Calculator · Dynamics & Control
Poles and Zeros Calculator
Enter G(s) = N(s)/D(s) in descending powers of s. The calculator returns the roots of both polynomials with their multiplicity, where each one sits in the complex plane, and what that means for stability.
Input
G(s) = (s + 1) / (s^2 + 2s + 5)
Roots
| Type | Value | Multiplicity | Region |
|---|---|---|---|
| × pole | -1 + 2j | 1 | Left half-plane |
| × pole | -1 − 2j | 1 | Left half-plane |
| ○ zero | -1 | 1 | Left half-plane |
Pole-zero map
| Root | Re(s) | Im(s) | Multiplicity |
|---|---|---|---|
| Pole × | -1 | 2 | 1 |
| Pole × | -1 | -2 | 1 |
| Zero ○ | -1 | 0 | 1 |
Poles are drawn as × and zeros as ○, so the marker shape — not colour — carries the meaning.
Stability summary
| Classification | Asymptotically stable |
|---|---|
| Poles in the right half-plane | 0 |
| Poles on the imaginary axis | 0 |
| Right-half-plane zeros (non-minimum phase) | 0 |
Technical reference
- LHP
- left half-plane, Re(s) < 0 — each such pole contributes a decaying mode
- RHP
- right half-plane, Re(s) > 0 — any such pole makes the system unstable
- jω-axis
- imaginary axis, Re(s) ≈ 0 — a bounded, non-decaying mode
Every pole in the LHP ⇒ asymptotically stable. Any pole in the RHP ⇒ unstable.
A SIMPLE (non-repeated) pole on the imaginary axis, with no RHP poles, is marginally stable: a bounded oscillation or offset that never decays or grows.
A REPEATED pole on the imaginary axis is unstable, not marginally stable: it produces a response growing like t·sin(ωt), which the engine flags explicitly rather than lumping in with the simple case.
The verdict is a pole-location classification of the internal response: all poles strictly LHP ⇒ asymptotically stable; simple jω-axis poles with no RHP poles ⇒ marginally stable; repeated jω-axis poles or any RHP pole ⇒ unstable. “Marginally stable” is not a claim of BIBO stability for every input — a sinusoid at that jω frequency drives an unbounded response.
For this calculator’s denominator-root stability classification, zeros do not change the verdict; right-half-plane zeros are reported separately as non-minimum-phase behaviour.
Assumptions
- · Continuous-time, single-input single-output, real coefficients.
- · Coincident roots are grouped into one entry carrying an explicit multiplicity.
- · Right-half-plane zeros are reported because they limit achievable control, not because they affect stability.
Continue
Time, frequency and algebraic stability analysis of the same transfer function.