Lyapunov stability

In mathematics, the notion of Lyapunov stability occurs in the study dynamical systems. In simple terms, if all points that start out near a point x stay near x forever, then x is Lyapunov stable. More strongly, if all points that start out near x converge to x, then x is asymptotically stable. 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.

Definition for continuous-time systems

A system is said to be (uniformly) stable about a trajectory x "in the sense of Lyapunov" if for every

\epsilon>0

, there is a

\delta>0

such that

\|y(t_0)-x(t_0)\| < \delta

implies that

\|y(t)-x(t)\| < \epsilon

for all $t \in \mathbb{R}^{+}.$ Here, $\|\cdot\|$ is a metric; the motion or flow occurs on a manifold M endowed with the metric $\|\cdot\|$ . The above statement holds for all points $y\in M$ . In plain language, if all points that start out near x stay near x forever, then x is Lyapunov stable.

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.)

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 $\epsilon>0$ , there is a $\delta>0$ such that for all $y\in X$ , if

d(x,y)<\delta

holds, and one has

d(f^n(x),f^n(y))<\epsilon

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 $\delta>0$ such that

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

whenever $d(x,y)<\delta$ .

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.

Lyapunov second theorem on stability

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

$V(x) > 0 : \forall{x} \neq 0$ (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. (An additional condition callled "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 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.

Stability for linear state space models

A linear state space model

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

is asymptotically stable if

A^{T}M + MA + N = 0

has a solution where $N = N^{T} > 0$ and $M = M^{T} > 0$ (positive definite matrices). (The relevant Lyapunov function is $V(x) = x^TMx$ .)

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.

Example

Consider the Van der Pol oscillator equation:

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

Let

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

so that the corresponding system is

\dot{x_{1}} = x_{2} , \dot{x_{2}} = -x_{1} + \epsilon \left( \frac^{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 $\epsilon$ 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.

If $\dot{f}(t) \to 0$ does not imply that $f(t)$ has a limit at $t\to\infty$

If $f(t)$ has a limit as $t \to \infty$ does not imply that $\dot{f}(t) \to 0$ .

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

Barbalat's Lemma says that If $f(t)$ 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 $t \to\infty$ .

But why do we need a Barbalat's lemma?

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.

What's the big deal about it? We have invariant set theorems when $\dot{V}$ is only NSD.

Agreed! 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.

But this flexibility is not available for *time-varying* systems.

This is where "Barbalat's lemma" comes into picture. It says:

IF $V(x,t)$ satisfies following conditions:

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

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

But how does it help in determining asymptotic stability?

There is a nice example on page 127 of "Slotine Li's book on 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. Let's assume that the input w(t) is bounded.

If we take $V=e^2+g^2$ then $\dot{V}=-2e^2 \le 0$

This says that $V(t)<=0$ 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, we can't apply invariant set theorem, because the dynamics is non-autonomous.

Now let's use 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 $e \to 0$ . If we are interested in error convergence, then our problem is solved.

Stability theory | Control theory

Stabilitätstheorie | Stabilité de Lyapunov | Stabilità secondo Lyapunov | リアプノフ安定 | Metody Lapunowa

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