Nyquist plot analysis in Simulink

Introduction

A Nyquist plot helps us see how a system behaves with different frequencies. It shows if the system is stable or not.

Checking if a control system will stay steady or become unstable.

Designing controllers for machines like robots or cars.

Understanding how a system reacts to changes in input signals.

Testing feedback loops in electrical circuits.

Analyzing frequency response of filters or amplifiers.

Syntax

Simulink

nyquist(sys)

% sys is the system model (transfer function or state-space) in Simulink or MATLAB

The nyquist function plots the frequency response of the system.

You can use it with transfer functions or state-space models.

Examples

This creates a simple transfer function and plots its Nyquist plot.

Simulink

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

This uses a state-space model and plots its Nyquist plot.

Simulink

sys = ss([0 1; -2 -3], [0; 1], [1 0], 0);
nyquist(sys)

Sample Program

This code creates a transfer function for a system and shows its Nyquist plot with a title.

Simulink

sys = tf([1], [1 2 1]);
nyquist(sys);
title('Nyquist Plot of Simple System');

Important Notes

Nyquist plots show complex values: real part on x-axis, imaginary part on y-axis.

Look for encirclements of the point (-1,0) to check stability.

Simulink uses MATLAB commands for plotting, so you can combine them.

Summary

Nyquist plots help check system stability by showing frequency response.

Use nyquist(sys) with system models in Simulink or MATLAB.

Look at how the plot circles the point (-1,0) to understand stability.

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