Transfer Function (control)
A transfer function is the ratio of the Laplace transform of the output to the Laplace transform of the input of a linear time-invariant (LTI) system, assuming zero initial conditions. It fully characterises a system's dynamic behaviour in the frequency domain, including its poles, zeros, gain, and stability margins. Transfer functions are the central tool in classical control theory for designing feedback controllers, analysing stability using root locus and Bode plots, and predicting transient and steady-state responses.
Key Formula
H(s) = Y(s)/X(s) = (b_m·s^m + ... + b_0) / (a_n·s^n + ... + a_0)
LaTeX: H(s) = \frac{Y(s)}{X(s)} = \frac{b_m s^m + \cdots + b_1 s + b_0}{a_n s^n + \cdots + a_1 s + a_0}
| Symbol | Meaning | Unit |
|---|---|---|
| H(s) | Transfer function | Dimensionless (or unit ratio) |
| Y(s) | Laplace transform of output signal | Varies |
| X(s) | Laplace transform of input signal | Varies |
| b_i | Numerator polynomial coefficients | Dimensionless |
| a_i | Denominator polynomial coefficients | Dimensionless |
Worked Example
Problem
A second-order system has transfer function H(s) = 100 / (s² + 10s + 100). Determine the natural frequency ωₙ, damping ratio ζ, and classify the response type.
Solution
Step 1: Standard second-order form: H(s) = ωₙ² / (s² + 2ζωₙ·s + ωₙ²). Step 2: Match coefficients with H(s) = 100 / (s² + 10s + 100). ωₙ² = 100 → ωₙ = 10 rad/s 2ζωₙ = 10 → ζ = 10 / (2 × 10) = 0.5 Step 3: Classify the response. Since 0 < ζ < 1 (ζ = 0.5), the system is underdamped. Step 4: Damped natural frequency: ω_d = ωₙ√(1 − ζ²) = 10√(1 − 0.25) = 10√0.75 ≈ 8.66 rad/s Step 5: The poles are at s = −ζωₙ ± jω_d = −5 ± j8.66.
Answer
ωₙ = 10 rad/s, ζ = 0.5 (underdamped), ω_d ≈ 8.66 rad/s, poles at −5 ± j8.66
Second-Order System Response Classification
| Damping Ratio ζ | Type | Pole Location | Step Response |
|---|---|---|---|
| ζ = 0 | Undamped | Imaginary axis ±jωₙ | Sustained oscillation |
| 0 < ζ < 1 | Underdamped | Complex conjugate pair | Decaying oscillation |
| ζ = 1 | Critically damped | Real, repeated −ωₙ | Fastest no-overshoot |
| ζ > 1 | Overdamped | Two distinct real poles | Sluggish, no oscillation |
| ζ < 0 | Unstable | Right-half s-plane | Diverging response |
Interactive Tools
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Related Terms
Laplace Transform
The Laplace Transform converts a time-domain function into the complex frequency domain (s-domain), enabling differential equations to be solved as algebraic equations. It generalises the Fourier Transform by including a real exponential damping term, making it applicable to a broader class of signals including those that grow with time. It is the primary mathematical tool in control systems engineering, circuit analysis, and linear system theory for analysing stability, transient response, and frequency behaviour.
Feedback Control
Feedback control is a control strategy in which the output of a system is measured and compared to a desired reference (setpoint), and the difference (error) is used to adjust the system input to reduce that error. Negative feedback — where the output is subtracted from the reference — is the basis of stable automatic control systems in engineering, biology, and economics. Feedback control enables systems to self-correct against disturbances and parameter variations, forming the foundation of servo systems, thermostats, autopilots, and industrial process control.
PID Controller
A PID (Proportional-Integral-Derivative) controller is a feedback control mechanism that calculates a control output based on three terms: the proportional term (current error), the integral term (accumulated past error), and the derivative term (rate of change of error). Together these three components allow the controller to eliminate steady-state error, respond rapidly to disturbances, and suppress oscillations. PID controllers are the most widely used feedback controllers in industrial automation, motor drives, temperature regulation, and robotic systems due to their effectiveness and relative simplicity of tuning.
The term "transfer function" was popularised in control engineering in the 1940s–1950s through the work of Harry Nyquist, Hendrik Bode, and others at Bell Laboratories. "Transfer" derives from Latin "transferre" (to carry across), describing how the mathematical function maps input to output. The concept was formalised using the Laplace transform framework developed by Norbert Wiener and others during WWII radar research.