Nyquist plot analysis in Simulink - Mini Project: Build & Apply

Nyquist Plot Analysis

📖 Scenario: You are working as a control engineer. You want to check if a system is stable by looking at its Nyquist plot. The Nyquist plot shows how the system behaves in the frequency domain. This helps you understand if the system will work well or if it might become unstable.

🎯 Goal: Write a simple MATLAB script to generate the Nyquist plot of a given transfer function. Then analyze the plot to decide if the system is stable.

📋 What You'll Learn

Create a transfer function with specific parameters using tf

Generate Nyquist plot data to visualize the system

Configure the frequency range for the Nyquist plot

Run the code and observe the Nyquist plot output

💡 Why This Matters

🌍 Real World

Nyquist plots are used by engineers to check if control systems will behave safely and stably before building physical devices.

💼 Career

Control engineers and system analysts use Nyquist plots daily to design and verify stable systems in industries like automotive, aerospace, and robotics.

1

Create the Transfer Function

Create a transfer function named sys with numerator coefficients [1] and denominator coefficients [1, 2, 1].

Simulink

% Create a transfer function named sys with numerator [1] and denominator [1 2 1]
% Your code here

Hint

Use the tf function to create the transfer function.

2

Set Frequency Range for Nyquist Plot

Create a variable called freq_range that contains frequencies from 0.1 to 10 radians per second with 100 points using the logspace function.

Simulink

sys = tf([1], [1 2 1]);
% Create freq_range from 0.1 to 10 with 100 points using logspace
% Your code here

Hint

Use logspace(log10(0.1), log10(10), 100) to create the frequency range.

3

Generate the Nyquist Plot Data

Use the nyquist function with sys and freq_range to get the Nyquist plot data. Store the output in variables re, im, and w.

Simulink

sys = tf([1], [1 2 1]);
freq_range = logspace(log10(0.1), log10(10), 100);
% Use nyquist to get re, im, w from sys and freq_range
% Your code here

Hint

Call nyquist(sys, freq_range) and assign outputs to re, im, and w.

4

Plot the Nyquist Diagram

Plot the Nyquist plot using the real part re and imaginary part im. Use plot with squeeze(re) on the x-axis and squeeze(im) on the y-axis. Add axis labels and a title.

Simulink

sys = tf([1], [1 2 1]);
freq_range = logspace(log10(0.1), log10(10), 100);
[re, im, w] = nyquist(sys, freq_range);
% Plot Nyquist plot with labels and title
% Your code here

Hint

Use plot with squeeze(re) and squeeze(im). Add xlabel, ylabel, and title.

Practice

1. What does a Nyquist plot primarily show in control system analysis?

easy

A. The error rate of a system over time

B. The time response of a system to a step input

C. The frequency response of a system in the complex plane

D. The physical layout of system components

Solution

1. Step 1: Understand Nyquist plot purpose

   A Nyquist plot maps the frequency response of a system onto the complex plane, showing gain and phase changes.

2. Step 2: Compare options to definition

   Only The frequency response of a system in the complex plane correctly describes this; others describe different system aspects.

3. Final Answer:

   The frequency response of a system in the complex plane -> Option C

4. Quick Check:

   Nyquist plot = frequency response [OK]

Hint: Nyquist plots map frequency response, not time or error [OK]

Common Mistakes:

• Confusing Nyquist plot with time-domain response
• Thinking Nyquist shows error rates
• Assuming it shows physical system layout

2. Which Simulink block is commonly used to generate a Nyquist plot?

easy

A. Step Response block

B. Bode Plot block

C. Frequency Response Estimator block

D. Nyquist Plot block

Solution

1. Step 1: Identify Simulink blocks for Nyquist plots

   Simulink provides a specific Nyquist Plot block to generate Nyquist plots directly.

2. Step 2: Eliminate other blocks

   Step Response shows time response, Bode Plot shows magnitude and phase separately, Frequency Response Estimator estimates frequency response but does not plot Nyquist directly.

3. Final Answer:

   Nyquist Plot block -> Option D

4. Quick Check:

   Nyquist plot uses Nyquist Plot block [OK]

Hint: Nyquist plots use the Nyquist Plot block in Simulink [OK]

Common Mistakes:

• Choosing Bode Plot block for Nyquist plot
• Confusing Step Response with frequency plots
• Selecting Frequency Response Estimator instead of Nyquist Plot block

3. Given the following MATLAB code snippet in Simulink environment:

sys = tf([1], [1, 2, 1]);
nyquist(sys)

What will the Nyquist plot show for this system?

medium

A. A plot showing no encirclement of the point -1+0i

B. A plot with multiple encirclements of the origin

C. A plot showing a clockwise encirclement of the point -1+0i

D. A plot showing a counterclockwise encirclement of the point -1+0i

Solution

1. Step 1: Analyze the transfer function

   The transfer function sys = 1/(s^2 + 2s + 1) has poles at s = -1 (double pole), stable system.

2. Step 2: Interpret Nyquist plot behavior

   For a stable system with no unstable poles, the Nyquist plot does not encircle the critical point -1+0i.

3. Final Answer:

   A plot showing no encirclement of the point -1+0i -> Option A

4. Quick Check:

   Stable system means no -1 encirclement [OK]

Hint: Stable poles mean no encirclement of -1 point [OK]

Common Mistakes:

• Assuming encirclement always occurs
• Confusing origin with -1+0i point
• Mixing clockwise and counterclockwise directions

4. You run a Nyquist plot in Simulink but the plot does not appear. Which of the following is the most likely cause?

medium

A. The system transfer function is not defined correctly

B. The Nyquist plot block is not connected to any input

C. The simulation time is set to zero

D. The plot window is minimized

Solution

1. Step 1: Check block connections in Simulink

   If the Nyquist plot block is not connected to a system input, it cannot generate data to plot.

2. Step 2: Evaluate other options

   Incorrect transfer function or zero simulation time may cause errors but usually show messages; minimized window does not prevent plot generation.

3. Final Answer:

   The Nyquist plot block is not connected to any input -> Option B

4. Quick Check:

   Block must be connected to input for plot [OK]

Hint: Always connect Nyquist block input to system [OK]

Common Mistakes:

• Ignoring block connections
• Assuming simulation time affects Nyquist plot
• Thinking minimized window stops plot generation

5. In a feedback control system modeled in Simulink, the Nyquist plot of the open-loop transfer function encircles the point -1+0i twice clockwise. What does this imply about the closed-loop system stability?

hard

A. The closed-loop system has two unstable poles

B. The closed-loop system is stable

C. The closed-loop system is marginally stable

D. The closed-loop system has no poles in the right half-plane

Solution

1. Step 1: Recall Nyquist stability criterion

   The number of clockwise encirclements of -1 equals the number of unstable closed-loop poles minus unstable open-loop poles.

2. Step 2: Apply to given scenario

   Assuming open-loop system has zero unstable poles, two clockwise encirclements mean two unstable closed-loop poles.

3. Final Answer:

   The closed-loop system has two unstable poles -> Option A

4. Quick Check:

   Clockwise encirclements = unstable poles count [OK]

Hint: Clockwise encirclements count unstable poles [OK]

Common Mistakes:

• Confusing clockwise with counterclockwise encirclements
• Ignoring open-loop pole count
• Assuming no encirclement means instability