📘 Laplace Transform in Control Systems – Transfer Function & Stability
Laplace Transform is the backbone of Control Systems. It converts differential equations into algebraic transfer functions. Understanding poles and zeros is critical for GATE.
🔹 1. Transfer Function
Transfer Function is defined as:
G(s) = Output(s) / Input(s)
It is valid for zero initial conditions.
🔹 2. Example – First Order System
Differential equation:
dy/dt + 5y = 10u(t)
Taking Laplace:
sY(s) + 5Y(s) = 10/s
Transfer Function:
G(s) = Y(s)/U(s) = 10 / (s + 5)
🔹 3. Poles and Zeros
- Zeros → Roots of numerator
- Poles → Roots of denominator
Example:
G(s) = 5 / (s² + 4s + 5)
Poles:
s = -2 ± j1
🔹 4. Stability Condition
- System stable if all poles have negative real part.
- Right-half plane poles → Unstable.
- Poles on imaginary axis → Marginally stable.
🔹 5. Step Response of First Order System
Given:
G(s) = 1/(s+3)
For unit step input:
Y(s) = 1/[s(s+3)]
Partial fraction:
1/[s(s+3)] = 1/3s − 1/3(s+3)
Inverse:
y(t) = 1/3 (1 − e^{-3t})
🔹 6. Second Order System
Standard Form:
G(s) = ωn² / (s² + 2ζωn s + ωn²)
- ζ → Damping ratio
- ωn → Natural frequency
🔹 7. Damping Cases
- ζ > 1 → Overdamped
- ζ = 1 → Critically damped
- 0 < ζ < 1 → Underdamped
- ζ = 0 → Undamped
🔹 8. Example – Pole Based Stability
G(s) = 1/(s² − 4s + 5)
Poles:
s = 2 ± j1
Since real part positive → System unstable.
🔹 9. Initial & Final Value in Control
Initial Value:
y(0⁺) = lim s→∞ [sY(s)]
Final Value:
y(∞) = lim s→0 [sY(s)]
🎯 GATE Important Points
- Poles determine stability.
- Right-half plane poles always unstable.
- Second order parameters frequently tested.
- Transfer function concept must be clear.
Poles Decide Stability – Laplace Makes It Visible
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