Bounded-Input Bounded-Output
 Bounded-Input Bounded-Output Classical Control Stability In order to prevent spam, users must register before they can edit or create articles.

## 1 Introduction to BIBO Stability

A system is defined to be BIBO Stable if every bounded input to the system results in a bounded output over the time interval $LaTeX: [t_0, \infty)$. This must hold for all initial times to. So long as we don't input infinity to our system, we won't get infinity output.

A system is defined to be uniformly BIBO Stable if there exists a positive constant k that is independent of t0 such that for all t0 the following conditions:

$LaTeX: \|u(t)\| \le 1$
$LaTeX: t \ge t_0$

implies that

$LaTeX: \| y(t) \| \le k$

## 2 Determining BIBO Stability

Mathematically a system f is BIBO stable if an arbitrary input x is bounded by two finite but large arbitrary constants M and -M:

$LaTeX: -M < x \le M$

Apply the input x, and the arbitrary boundries M and -M to the system to produce three outputs:

$LaTeX: y_x = f(x)$
$LaTeX: y_M = f(M)$
$LaTeX: y_{-M} = f(-M)$

Now, all three outputs should be finite for all possible values of M and x, and they should satisfy the following relationship:

$LaTeX: y_{-M} \le y_x \le y_M$

If this condition is satisfied, then the system is BIBO stable.