, 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.
Consider the system given by

Here, and are Lipschitz continuous in uniformly with respect to the inputs from the -th subsystem.
For the -th subsystem of the definition of an ISS-Lyapunov function can be written as follows.
A smooth function is an ISS-Lyapunov function
for the -th subsystem of, if there exist
functions,,
,, and a positive definite function, such that:
and it holds

Cascade interconnections

Cascade interconnections are a special type of interconnection, where the dynamics of the -th subsystem does not depend on the states of the subsystems. Formally, the cascade interconnection can be written as
If all subsystems of the above system are ISS, then the whole cascade interconnection is also ISS,
In contrast to cascades of ISS systems, the cascade interconnection of 0-GAS systems is in general not 0-GAS. The following example illustrates this fact. Consider a system given by

Both subsystems of this system are 0-GAS, but for sufficiently large initial states and for a certain finite time it holds for , i.e. the system exhibits finite escape time, and thus is not 0-GAS.

Feedback interconnections

The interconnection structure of subsystems is characterized by the internal Lyapunov gains. The question, whether the interconnection is ISS, depends on the properties of the gain operator defined by
The following small-gain theorem establishes a sufficient condition for ISS of the interconnection of ISS systems. Let be an ISS-Lyapunov function for -th subsystem of with corresponding gains,. If the nonlinear small-gain condition
holds, then the whole interconnection is ISS,.
Small-gain condition holds iff for each cycle in and for all it holds
The small-gain condition in this form is called also cyclic small-gain condition.

Related stability concepts

Integral ISS (iISS)

System is called integral input-to-state stable if there exist functions and so that for all initial values, all admissible inputs and all times the following inequality holds
In contrast to ISS systems, if a system is integral ISS, its trajectories may be unbounded even for bounded inputs. To see this put for all and take. Then the estimate 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.
A smooth function is called an iISS-Lyapunov function for, if, and positive definite function , such that:
and
it holds:
An important result due to D. Angeli, E. Sontag and Y. Wang is that system is integral ISS if and only if there exists an iISS-Lyapunov function for it.
Note that in the formula above is assumed to be only positive definite.
It can be easily proved, that if is an iISS-Lyapunov function with, then is actually an ISS-Lyapunov function for a system.
This shows in particular, that every ISS system is integral ISS.
The converse implication is not true, as the following example shows. Consider the system
This system is not ISS, since for large enough inputs the trajectories are unbounded. However, it is integral ISS with an iISS-Lyapunov function defined by

Local ISS (LISS)

An important role are also played by local versions of the ISS property. A system is called locally ISS if there exist a constant and functions
and so that for all, all admissible inputs and all times it holds that
An interesting observation is that 0-GAS implies LISS.

Other stability notions

Many other related to ISS stability notions have been introduced: incremental ISS, input-to-state dynamical stability, input-to-state practical stability, input-to-output stability etc.

ISS of time-delay systems

Consider the time-invariant time-delay system
Here is the state of the system at time, and satisfies certain assumptions to guarantee existence and uniqueness of solutions of the system.
System is ISS if and only if there exist functions and such that for every, every admissible input and for all, it holds that
In the ISS theory for time-delay systems two different Lyapunov-type sufficient conditions have been proposed: via ISS Lyapunov-Razumikhin functions and by ISS Lyapunov-Krasovskii functionals. For converse Lyapunov theorems for time-delay systems see.

ISS of other classes of systems

Input-to-state stability of the systems based on time-invariant ordinary differential equations is a quite developed theory. However, ISS theory of other classes of systems is also being investigated: time-variant ODE systems, hybrid systems. In the last time also certain generalizations of ISS concepts to infinite-dimensional systems have been proposed.