Chapter 8 Root Locus Techniques

BMFA3453 Advance Control System Chapter 8: Root Locus Techniques

Overview ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Muhamad Arfauz Bin A Rahman [email_address] Robotic & Automation, FKP UTeM

Objectives ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Muhamad Arfauz Bin A Rahman [email_address] Robotic & Automation, FKP UTeM

Introduction ,[object Object],[object Object],[object Object],[object Object],[object Object],Muhamad Arfauz Bin A Rahman [email_address] Robotic & Automation, FKP UTeM

The Control System Problem ,[object Object],[object Object],[object Object],[object Object],[object Object],Muhamad Arfauz Bin A Rahman [email_address] Robotic & Automation, FKP UTeM

[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Muhamad Arfauz Bin A Rahman [email_address] Robotic & Automation, FKP UTeM

Vector representation of complex number ,[object Object],[object Object],[object Object],Muhamad Arfauz Bin A Rahman [email_address] Robotic & Automation, FKP UTeM

[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Muhamad Arfauz Bin A Rahman [email_address] Robotic & Automation, FKP UTeM

Draw the vector representation of (s + 7)| s  5 + j2 ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Muhamad Arfauz Bin A Rahman [email_address] Robotic & Automation, FKP UTeM (s +z 1 )(s +z 2 )… (s + p 1 )(s + p 2 )… F(s) = ∏ (s + z i ) / ∏ (s + p j ) m n I =1 j =1

Summary ,[object Object],[object Object],Muhamad Arfauz Bin A Rahman [email_address] Robotic & Automation, FKP UTeM

Given F(s) = , Find F(s) at the point s = -3 + j4 ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Muhamad Arfauz Bin A Rahman [email_address] Robotic & Automation, FKP UTeM (s + 1) s(s + 2)

Defining the Root Locus ,[object Object],[object Object],[object Object],[object Object],[object Object],Muhamad Arfauz Bin A Rahman [email_address] Robotic & Automation, FKP UTeM

Muhamad Arfauz Bin A Rahman [email_address] Robotic & Automation, FKP UTeM

Observations: ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Muhamad Arfauz Bin A Rahman [email_address] Robotic & Automation, FKP UTeM

Properties of the Root Locus ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Muhamad Arfauz Bin A Rahman [email_address] Robotic & Automation, FKP UTeM

Given a unity feedback system that has a the following forward transfer function: ,[object Object],[object Object],[object Object],Muhamad Arfauz Bin A Rahman [email_address] Robotic & Automation, FKP UTeM

[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Muhamad Arfauz Bin A Rahman [email_address] Robotic & Automation, FKP UTeM j3 -j3 -3 -2 -1 θ 2 θ 1 S-Plane j ω σ

Sketching the Root Locus ,[object Object],Muhamad Arfauz Bin A Rahman [email_address] Robotic & Automation, FKP UTeM Number of Branches Starting & Ending Point Behavior at infinity Real Axis segment Symmetry RULES

No. of branches ,[object Object],Muhamad Arfauz Bin A Rahman [email_address] Robotic & Automation, FKP UTeM 1st 2nd

Symmetry ,[object Object],Muhamad Arfauz Bin A Rahman [email_address] Robotic & Automation, FKP UTeM Symmetrical about real axis

Real-axis segment ,[object Object],Muhamad Arfauz Bin A Rahman [email_address] Robotic & Automation, FKP UTeM To the left of an odd number

Starting & Ending Points ,[object Object],Muhamad Arfauz Bin A Rahman [email_address] Robotic & Automation, FKP UTeM Ending Starting

Concept of Infinite pole & zero ,[object Object],[object Object],[object Object],[object Object],[object Object],Muhamad Arfauz Bin A Rahman [email_address] Robotic & Automation, FKP UTeM

Behavior at infinity ,[object Object],[object Object],[object Object],Muhamad Arfauz Bin A Rahman [email_address] Robotic & Automation, FKP UTeM

