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Encyclopedia > Lyapunov stability

In mathematics, the notion of Lyapunov stability occurs in the study of dynamical systems. In simple terms, if all solutions of the dynamical system that start out near an equilibrium point $x e$ stay near $x e$ forever, then $x e$ is Lyapunov stable. More strongly, if all solutions that start out near $x e$ converge to $x e$ , then $x e$ 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, which concerns the behaviour of different but "nearby" solutions to differential equations. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. ... Structural stability is a mathematical concept concerning whether a given function is sensitive to a small perturbation. ...

1 Definition for continuous-time systems
2 Definition for iterated systems
3 Lyapunov stability theorems
- 3.1 Lyapunov second theorem on stability
4 Stability for linear state space models
5 Stability for systems with inputs
6 Example
7 Barbalat's lemma and stability of time-varying systems
8 References

Definition for continuous-time systems

Consider an autonomous nonlinear dynamical system

$dot{x} = f(x(t)), ;;;; x(0) = x_0$ ,

where $x(t) in mathcal{D} subseteq mathbb{R}^n$ denotes the system state vector, $mathcal{D}$ an open set containing the origin, and $f: mathcal{D} rightarrow mathbb{R}^n$ continuous on $mathcal{D}$ . Without loss of generality, we may assume that the origin is an equilibrium.

The origin of the above system is said to be Lyapunov stable, if, for every $ε > 0$ , there exists a $δ = δ(ε) > 0$ such that, if $|x(0)| < delta$ , then $|x(t)| < epsilon$ , for every $t geq 0$ .
The origin of the above system is said to be asymptotically stable if it is Lyapunov stable and if there exists $δ > 0$ such that if $|x(0) |< delta$ , then $lim_{t rightarrow infty}x(t) = 0$ .
The origin of the above system is said to be exponentially stable if it is asymptotically stable and if there exist $α,β,δ > 0$ such that if $|x(0)| < delta$ , then $|x(t)| leq alpha|x(0)|e^{-beta t}$ , for $t geq 0$ .

Conceptually, the meanings of the above terms are the following:

Lyapunov stability of an equilibrium means that solutions starting "close enough" to the equilibrium (within a distance $δ$ from it) remain "close enough" forever (within a distance $ε$ from it). Note that this must be true for any $ε$ that one may want to choose.
Asymptotic stability means that solutions that start close enough not only remain close enough but also eventually converge to the equilibrium.
Exponential stability means that solutions not only converge, but in fact converge faster than or at least as fast as a particular known rate $alpha|x(0)|e^{-beta t}$ .

The trajectory x is (locally) attractive if

$|y(t)-x(t)| rightarrow 0$

for $t rightarrow infty$ 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. 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.) 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. ... A 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. ...

Definition for iterated 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 $(X, d)$ be a metric space and $fcolon Xto X$ a continuous function. A point $xin X$ is said to be Lyapunov stable, if, for each $ε > 0$ , there is a $δ > 0$ such that for all $yin X$ , if In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ...

d (x, y) < δ

holds, and one has

d (f n (x), f n (y)) < ε

for all $nin mathbb{N}$ .

We say that $x$ is asymptotically stable if it belongs to the interior of its stable set, i.e. if there is a $δ > 0$ such that 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. ...

$lim_{ntoinfty} d(f^n(x),f^n(y))=0$

whenever $d (x, y) < δ$ .

Lyapunov stability theorems

The general study of the stability of solutions of differential equations is known as stability theory. Lyapunov stability theorems give only sufficient condition. In mathematics, stability theory deals with the stability of the solutions of differential equations and dynamical systems. ...

Lyapunov second theorem on stability

Consider a function V(x) : Rⁿ → R such that

$V(x) ge 0$ with equality if and only if $x = 0$ (positive definite)
$dot{V}(x) < 0$ (negative definite)

Then V(x) is called a Lyapunov function candidate and the system is asymptotically stable in the sense of Lyapunov (i.s.L.). (Note that $V (0) = 0$ is required; otherwise $V (x) = 1 / (1 + | x | )$ would "prove" that $dot x(t) = x$ is locally stable. An additional condition called "properness" or "radial unboundedness" is required in order to conclude global asymptotic stability.) In the theory of dynamical systems, and control theory, Lyapunov functions, named after Aleksandr Mikhailovich Lyapunov, are a family of functions that can be used to demonstrate the stability of some state points of a system. ...

It is easier to visualise this method of analysis by thinking of a physical system (e.g. vibrating spring and mass) and considering the energy 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. 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. In dynamical systems, an attractor is a set to which the system evolves after a long enough time. ...

Lyapunov's realisation was that stability can be proven without requiring knowledge of the true physical energy, providing a Lyapunov function can be found to satisfy the above constraints. In the theory of dynamical systems, and control theory, Lyapunov functions, named after Aleksandr Mikhailovich Lyapunov, are a family of functions that can be used to demonstrate the stability of some state points of a system. ...

Stability for linear state space models

A linear state space model 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. ...

$dot{textbf{x}} = Atextbf{x}$

is asymptotically stable if This article needs to be cleaned up to conform to a higher standard of quality. ...

A T M + M A + N = 0

has a solution where $N = N T > 0$ and $M = M T > 0$ (positive definite matrices). (The relevant Lyapunov function is $V (x) = x T M x$ .) In linear algebra, a positive-definite matrix is a Hermitian matrix which in many ways is analogous to a positive real number. ...

