control systems - lehigh

1

Classical Control – Prof. Eugenio Schuster – Lehigh University 1

Control Systems

Lecture 8 Root Locus

Classical Control – Prof. Eugenio Schuster – Lehigh University 2

Root Locus Controller

)(sC-

+ )(sR )(sY)(sG)(sE )(sU

)(1)()(

)()()(1)()(

)()(

sKLsGsC

sHsGsCsGsC

sRsY

+=

+=

)(sH

Plant

Sensor

Writing the loop gain as KL(s) we are interested in tracking the closed-loop poles as �gain��K varies

⇒= )()( sKDsC

2

Classical Control – Prof. Eugenio Schuster – Lehigh University

Root Locus Characteristic Equation:

0)(1 =+ sKL

The roots (zeros) of the characteristic equation are the closed-loop poles of the feedback system!!!

The closed-loop poles are a function of the �gain��K

Writing the loop gain as

nnnn

mmmm

asasasbsbsbs

sasbsL

++++++++

==−

−−

−

11

1

11

1

)()()(

The closed loop poles are given indistinctly by the solution of:

KsLsKbsa

sasbKsKL 1)(,0)()(,0)()(1,0)(1 −==+=+=+

3


Root Locus

{ } { })()()(1 LKLsKL numdenrootszerosRL +=+=when K varies from 0 to ∞ (positive Root Locus) or

from 0 to -∞ (negative Root Locus)

!"

!#$

=∠

=⇔−=>

180)(

1)(1)(:0sL

KsL

KsLK

Phase condition

Magnitude condition

!"

!#$

=∠

−=⇔−=<

0)(

1)(1)(:0sL

KsL

KsLK

Phase condition

Magnitude condition

4

3


Root Locus by Characteristic Equation Solution

Example: K-

+ )(sR )(sY( )( )110

1++ ss

)(sE )(sU

)10(11)()(

2 KssK

sRsY

+++=

Closed-loop poles: 0)10(110)(1 2 =+++⇔=+ KsssL

24815.5 Ks −

±−=

28145.5

5.524815.5

10,1

−±−=

−=

−±−=

−−=

Kis

s

Ks

s

0481048104810

<−

=−

>−

=

KKKK

5


We need a systematic approach to plot the closed-loop poles as function of the gain K → ROOT LOCUS

Root Locus by Characteristic Equation Solution

6

4


Phase and Magnitude of a Transfer Function

011

1

011

1)(asasasbsbsbsbsG n

nn

mm

mm

++++++++

= −−

−−

The factors K, (s-zj) and (s-pk) are complex numbers:

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

21

21

n

m

pspspszszszsKsG

−−−−−−

=

K

pk

zj

i

ipkk

izjj

eKK

pkerps

mjerzs

φ

φ

φ

=

==−

==−

…

1,)(

1,)(

pn

pp

zm

zzK

ipn

ipip

izm

izizi

erererererereKsG

φφφ

φφφφ

21

21

21

21)( =

7


Phase and Magnitude of a Transfer Function

Now it is easy to give the phase and magnitude of the transfer function:

( )

( )

( ) ( )[ ]pnppzm

zzK

pn

pp

zm

zzK

pn

pp

zm

zzK

ipn

pp

zm

zz

ipn

pp

izm

zzi

ipn

ipip

izm

izizi

errrrrrK

errrerrreK

erererererereKsG

φφφφφφφ

φφφ

φφφφ

φφφ

φφφφ

+++−++++

+++

+++

=

=

=

2121

21

21

21

21

21

21

21

21

21

21)(

( ) ( )pnppzm

zzK

pn

pp

zm

zz

sG

rrrrrrKsG

φφφφφφφ +++−++++=∠

=

2121

21

21

)(

,)(

8

5


Root Locus by Phase Condition

Example: K-

+ )(sR )(sY( )( )845

12 ++++

sssss)(sE )(sU

( )( )

( )( )( )isissssssss

ssL

222251845

1)( 2

−+++++

=

++++

=

iso 31+−=

belongs to the locus?

9



43.10887.36

45

70.78

90

[ ] 18070.784587.3643.10890 −≈+++− iso 31+−=⇒ belongs to the locus!

