Nyquist plot analysis in Simulink - Deep Dive

Overview - Nyquist plot analysis

What is it?

Nyquist plot analysis is a way to check if a system is stable by looking at a special graph. This graph shows how the system reacts to different frequencies, using a curve on a complex plane. It helps engineers see if the system will behave well or become unstable. The plot is made by plotting the system's response as frequency changes from low to high.

Why it matters

Without Nyquist plot analysis, engineers might not know if a system will suddenly fail or oscillate wildly. This could cause machines or processes to break down, leading to costly repairs or unsafe conditions. Using this analysis helps design systems that work reliably and safely in real life, like in cars, airplanes, or factories.

Where it fits

Before learning Nyquist plot analysis, you should understand basic control systems and frequency response concepts like Bode plots. After mastering Nyquist plots, you can explore advanced stability criteria, controller design, and robust control methods. It fits in the journey of learning how to design and analyze systems that react to signals over time.

Mental Model

Core Idea

Nyquist plot analysis shows system stability by mapping how the system's output changes with frequency on a complex plane and checking if the plot encircles a critical point.

Think of it like...

Imagine walking around a lake while holding a compass. The path you take around the lake tells you if you have circled a special landmark. Similarly, the Nyquist plot traces a path around a point that tells if the system is stable or not.

Frequency (ω) increases →

Complex Plane:

          Imaginary Axis (jω)
               ↑
               │          ● Critical Point (-1,0)
               │         ╱
               │        ╱
               │       ╱  Nyquist Curve
               │      ╱
──────────────┼────────────→ Real Axis
               │
               │
               │

Build-Up - 7 Steps

1

Foundation

Understanding frequency response basics

Concept: Learn how systems respond to different frequencies and what frequency response means.

A system's frequency response shows how it changes input signals of different frequencies. For example, a speaker might play low sounds louder than high sounds. We measure this by sending sine waves at various frequencies and seeing the output amplitude and phase shift.

Result

You can describe a system's behavior using amplitude and phase for each frequency.

Understanding frequency response is key because Nyquist plots are built from these responses, showing how the system behaves across all frequencies.

2

Foundation

Complex plane and system transfer function

Concept: Introduce the complex plane and how system transfer functions map frequency to complex numbers.

The complex plane has a horizontal axis (real) and vertical axis (imaginary). A system's transfer function at a frequency gives a complex number showing gain and phase. Plotting these numbers for all frequencies creates the Nyquist plot.

Result

You can represent system behavior as points on the complex plane for each frequency.

Seeing system response as complex numbers helps visualize stability and gain/phase relationships clearly.

3

Intermediate

Constructing the Nyquist plot step-by-step

🤔Before reading on: do you think the Nyquist plot includes frequencies from zero to infinity or only a few select frequencies? Commit to your answer.

Concept: Learn how to plot the system's transfer function over a full range of frequencies to form the Nyquist curve.

Start at frequency zero, calculate the transfer function's complex value, plot it on the complex plane. Increase frequency gradually to very high values, plotting each point. Connect points smoothly to see the full curve. This curve shows how the system's gain and phase change with frequency.

Result

A continuous curve on the complex plane representing system response from low to high frequencies.

Plotting the full frequency range reveals how the system behaves overall, not just at isolated points.

4

Intermediate

Interpreting encirclements and stability

🤔Before reading on: do you think encircling the point (-1,0) on the Nyquist plot means the system is stable or unstable? Commit to your answer.

Concept: Understand how the number of times the Nyquist plot circles the point (-1,0) relates to system stability.

The Nyquist stability criterion says: count how many times the plot encircles (-1,0) clockwise. If the number matches the number of unstable poles in the system, the system is stable. If not, it is unstable. This helps predict if feedback will cause oscillations or failure.

Result

You can decide system stability by counting encirclements around (-1,0).

This rule connects the plot's shape directly to system behavior, making stability checks visual and intuitive.

5

Intermediate

Using Simulink for Nyquist plots

Concept: Learn how to generate Nyquist plots using Simulink tools for real systems.

