Root locus analysis in Simulink - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is the purpose of root locus analysis in control systems?

Root locus analysis helps us see how the roots of a system's characteristic equation change as a parameter (usually gain) changes. This shows system stability and response behavior.

beginner

In root locus plots, what do the 'branches' represent?

Branches show the paths that the system poles follow on the complex plane as the gain changes from zero to infinity.

intermediate

How does root locus analysis help in designing a controller in Simulink?

It helps choose the right gain to place poles in locations that make the system stable and meet performance goals like speed and overshoot.

intermediate

What does it mean if a root locus branch crosses into the right half of the complex plane?

It means the system becomes unstable because poles in the right half-plane cause outputs to grow without bound.

beginner

Which Simulink tool can you use to plot the root locus of a system?

You can use the Control System Toolbox in MATLAB, specifically the 'rlocus' function or the Root Locus Editor block in Simulink.

What does the root locus plot show as gain increases?

A Step response overshoot

B System output over time

C Frequency response magnitude

D Paths of system poles on the complex plane

Root locus plots the movement of system poles as gain changes.

If poles move to the right half-plane in root locus, what happens?

A System becomes unstable

B System becomes faster

C System output is zero

D System is overdamped

Poles in the right half-plane cause instability.

Which parameter is usually varied in root locus analysis?

A Input signal frequency

B Gain

C Time delay

D Sampling time

Gain is the parameter varied to see pole movement.

What is the main goal of using root locus in controller design?

A To calculate system bandwidth

B To measure noise in the system

C To place poles for desired stability and performance

D To find the system's steady-state error

Root locus helps select gain to place poles properly.

Which Simulink toolbox provides root locus plotting functions?

A Control System Toolbox

B Signal Processing Toolbox

C Image Processing Toolbox

D Simscape Toolbox

Control System Toolbox includes root locus tools.

Explain what root locus analysis is and why it is useful in control system design.

Think about how changing gain affects system poles and stability.

Describe how you would use Simulink to perform root locus analysis on a system.

Consider the tools and steps inside Simulink.

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