Note: Check code rlocus_phasecondition.m 10

6



We need a systematic approach to plot the closed-loop poles as function of the gain K → ROOT LOCUS

iso 31+−=

11


Root Locus

Basic Properties: •  Number of branches = number of open-loop poles •  RL begins at open-loop poles •  RL ends at open-loop zeros or asymptotes •  RL symmetrical about Re-axis

{ } { })()()(1 LKLsKL numdenrootszerosRL +=+=

0)(0 =⇒= saK

0)()(1)(0)(1 =+⇔−=⇔=+ sKbsaK

sLsKL

!"#

>−∞→

=⇔=⇒∞=

)0(0)(

0)(mns

sbsLK

when K varies from 0 to ∞ (positive Root Locus) or from 0 to -∞ (negative Root Locus)

12

7


Root Locus

Rule 1: The n branches of the locus start at the poles of L(s) and m of these branches end on the zeros of L(s). n: order of the denominator of L(s) m: order of the numerator of L(s)

Rule 2: The locus is on the real axis to the left of and odd number of poles and zeros. In other words, an interval on the real axis belongs to the root locus if the total number of poles and zeros to the right is odd. This rule comes from the phase condition!!!

13


Root Locus

Rule 3: As K→∞, m of the closed-loop poles approach the open-loop zeros, and n-m of them approach n-m asymptotes with angles

1,,1,0,)12( −−=−

+= mnlmn

ll … πφ

and centered at

1,,1,0,11 −−=−

−=

−−

= ∑∑ mnlmnmn

ab…

zerospolesα

14

8


Root Locus

Rule 4: The locus crosses the jω axis (looses stability) where the Routh criterion shows a transition from roots in the left half-plane to roots in the right-half plane.

Example:

)54(5)( 2 ++

+=

sssssG

5,20 jsK ±==

15


Root Locus

Example: 4673

1)( 234 +++++

=ssss

ssG

16

9


Root Locus

Design dangers revealed by the Root Locus: •  High relative degree: For n-m≥3 we have closed loop instability due to asymptotes. •  Nonminimum phase zeros: They attract closed loop poles into the RHP

46731)( 234 ++++

+=

ssssssG

11)( 2 ++

−=

ssssG

Note: Check code rootlocus.m 17


Root Locus

Viete�s formula: When the relative degree n-m≥2, the sum of the closed loop poles is constant

∑−= poles loop closed1a

nnnn

mmmm

asasasbsbsbs

sasbsL

++++++++

==−

−−

−

11

1

11

1

)()()(

18

10


Phase and Magnitude of a Transfer Function Example:

( )( ) )20(51)735.6()(+++

+=

sssssG

p1φ

p2φ

p3φ

z1φ

pr3zr1 pr2

pr1 ( )pppz

ppp

z

sG

rrrrsG

3211

321

1

)(

)(

φφφφ ++−=∠

=iss o 57 +−==

19


Root Locus- Magnitude and Phase Conditions

{ } { })()()(1 LKLsKL numdenrootszerosRL +=+=when K varies from 0 to ∞ (positive Root Locus) or

from 0 to -∞ (negative Root Locus)

⇔−=>K

sLK 1)(:0

⇔−=<K

sLK 1)(:0

( ) ( )[ ]pnppzm

zzpKipn

pp

zm

zz

pn

mp e

rrrrrrK

pspspszszszsKsL φφφφφφφ +++−++++=

−−−−−−

=

2121

21

21

21

21

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

( ) ( )

180)(

1)(

2121

21

21

=+++−++++=∠

==

pn

ppzm

zzK

pn

pp

zm

zz

p

psL

KrrrrrrKsL

φφφφφφφ

( ) ( )

0)(

1)(

2121

21

21

=+++−++++=∠

−==

pn

ppzm

zzK

pn

pp

zm

zz

p

psL

KrrrrrrKsL

φφφφφφφ

20

11


Root Locus

Selecting K for desired closed loop poles on Root Locus: If so belongs to the root locus, it must satisfies the characteristic equation for some value of K Then we can obtain K as

KsL o

1)( −=

)(1)(

1

o

o

sLK

sLK

=

−=

21


Root Locus

Example: ( )( )511)()(++

==ss

sGsL

( ) ( ) 204242

54314351)(

143

2222 =++−=

++−++−=++==⇒+−=

iisssL

Kis ooo

o

sys=tf(1,poly([-1 -5])) so=-3+4i [K,POLES]=rlocfind(sys,so)

Using MATLAB:

22

12


Root Locus Example: ( )( )51

1)()(++

==ss

sGsL

06.42)(

1

57

==

⇓

+−=

o

o

sLK

is

57 iso +−=

43 iso +−=

When we use the absolute value formula we are assuming that the point belongs to the Root Locus!

