Poles and Zeros in Dynamic Systems: Functions, Examples, and How to Determine Them

 What Are Poles and Zeros in Dynamic Systems? A Beginner's Guide

In the world of dynamic systems, particularly in control engineering, two very important and frequently used terms are poles and zeros. While both may sound mathematical, understanding them is vital in designing and analyzing systems, whether mechanical, electronic, or other automatic control systems.

This article will explain in a simple, step-by-step manner what poles and zeros are, why they are important, how to find them, and how they affect the behavior of dynamic systems.

2. Transfer Function

A transfer function is a mathematical representation of the relationship between a system's inputs and outputs in the Laplace domain. The general form of a transfer function is:

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.
• s is a complex variable (in the Laplace domain).
• The denominator is the system characteristic, which is related to the pole.
• The numerator is related to the zero.

3. What is Zero?

Zero is the value of the variable 𝑠 , s that causes the transfer function to be zero.

In other words, zero is the root of the numerator of the transfer function. Mathematically:

then 

𝑧1 and 𝑧2 are the zeros of the system

Physical interpretation:

Zero is the frequency at which the system's output becomes zero, despite the presence of input. This can be thought of as "filtering" certain frequencies from the input.

4. What is a Pole?

A pole is the value of 𝑠

s that causes the transfer function to reach infinity.

Mathematically, a pole is the root of the denominator of the transfer function. In the form:

Then 

𝑝1 and 𝑝2 are the poles of the system.

Physical interpretation:

Poles determine the natural behavior of a system, specifically how the system responds to inputs and how the system reacts over time—whether it will be stable, oscillatory, or unstable.

5. Simple Example

Let's look at a simple example:

• Numerator : 
• Denominator:  
• Pole at 𝑠 $=-1$ and $s = -2$
  

Here, we have one zero at s=−2, and two poles at s=−1 and s=−2. Interestingly, the zero and pole can be at the same point, and when that happens, we say they cancel each other out.

6. Why Are Poles and Zeros Important?

a. System Stability

Poles significantly determine system stability. Generally, if all the system poles are in the left half of the complex plane (Re(s) < 0), then the system is stable.

If there are poles in the right half or on the imaginary axis, then the system can be unstable or oscillate continuously.

b. System Response Time

• Poles with a large negative real part → fast response (the system quickly returns to steady state).
• Poles near the imaginary axis → slow response and may oscillate.
• Complex poles → cause oscillation or vibration.

c. Filters and Frequency

Zeros affect the frequencies that a system passes or rejects.

For example, in electronic filter design, zeros are used to remove certain frequencies from a signal.

7. Visualization in the S-Plane

Poles and zeros can be depicted in the complex plane, called the s-plane.

• Poles are usually marked with the symbol 'X'.
• Zeros are usually marked with the symbol 'O'.
• The horizontal axis is the real part.
• The vertical axis is the imaginary part.

The location of the poles and zeros in this plane indicates much about the system's characteristics, including stability and response speed.

8. Relationship to Discrete Systems

In discrete systems (such as digital systems), poles and zeros also exist, but the analysis is performed in the z-domain (z-plane), not the s-plane. Although the concepts are similar, the stable locations of poles and zeros in discrete systems are within the unit circle on the z-plane.

9. How to Find Poles and Zeros?

Typically, we use:

• The Laplace transform to obtain the transfer function from the differential equation
• The transfer function directly from the system model
• Software such as MATLAB or Python (scipy.signal) to calculate and plot them

Example in Python:

from scipy import signal

numerator = [1, 2]  # s + 2

denominator = [1, 3, 2]  # s^2 + 3s + 2

system = signal.TransferFunction(numerator, denominator)

print("Zero:", system.zeros)

print("Pole:", system.poles)

Conclusion

Poles and zeros are not just mathematical concepts, but also essential tools in engineering. By understanding them, we can design systems that are more reliable, efficient, and tailored to our needs.

For beginners, it's important to familiarize yourself with these concepts through practice, visualization, and simulation. The more you work with dynamic systems, the more intuitive your understanding of poles and zeros will become.

So, don't be intimidated by these terms—with the right approach, poles and zeros can become your best friends in the world of systems engineering!