In Simulink, use the Control System Toolbox blocks to model your system. Then use the 'nyquist' function or plot blocks to generate the Nyquist plot automatically. This saves time and reduces errors compared to manual plotting.

Result

A Nyquist plot generated from your Simulink model, ready for analysis.

Using Simulink integrates modeling and analysis, making it easier to test real systems and iterate designs.

6

Advanced

Handling systems with poles on imaginary axis

🤔Before reading on: do you think Nyquist plots can be directly used if the system has poles exactly on the imaginary axis? Commit to your answer.

Concept: Learn how to adjust Nyquist analysis when the system has poles on the imaginary axis, which complicates stability checks.

Systems with poles on the imaginary axis cause the Nyquist plot to pass through infinity, making direct plotting impossible. To handle this, use a small semicircular detour around the pole in the complex plane or apply modified Nyquist criteria. This ensures the plot remains valid for stability analysis.

Result

A corrected Nyquist plot that accounts for imaginary axis poles, allowing proper stability assessment.

Knowing how to handle these special cases prevents wrong conclusions about system stability in critical designs.

7

Expert

Nyquist plot in robust control design

🤔Before reading on: do you think Nyquist plots are only useful for simple systems or also for complex, uncertain systems? Commit to your answer.

Concept: Explore how Nyquist plots help design controllers that keep systems stable even with uncertainties and disturbances.

In robust control, Nyquist plots are used to check gain and phase margins, which measure how much uncertainty a system can handle before becoming unstable. Engineers use these margins to design controllers that maintain stability despite changes in system parameters or external noise.

Result

Controllers designed with Nyquist analysis that ensure stability under real-world uncertainties.

Applying Nyquist plots beyond simple checks enables building reliable systems that work safely in unpredictable environments.

Under the Hood

Nyquist plot analysis is based on the Nyquist stability criterion, which uses complex analysis and contour integration. The system's open-loop transfer function is evaluated along a contour that encircles the right half of the complex plane. The plot maps frequency response points, and the number of encirclements of the critical point (-1,0) relates to the number of unstable poles in the closed-loop system. This connection comes from the argument principle in complex analysis.

Why designed this way?

Nyquist plots were developed to provide a graphical method to assess stability without solving complex equations. Before computers, engineers needed visual tools to understand system behavior. The method balances mathematical rigor with practical usability, allowing quick stability checks. Alternatives like root locus or Bode plots exist but Nyquist uniquely combines gain and phase information in one plot.

Open-loop Transfer Function G(s)H(s) → Frequency sweep along jω axis → Calculate G(jω)H(jω) complex values → Plot points on complex plane → Count encirclements of (-1,0) → Determine closed-loop stability

┌───────────────┐
│ Frequency ω   │
└──────┬────────┘
       │
       ▼
┌─────────────────────────┐
│ Compute G(jω)H(jω)       │
└──────┬──────────────────┘
       │
       ▼
┌─────────────────────────┐
│ Plot on Complex Plane    │
│ (Nyquist Plot)           │
└──────┬──────────────────┘
       │
       ▼
┌─────────────────────────┐
│ Count Encirclements of   │
│ (-1,0)                  │
└──────┬──────────────────┘
       │
       ▼
┌─────────────────────────┐
│ Stability Decision       │
└─────────────────────────┘

Myth Busters - 4 Common Misconceptions

Quick: Does a Nyquist plot encircling (-1,0) always mean the system is unstable? Commit to yes or no.

Common Belief: If the Nyquist plot encircles the point (-1,0), the system is always unstable.

Reality: Encirclement must be counted with respect to the number of unstable poles in the open-loop system. The system is stable if the number of clockwise encirclements equals the number of open-loop unstable poles.

Why it matters: Misinterpreting encirclements can lead to wrongly declaring a stable system unstable or vice versa, causing poor design decisions.

Quick: Can Nyquist plots be used directly if the system has poles on the imaginary axis? Commit to yes or no.

Common Belief: Nyquist plots can be used directly for any system without adjustments.

Reality: Systems with poles on the imaginary axis require special handling, like detours in the contour, to avoid infinite values and incorrect stability conclusions.

