Lyapunov stability

From Wikipedia, the free encyclopedia

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.

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

[edit] 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.)

[edit] 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 $f\colon X\to X$ a continuous function. A point $x\in X$ is said to be Lyapunov stable, if, for each $ε > 0$ , there is a $δ > 0$ such that for all $y\in X$ , if

d (x, y) < δ

holds, and one has

d (f n (x), f n (y)) < ε

for all $n\in \mathbb{N}$ .

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

$\lim_{n\to\infty} d(f^n(x),f^n(y))=0$

whenever $d (x, y) < δ$ .

[edit] 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.

[edit] 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.)

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.

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.

[edit] Stability for linear state space models

A linear state space model

$\dot{\textbf{x}} = A\textbf{x}$

is asymptotically stable if

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 $V (x) = x T M x$ .)

[edit] 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.

[edit] Example

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

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

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

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

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

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

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

[edit] 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 $t\to\infty$

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 $t \to\infty$ .

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

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

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

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:

$\ddot{V}= -4e(-e+g\cdot w)$ .

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