CHAPTER 6 Root Locus - cse.zju.edu.cncse.zju.edu.cn/eclass/attachments/2013-05/01-1367831465-6865.pdf · Basic Concepts of Root Locus Method ROOT LOCUS Walter Richard Evans (January

Basic Concepts of Root Locus Method

Principles of Automatic Control

CHAPTER 6CHAPTER 6CHAPTER 6CHAPTER 6Root LocusRoot LocusRoot LocusRoot Locus

浙江大学控制科学与工程学系


Review (Chapter 1-5)Review (Chapter 1 5) Basic Concepts of Control System

Definitions, terms, block diagram, …… System modeling and representation

Dynamics of system----modeling y y gVarious models: differential equation; transfer function;

state space model; signal flow graphs…… ;Relationship between various models

Linearization of nonlinear system Control system characteristics

Solution of linear differential equations: time responseSolution of linear differential equations: time responseSolution of the state EquationRouth’s stability criterion and steady-state error

…………

2


Outline of Chapter 6p

Introduction Introduction Basic Concepts of Root Locus

MethodMethod Geometrical Properties

(Construction Rules) Generalized Root Locus Performance characteristics ………

3


ROOT LOCUS1. Introduction

For a system designer: two things are very important.1) The stability-----determined by the roots obtained from

i i i 1 G( ) ( ) 0 (S ithe characteristic equation 1+G(s)H(s)=0 (Solving the equation or applying Routh’s criterion to this equation).

2) Th d f t bilit i th t f h t2) The degree of stability-----i.e., the amount of overshoot, the settling time of the controlled variable (Specifications).

The graphical methods in this text bookthe root-locus method------in this chapterthe frequency response approach----- in the next chapter

4



The Root Locus Method:

– The Root Locus ConceptWhat is root loc s? Wh se it?–What is root locus? Why use it?

– Root Locus Sketching–How to plot root locus?–How to plot root locus?

– Performance characteristics–Where and how to use it?Where and how to use it?

5


ROOT LOCUSWalter Richard Evans (January 15, 1920 -July 10, 1999) was a noted American control theorist and the inventor of the root locus method in 1948. He was the recipient of the 1987 American pSociety of Mechanical Engineers Rufus Oldenburger Medal and the 1988 AACC Richard E. Bellman Control Heritage Award.g

He was born on January 15, 1920, and received his B.E. in Electrical Engineering from Washington Uni ersit in St Lo is in 1941 andWashington University in St. Louis in 1941 and his M.E. in Electrical Engineering from the University of California, Los Angeles in 1951.E k d i lEvans worked as an engineer at several companies, including General Electric, Rockwell International, and Ford Aeronautic Company.

6



Definition:

The root locus is a plot of the roots of the characteristic pequation of the closed-loop system as a function of one system parameter varies, such as the gain of the open-loop transfer function.

It is a method that determines how the poles move around the S-plane as we change one control parameter.

This plot was introduced by Evans in 1948 and has been p ydeveloped and used extensively in control engineering.

7



The difference between the root locus method and the Routh-Hurwitz method:

The Root Locus Method ----- tells us the position of the i f fpoles in the s-plane for each value of a control parameter.

The Routh-Hurwitz Method ---- could only tell us forThe Routh-Hurwitz Method ---- could only tell us for which values of the control parameter the poles would be to the left of a given vertical axis in the s-plane.g p

8



For a control system underlying: overall transfer function

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

)()(

)()(

DsN

RsCControl

ratio

f ( i d ) l d

H(s)-)()( sDsRratio

The poles of C(s)/R(s)(transient-response modes) are related to the Zeros((zzii)) and the Poles ((ppjj)) of the open-loop transfer function G(s)H(s) and also the gain----static loop sensitivityfunction G(s)H(s) and also the gain----static loop sensitivity.

