Lyapunov stability

# Lyapunov stability

Discussion

Encyclopedia
Various types of stability may be discussed for the solutions of differential equations describing dynamical systems. The most important type is that concerning the stability of solutions near to a point of equilibrium. This may be discussed by the theory of Lyapunov. In simple terms, if all solutions of the dynamical system that start out near an equilibrium point stay near forever, then is Lyapunov stable. More strongly, if is Lyapunov stable and all solutions that start out near converge to , then is asymptotically stable. The notion of exponential stability guarantees a minimal rate of decay, i.e., an estimate of how quickly the solutions converge. The idea of Lyapunov stability can be extended to infinite-dimensional manifolds, where it is known as structural stability
Structural stability
In mathematics, structural stability is a fundamental property of a dynamical system which means that the qualitative behavior of the trajectories is unaffected by C1-small perturbations....

, which concerns the behavior of different but "nearby" solutions to differential equations. Input-to-state stability (ISS) applies Lyapunov notions to systems with inputs.

## History

Lyapunov stability is named after Aleksandr Lyapunov, a Russian mathematician who published his book "The General Problem of Stability of Motion" in 1892. Lyapunov was the first to consider the modifications necessary in nonlinear systems to the linear theory of stability based on linearizing near a point of equilibrium. His work, initially published in Russian and then translated to French, received little attention for many years. Interest in it started suddenly during the Cold War (1953-1962)
Cold War (1953-1962)
The Cold War discusses the period within the Cold War from the death of Soviet leader Joseph Stalin in 1953 to the Cuban Missile Crisis in 1962...

period when the so-called "Second Method of Lyapunov" was found to be applicable to the stability of aerospace guidance system
Guidance system
A guidance system is a device or group of devices used to navigate a ship, aircraft, missile, rocket, satellite, or other craft. Typically, this refers to a system that navigates without direct or continuous human control...

s which typically contain strong nonlinearities not treatable by other methods. A large number of publications appeared then and since in the control and systems literature. More recently the concept of Lyapunov exponent
Lyapunov exponent
In mathematics the Lyapunov exponent or Lyapunov characteristic exponent of a dynamical system is a quantity that characterizes the rate of separation of infinitesimally close trajectories...

(related to Lyapunov's First Method of discussing stability) has received wide interest in connection with chaos theory
Chaos theory
Chaos theory is a field of study in mathematics, with applications in several disciplines including physics, economics, biology, and philosophy. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions, an effect which is popularly referred to as the...

. Lyapunov stability methods have also been applied to finding equilibrium solutions in traffic assignment problems following work by MJ Smith and MB Wisten.

## Definition for continuous-time systems

Consider an autonomous nonlinear dynamical system
,

where denotes the system state vector, an open set containing the origin, and continuous on . Suppose has an equilibrium .
1. The equilibrium of the above system is said to be Lyapunov stable, if, for every , there exists a such that, if , then , for every .
2. The equilibrium of the above system is said to be asymptotically stable if it is Lyapunov stable and if there exists such that if , then .
3. The equilibrium of the above system is said to be exponentially stable if it is asymptotically stable and if there exist such that if , then , for .

Conceptually, the meanings of the above terms are the following:
1. Lyapunov stability of an equilibrium means that solutions starting "close enough" to the equilibrium (within a distance from it) remain "close enough" forever (within a distance from it). Note that this must be true for any that one may want to choose.
2. Asymptotic stability means that solutions that start close enough not only remain close enough but also eventually converge to the equilibrium.
3. Exponential stability means that solutions not only converge, but in fact converge faster than or at least as fast as a particular known rate .

The trajectory x is (locally) attractive if

for for all trajectories that start close enough, and globally attractive if this property holds for all trajectories.

That is, if x belongs to the interior of its stable manifold
Stable manifold
In mathematics, and in particular the study of dynamical systems, the idea of stable and unstable sets or stable and unstable manifolds give a formal mathematical definition to the general notions embodied in the idea of an attractor or repellor...

. It is asymptotically stable if it is both attractive and stable. (There are counterexamples showing that attractivity does not imply asymptotic stability. Such examples are easy to create using homoclinic connections
Homoclinic orbit
In mathematics, a homoclinic orbit is a trajectory of a flow of a dynamical system which joins a saddle equilibrium point to itself. More precisely, a homoclinic orbit lies in the intersection of the stable manifold and the unstable manifold of an equilibrium.Homoclinic orbits and homoclinic points...

.)

### Lyapunov's second method for stability

Lyapunov, in his original 1892 work proposed two methods for demonstrating stability. The first method developed the solution in a series which was then proved convergent within limits. The second method, which is almost universally used nowadays, makes use of a Lyapunov function V(x) which has an analogy to the potential function of classical dynamics. It is introduced as follows. Consider a function such that
• with equality if and only if (positive definite)
• with equality if and only if (negative definite).

Then V(x) is called a Lyapunov function
Lyapunov function
In the theory of ordinary differential equations , Lyapunov functions are scalar functions that may be used to prove the stability of an equilibrium of an ODE. Named after the Russian mathematician Aleksandr Mikhailovich Lyapunov, Lyapunov functions are important to stability theory and control...

candidate and the system is asymptotically stable in the sense of Lyapunov (i.s.L.). (Note that is required; otherwise for example would "prove" that is locally stable. An additional condition called "properness" or "radial unboundedness" is required in order to conclude global asymptotic stability.)

It is easier to visualize this method of analysis by thinking of a physical system (e.g. vibrating spring and mass) and considering the energy
Energy
In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...

of such a system. If the system loses energy over time and the energy is never restored then eventually the system must grind to a stop and reach some final resting state. This final state is called the attractor
Attractor
An attractor is a set towards which a dynamical system evolves over time. That is, points that get close enough to the attractor remain close even if slightly disturbed...

