NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 3- Lecture 17

Root Locus Method Part 2

D Bi h kh Bh tt hDr. Bishakh Bhattacharya

Professor, Department of Mechanical Engineering

IIT Kanpur

Joint Initiative of IITs and IISc - Funded by MHRD

NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 3- Lecture 17

This Lecture Contains

Angle of the Asymptotes

Imaginary Axis Intercept

Departure and Arrival Angle

Numerical Simulation

Joint Initiative of IITs and IISc - Funded by MHRD

NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 3- Lecture 17

Angle of the AsymptotesAngle of the Asymptotes

• The root loci approaches the zeros at infinity along asymptotes. The real axis i t t d l f hi h i b th f ll i lintercept and angle of which are given by the following rules:

nnmZerosPoles

R

)1..(2,1,0,180)12(

,

0int

int

mimi

nnmm

R zp

NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 3- Lecture 17

JJωω intercept:intercept: Use 1+KG(jUse 1+KG(jωω)H(j)H(jωω) = 0) = 0JJωω intercept: intercept: Use 1+KG(jUse 1+KG(jωω)H(j)H(jωω) 0) 0

Find Find ωω and   and   K and for any Complex Pole Find AngleK and for any Complex Pole Find Angle

Another way: UseAnother way: Use Routh’sRouth’s Array find stability condition for a complete zero rowArray find stability condition for a complete zero rowAnother way: Use Another way: Use Routh sRouth s Array, find stability condition for a complete zero row,Array, find stability condition for a complete zero row,

Obtain, K, go back to the upper row, obtain Obtain, K, go back to the upper row, obtain ωω

Take points along the imaginary axis and check the phase conditionTake points along the imaginary axis and check the phase condition

Try the problem T=K(s+3)/(s(s+1)(s+2)(s+4)) (OLTF with unity feedback)Try the problem T=K(s+3)/(s(s+1)(s+2)(s+4)) (OLTF with unity feedback)22‐‐KK22 –– 65K +720=0 => K=9.6565K +720=0 => K=9.65

NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 3- Lecture 17

Angle Checking at a Test PointAngle Checking at a Test Point

NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 3- Lecture 17

Departure and Arrival AngleDeparture and Arrival Angle

• Angle of Departure from a complex Pole with ‘r’ multiplicity:

r..l)),l((r oo

liiidep,l 11360180

φψφ

• Angle of Arrival at a complex Pole with ‘r’ multiplicity:

li

r..l)),l((r ooiiarr,l 11360180 ψφψ

li

NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 3- Lecture 17

E lE l Example:Example:

K )()3)(2(

)()( sKFsss

KsHsG

)3)(2(1)(

ssssF

)3)(2( sss

320(1) 3 Loci Starting From (1) 3 Loci Starting From  3,2,0 321 PPP(2) All Roots Loci Terminate at (2) All Roots Loci Terminate at 

Open Loop Zeros Open Loop Zeros 

NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 3- Lecture 17

i)i()(F o 2101801232

1 K

..,,i,)i(sss)s(F o 2101801232

1)3)(2(

,,1)(

sss

KOrK

sF

( ) R L i Th R l A i( ) R L i Th R l A i(a) Root Loci on The Real Axis(a) Root Loci on The Real Axis

(i) Positive Real Axis (i) Positive Real Axis  osss 032 (ii)(ii)

(iii)(iii)

03218018020 ss,s,,:s o

03180218036032 s,s,s,,:s o

(iv)(iv) 0180318021805403 s,s,s,,:s o

NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 3- Lecture 17

Six basic rules of Root-Locus ConstructionSix basic rules of Root Locus Construction

1. ‘n’ branches of root locus starts at the open loop p ppoles and ‘m’ of them meet the zeroes of the same

2. Loci are on the real axis to the left of odd number of poles and zeroespoles and zeroes

3. For large s and K, n-m of the root loci are asymptotic. Get Asymptote angle and Real intercept by using the earlier equationsby using the earlier equations

4. Calculate the Angle of Departure from poles and arrival at zeroes by using the rule derived

5. Calculate the ‘jω’ crossing using Routh’s stability6. Calculate the break away and break-in points using

dK/d = 0dK/d 0

NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 3- Lecture 17

Consider another system with unity feedback.

1,210

1

Hpss

GH 2 1 pss

CLTKGH 101010

)ss(

)s(pssps)p(sCLT,

pssKGH

10210

210210222

2

12

112

1

)ss()s(p

)ss(

10221

102

21

Find the root locus for gain pFind the root locus for gain p11 and TF: (s+2)/(sand TF: (s+2)/(s22+2s+10)+2s+10)

NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 3- Lecture 17

Assignment: Find the position of the leftmost pole by using rootAssignment: Find the position of the leftmost pole by using rootAssignment: Find the position of the leftmost pole by using root Assignment: Find the position of the leftmost pole by using root locus corresponding to the Control Gains K= 7 and 40locus corresponding to the Control Gains K= 7 and 40

NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 3- Lecture 17

Special References for this lectureSpecial References for this lecture

Feedback Control of Dynamic Systems, Frankline, Powell and Emami, Pearson

Control Systems Engineering – Norman S Nise, John Wiley & Sons

Design of Feedback Control Systems – Stefani Shahian Savant Hostetter Design of Feedback Control Systems – Stefani, Shahian, Savant, Hostetter

Oxford

Joint Initiative of IITs and IISc ‐ Funded by MHRD 12