Lyapunov Stability - Barbalat's Lemma and Stability of Time-varying Systems

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 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.