)())(()())(()()( 21 mzszszsKsHsG

)()(1)(

)()(

HGsG

RsC

)())((

)()(21 npspsps )()(1)( sHsGsR

0)())(()())(()()(1 2121 zszszsKpspspssHsG

9

0)())(()())(()()(1 2121 mn zszszsKpspspssHsG



The advantage of the root locus method:

Th t f th h t i ti ti f th t b• The roots of the characteristic equation of the system can be obtained directly.

i i f• The characteristic of the system can be completely and accurately yielded.

• Design a system with relative ease. Such as using MATLAB.

10

Basic Concepts of Root LocusBasic Concepts of Root Locus

控制科学与工程学系


ROOT LOCUSPlotting roots of a characteristic equation

C( )G(s)

-

R(s) C(s)Example 6-1:

)( KsGThe open-loop transfer function:

)2()(

sssGp p

The closed-loop transfer function or control ratio:

22

2

2 22)2()()(

nn

n

ssKssK

KssK

sRsC

Ks 112,1

The roots of the characteristic equation are:

12

,


ROOT LOCUS Gain of the open-loop transfer function

A l i i i f i f KK 11

Plotting roots of a characteristic equationtransfer function

Analysis: , it is a function of K

when K=0---- the roots are s1=0 and s2= -2 (which also

Ks 112,1

are the poles of the open loop transfer function)

K=1---- the roots s1= s2= -1)2(

)(

KsG1 2

0<K<1---- the roots s1,2 are real and lie on the negative real axis of the S-plans between –2 and

)2( ss

eg ve e s o e S p s be wee d–1 and 0 to –1, respectively

K>1---- the roots are complex and are given by

111 22,1 Kjjs nn

K>1 the roots are complex and are given by

13


ROOT LOCUS)2(

)( KsG

Plotting roots of a characteristic equation )2(

)(ss

K=0---- s1=0 and s2= -2 (which also are the poles of the open loop transfer function)0<K<1---- the roots s1,2 are real and lie on the negative real axis of the S-plans

between –2 and –1 and 0 to –1, respectively, p yK=1---- the roots s1= s2= -1K>1---- the roots are complex and are given by 111 2

21 Kjjs nn ,

Im[s]

Sketch above roots now.

K=1K

ReNote: each point in the locus is closed-loop Re

-2 0-1

O l l (K 0)

root, so the locus is called root locus.

K

14

Open-loop poles (K=0)K


ROOT LOCUS )( KsG

’ i f i i f

Plotting roots of a characteristic equation)2(

)(ss

Let’s look the location of closed-loop roots in detail. The roots of the characteristic equation are determined for a number of values of K: such as

K s1 s2

0 -0+j0 -2.0-j01,2 1 1s j K 0.5 -0.293+j0 -1.707-j0

0.75 -0.5+j0 -1.5-j0

,

j j

1.0 -1.0+j0 -1.0-j0

2 -1.0+j1.0 -1.0-j1.0j j

3 -1.0+j1.414 -1.0-j1.414

50.0 -1.0+j7.0 -1.0-j7.0

15

j j

：：：


ROOT LOCUS )( KsG

Plotting roots of a characteristic equation)2(

)(ss

sG

j2.0jω

[s]The values of K at point on the

locus are underlined3.0

∞↑K

j1.02.0

K

Contain two branches(K as a parameter)

-2 0 σ-1

K=0K=0

0.50.5

0.750.75

1.0Lie all possible roots of the

(K as a parameter)

-j1.02.0

3.0

pcharacteristic equation for all the values of K from zero to infinity (K>=0)

-j2.03.0

K↓∞

infinity (K> 0)

Kthe roots the time response

16The root-locus plot



G(s)-


)2()( sKsG The open-loop transfer function:

2)(s

sG

The closed-loop transfer function:

;)(

)()()(

22

2

2 22

22

nn

nn

sss

sKssK

sRsC

When K>0, system stable.