23


Root Locus - Compensators

Example: ( )( )511)()(++

==ss

sGsL

Can we place the closed loop pole at so=-7+i5 only varying K? NO. We need a COMPENSATOR.

( )( )( )51

110)()()(++

+==ss

ssGsDsL( )( )51

1)()(++

==ss

sGsL

The zero attracts the locus!!! 24

13


Root Locus – Phase lead compensator

Pure derivative control is not normally practical because of the amplification of the noise due to the differentiation and must be approximated:

( )( )zp

sspszssGsDsL >

++++

== ,51

1)()()(

How do we choose z and p to place the closed loop pole at so=-7+i5?

zppszssD >

++

= ,)( Phase lead COMPENSATOR

When we study frequency response we will understand why we call �Phase Lead� to this compensator.


19.14080.11104.21

?1zφ

Root Locus – Phase lead compensator Example:

Phase lead COMPENSATOR

( )( )zp

sspszssGsDsL >

+++

+== ,

511)()()(

03.9303.45304.2180.11119.1401801 ==+++=zφ 735.6−=⇒ z( ) ( ) 180)( 2121 =+++−++++=∠ p

nppz

mzzK psL φφφφφφφ

26

Let us choose p=20

14


Root Locus – Phase lead compensator Example:

Phase lead COMPENSATOR

( )( )511

20735.6)()()(

++++

==sss

ssGsDsL

11757

=

+−=

Kiso

27


Root Locus – Phase lead compensator

Selecting z and p is a trial an error procedure. In general:

•  The zero is placed in the neighborhood of the closed-loop natural frequency, as determined by rise-time or settling time requirements. •  The poles is placed at a distance 5 to 20 times the value of the zero location. The pole is fast enough to avoid modifying the dominant pole behavior.

The exact position of the pole p is a compromise between:

•  Noise suppression (we want a small value for p) •  Compensation effectiveness (we want large value for p)

28

zppszs

sD > ,++

=)( Phase lead COMPENSATOR

15


Root Locus – Phase lag compensator Example:

Phase lag COMPENSATOR

( )( )511

20735.6)()()(

++++

==sss

ssGsDsL

( )( )2

00010735.6

511

20735.6lim)()(lim)(lim −

→→→×=

++++

===sss

ssGsDsLKsssp

What can we do to increase Kp? Suppose we want Kp=10.

( )( )zp

sssspszssGsDsL <

++++

++

== ,51

120735.6)()()(

We choose: 48.14810735.61 3 =×=

pz

29


Root Locus – Phase lag compensator Example:

( )( )511

20735.6

001.014848.0)()()(

++++

++

==sss

ssssGsDsL

31.1803.594.6

=

+−=

Kiso

30

16


Root Locus – Phase lag compensator

Selecting z and p is a trial an error procedure. In general:

•  The ratio zero/pole is chosen based on the error constant specification. •  We pick z and p small to avoid affecting the dominant dynamic of the system (to avoid modifying the part of the locus representing the dominant dynamics) •  Slow transient due to the small p is almost cancelled by an small z. The ratio zero/pole cannot be very big.

The exact position of z and p is a compromise between:

•  Steady state error (we want a large value for z/p) •  The transient response (we want the pole p placed far from the origin)

31

zppszs

sD < ,++

=)( Phase lag COMPENSATOR


Root Locus - Compensators

Phase lead compensator: pzpszssD <

++

= ,)(

pzpszssD >

++

= ,)(Phase lag compensator:

We will see why we call �phase lead� and �phase lag� to these compensators when we study frequency response

32

control systems - lehigh

Documents