Properties of Root Locus

Properties of Root Locus Lesson Objectives : After finish this lesson, students will be able to be aware of the control system problem be aware of the vector representation of complex number define the root locus determine the properties of root locusNPIC Faculty of ElectricityControl Engineering

1Lecturer : IN SOKVANLesson Objectives 01Aware of Control System Problem 2Lecturer : IN SOKVANThe Prerequisites To Root Locus TechniqueUnderstanding the control system problemsVector representation of complex numbers3Lecturer : IN SOKVANThe Control System Problem

General Feedback Control SystemOpen Loop Transfer FunctionAlso called, Loop Gain

Closed-loop Transfer Function

4Lecturer : IN SOKVANThe Control System ProblemWhat is the control system problem?Difficult to obtain the polesPoles location varied with the gain

Lets find the closed-loop transfer function

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OL poles are roots of: Forward transfer function denominatorFeedback transfer function denominatorCL poles are roots of: Combinations of numerator and denominator of forward and feedback transfer functionsPoles depend on gain, KComparison Between Open Loop and Closed Loop SystemsTherefore, with CL system, the poles are not easily obtained and change with the value of KThe Control System Problem6Lecturer : IN SOKVAN

Given:

Open Loop SystemPoles: 0, -2 and -4

Closed-loop SystemPoles: We have to factor and also depends on K. Root locus technique help to find poles!The Control System Problem - Example7Lecturer : IN SOKVANAware of the Vector Representation of Complex Number 8Lecturer : IN SOKVANLesson Objectives 02

What is complex number?Complex number is a vectorVector has magnitude and direction

Therefore, we can also represent complex number s as:

Vector Representation of Complex Number9Lecturer : IN SOKVANBut, if s is a variable in a function, how to represent the complex number. For example,

Replacing s,

Another complex numberGraphically,

Vector Representation of Complex Number10Lecturer : IN SOKVAN

Vector Representation of Complex Number11Lecturer : IN SOKVAN

Same Vector is a complex number and can be represented by a vector drawn from the zero of the function to the point s.Vector Representation of Complex Number Same Vector

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Vector Representation of Complex Number General13Lecturer : IN SOKVAN

What is F(s)?

Vector Representation of Complex Number Example14Lecturer : IN SOKVANdefine the root locus 15Lecturer : IN SOKVANLesson Objectives 03What is Root Locus?The root locus is the path of the roots of the characteristic equation shown out in the s-plane as a system parameter is changed.16Defining Root Locus

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Defining Root Locus18Lecturer : IN SOKVAN

Root LocusDefining Root Locus19Lecturer : IN SOKVAN

Root LocusRepresentation of the paths of closed-loop poles as the gain is variedWhat can we learn from this graphic?0