KKKs 821

22

2,1 The roots of the characteristic equation are:

17



K 1Analysis: , it is a function of K

when K=0---- the roots are s1=s2=0 (which also are the poles of

KKKs 821

22

2,1

1 2 ( pthe open loop transfer function)

K=8---- the roots s1= s2= -4 2

)2()(ssKsG

K 8 the roots s1 s2 4

K>8---- the roots are real and lie on the negative real axis of the s-plans and s - s -2 respectively

s

of the s-plans, and s1 -, s2 -2 respectively

0<K<8---- the roots s1,2 are complex-conjugate poles and lie on the left half of the s plans

222,1 8

21

21 KKjKjs nn

lie on the left half of the s-plans

18

22


ROOT LOCUS KKKs 821

22

2,1 2

)2()(ssKsGopen

Plotting roots of a characteristic equation222,1s

K=0---- the roots are s1=s2=0 (which also are open-loop poles)K=8---- the roots s1= s2= -4K>8---- the roots are real and lie on the negative real axis of the s-plans, andK 8 the roots are real and lie on the negative real axis of the s plans, and

s1 -, s2 -2 respectively0<K<8---- the roots s1,2 are complex-conjugate poles and lie on the LHP

Im[s]

Note: each point in

Sketch above roots now.

R

KNote: each point in the locus is closed-loop root.

Re-2 0-4

2 Open-loop Poles(K=0)Open-loop Zero(K= )


ROOT LOCUSPlotting roots of a characteristic equation:

Im[s]K

Start: open-loop Poles(K=0)

Im[s]Re

-2 0-1 [ ]K

-2 0-1

Re-2 0-4

--

There are 2 branches locus (2 closed-loop roots). E d l i fi it i t Z (K )

20

( p ) End: open-loop or infinity point Zero(K=)



G(s)-


)(1)(T

sG If K=1 in Ex. 6-1, the parameter T is adjustable, the l f f i )(

)(Tss open-loop transfer function:

11)(

sCThe closed-loop transfer function:

11)()( 2 TssTsssR

The characteristic equation is: 012 Tss

01 2 Ts )( 2

TssGeqo41 221 TTs

21

01

1 2 s 1

)( 2 sGeqo222,1


ROOT LOCUS Plotting roots of a characteristic equation

h h i i i i 2 1TThe characteristic equation is: 012 Tss

01 Ts

1)( 2

TssGeqo

421

22

2,1 TTs

Im[s]

01

1 2 s1

)( 2 seqo

Sketching the roots locus: 2 branches ssT 12

[s]

T=2T

j T=0Open Poles are: +j, -j Open Zero is: 0

T=0---- the roots are s1=s2=±j; Zero:0Re

T=2

-1

T

- 0

T 0 the roots are s1 s2 ±j; Zero:0

T=2---- the roots s1= s2= -1

0<T<2 the roots s are complex-j T=0

So, not only K can be adjusted.

0<T<2---- the roots s1,2 are complex-conjugate poles and lie on the LHP

T>2---- the roots are real and s1 -,

22

s2 0 respectively



j

The complex-conjugate poles: dnn jj 21

jω

The root with a dampings1

djj 21ωnratio ζ lies on a line

making the angle η=cos-1 ζ

dn jj 1nη=cos-1ζ

σwith the negative real axis.

s

-ζωn=σ

s2

23


ROOT LOCUS

The analysis of the root locus with increase of K: KPlotting roots of a characteristic equation:Ex.6-1

The analysis of the root locus with increase of K: K

1) A decrease in the damping ratio ζ. This increases the overshoot of the time response. ζ Mp , ts

jω

jK2) An increase in the undamped natural frequency ωn.

The value of ωn is the distance from the origin to the complex root. ωnjωd

jωd δ

K

Kδ

co p e oo . ωn

3) An increase in the damped natural frequency ωd. The value ωd is the imaginary component of the complex root and is equal to the frequency of the transient

jωd β

jωd γ

Kβ

Kγ

root and is equal to the frequency of the transient response. ωd

4) No effect on the rate of decay σ: that is , it remains -a σηβ ηγ ηδ

K=0 K=0Kα

constant for all values of gain equal to or greater than Kα.

