root location in the s-plane

7/26/2019 Root Location in the S-plane

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Root location in the s-plane


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Importance of Root Locations

Transient response is closely related to

the location of roots (poles and zeros).

Graphical presentation of root locations: relationships between poles and zeros

dominant and insignificant roots

stability

Design tools and methods: root locus (root change vs. parameters)

compensation to modify root location


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Step Response vs s-plane

location


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Addition of a pole to 1-st

order system Transfer function with one pole :

Transfer function with additional pole :


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Step response with added

pole


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Dominant and Insignicant

Pole


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Addition of a pole to 2-st

order system Transfer function with two pole :

Transfer function with additional pole :

Underdamped


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Step response with added

pole


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Dominant and InsignicantPole


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Dominant and InsignicantPole


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Pole-ero !ancellation

Assume a three pole system with a zero as shown in

equation below. Ifthe zero at z is very close to the

pole at p3, then the partial fraction of the equation

show that the residue of the eponential decay ismuch smaller than the amplitude of the second!order

response. In other word, the zero and the pole

cancel out.

))(()()(

2

3 basspszsKsT

+++

+=


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"a#ing second orderappro$imation


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"a#ing second orderappro$imation


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%$ample 1

"ind the step response

for:

542.244

542.24)(

21++

=ss

sT

( ) ( )542.2441042.245

)(22

+++=

ssssT

( ) ( )542.2443626.73)(

23+++

=sss

sT


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%$ample 2

"ind the step response for:

The #spansion of those equation:

)6)(5)(01.4()4(25.26)(

)6)(5)(5.3(

)4(25.26)(

2

1

+++

+=

+++

+=

Ssss

ssC

Ssss

ssC

)01.4(

033.0

)6(

4.4

)5(

3.587.0)(

)5.3(1

)6(5.3

)5(5.31)(

2

1

+

+

+

+

=

+

+

+

+

=

sssssC

sssssC


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ReviewSecond 'rder System

Specication

122,1 = nns

cos=


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ReviewSecond 'rder System

Specication

1cos=

cos=

d

pT

=

d

pT

=

n

sT

4=

!"# $ !"%

Tp% $ Tp#

Ts% $ Ts#


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Specication on the s-plane

ma&


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'verdamped system (withinspecs)

ma&


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*nderdamped system(within specs)

ma&


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+-order system (withinspecs)

ma&


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+-order system (o,t of specs)


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-order system (withinspecs)

ma&


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-order system (o,t of specs)

ma&


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Transient Response Design via GainAdjustment

%$ample. /hird-order gain design $esi%n the value of %ain, &, to yield '.()*overshoot. Also estimate the settlin% time,

pea+ time, and steady!state error.


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%$ample. /hird-order gain designOriginal Root Locus

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1

-6

-4

-2

0

2

4

6

Real Axis

ImagAxis


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%$ample. /hird-order gain design-line

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1

-6

-4

-2

0

2

4

6

Real Axis

ImagAxis

)100/(%ln

)100/ln(%

22 OS

OS

+

=

' #.% overshoot

corresponds to a

damping ratio of .*.


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%$ample. /hird-order gain designClosed-loop Poles

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1

-6

-4

-2

0

2

4

6

Real Axis

ImagAxis

"econd+order

system

appro&imation


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%$ample. /hird-order gain designClosed-loop Poles & Characteristic

of !stem

ase ' and ) yield third poles that are relatively far from theclosed!loop zero. "or this two cases there is no pole!zerocancellation, and the second!order system approimation is not

valid. In case 3 yield, the third closed!loop pole and the closed!loop

zero are relatively close to each other, and the second!ordersystem approimation can be considered valid.


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%$ample. /hird-order gain design


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%$ample. /hird-order gain design

root location in the s-plane

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