Why it matters: Ignoring this leads to invalid plots and wrong stability assessments, risking system failure.

Quick: Does the Nyquist plot show time-domain behavior directly? Commit to yes or no.

Common Belief: Nyquist plots directly show how the system behaves over time.

Reality: Nyquist plots show frequency-domain behavior, not time-domain responses. They help infer stability but do not show exact time responses like step or impulse plots.

Why it matters:

Quick: Is the Nyquist plot the only way to check stability? Commit to yes or no.

Common Belief: Nyquist plot is the only method to check system stability.

Reality: Other methods like root locus and Bode plots also check stability, each with different strengths and use cases.

Why it matters: Relying solely on Nyquist plots limits flexibility and understanding of system behavior.

Expert Zone

1

The direction of frequency sweep (low to high or high to low) affects the Nyquist plot orientation and must be consistent for correct encirclement counting.

2

Phase wrapping in Nyquist plots can cause confusing loops; understanding how phase shifts beyond ±180° affect the plot is crucial for accurate interpretation.

3

In multi-input multi-output (MIMO) systems, Nyquist plots extend to Nyquist arrays, requiring matrix analysis rather than simple curves.

When NOT to use

Nyquist plot analysis is less effective for nonlinear or time-varying systems where frequency response is not well-defined. In such cases, time-domain simulations or Lyapunov methods are better alternatives.

Production Patterns

Engineers use Nyquist plots in feedback controller design to ensure gain and phase margins meet safety requirements. They combine Nyquist analysis with automated tools in Simulink for iterative tuning and robustness checks before deployment.

Connections

Bode plot

Builds-on

Both Nyquist and Bode plots use frequency response but Nyquist combines gain and phase in one complex plot, helping understand stability more directly.

Complex analysis (Argument Principle)

Same pattern

Nyquist stability criterion is a practical application of the argument principle from complex analysis, linking math theory to engineering practice.

Navigation and path encirclement

Analogous pattern

Counting encirclements in Nyquist plots is similar to how sailors use compass bearings to determine if they have circled an island, showing how abstract math connects to real-world navigation.

Common Pitfalls

#1 Miscounting encirclements around (-1,0) leading to wrong stability conclusion.

Wrong approach:

Assuming any encirclement means instability without considering open-loop poles.

// Incorrect conclusion
if (nyquistPlot.encirclements > 0) {
system.stable = false;
}

Correct approach:

Count encirclements and compare with number of open-loop unstable poles.

// Correct stability check
if (nyquistPlot.encirclements == openLoopUnstablePoles) {
system.stable = true;
} else {
system.stable = false;
}

Root cause:

Not understanding the Nyquist stability criterion fully, especially the role of open-loop poles.

#2 Plotting Nyquist plot without handling poles on imaginary axis, causing infinite values.

Wrong approach:

// Direct plot without detour
for (ω in frequencies) {
plotPoint = G(j*ω);
plot(plotPoint);
}

Correct approach:

// Modify contour to avoid poles
for (ω in frequencies) {
if (ω near pole) {
use semicircular detour;
} else {
plotPoint = G(j*ω);
plot(plotPoint);
}
}

Root cause:

Ignoring special cases in system poles leads to invalid plots.

#3 Confusing Nyquist plot with time-domain response and expecting direct time behavior.

Wrong approach:

// Using Nyquist plot to predict exact time response
systemResponse = nyquistPlot;
plot(systemResponse.time);

Correct approach:

// Use step or impulse response for time-domain behavior
systemResponse = step(system);
plot(systemResponse.time);

Root cause:

Mixing frequency-domain and time-domain concepts.

Key Takeaways

Nyquist plot analysis is a powerful graphical tool to assess system stability by mapping frequency response on the complex plane.

The key to stability is counting how many times the Nyquist plot encircles the critical point (-1,0) relative to unstable poles in the system.

Handling special cases like poles on the imaginary axis is essential for accurate Nyquist analysis.

Using tools like Simulink simplifies generating Nyquist plots and applying stability criteria in real-world systems.

Nyquist plots connect deep mathematical principles with practical engineering, enabling robust and reliable system design.

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