5) For K>= K , σ= -ζω is constant, the system can never

24

5) For K> Kα, σ ζωn is constant, the system can never become unstable (P204)


ROOT LOCUSQualitative analysis of the root locus

(1) A i dd d h i l d d(1) A zero is added to the simple second-order system:

)()(

assKsG

)()()(assbsKsG

jω

∞↑

Conclusion:The branches of the root locus have

)( ass )( ass

-a σ

K=0 K=0Kα

K

K↓

0The branches of the root locus have

been “pulled to the left”, or farther from the imaginary axis. For values of t ti l iti it t th K jω

↓∞

static loop sensitivity greater than Ka, the roots are farther to the left than for the original system.

K=0∞←K

jω

bTherefore, the transients will decay

faster, yielding a more stable system.-a

K=0 K=0Kα∞←K

K=∞σ

-b0

25



jω

(2) A pole is added to the simple second-order system:

)()( KsG

))(()( KsG K=0 K=0Kα

∞↑K

Conclusion:

)()(

ass ))(()(

csass -a σK↓∞

0

The branches of the root locus have been “pulled to the right”, or closer to the imaginary axis For values of static

jωK=∞the imaginary axis. For values of static

loop sensitivity greater than Ka, the roots are closer to the imaginary axis than for the original system 0 σ

K=0 K=0Kα∞←K K=0

the original system.Therefore, the transients will result in

a less stable system.

0 σ-c

K=∞

26



General conclusion:the addition of a Zero to a system has the general y geffect of pulling the root locus to the left, tending to make it a more stable and faster-responding system (shorter settling time ts)The addition of a Pole to a system has the effect of

lli th t l t th i ht t di t k itpulling the root locus to the right, tending to make it a less stable and slower-responding system

For more complex transfer functions root-locus see Fig. 7.5, P206.

27


ROOT LOCUSProcedure outline of plotting root locus

Step 1: Derive the open-loop transfer function G(s)H(s) of the systemStep 2: Factor the numerator and denominator of G(s)H(s) St 3 Pl t th d l f G( )H( ) i th lStep 3: Plot the zeros and poles of G(s)H(s) in the s-planeStep 4: Use the geometrical shortcuts summarized in Sec.7.8 to determine

the locus that describes the roots of closed-loop characteristicthe locus that describes the roots of closed loop characteristic equation

Step 5: Calibrate the locus in terms of K(or other parameter we care about).If K is predetermined, the location of the exact roots of 1+G(s)H(s)

are known. If the location of the roots is specified, K can be d t i ddetermined

Step 6: Once roots have been found, c(t) can be gotten.Step 7: Does the response meet the desired specification?

28

Step 7: Does the response meet the desired specification?Adjustable? Compensation?


ROOT LOCUSOpen-loop transfer functionThe open-loop transfer function is of the form:

w

hzsKK

)()()(

n

jm

hh

um

w

pssbsbssasasKsHsG 1

1

1

)(

)(

)()()()()()(

mj 1

Where: i i f f ( )ai and bj-----real or complex, may lie in the left-half plane (LHP) or

right-half plane (RHP)K-------defined as the static loop sensitivity, positive/negativeK defined as the static loop sensitivity, positive/negativezh ------- the zeros of G(s)H(s), z1=-a1,z2=-a2,…, zw=-aw

pj ------- the poles of G(s)H(s), p1=…=pm=0, pm+1=-b1,…,pn=-bu

29

pj p ( ) ( ) p1 pm pm+1 1 pn u


ROOT LOCUSPoles of Closed-loop transfer function

(1) Th l ti hi b t th l d f l d l t f(1) The relationship between the poles and zeros of closed-loop transfer function and the open-loop transfer function:

G(s)-

R(s) C(s)

H(s)

The closed-loop transfer function (Control Ratio):

