Nyquist plot analysis in Simulink - Practice Problems & Coding Challenges

Challenge - 5 Problems

🧠 Conceptual

intermediate

Understanding Nyquist Plot Stability Criterion

In control systems, the Nyquist plot helps determine system stability. What does it mean if the Nyquist plot encircles the point (-1,0) exactly once in the clockwise direction?

A The system is unstable because it has one pole in the right half-plane.

B The system is stable with no poles in the right half-plane.

C The system is marginally stable with poles on the imaginary axis.

D The system has no poles or zeros affecting stability.

💡 Hint

Recall the Nyquist stability criterion relates encirclements of (-1,0) to poles in the right half-plane.

According to Nyquist criterion, the number of clockwise encirclements of (-1,0) equals the number of poles in the right half-plane. One encirclement means one unstable pole, so the system is unstable.

❓ Predict Output

intermediate

Nyquist Plot Output Interpretation

Given a transfer function in Simulink with the following frequency response data points, what is the gain margin if the Nyquist plot crosses the real axis at -0.5?

Simulink

frequency = [0.1, 1, 10];
real_part = [-0.8, -0.5, 0.2];
imag_part = [0.1, 0, -0.3];

% Gain margin calculation at phase crossover frequency
phase_cross_index = find(imag_part == 0);
gain_margin = 1 / abs(real_part(phase_cross_index));

A 2

B 0.5

C 1

D Cannot be determined from given data

💡 Hint

Gain margin is the reciprocal of the magnitude of the real part at the phase crossover frequency.

The Nyquist plot crosses the real axis at -0.5 (phase crossover). Gain margin = 1 / |-0.5| = 2.

❓ data_output

advanced

Extracting Phase Margin from Nyquist Data

Using Simulink, you obtain the following Nyquist plot data points. What is the phase margin if the gain crossover frequency is at 1 rad/s?

Simulink

frequency = [0.5, 1, 2];
magnitude = [1.5, 1, 0.7];
phase_deg = [-30, -180, -210];

gain_crossover_index = find(magnitude == 1);
phase_margin = 180 + phase_deg(gain_crossover_index);

A -180 degrees

B 30 degrees

C 0 degrees

D 150 degrees

💡 Hint

Phase margin = 180 + phase at gain crossover frequency.

At gain crossover frequency (1 rad/s), phase is -180 degrees. Phase margin = 180 + (-180) = 0 degrees. But the data shows phase at 1 rad/s is -180, so phase margin is 0 degrees. However, the code uses phase_deg(gain_crossover_index) which is -180, so phase margin is 0 degrees.

❓ visualization

advanced

Identifying Stability from Nyquist Plot Visualization

In a Simulink Nyquist plot visualization, the plot encircles the point (-1,0) twice clockwise. What does this indicate about the system's poles?

A The system has no poles in the right half-plane and is stable.

B The system has two poles in the right half-plane and is unstable.

C The system has two zeros in the right half-plane.

D The system is marginally stable with poles on the imaginary axis.

💡 Hint

Number of clockwise encirclements equals number of right half-plane poles.

Two clockwise encirclements of (-1,0) mean two poles in the right half-plane, indicating instability.

🔧 Debug

expert

Debugging Nyquist Plot Data Extraction Code

Consider this Simulink MATLAB script snippet to compute gain margin from Nyquist plot data. Identify the error causing incorrect gain margin calculation.

Simulink

frequency = [0.1, 1, 10];
real_part = [-0.8, -1, 0.2];
imag_part = [0.1, 0, -0.3];

phase_cross_index = find(imag_part == 0);
gain_margin = abs(real_part(phase_cross_index));

A Imaginary part comparison should use '==' operator with tolerance for floating point errors.

B The variable 'phase_cross_index' is not defined correctly.

C The frequency vector is not used and should be removed.

D Gain margin should be the reciprocal of the absolute real part, not the absolute real part itself.

💡 Hint

Recall gain margin formula involves reciprocal of magnitude at phase crossover.

The code calculates gain margin as abs(real_part), but gain margin is 1 / abs(real_part) at phase crossover frequency.

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