Ahmed Abu Hajar, Ph.D. - Faculty Server Contact |...

Root LocusRoot Locus

Ahmed Abu‐Hajar, Ph.D.


The Characteristic EquationThe Characteristic Equation• The Transfer Function -

G(s)+ C(s)R(s)

• The locations of the poles affect the stability of the system

H(s)

• The locations of the poles affect the stability of the system. • Adjusting the location of the poles requires adjusting some

parameters‐ usually the gain of the controllerTh Ch t i ti E ti f th l d L S t• The Characteristic Equation of the closed Loop System

• Two conditions that full fill the closed loop polesp p– Angle Condition

– The Magnitude Conditiong


The Concept of Root LocusThe Concept of Root Locus

• Root Locus: it is a graph that illustrates the oot ocus: t s a g ap t at ust ates t elocation of the poles by adjusting some parameter. – Mostly it is the gain K.

Wh K i f 0 t– Where K varies form 0 to ∞– Example

– The root loci start at the Poles and at the zeros.


Examples of root locusExamples of root locus


Steps for Root LocusSteps for Root Locus1‐Write the characteristic

equation with the parameter k as follows:

Locate the poles and zeros of G(s)H(s) on the s‐plane (loci

-5 -4 -2

G(s)H(s) on the s plane (loci begin @ the poles and they end @ the zeros)

3‐ Locate the segments of the3 Locate the segments of the real axis that are root loci.– Lie on the left to the odd

number of poles. p


Steps for Root LocusSteps for Root Locus

• 4‐ Determine the4 Determine the number of separate branches = # of poles

-3+5j

5‐ Root loci is symmetric around the horizontal axis. Th b f l i

-5 -1

The number of loci asymptotes move to infinity is

-3+5j

y# finite poles ‐ # finite zeros



6‐ The loci proceed to6 The loci proceed to zero at infinity along asymptotes centeredasymptotes centered at σ x and an angle ϕx

wherewhere



7‐ Find the points that7 Find the points that cross the imaginary axis.



8‐ determine the8 determine the breaking point on the real axis– Set K = p(s)– Obtain dK/ds = 0– Determine the roots – Or determine the max

f ( ) h llof p(s) graphically.



9‐ determine the angle9 determine the angle of departure from the complex pole and thecomplex pole and the angle of arrival at the complex zero usingcomplex zero, using the phase criterion



10‐Determine k at a given s10 Determine kx at a given sx


Changing the shape of Root Locus by adding Poles and zeros.

Adding a pole to the system will pull root locus to the right


Changing the shape of Root Locus by adding Poles and zeros.

Adding a zero to the system will pull root locus to the left


Control System Design Using Root Locus

• The Specification of the system (Most


The Specification of the system (Most common)– Overshoot MP– Overshoot MP, – Steady State error (velocity error constant) Time rise– Time rise.



General steps


General steps – Determines the location of the desired poles of the feedback system from the specificationsthe feedback system from the specifications

– If the root locus will pass through the desired poles.poles.

– Determine the gain K @ the desired pole using this equation: q



General steps: Continue


General steps: Continue– If the root locus does not pass through the desired polespoles. • Re‐shape the root locus by adding compensator (Controller)


Configurations of CompensatorsConfigurations of CompensatorsG(s)

C(s)R(s)Typical Feedback System

H(s)

Series Configuration

G(s)

H(s)

C(s)R(s)Gc(s)

g

Parallel (Feedback) Configuration ( )

C(s)R(s)G1(s) G2(s)

Parallel (Feedback) Configuration

H(s)

Gc(s)


H(s)

Types of CompensatorsTypes of Compensators

• Lead CompensatorsLead Compensators• Lag Compensators

d C• Lead‐Lag Compensators • PID CompensatorsAdvantage:

May be implemented using RC Op‐AmpMay be implemented using RC Op Amp Circuits (Active Filters)


Ahmed Abu Hajar, Ph.D. - Faculty Server Contact |...

Documents

Transcript of Ahmed Abu Hajar, Ph.D. - Faculty Server Contact |...