Definition
BIBO stability (Bounded‑Input Bounded‑Output stability) is a property of linear time‑invariant (LTI) systems in control theory and signal processing. A system is said to be BIBO stable if every bounded input signal produces a bounded output signal. Formally, if an input $x(t)$ satisfies $|x(t)|\infty = \sup{t}|x(t)| < \infty$, then the corresponding output $y(t)$ must also satisfy $|y(t)|_\infty < \infty$.
Mathematical criterion for LTI systems
For an LTI system with impulse response $h(t)$ (continuous time) or $h[n]$ (discrete time):
- Continuous‑time: The system is BIBO stable ⇔ $\displaystyle \int_{-\infty}^{\infty} |h(t)| , dt < \infty$.
- Discrete‑time: The system is BIBO stable ⇔ $\displaystyle \sum_{k=-\infty}^{\infty} |h[k]| < \infty$.
These conditions guarantee that the convolution integral (or sum) $y(t)=\int h(\tau)x(t-\tau)d\tau$ (or $y[n]=\sum h[k]x[n-k]$) yields a bounded result for any bounded input.
Relation to pole locations
In the Laplace (continuous) or Z‑transform (discrete) domains, BIBO stability can be assessed by the location of system poles:
- Continuous‑time: All poles of the transfer function $H(s)$ must lie strictly in the left half‑plane ($\Re{s}<0$).
- Discrete‑time: All poles of $H(z)$ must lie strictly inside the unit circle ($|z|<1$).
Poles on the imaginary axis (continuous) or on the unit circle (discrete) generally cause marginal stability; such systems are not BIBO stable because they can produce unbounded outputs for certain bounded inputs (e.g., resonant sinusoidal inputs).
Examples
| System | Impulse response $h(t)$ / $h[n]$ | BIBO stable? |
|---|---|---|
| First‑order low‑pass RC filter (continuous) | $h(t)=\frac{1}{RC}e^{-t/(RC)}u(t)$ | Yes (integral of $ |
| Ideal integrator (continuous) | $h(t)=u(t)$ | No (integral diverges) |
| Unit delay (discrete) | $h[n]=\delta[n-1]$ | Yes (finite sum) |
| Discrete-time accumulator | $h[n]=u[n]$ | No (sum diverges) |
Importance
BIBO stability is a practical criterion because it directly addresses whether a system will respond acceptably to real‑world signals, which are inherently bounded in amplitude. It is distinct from internal (asymptotic) stability, which concerns the behavior of the system’s homogeneous solution (e.g., state trajectories) and may be stricter.
Testing procedures
- Impulse‑response test – Compute or measure $h(t)$ / $h[n]$ and evaluate the absolute integrability (continuous) or absolute summability (discrete).
- Pole‑location test – Determine the poles of the transfer function and verify they satisfy the half‑plane or unit‑circle criteria.
- Simulation – Apply a set of bounded test inputs (e.g., step, sinusoid, random bounded sequence) and observe whether the output remains bounded.
Limitations
- The BIBO criterion applies only to linear time‑invariant systems; nonlinear or time‑varying systems require alternative stability concepts.
- It does not guarantee performance metrics such as speed of response, overshoot, or robustness to parameter variations.
References
- Ogata, K. Modern Control Engineering, 5th ed., Prentice Hall, 2010.
- Oppenheim, A. V., Schafer, R. W., & Buck, J. R. Discrete‑Time Signal Processing, 3rd ed., Pearson, 2010.
- Franklin, G. F., Powell, J. D., & Emami‑Nasr, S. Feedback Control of Dynamic Systems, 7th ed., Pearson, 2015.
