Nyquist plot analysis in Simulink - Step-by-Step Execution

Concept Flow - Nyquist plot analysis

Start with Transfer Function

↓

Calculate Frequency Response

↓

Plot Real vs Imaginary

↓

Trace Nyquist Curve

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Analyze Encirclements of -1+0j

↓

Determine Stability

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End

The Nyquist plot analysis starts from a system's transfer function, computes its frequency response, plots the real and imaginary parts, traces the Nyquist curve, checks encirclements of the critical point -1+0j, and concludes system stability.

Execution Sample

Simulink

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

This code creates a transfer function and plots its Nyquist plot with a grid.

Execution Table

Step	Action	Frequency (rad/s)	Real Part	Imaginary Part	Plot Point	Encirclement Count
1	Start frequency sweep	0.01	0.9997	-0.0200	(0.9997, -0.0200)	0
2	Frequency increase	0.1	0.9703	-0.1960	(0.9703, -0.1960)	0
3	Frequency increase	1	0.0000	-0.5000	(0.0000, -0.5000)	0
4	Frequency increase	10	-0.0097	-0.0020	(-0.0097, -0.0020)	0
5	Frequency increase	100	-0.0001	-0.0000	(-0.0001, -0.0000)	0
6	Check encirclement of -1+0j	-	-	-	-	0 (No encirclement)
7	Determine stability	-	-	-	-	Stable (No encirclement)

💡 Frequency sweep completed; no encirclement of critical point -1+0j detected, system is stable.

Variable Tracker

Variable	Start	After Step 1	After Step 2	After Step 3	After Step 4	After Step 5	Final
Frequency (rad/s)	N/A	0.01	0.1	1	10	100	100
Real Part	N/A	0.9997	0.9703	0.0000	-0.0097	-0.0001	-0.0001
Imaginary Part	N/A	-0.0200	-0.1960	-0.5000	-0.0020	-0.0000	-0.0000
Encirclement Count	0	0	0	0	0	0	0

Key Moments - 3 Insights

Why does the Nyquist plot trace a curve instead of discrete points?

Because the frequency response is calculated continuously over a range of frequencies, the plot connects these points to form a curve, as shown in the execution_table steps 1 to 5.

What does it mean if the Nyquist plot encircles the point -1+0j?

Encirclement of -1+0j indicates potential instability. In the execution_table at step 6, no encirclement means the system is stable.

Why do the real and imaginary parts change with frequency?

The transfer function's response varies with frequency, changing both magnitude and phase, which affects real and imaginary parts as seen in the variable_tracker across steps.

Visual Quiz - 3 Questions

Look at the execution_table at step 3. What are the real and imaginary parts of the frequency response?

A Real: 0.9703, Imaginary: -0.1960

B Real: 0.0000, Imaginary: -0.5000

C Real: -0.0097, Imaginary: -0.0020

D Real: 0.9997, Imaginary: -0.0200

💡 Hint

Check the row labeled step 3 in the execution_table under Real Part and Imaginary Part columns.

At which step does the Nyquist plot check for encirclement of the critical point -1+0j?

A Step 6

B Step 2

C Step 4

D Step 1

💡 Hint

Look for the action mentioning encirclement in the execution_table.

If the frequency range was extended beyond 100 rad/s, how would the variable_tracker change?

A The real and imaginary parts would remain constant.

B The encirclement count would reset to zero.

C More columns would be added with higher frequency values and corresponding real and imaginary parts.

D The frequency values would decrease.

💡 Hint

Variable_tracker columns represent frequency steps; adding frequencies adds columns.

Concept Snapshot

Nyquist Plot Analysis:
- Input: Transfer function of system
- Compute frequency response over range
- Plot real vs imaginary parts as curve
- Check encirclements of -1+0j point
- Encirclements indicate stability or instability
- Used to assess control system stability visually

Full Transcript

Nyquist plot analysis starts by taking a system's transfer function. We calculate its frequency response at many frequencies. For each frequency, we find the real and imaginary parts of the response. We plot these points on a graph, connecting them to form the Nyquist curve. We then check if this curve encircles the critical point -1 plus 0j. If it does, the system may be unstable. If not, the system is stable. This method helps us visually understand system stability from frequency data.

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