Stability analysis using Simulink

Introduction

Stability analysis helps us check if a system will behave well over time or if it will go out of control. Using Simulink makes this easy by letting us build and test system models visually.

When designing a control system for a robot to ensure it moves smoothly without wobbling.

When testing an electrical circuit model to see if voltages stay steady.

When simulating a car suspension system to check if it returns to normal after bumps.

When verifying if a temperature control system keeps the temperature stable.

When learning how different system settings affect overall system behavior.

Syntax

Simulink

1. Open Simulink and create a new model.
2. Add system blocks (like Transfer Function, Gain, Sum).
3. Connect blocks to build your system.
4. Use the 'Linear Analysis' tool from the Analysis menu.
5. Select points in your model to analyze stability (e.g., input/output).
6. Run the analysis to get stability plots like Bode or Root Locus.
7. Interpret plots to decide if the system is stable.

You do not write code line-by-line; instead, you build models with blocks.

The Linear Analysis tool helps find poles and zeros which tell about stability.

Examples

This example shows a basic system with feedback to test if it stays stable.

Simulink

Create a simple feedback system:
- Add a Transfer Function block with numerator [1] and denominator [1 2 1].
- Add a Feedback block.
- Connect output back to input through feedback.
- Use Linear Analysis to check stability.

Bode plots help visualize frequency response and stability margins.

Simulink

Use Bode plot analysis:
- Select input and output points.
- Run Bode plot in Linear Analysis.
- Check gain and phase margins to assess stability.

Sample Program

This code builds a simple feedback system in Simulink programmatically, linearizes it, and finds the poles to check stability. Poles with negative real parts mean the system is stable.

Simulink

% Open Simulink and create a new model
model = 'stability_example';
new_system(model);
open_system(model);

% Add Transfer Function block
add_block('simulink/Commonly Used Blocks/Transfer Fcn', [model '/Plant']);
set_param([model '/Plant'], 'Numerator', '[1]', 'Denominator', '[1 2 1]');

% Add Feedback block
add_block('simulink/Commonly Used Blocks/Sum', [model '/Sum']);
set_param([model '/Sum'], 'Inputs', '+-');

% Add Gain block for feedback
add_block('simulink/Commonly Used Blocks/Gain', [model '/Gain']);
set_param([model '/Gain'], 'Gain', '1');

% Connect blocks
add_line(model, 'Sum/1', 'Plant/1');
add_line(model, 'Plant/1', 'Gain/1');
add_line(model, 'Gain/1', 'Sum/2');

% Open Linear Analysis Tool
linearAnalysis = linearize(model);

% Get poles of the linearized system
poles = pole(linearAnalysis);

% Display poles
disp('Poles of the system:');
disp(poles);

% Close the model without saving
close_system(model, 0);

Important Notes

Stable systems have poles with negative real parts (left side of the complex plane).

Simulink's Linear Analysis tool helps visualize and calculate stability without manual math.

Always test your model with different parameters to see how stability changes.

Summary

Stability analysis checks if a system behaves well over time.

Simulink lets you build system models visually and analyze stability easily.

Use poles and plots from Linear Analysis to decide if your system is stable.

Practice

1. What does stability analysis in Simulink primarily check for in a system?

easy

A. Whether the system output settles to a steady value over time

B. The color scheme of the Simulink blocks

C. The speed of the computer running Simulink

D. The number of blocks used in the model

Solution

1. Step 1: Understand stability concept

   Stability means the system output does not grow uncontrollably but settles to a steady value.

2. Step 2: Relate to Simulink analysis

   Simulink stability analysis checks if outputs stabilize after simulation with inputs like Step.

3. Final Answer:

   Whether the system output settles to a steady value over time -> Option A

4. Quick Check:

   Stability = output settles [OK]

Hint: Stability means output settles, not grows [OK]

Common Mistakes:

• Confusing stability with simulation speed
• Thinking stability depends on block colors
• Assuming stability is about model size

2. Which Simulink block is commonly used to provide a sudden change input for stability testing?

easy

A. Integrator

B. Gain

C. Scope

D. Step

Solution

1. Step 1: Identify input blocks for testing

   The Step block provides a sudden change from one value to another, ideal for testing system response.

2. Step 2: Confirm block purpose

   Integrator accumulates signals, Gain multiplies, Scope displays output; only Step creates sudden input change.

3. Final Answer:

   Step -> Option D

4. Quick Check:

   Step block = sudden input [OK]

Hint: Step block creates sudden input change [OK]

Common Mistakes:

• Using Scope as input block
• Confusing Integrator with input source
• Thinking Gain creates input changes

3. Given a Simulink model with a Step input and a Scope showing output, what does an output graph that grows without bound indicate?

medium

A. The simulation stopped early

B. The system is stable

C. The system is unstable

D. The Step input is not working

Solution

1. Step 1: Analyze output behavior

   If the output graph grows without bound, it means the system output is increasing endlessly over time.

2. Step 2: Interpret stability meaning

   A stable system output settles or stays bounded; unbounded growth means instability.

3. Final Answer:

   The system is unstable -> Option C

4. Quick Check:

   Unbounded output = unstable system [OK]

Hint: Output grows endlessly means unstable system [OK]

Common Mistakes:

• Assuming growing output means stable
• Thinking simulation error caused growth
• Confusing input failure with output behavior

4. You run a Simulink simulation for stability but the Scope shows a flat line at zero despite a Step input. What is the likely error?

medium

A. The Step block is not connected to the system input

B. The Scope block is missing

C. The simulation time is too long

D. The system is perfectly stable

Solution

1. Step 1: Check signal connections

   If the Scope shows zero output despite Step input, the input may not be connected to the system properly.

2. Step 2: Rule out other causes

   Scope presence is confirmed; simulation time too long would show output; perfect stability would not cause zero output if input changes.

3. Final Answer:

   The Step block is not connected to the system input -> Option A

4. Quick Check:

   No input connection = zero output [OK]

Hint: Check if Step input is connected to system [OK]

Common Mistakes:

• Assuming zero output means perfect stability
• Ignoring block connections
• Thinking simulation time affects output zero

5. You want to analyze the stability of a feedback control system in Simulink. Which combination of blocks and steps will best help you determine if the system is stable?

hard

A. Use only a Gain block and check its value without simulation

B. Use a Step input block, connect it to the system input, add a Scope to observe output, and simulate to see if output settles

C. Add multiple Scope blocks without any input and run simulation

D. Use a Constant block as input and ignore output behavior

Solution

1. Step 1: Set up input and output blocks

   Use a Step block to provide a sudden input change and a Scope block to observe system output over time.

2. Step 2: Run simulation and analyze output

   Simulate the model and check if the output settles to a steady value, indicating stability.

3. Step 3: Confirm feedback system setup

   Ensure feedback connections are correct to reflect real system behavior during simulation.

4. Final Answer:

   Use a Step input block, connect it to the system input, add a Scope to observe output, and simulate to see if output settles -> Option B

5. Quick Check:

   Step + Scope + simulate = check stability [OK]

Hint: Step input + Scope output + simulate to check stability [OK]

Common Mistakes:

• Ignoring input type and output observation
• Not simulating the model
• Using constant input without checking output