Stability Fundamentals

A system is stable if bounded inputs produce bounded outputs; for LTI systems this reduces to all poles lying in the left half-plane.

What Stability Means

Stability is the most basic requirement of any control system: it must not blow up. The precise meaning depends on the framing, but the practical question is always whether disturbances and inputs produce responses that stay bounded and return toward equilibrium.

BIBO stability

A system is bounded-input bounded-output (BIBO) stable if every bounded input produces a bounded output. For an LTI system this holds if and only if every pole of the transfer function lies strictly in the open left half of the s-plane, that is, has negative real part.

The pole-location test

• All poles in the left half-plane: asymptotically stable, transients decay to zero.
• Any pole in the right half-plane: unstable, some response grows without bound.
• Simple poles on the imaginary axis: marginally stable, bounded oscillation that neither grows nor decays.
• Repeated poles on the imaginary axis: unstable, response grows polynomially.

Internal versus input-output

BIBO stability concerns only the input-output map. A hidden unstable mode that a pole-zero cancellation removes from the transfer function can still grow internally. Internal stability, checked on the full state-space model, is the stronger and safer condition, requiring every internal mode to decay.

Discrete-time counterpart

For sampled-data systems the stability region changes: poles must lie inside the unit circle of the z-plane rather than in the left half of the s-plane. The mapping z = e^(s*T) sends the left half-plane to the unit disk.

Because directly computing poles can be laborious, algebraic tests such as Routh-Hurwitz and graphical tests such as Nyquist let engineers judge stability without solving for the roots. Stability is necessary but not sufficient: a stable loop can still be sluggish or ring badly, which is why margins and transient specifications follow.