. However, finding a function that gives the precise energy of a physical system can be difficult, and for abstract mathematical systems, economic systems or biological systems, the concept of energy may not be applicable.

Lyapunov's realization was that stability can be proven without requiring knowledge of the true physical energy, providing a Lyapunov function
Lyapunov function
In the theory of ordinary differential equations , Lyapunov functions are scalar functions that may be used to prove the stability of an equilibrium of an ODE. Named after the Russian mathematician Aleksandr Mikhailovich Lyapunov, Lyapunov functions are important to stability theory and control...

can be found to satisfy the above constraints.

## Definition for discrete-time systems

The definition for discrete-time systems is almost identical to that for continuous-time systems. The definition below provides this, using an alternate language commonly used in more mathematical texts.

Let be a metric space
Metric space
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...

and a continuous function
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...

. A point is said to be Lyapunov stable, if, for each , there is a such that for all , if

then

for all .

We say that is asymptotically stable if it belongs to the interior of its stable set
Stable manifold
In mathematics, and in particular the study of dynamical systems, the idea of stable and unstable sets or stable and unstable manifolds give a formal mathematical definition to the general notions embodied in the idea of an attractor or repellor...

, i.e. if there is a such that

whenever .

## Stability for linear state space models

A linear state space
State space (controls)
In control engineering, a state space representation is a mathematical model of a physical system as a set of input, output and state variables related by first-order differential equations...

model

is asymptotically stable (in fact, exponentially stable) if all real parts of the eigenvalues of are negative. This condition is equivalent to the following one:

has a solution where and (positive definite
Positive-definite matrix
In linear algebra, a positive-definite matrix is a matrix that in many ways is analogous to a positive real number. The notion is closely related to a positive-definite symmetric bilinear form ....

matrices). (The relevant Lyapunov function is .)
Correspondingly, a time-discrete linear state space
State space (controls)
In control engineering, a state space representation is a mathematical model of a physical system as a set of input, output and state variables related by first-order differential equations...

model

is asymptotically stable (in fact, exponentially stable) if all the eigenvalues of have a modulus smaller than one.

This latter condition has been generalized to switched systems: a linear switched discrete time system (ruled by a set of matrices
)

is asymptotically stable (in fact, exponentially stable) if the joint spectral radius
In mathematics, the joint spectral radius is a generalization of the classical notion of spectral radius of a matrix, to sets of matrices. In recent years this notion has found applications in a large number of engineering fields and is still a topic of active research.-General description:The...

of the set is smaller than one.

## Stability for systems with inputs

A system with inputs (or controls) has the form

where the (generally time-dependent) input u(t) may be viewed as a control, external input,
stimulus, disturbance, or forcing function. The study of such systems is the subject
of control theory
Control theory
Control theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems. The desired output of a system is called the reference...

and applied in control engineering
Control engineering
Control engineering or Control systems engineering is the engineering discipline that applies control theory to design systems with predictable behaviors...

. For systems with inputs, one must
quantify the effect of inputs on the stability of the system. The main two approaches to this
analysis are BIBO stability
BIBO stability
In electrical engineering, specifically signal processing and control theory, BIBO stability is a form of stability for linear signals and systems that take inputs. BIBO stands for Bounded-Input Bounded-Output...

(for linear system
Linear system
A linear system is a mathematical model of a system based on the use of a linear operator.Linear systems typically exhibit features and properties that are much simpler than the general, nonlinear case....

s) and input-to-state (ISS) stability (for nonlinear systems)

## Example

Consider an equation, where compared to the Van der Pol oscillator equation the friction term is changed:

The equilibrium is at :

Here is a good example of an unsuccessful try to find a Lyapunov function that proves stability:

Let

so that the corresponding system is

Let us choose as a Lyapunov function

which is clearly positive definite. Its derivative is

It seems that if the parameter is positive, stability is asymptotic for But this is wrong, since does not depend on , and will be 0 everywhere on the axis.

## Barbalat's lemma and stability of time-varying systems

Assume that f is function of time only.
• Having does not imply that has a limit at . For example, .

• Having approaching a limit as does not imply that . For example, .

• Having lower bounded and decreasing () implies it converges to a limit. But it does not say whether or not as .

Barbalat's Lemma
Lemma (mathematics)
In mathematics, a lemma is a proven proposition which is used as a stepping stone to a larger result rather than as a statement in-and-of itself...

says:
If has a finite limit as and if is uniformly continuous (or is bounded), then as .

Usually, it is difficult to analyze the asymptotic stability of time-varying systems because it is very difficult to find Lyapunov functions with a negative definite derivative.

We know that in case of autonomous (time-invariant) systems, if is negative semi-definite (NSD), then also, it is possible to know the asymptotic behaviour by invoking invariant-set theorems. However, this flexibility is not available for time-varying systems.
This is where "Barbalat's lemma" comes into picture. It says:
IF satisfies following conditions:
1. is lower bounded
2. is negative semi-definite (NSD)
3. is uniformly continuous in time (satisfied if is finite)
then as .

The following example is taken from page 125 of Slotine and Li's book Applied Nonlinear Control.

Consider a non-autonomous system

This is non-autonomous because the input is a function of time. Assume that the input is bounded.

Taking gives

This says that by first two conditions and hence and are bounded. But it does not say anything about the convergence of to zero. Moreover, the invariant set theorem cannot be applied, because the dynamics is non-autonomous.

Using Barbalat's lemma:
.

This is bounded because , and are bounded. This implies as and hence . This proves that the error converges.