Input-to-state stability (ISS)
[Eduardo D. Sontag. Mathematical Control Theory: Finite-Dimensional Systems. Springer-Verlag, London, 1998][Hassan K. Khalil. Nonlinear Systems. Prentice Hall, 2002.][Andrii Mironchenko. Input-to-state stability]
Springer, 2023. is a stability notion widely used to study stability of nonlinear
control systems
A control system manages, commands, directs, or regulates the behavior of other devices or systems using control loops. It can range from a single home heating controller using a thermostat controlling a domestic boiler to large industrial c ...
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
Eduardo Daniel Sontag (born April 16, 1951, in Buenos Aires, Argentina) is an Argentine-American mathematician, and distinguished university professor at Northeastern University, who works in the fields control theory, dynamical systems, syste ...
in 1989.
[Eduardo D. Sontag. Smooth stabilization implies coprime factorization. IEEE Trans. Autom. Control, 34(4):435–443, 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.
[A. Mironchenko, Ch. Prieur. Input-to-state stability of infinite-dimensional systems: recent results and open questions]
SIAM Review, 62(3):529–614, 2020.
Definition
Consider a time-invariant system of
ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast w ...
s of the form
where
is a
Lebesgue measurable
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides wit ...
essentially bounded
Essence ( la, essentia) is a polysemic term, used in philosophy and theology as a designation for the property or set of properties that make an entity or substance what it fundamentally is, and which it has by necessity, and without which it ...
external input and
is a
Lipschitz continuous
In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there exis ...
function w.r.t. the first argument uniformly w.r.t. the second one. This ensures that there exists a unique
absolutely continuous
In calculus, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship between the two central oper ...
solution of the system ().
To define ISS and related properties, we exploit the following classes of
comparison function In applied mathematics, comparison functions are several classes of continuous functions, which are used in stability theory to characterize the stability properties of control systems as Lyapunov stability, uniform asymptotic stability etc. 1 + 1 e ...
s. 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
Various types of stability may be discussed for the solutions of differential equations or difference equations describing dynamical systems. The most important type is that concerning the stability of solutions near to a point of equilibrium. T ...
, 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
[Eduardo D. Sontag and Yuan Wang. New characterizations of input-to-state stability]
IEEE Trans. Autom. Control, 41(9):1283–1294, 1996.
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.
[Andrii Mironchenko and Fabian Wirth. Characterizations of input-to-state stability for infinite-dimensional systems. IEEE Trans. Autom. Control, 63(6): 1602-1617, 2018.]
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
In mathematics, a positive-definite function is, depending on the context, either of two types of function.
Most common usage
A ''positive-definite function'' of a real variable ''x'' is a complex-valued function f: \mathbb \to \mathbb such th ...
, 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
In the theory of ordinary differential equations (ODEs), Lyapunov functions, named after Aleksandr Lyapunov, are scalar functions that may be used to prove the stability of an equilibrium of an ODE. Lyapunov functions (also called Lyapunov’s se ...
.
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.
[Eduardo D. Sontag and Yuan Wang. On characterizations of the input-to-state stability property]
. Systems Control Lett., 24(5):351–359, 1995.
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 (ISS-LF)
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.
[Eduardo D. Sontag. Input to state stability: basic concepts and results. In Nonlinear and optimal control theory, volume 1932 of Lecture Notes in Math., pages 163–220, Berlin, 2008. Springer]
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
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marked ...
, 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
(that is for all
, where
) 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 (ISS) 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
In mathematics, a positive-definite function is, depending on the context, either of two types of function.
Most common usage
A ''positive-definite function'' of a real variable ''x'' is a complex-valued function f: \mathbb \to \mathbb such th ...
, 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 In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive-definite. See, in particular:
* Positive-definite bilinear form
* Positive-definite f ...
.
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 (LISS) 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 (ISDS),
input-to-state practical stability (ISpS),
input-to-output stability (IOS) etc.
ISS of time-delay systems
Consider the time-invariant
time-delay system
Here
is the state of the system () at time
,