BIBOstabiilsus

BIBOstabiilsus, or BIBO stability, is a property of a dynamical system stating that every bounded input produces a bounded output. It is a central concept in linear and nonlinear system theory, used to guarantee predictable input–output behavior.

For a continuous-time Linear Time-Invariant (LTI) system with impulse response h(t), BIBO stability requires that the impulse response is absolutely integrable: the integral of the absolute value over all time is finite, ∫_{-∞}^{∞} |h(t)| dt < ∞. For a discrete-time LTI system with impulse response h[n], the corresponding condition is that the sum of the absolute values is finite: ∑_{n=-∞}^{∞} |h[n]| < ∞. These conditions ensure that any bounded input leads to a bounded output.

In the frequency domain, BIBO stability for continuous-time systems is often linked to the location of poles of the transfer function H(s): all poles must lie in the open left half-plane. In discrete time, all poles must lie inside the unit circle. Poles on the imaginary axis or outside these regions imply potential unbounded outputs for bounded inputs.

State-space representations connect BIBO stability to the system matrix: for a controllable and observable LTI system, BIBO stability is equivalent to A being Hurwitz (continuous time) or Schur stable (discrete time).

Examples include a first-order stable system with h(t) = e^{-at}u(t), a > 0, which is BIBO stable; its transfer function 1/(s+a) has a left-half-plane pole. An integrator with h(t) = 1 is not BIBO stable, since a constant input yields an unbounded output.

BIBO stability concerns bounded inputs and outputs and is distinct from internal stability or energy-based notions, though it often supports robust input–output design.