Root locus analysis in Simulink - Mini Project: Build & Apply

Root locus analysis

📖 Scenario: You are working as a control engineer. You want to understand how changing a system parameter affects the stability of a control system. Root locus analysis helps you see how the system's poles move in the complex plane as you vary a gain.

🎯 Goal: Build a simple Simulink model to perform root locus analysis on a given transfer function. You will create the system, set a gain variable, generate the root locus plot, and display the result.

📋 What You'll Learn

Create a transfer function block with exact numerator and denominator coefficients

Define a gain variable to vary the system gain

Use the root locus function to analyze the system poles

Display the root locus plot as output

💡 Why This Matters

🌍 Real World

Root locus analysis is used by engineers to design and tune control systems in industries like automotive, aerospace, and manufacturing.

💼 Career

Understanding root locus helps control engineers ensure system stability and performance by selecting appropriate controller gains.

1

Create the transfer function

Create a transfer function in Simulink with numerator coefficients [1] and denominator coefficients [1, 3, 2]. Name the transfer function block sys.

Simulink

% Create transfer function sys with numerator [1] and denominator [1 3 2]
% Your code here

Hint

Use the tf function with numerator and denominator vectors.

2

Define the gain variable

Create a variable called K and set it to 1 to represent the system gain.

Simulink

sys = tf([1], [1 3 2]);
% Define gain variable K and set it to 1
% Your code here

Hint

Simply assign the value 1 to the variable K.

3

Generate the root locus plot

Use the rlocus function with the product of K and sys to generate the root locus plot.

Simulink

sys = tf([1], [1 3 2]);
K = 1;
% Generate root locus plot for K*sys
% Your code here

Hint

Call rlocus with K*sys as argument.

4

Display the root locus plot

Use title to add the title 'Root Locus Plot' to the plot and use grid on to show the grid.

Simulink

sys = tf([1], [1 3 2]);
K = 1;
rlocus(K*sys);
% Add title 'Root Locus Plot' and turn grid on
% Your code here

Hint

Use title and grid on commands after plotting.

Practice

1. What does the root locus plot primarily show in control systems analysis?

easy

A. The noise level in the system output

B. The frequency response of the system

C. How the system poles move as the gain changes

D. The time response of the system to a step input

Solution

1. Step 1: Understand root locus purpose

   The root locus plot shows how the locations of system poles change as the gain parameter varies.

2. Step 2: Differentiate from other plots

   Frequency response and time response plots show different system behaviors, not pole movement.

3. Final Answer:

   How the system poles move as the gain changes -> Option C

4. Quick Check:

   Root locus = pole movement with gain [OK]

Hint: Root locus tracks poles as gain varies [OK]

Common Mistakes:

• Confusing root locus with frequency response
• Thinking it shows output noise
• Mixing it up with time response plots

2. Which Simulink command is used to plot the root locus of a system with transfer function G = tf([1], [1, 3, 2])?

easy

A. rlocus(G)

B. step(G)

C. bode(G)

D. pzmap(G)

Solution

1. Step 1: Identify root locus command

   In Simulink and MATLAB, rlocus() is the function to plot root locus.

2. Step 2: Differentiate from other commands

   step() plots step response, bode() plots frequency response, and pzmap() plots poles and zeros but not root locus.

3. Final Answer:

   rlocus(G) -> Option A

4. Quick Check:

   Root locus plot = rlocus() [OK]

Hint: Use rlocus() to plot root locus in Simulink [OK]

Common Mistakes:

• Using step() instead of rlocus()
• Confusing bode() with root locus
• Using pzmap() which shows poles but no gain variation

3. Given the transfer function G = tf([2], [1, 4, 5]), what is the number of branches in the root locus plot?

medium

A. 1

B. 2

C. 3

D. 4

Solution

1. Step 1: Determine number of poles

   The denominator polynomial [1, 4, 5] is second order, so there are 2 poles.

2. Step 2: Relate poles to root locus branches

   Root locus branches equal the number of poles, so there are 2 branches.

3. Correction:

   The denominator polynomial [1, 4, 5] is of order 2, so there are 2 poles, but the number of branches in root locus equals the number of poles, which is 2, not 3.

4. Final Answer:

   2 -> Option B

5. Quick Check:

   Branches = number of poles = 2 [OK]

Hint: Count denominator order for branches [OK]

Common Mistakes:

• Counting zeros instead of poles
• Confusing numerator order with poles
• Assuming branches equal zeros

4. Identify the error in this Simulink command for root locus: rlocus(tf([1 2], [1 3]))

medium

A. No error, command is correct

B. The numerator vector is invalid

C. The transfer function is not proper

D. The denominator vector length is less than numerator

Solution

1. Step 1: Check transfer function order

   Numerator order is 1 (degree 1), denominator order is 1 (degree 1). Transfer function is proper if denominator order >= numerator order.

2. Step 2: Analyze properness

   Here numerator and denominator orders are equal, so transfer function is proper, not improper.

3. Step 3: Check command validity

   The command is syntactically correct and will run without error.

4. Final Answer:

   No error, command is correct -> Option A

5. Quick Check:

   Proper TF and syntax = no error [OK]

Hint: Check numerator and denominator orders for proper TF [OK]

Common Mistakes:

• Assuming numerator length must be less than denominator
• Thinking equal order is improper
• Confusing vector length with polynomial degree

5. You have a system with transfer function G = tf([1], [1, 2, 2, 1]). You want to design a controller gain K so that the system poles lie on the left half of the complex plane for stability. Using root locus analysis, which gain range should you choose?

hard

A. K = 0 only

B. Any positive K value

C. K < 0 only

D. K > 0 and less than the gain where poles cross imaginary axis

Solution

1. Step 1: Understand root locus stability

   Root locus shows pole movement as gain K changes. Poles in left half-plane mean stable system.

2. Step 2: Identify gain limits for stability

   Poles start stable at K=0 and move as K increases. Stability is maintained until poles cross imaginary axis at a critical gain.

3. Step 3: Choose gain range

   Gain K must be positive and less than the critical gain where poles cross imaginary axis to keep system stable.

4. Final Answer:

   K > 0 and less than the gain where poles cross imaginary axis -> Option D

5. Quick Check:

   Stable poles = gain below crossing point [OK]

Hint: Pick gain before poles cross imaginary axis [OK]

Common Mistakes:

• Assuming any positive gain is stable
• Choosing negative gain for stability
• Ignoring pole crossing points