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Effect of pole location on stability

Short answer

Poles in the left half-plane (σ<0) produce stable decay; simple poles on the jω axis give sustained oscillation (marginal stability); repeated jω axis poles or right half-plane poles (σ>0) cause instability. Poles farther left decay faster; dominant poles near the jω axis govern transient response.

How does the location of poles affect the stability in control system?

The stability of a system depends only on where the poles of its closed-loop transfer function (the roots of the characteristic equation) lie in the s-plane. Each pole s=σ±jωs = \sigma \pm j\omega adds a term eσtsin⁡(ωt)e^{\sigma t}\sin(\omega t) to the response.

Pole locationResponse termStability
Left half plane (σ<0\sigma<0)Decays to zeroStable
Simple poles on jωj\omega axisConstant or steady oscillationMarginally stable
Repeated poles on jωj\omega axistsin⁡ωtt\sin\omega t, growsUnstable
Right half plane (σ>0\sigma>0)Grows exponentiallyUnstable
  • The further left the poles are, the faster the response decays. This means greater relative stability.
  • Poles close to the jωj\omega axis (dominant poles) decay slowly and decide the transient response.
  • Imaginary part ω\omega sets the frequency of oscillation; real poles give no oscillation.

✓ Answer checked by Claude Opus 5.5 (high effort)

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