Stability for systems with inputs

A system with inputs (or controls) has the form

$dot{textbf{x}} = textbf{f(x,u)}$

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 and applied in control engineering. 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 and input to state stability. For the sociological theory of deviant behavior, see control theory (sociology). ... Control engineering is the engineering discipline that focuses on the mathematical modelling systems of a diverse nature, analysing their dynamic behaviour, and using control theory to make a controller that will cause the systems to behave in a desired manner. ... In electrical engineering, specifically signal processing and control theory, BIBO Stability is a form of stability for signals and systems. ...

Example

Consider an equation, where compared to the Van der Pol oscillator equation the friction term is changed: Phase portrait of the unforced Van der Pol oscillator, showing a limit cycle. ...

$ddot{y} + y -epsilon left( frac{dot{y}^{3}}{3} - dot{y}right) = 0$

Let

$x_{1} = y , dot{x_{1}} = x_{2}$

so that the corresponding system is

$dot{x_{2}} = -x_{1} + epsilon left( frac{{x_{2}}^{3}}{3} - {x_{2}}right)$

Let us choose as a Lyapunov function

$V = frac {1}{2} left(x_{1}^{2}+x_{2}^{2} right)$

which is clearly positive definite. Its derivative is In mathematics, a definite bilinear form B is one for which B(v,v) has a fixed sign (positive or negative) when it is not 0. ...

$dot{V} = x_{1} dot x_{1} +x_{2} dot x_{2}$

$= x_{1} x_{2} - x_{1} x_{2}+epsilon left(frac{x_{2}^4}{3} -{x_{2}^2}right)$

$= -epsilon left({x_{2}^2} - frac{x_{2}^4}{3}right)$

If the parameter $ε$ is positive, stability is asymptotic for $x_{2}^{2} < 3.$

Barbalat's lemma and stability of time-varying systems

Assume that f is function of time only.

Having $dot{f}(t) to 0$ does not imply that $f (t)$ has a limit at $ttoinfty$

Having $f (t)$ approaching a limit as $t to infty$ does not imply that $dot{f}(t) to 0$ .

Having $f (t)$ lower bounded and decreasing ( $dot{f}le 0$ ) implies it converges to a limit. But it does not say whether or not $dot{f}to 0$ as $t to infty$ .

Barbalat's Lemma says

f (t)

has a finite limit as $t to infty$ and if $dot{f}$ is uniformly continuous (or $ddot{f}$ is bounded), then $dot{f}(t) to 0$ as $t toinfty$ .

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 $dot{V}$ 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:

V (x, t)

satisfies following conditions:

$V (x, t)$ is lower bounded
$dot{V}(x,t)$ is negative semi-definite (NSD)
$dot{V}(x,t)$ is uniformly continuous in time (i.e, $ddot{V}$ is finite)

then $dot{V}(x,t)to 0$ as $t to infty$ .

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

Consider a non-autonomous system

$dot{e}=-e + gcdot w(t)$

$dot{g}=-e cdot w(t)$

This is non-autonomous because the input $w$ is a function of time. Assume that the input $w (t)$ is bounded.

Taking $V = e 2 + g 2$ gives $dot{V}=-2e^2 le 0$

This says that $V (t) < = V (0)$ by first two conditions and hence $e$ and $g$ are bounded. But it does not say anything about the convergence of $e$ to zero. Moreover, the invariant set theorem cannot be applied, because the dynamics is non-autonomous.

Using Barbalat's lemma:

$ddot{V}= -4e(-e+gcdot w)$ .

This is bounded because $e$ , $g$ and $w$ are bounded. This implies $dot{V} to 0$ as $ttoinfty$ and hence $e to 0$ . This proves that the error converges.

References

Lyapunov A.M. Stability of motion, Academic Press, New-York and London,1966

This article incorporates material from asymptotically stable on PlanetMath, which is licensed under the GFDL. Aleksandr Lyapunov Aleksandr Mikhailovich Lyapunov (ÐÐ»ÐµÐºÑÐ°Ð½Ð´Ñ€ ÐœÐ¸Ñ…Ð°Ð¹Ð»Ð¾Ð²Ð¸Ñ‡ Ð›ÑÐ¿ÑƒÐ½Ð¾Ð²) (June 6, 1857 â€“ November 3, 1918, all new style) was a Russian mathematician, mechanician and physicist. ... PlanetMath is a free, collaborative, online mathematics encyclopedia. ...

Categories: PlanetMath sourced articles | Stability theory | Control theory | Dynamical systems

Results from FactBites:

Aleksandr Lyapunov - Wikipedia, the free encyclopedia (1661 words)
	Lyapunov had already begun to study this stability in his previous two-years attempts at solving the task.
	Lyapunov lectured at the university on themes from theoretical mechanics, integrals of differential equations and the theory of probability.
	His main preoccupations were the stability of equilibria and the motion of mechanical systems, the model theory for the stability of uniform turbulent liquid, and particles under the influence of gravity.

Lyapunov stability - Wikipedia, the free encyclopedia (841 words)
	The idea of Lyapunov stability can be extended to infinite-dimensional manifolds, where it is known as structural stability, which concerns the behaviour of different but "nearby" solutions to differential equations.
	The general study of the stability of solutions of differential equations is known as stability theory.
	Lyapunov's realisation was that stability can be proven without requiring knowledge of the true physical energy, providing a Lyapunov function can be found to satisfy the above constraints.

More results at FactBites »

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