)()( GY)()(

)()()()()(

sHsGsG

sRsYssGB

1

30



The forward transfer function G(s) and the feedback transfer function H(s):

f

jj

f

jj zss

11

11 )()1(

q

ii

m

jrq

ii

m

j

pssK

sTsKsG

11

11

11

11

)()1()(

ll

h

l

jj

rh

l

jj zs

KT

sKsH 1

2

21

2

2

)(

)(

)1(

)1()(

Where:K1 -------the gain of the forward paths, K1r -----the root locus gain of

i

ii

i pssT1

21

2 )()1(

the forward path K2 -------the gain of the feedback paths, K2r -----the root locus gain of

the feedback path

31

p



The open-loop transfer function G(s)H(s):

lfww

zszszss 1 )()()()(

h

i

q

im

jj

jj

rmn

im

jj

rmn

im

jj

pspss

zszsK

pss

zsK

sTs

sKsHsG

21

12

11

11

1

1

)()(

)()(

)(

)(

)(

)()()(

i

ii

ii

ii

i pp1

21

111

)()(

Where:

K=K1*K2------ the open-loop gain of the system

Kr=K1r*K2r ------ the root locus gain of the open-loop system

w=f+l ------ the zero number of open-loop system

n=m+q+h ------ the pole number of open-loop system

32


ROOT LOCUS

The closed loop transfer function:Poles of Closed-loop transfer function

h

j

f

ir pszsKsG

211 )()()(

The closed-loop transfer function:

w

jir

mn

ii

m

ji

zsKpsssHsG

sGs

11

11

)()()()(1

)()(

ji 11

1) The root locus gain of closed-loop system is equal to the root locus gain K1r of the forward path of the open-loop system. g 1r p p p y

2) The closed-loop zeros are composed of the zeros of forward path and the poles of feedback.p

3) The closed-loop poles are related to open-loop zeros-poles and the root locus gain Kr.

33


ROOT LOCUSPoles of Closed-loop transfer function(2)The characteristic equation of the system Δ(s):

0)()(1)( sHsGs )()()(

1)]()([)()()()( 11

sHsGKzszsKsHsG w

Then:

The root locus of closed-loop transfer function Φ(s):

1)]()([)()(

)()( 11

sHsGKpspss

sHsGu

m

p ( )The corresponding values of s that satisfies the above equation for

any value of K from zero to infinity

)21()()( hjj eFesHsG

34


ROOT LOCUS 1)]()([)()()()()()( 1

1

1

sHsGKpspsszszsKsHsGu

mw

Poles of Closed-loop transfer function

)()( 1 pp u

For K>0)21()()( hjesHsG

210180)21()()( hhG

1)()( sHsGMagnitude condition:

For K<0

,2,1,0180)21()()( hhsHsGAngle condition:

For K<0

1)()( sHsGMagnitude condition: )()(

,2,1,0360)()( hhsHsGAngle condition:

g

35


ROOT LOCUS

E l 6 4Poles of Closed-loop transfer function

Example 6-4:In the case of the following single loop system we have

(with K a variable parameter K>0)(with K a variable parameter, K>0)

G(s)r y

K ( )_ K

1)(0)(1)( sKGsKGs

The characteristic equation Δ(s):

1)(0)(1)( sKGsKGs

)21()(,1)( hsKGsKG

36


ROOT LOCUSPoles of Closed-loop transfer function: Conclusion

The only possible locations for the poles in the s-plane are the ones that verify the angle condition above. K is then obtained from the magnitude condition.

From s(angle)K(magnitude)g g

All that needs to be done then is to identify the locations i th l th t lid l ti f th l fin the s-plane that are valid locations for the poles for a given value of the parameter K.

From given K s location

37


ROOT LOCUSApplication of the magnitude and angle conditions

O th l t f f ti G( )H( ) h b d t i d dOnce the open-loop transfer function G(s)H(s) has been determined and put into the proper form, the poles and zeros of this function can be plotted in the s-plane. For example:

))()(()(

)2)(/1()/1()()(

321

21

222

21

pspspsszsK

ssTssTsKsHsG

nn

For the quadratic factor s2+2ωs+ ωn2 with the damping ratio <1:

jjp 21 dnn jjp 3,2 1Notes:1) The complex poles or zeros always occur in conjugate pairs.2) σ is the damping constant, ωd is the damping natural frequency of

oscillation

38



jω

p2=σ+jωd s-z1

s-p2s))()((

)()()(321

21

pspspsszsKsHsG

s

s-p1

s z1

s-p3There are 4 Poles and 2 Zeros

σz1]2 p0

p3=σ-jωd

p1

Notes:

