Nyquist plot analysis in Simulink - Time & Space Complexity

Time Complexity: Nyquist plot analysis

O(n)

Understanding Time Complexity

When analyzing Nyquist plots using MATLAB, we want to understand how the time to generate the plot changes as the system or input size grows.

We ask: How does the computation time grow when we increase the frequency points or system complexity?

Scenario Under Consideration

Analyze the time complexity of this MATLAB code snippet for Nyquist plot generation.

% MATLAB code snippet for Nyquist plot
sys = tf([1],[1 3 2]);
freqs = logspace(-1,2,1000);
[re,im,w] = nyquist(sys,freqs);
plot(squeeze(real(re)), squeeze(imag(re)));

This code computes the Nyquist plot by evaluating the system's frequency response at 1000 points and then plots the real and imaginary parts.

Identify Repeating Operations

Look for repeated calculations in the code.

• Primary operation: Calculating system response at each frequency point.
• How many times: Once for each frequency in the list (1000 times here).

How Execution Grows With Input

As the number of frequency points increases, the number of calculations grows proportionally.

Input Size (n)	Approx. Operations
10	10 calculations
100	100 calculations
1000	1000 calculations

Pattern observation: Doubling the frequency points roughly doubles the work needed.

Final Time Complexity

Time Complexity: O(n)

This means the time to compute the Nyquist plot grows linearly with the number of frequency points.

Common Mistake

[X] Wrong: "The time to generate the Nyquist plot stays the same no matter how many frequency points we use."

[OK] Correct: Each frequency point requires a separate calculation, so more points mean more work and longer time.

Interview Connect

Understanding how computation time grows with input size helps you explain performance in real projects, showing you can think about efficiency clearly.

Self-Check

"What if we increased the system order, making each frequency calculation more complex? How would the time complexity change?"

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