Time-Domain Stability

Our original stability criterion is only available for LTI systems which allow us to convert to the frequency domain. However, sampled-data systems are not LTI, so we need to come up with a notion of stability in the time domain.

Bounded signal

Let $u(t)$/$u[k]$ be a real-valued signal defined for $t\ge 0$/$k\ge 0$. Then $u$ is bounded if there exists some constant $\overline{u}\in \mathbb{R},\overline{u}>0$ such that

• $|u(t)|\le \overline{u}\forall t\ge 0$, or
• $|u[k]|\le \overline{u}\forall k\in {\mathbb{Z}}_{\ge 0}$.

Only signals are defined to be bounded, not transfer functions!

• See also: BIBO Stability