Pole-zero diagram:

Notes:1) a multiple pole or zeros is indicated on the pole-zero diagram

by ×]q or ⊙]q q is the multiplicity of the pole or zero.y ]q ⊙]q ,q p y p2) all angles are considered positive, measured in the counter-

clockwise(CCW) sense

39



Pole-zero diagram:

s-p2s

jω For example:

s = -4+j4, p1= -1, then

s

s-p1

s-z1

s-p3

sp2=σ+jωd

431 orjps

z1]2 p0

j

σ

8.126)(54)3(

1

221

psps

p3=σ-jωd

40


ROOT LOCUS

Th t f th h t i ti ti 1+G( )H( ) 0 t ti f th

Application of the magnitude and angle conditionsThe roots of the characteristic equation 1+G(s)H(s)=0 must satisfy the two condition:

zszszsK1

21

21

mnm

w

pspspss

zszszsK

Magnitude condition:

)()()()()()( HG

03600180)12(

)()()()()()( 11

KforhKforh

pspssmzszssHsG mnw Angle condition:

ysensitivitloopzszszs

pspspssK

w

mnm

21

21

03600180)12(

)()()()(

KforhKforh

termsrdenominatoofanglestermsnumeratorofanglessHsG

41

0360 Kforh


ROOT LOCUSApplication of the magnitude and angle conditionsExample 6-5:

)25.01()()( 0 sKsHsG

( 4)( ) ( )( 1)( 2)( 5)

K sG s H ss s s

)2.01)(5.01)(1()()(

ssssHsG

0

( 1)( 2)( 5)2.5

s s sK K

Step1 The poles and zeros are plottedStep1. The poles and zeros are plotted

Step2. Φ’s are denominator angles and ψ’s are numerator angles the l’s are the

jωSearch point

ψ s are numerator angles, the l s are the lengths of the directed segments stemming from the denominator factors

Φ Φ1Φ2ψ1

l3 (l)1 l2l1

and the (l)’s are the lengths of the directed segments stemming from the numerator factors

σ-1-2-4-5

Φ3 1Φ2ψ1

42

numerator factors



0180)12( Kforh

Step3. Arbitrarily choose a search point. If the following angles condition is satisfy, this search point is a point on the locus.

03600180)12(

)()( 3211 KforhKforh

sHsG

Once the complete locus has been determined. The locus can be pcalibrated in terms of the root-locus gain for any root s1 as following:

321

)(llllK 4)(521 11131211 slslslsl1)(l

1) the appropriate sign must be given to the magnitude of K, compatible with the particular angle condition that is utilized to obtain the root locus

Notes:

the particular angle condition that is utilized to obtain the root locus.

2) Since complex roots must occur in conjugate pairs, the locus is symmetrical about the real axis.

43

symmetrical about the real axis.


ROOT LOCUS

)4( K

Application of the magnitude and angle conditions

K=∞

The root locus for this system:)5)(2)(1(

)4()()(

ssssKsHsG

jω

K

jω

K=-2.5

KaKb Ka Ka

Kb

K=0 K=0 K=0K=∞σ-1-2-4-5

K=0 K=-∞K=-∞ K=0 K=0K=-∞

σ-1-2-4-5

Kb K<0Notes:

K=∞For K<0, the three branches of the root locus lie in the left half plane for

Notes:

44K>0

pK<-2.5


ROOT LOCUS

)4( K

Application of the magnitude and angle conditions

)5)(2)(1()4()()(

ssssKsHsGThe root locus plotted by MATLAB

K 0K>0

45


The End

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CHAPTER 6 Root Locus - cse.zju.edu.cncse.zju.edu.cn/eclass/attachments/2013-05/01-1367831465-6865.pdf · Basic Concepts of Root Locus Method ROOT LOCUS Walter Richard Evans (January