Step-by-Step Understanding of Control System Transfer Functions (Complete with Examples)

introduction

 In engineering, particularly in the field of control systems, we often encounter the term transfer function. The transfer function is a fundamental concept used to analyze and design dynamic systems, whether electrical, mechanical, thermal, or hydraulic. Transfer functions allow us to understand how a system responds to a given input, and are thus crucial in the process of designing stable, fast, and accurate systems.

However, many students and novice practitioners struggle to grasp the concept of transfer functions because they involve Laplace transforms, dynamic systems, and complex mathematical representations. Therefore, this material will explain in a simple and step-by-step manner what a transfer function is, why it is important, how to determine it, and how to use it in control system analysis.

What Is a Transfer Function?

Basic Definition

Simply put, a transfer function is the relationship between the inputs and outputs of a system in the Laplace domain (s-domain). Mathematically, the transfer function is written as:

where:

• G(s) is the system transfer function,
• Y(s) is the Laplace transform of the output,
• U(s) is the Laplace transform of the input.

The transfer function only applies to linear and time-invariant (LTI) systems.

A Simple Analogy

Imagine controlling the temperature of a room with an air conditioner. You give a temperature command (say, 22°C), and a sensor measures the actual temperature. The transfer function is how the air conditioner and the room respond to the change in temperature command. Whether it's fast, slow, overly responsive, or stable—all of this can be analyzed through the transfer function.

Why Are Transfer Functions Important?

Transfer functions have many uses in control systems, including:

• System Stability Analysis

            We can evaluate whether a system will be stable by looking at the poles of the transfer function.

• System Dynamic Characteristics

        System responses such as rising time, overshoot, and settling time can be determined from the transfer function.

• Controller Design

        Transfer functions make it easier to design controllers such as PID, lead-lag, or other compensators.

• Simulation and Prediction

        With transfer functions, we can simulate system output without building the physical system.

How is the Transfer Function Obtained?

To obtain the transfer function of a physical system, we typically use the following methods:

1. Determining the Physical Model of the System

The first step is to create a mathematical model of the physical system, usually in the form of a differential equation.

    2. Laplace Transform

The differential equation is transformed into an algebraic form in the Laplace domain.

    3.Determining the Ratio of Output to Input

From the transformation equation, we take the ratio 

Example 1: Mass-Spring-Damping Mechanical System

Suppose we have the system:

Where:

• m: mass (kg)
• c: damping coefficient (Ns/m)
• k: spring constant (N/m)
• x(t): position (m)
• x(t): input force (N)

Step 1: Laplace Transform

Step 2: Take Over Function

General Form of Transfer Function

Transfer functions are generally written in the form:

Important components:

• Zer0 (root of the numerator): Determines the frequency at which the system output becomes zero even if the input is not zero.
• Pole (root of the denominator): Determines the stability and dynamics of the system.
• System Order: Determined by the highest power of s in the denominator.

System Response from the Transfer Function

1. Impulse Response

Input: 

δ(t) → 𝑈(𝑠)=1 Output: Y(s)=G(s)⋅1=G(s)

Analysis of System Properties Based on Transfer Function

1. Stability

A system is stable if all poles of the transfer function have negative real parts (located to the left of the imaginary axis in the s-plane).

2. Response Speed

• Poles close to the imaginary axis → slow system
• Poles far from the imaginary axis → fast system

3. Damping and Overshoot

If the system has complex poles:

𝑠=−𝜁𝜔𝑛±𝑗𝜔𝑑

Then we can calculate:

• Damping ratio (𝜁)
• Natural frequency (𝜔𝑛)
• Overshoot and settling time

Example of Using Transfer Functions in Electrical Circuits

RC Circuits

For example, we want to know the transfer function of the output voltage Vout​(t)  of an RC circuit with respect to the input voltage Vin​(t).

This shows that the system acts like a low-pass filter, where the response is slow to fast changes in the input.

Transfer Function Block Representation

In control systems, transfer functions are often displayed in block diagram form to show the relationships between multiple subsystems.

Example:

[ U(s) ] ---> [ G1(s) ] ---> [ G2(s) ] ---> [ Y(s) ]

If two systems are sequential:

If the system is in negative feedback:

Practical Use of Transfer Functions (Simulation)

In the real world, we often use software such as MATLAB/Simulink to analyze and simulate transfer functions.

Example MATLAB 

commands:

num = [1];

den = [1 3 2];

G = tf(num, den);

step(G); % Untuk melihat respon tangga

Conclusion

Transfer functions are an essential tool in control systems for understanding how a system responds to an input. By converting a mathematical model of a system to the Laplace domain and constructing its transfer function, we can analyze various system characteristics such as stability, speed, and accuracy. While this concept may seem complex at first, it becomes very useful and easy to understand step by step, especially with the help of simulations and real-world case studies.