Root locus analysis in Simulink

Introduction

Root locus analysis helps us see how a system's behavior changes when we adjust a control setting. It shows the paths of system stability points as we change a parameter.

When designing a controller to make a machine stable and responsive.

To check how changing a gain affects system stability in a feedback loop.

When tuning parameters to avoid system oscillations or slow responses.

To visualize how poles move in the complex plane as gain varies.

When learning how control systems respond to changes in settings.

Syntax

Simulink

rlocus(sys)
rlocus(sys, k)
rlocus(sys, k, plotoptions)

sys is the system model, usually a transfer function or state-space model.

k is an optional vector of gain values to plot the root locus for.

Examples

Plot root locus for a simple transfer function with numerator 1 and denominator s² + 3s + 2.

Simulink

sys = tf([1], [1, 3, 2]);
rlocus(sys)

Plot root locus for gain values from 0 to 10 in steps of 0.1.

Simulink

sys = tf([1], [1, 3, 2]);
k = 0:0.1:10;
rlocus(sys, k)

Sample Program

This code creates a transfer function model and plots its root locus. The plot shows how the system poles move as gain changes.

Simulink

sys = tf([1], [1, 3, 2]);
figure;
rlocus(sys);
title('Root Locus of the System');

Important Notes

Root locus plots help you understand system stability visually.

Poles on the left half of the complex plane mean the system is stable.

Use rlocus in Simulink by linking to control system blocks or using MATLAB commands.

Summary

Root locus shows how system poles move when gain changes.

It helps design stable and well-behaved control systems.

Simulink uses rlocus to plot these paths easily.

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