Example: Sketch the root locus for the system shown ,[object Object],[object Object],[object Object],[object Object],Muhamad Arfauz Bin A Rahman [email_address] Robotic & Automation, FKP UTeM

Root locus and asymptotes for the system of previous example Muhamad Arfauz Bin A Rahman [email_address] Robotic & Automation, FKP UTeM Real axis intercept Π /3 5 Π /3

Muhamad Arfauz Bin A Rahman [email_address] Robotic & Automation, FKP UTeM Finding the j ω axis crossings Angle of departure & arrival from complex poles & zeros Real-Axis Breakaway & Break-in Points Refining the Sketch

Real-Axis Breakaway & Break-in Points ,[object Object],[object Object],[object Object],[object Object],Muhamad Arfauz Bin A Rahman [email_address] Robotic & Automation, FKP UTeM

Finding the Breakaway & Break-in points via differentiation ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Muhamad Arfauz Bin A Rahman [email_address] Robotic & Automation, FKP UTeM

Find the breakaway & break-in points for the root locus shown Muhamad Arfauz Bin A Rahman [email_address] Robotic & Automation, FKP UTeM

Finding Breakaway & Break-in Points by transition method ,[object Object],[object Object],[object Object],[object Object],[object Object],Muhamad Arfauz Bin A Rahman [email_address] Robotic & Automation, FKP UTeM

Finding the j ω a xis crossings ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Muhamad Arfauz Bin A Rahman [email_address] Robotic & Automation, FKP UTeM

For the system shown, find the frequency & gain, K, for which the root locus crosses the imaginary axis. For what range of K is the system stable? Muhamad Arfauz Bin A Rahman [email_address] Robotic & Automation, FKP UTeM

Construction of Routh table Muhamad Arfauz Bin A Rahman [email_address] Robotic & Automation, FKP UTeM

Continuation of Previous Problem Solving ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Muhamad Arfauz Bin A Rahman [email_address] Robotic & Automation, FKP UTeM

Angle of departure & arrival from complex poles & zeros ,[object Object],[object Object],[object Object],[object Object],[object Object],Muhamad Arfauz Bin A Rahman [email_address] Robotic & Automation, FKP UTeM

Angle of departure & arrival ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Muhamad Arfauz Bin A Rahman [email_address] Robotic & Automation, FKP UTeM

Example: Given the unity feedback system, find the angle of departure from the complex poles & sketch the root locus Muhamad Arfauz Bin A Rahman [email_address] Robotic & Automation, FKP UTeM Root locus for the system showing angle of departure

Continuation of Previous Problem Solving Muhamad Arfauz Bin A Rahman [email_address] Robotic & Automation, FKP UTeM

Skill assessment Exercise 8.4: Given a unity feedback system that has the forward transfer function ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Muhamad Arfauz Bin A Rahman [email_address] Robotic & Automation, FKP UTeM

An Example ,[object Object],[object Object],[object Object],[object Object],[object Object],Muhamad Arfauz Bin A Rahman [email_address] Robotic & Automation, FKP UTeM

Transient Response Design via Gain Adjustment ,[object Object],[object Object],[object Object],[object Object],Muhamad Arfauz Bin A Rahman [email_address] Robotic & Automation, FKP UTeM

Please Read!!! ,[object Object],[object Object],Muhamad Arfauz Bin A Rahman [email_address] Robotic & Automation, FKP UTeM

Pole Sensitivity ,[object Object],[object Object],[object Object],[object Object],[object Object],Muhamad Arfauz Bin A Rahman [email_address] Robotic & Automation, FKP UTeM

Example: Root Sensitivity of a closed loop system to gain variations ,[object Object],[object Object],Muhamad Arfauz Bin A Rahman [email_address] Robotic & Automation, FKP UTeM

Chapter 8 Root Locus Techniques

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Chapter 8 Root Locus Techniques