Pole-Zero & System Stability
Interactive control-system demonstration
1. Pole-Zero Concept
A pole is a value of s that makes the denominator of a transfer function zero. A zero is a value that makes the numerator zero.
Continuous-time stability rule:
Pole in the Left Half-Plane (LHP) → Stable
Pole on the Imaginary Axis → Marginal case
Pole in the Right Half-Plane (RHP) → Unstable
Pole in the Left Half-Plane (LHP) → Stable
Pole on the Imaginary Axis → Marginal case
Pole in the Right Half-Plane (RHP) → Unstable
2. Move the Pole
σ = -1.0
ω = 2.0
1.0 ×
✓ STABLE SYSTEM
Real Part
-1.0
Imaginary Part
2.0
Region
LHP
System
Stable
3. Pole-Zero Plot
S-Plane
× = Pole
○ = Zero
The vertical line is the imaginary axis. The horizontal line is the real axis.
The vertical line is the imaginary axis. The horizontal line is the real axis.
4. System Response
Impulse Response
A stable system produces a response that gradually returns toward zero.
An unstable system produces a response whose amplitude grows with time.
5. Simple Mathematical Model
For demonstration, the system uses a pole pair:
s = σ ± jω
The approximate impulse response is represented by:
h(t) = eσt cos(ωt)
The important factor is σ:
σ < 0 → exponential decay → Stable
σ = 0 → constant-amplitude oscillation → Marginal
σ > 0 → exponential growth → Unstable
s = σ ± jω
The approximate impulse response is represented by:
h(t) = eσt cos(ωt)
The important factor is σ:
σ < 0 → exponential decay → Stable
σ = 0 → constant-amplitude oscillation → Marginal
σ > 0 → exponential growth → Unstable
6. Practical Observation
- Click Stable Example and observe the pole in the left half-plane.
- Click Unstable Example and observe the pole move to the right half-plane.
- Watch the impulse response change from decay to growth.
- Move the pole slowly using the Real Part slider.
- When the pole crosses the imaginary axis, the system changes from stable to unstable.