How do poles affect system stability in the s-plane?

Poles in the Left-Half Plane (LHP, real part 0) yield unstable, growing oscillations.

Laplace Transform Simulator - AI Math

System Analysis

Adjust continuous system poles to observe damping ratios, frequency response changes, and overall stability bounds.

Pole (Active): Determines damping & frequency.

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Zero (Inactive): Places zeroes to filter out frequencies (not present in this model).

Damping Ratio (ζ)0.316

Natural Freq (ωn)3.16 rad/s

System Stability Status

🟢 Stable (Decaying Response)

Variable Adjuster

Real Part (σ)-1

-44

Imaginary Part (ωd)3

Cycle Time (Tc)8s

220

Auto-Sweep Engine

Target Param:

Sweep Speed: 2x

Laplace Transform Explorer

LAPLA

The Laplace transform maps continuous-time functions to the s-domain (s = σ + jω). By plotting system poles (denominator roots of the transfer function) in the complex s-plane, we can evaluate stability. Left-Half Plane (σ < 0) poles generate decaying stable steps, while Right-Half Plane poles cause unstable exponential runaway.

F(s) = \int_{0}^{\infty} f(t) e^{-st} dt

Whiteboard Solver Steps

Step 1

s-Domain System Transfer Function

Centroid Poles: - Poles are located at $s = \sigma \pm j\omega_d = -1.00 \pm 3.00j$ . - Natural Frequency $\omega_n = \sqrt{\sigma^2 + \omega_d^2} = 3.16$ rad/s. - Damping Ratio $\zeta = \frac{-\sigma}{\omega_n} = 0.316$ .

H(s) = \frac{\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2} = \frac{10.00}{s^2 + 2.00s + 10.00}

Step 2

Time-Domain Step Response Derivation

System Behavior: - **Damping ratio ( $\zeta = 0.32$ )**: Since $\zeta < 1$ , the system is underdamped, resulting in overshoot and oscillations. - The real part $\sigma = -1.00$ acts as the exponential envelope ( $e^{\sigma t}$ ). Because $\sigma < 0$ , this envelope decays to zero, making the system stable.

Y(s) = H(s)X(s) = H(s)\frac{1}{s} \quad \xrightarrow{\mathcal{L}^{-1}} \quad y(t) = 1 - e^{-1.00t} \left( \cos(3.00t) - \frac{-1.00}{3.00} \sin(3.00t) \right)

Step 3

Laplace Transform Utility in Control & AI

Real-World Utility: - Control Systems: Feedback controllers (like PID loops regulating motor speed or robot joints) use Laplace transforms to configure poles in the Left-Half Plane, ensuring stable performance without runaway oscillations. - Analog Filter Circuits: Audio processing hardware (op-amps, passive filters) are designed in the s-domain to shape frequency responses.

G(s) = \frac{Y(s)}{U(s)} \implies \text{Poles LHP (Stable) vs RHP (Unstable)}

RV Anveshana