BIBO Stability

An LTI system $P(s)$/$G[z]$ is Bounded-Input Bounded-Output Stable (BIBO stable) if every bounded input produces a bounded output.

In continuous time:

$$|u(t)|\le \overline{u}\forall t\ge 0⟹\exists \overline{y}\in \mathbb{R},\overline{y}>0\ni |y(t)|\le \overline{y}\forall t\ge 0$$

In continuous time:

$$|u(k)|\le \overline{u}\forall k\in {\mathbb{Z}}_{\ge 0}⟹\exists \overline{y}\in \mathbb{R},\overline{y}>0\ni |y[k]|\le \overline{y}\forall k\ge 0$$

Notation

• $\forall$ = “For all”
• $\exists$ = “Exists”
• $\ni$ = “Such that”

Examples of simple bounded inputs:

• Unit step $1(t)$ or $1[k]$
• Sinusoid $\sin (t)$ or $\sin [k]$
• But it can be anything, as long as it’s bounded by some values!

BIBO Stability Theorem (theorem from class)

Assume $P(s)$/$G[z]$ is real, rational and proper. Then $P(s)$/$G[z]$ is stable if and only if it is BIBO stable.

$$\text{Stable}⟺\text{BIBO stable}$$

• That means if something is unstable, it cannot be BIBO stable either