BIBOstabil

BIBOstabil, or BIBO stability, is a term used in control theory and signal processing to describe a property of a dynamical system: every bounded input yields a bounded output. BIBO stands for bounded input, bounded output.

A system is BIBO stable if, for any input signal whose magnitude is bounded by a finite value, the output signal remains bounded. This concept applies to both linear and nonlinear systems, but it is most commonly discussed in the context of linear time-invariant (LTI) systems.

For linear time-invariant systems, BIBO stability is characterized by the impulse response. In continuous time, an LTI system is BIBO stable if the impulse response h(t) is absolutely integrable: ∫0^∞ |h(t)| dt < ∞ for causal systems. In discrete time, the impulse response h[n] must be absolutely summable: ∑n=0^∞ |h[n]| < ∞ for causal systems. Roughly speaking, the impulse response must decay sufficiently fast.

Practically, many common stable circuits and filters satisfy BIBO stability. For example, a first-order RC low-pass filter has impulse response h(t) = (1/RC) e^{-t/RC} u(t), which is absolutely integrable and thus BIBO stable.

In more general settings, including nonlinear or time-varying systems, BIBO stability means there exists a bound on the output that depends only on the bound of the input. It remains a fundamental criterion in analysis and design, alongside other notions of stability such as internal or Lyapunov stability.