BIBO stability

BIBO stability (from English bounded input, bounded output ), also input / output stability , is a term from systems theory or control engineering . The stability of a system is ensured if the output signal does not exceed all limits with a limited input signal. Stable systems can be physically implemented on the basis of energy conservation without showing saturation effects.

There are various methods of defining stability, whereby the so-called BIBO stability is common in systems theory. There is a restriction to linear time-invariant systems , so-called LTI systems, in the form of continuous and time-discrete systems.

definition

An LTI system is BIBO-stable if it reacts to every limited input function or every limited input sequence with a limited output function or with a limited output sequence . A function is restricted if its amount is less than a fixed limit for all times : ${\displaystyle x(t)}$ ${\displaystyle x[k]}$ ${\displaystyle y(t)}$ ${\displaystyle y[k]}$ ${\displaystyle t}$ ${\displaystyle M}$

${\displaystyle |x(t)|<M<\infty }$

Similarly, a restricted sequence of the condition suffices:

${\displaystyle |x[k]|<M<\infty }$

A system is BIBO-stable if, with a finite constant, the following applies: ${\displaystyle C}$

${\displaystyle |y(t)|<C\cdot M<\infty }$

or for discrete systems:

${\displaystyle |y[k]|<C\cdot M<\infty }$

Impulse response

The BIBO stability of an LTI system can also be expressed in terms of its impulse response if it can be absolutely integrated :

${\displaystyle \int _{-\infty }^{\infty }{\left|g(t)\right|\,{\mathord {\operatorname {d} }}t}=\|g\|_{1}<\infty }$

The same applies to discrete-time LTI systems:

${\displaystyle \sum _{k=-\infty }^{\infty }{\left|g[k]\right|}=\|g\|_{1}<\infty }$

literature

• Bernd Girod, Rudolf Rabenstein, Alexander Stenger: Introduction to systems theory . 4th edition. Teubner, Wiesbaden 2007, ISBN 978-3-8351-0176-0 .