Root Location in the S-plane

Root location in the splane

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

Step Response vs s-plane location

 Addition of a pole to 1-st order system 

Transfer function with one pole :



Transfer function with additional pole :

Step response with added pole

Dominant and Insignicant Pole

 Addition of a pole to 2-st order system 

Transfer function with two pole : Underdamped



Transfer function with additional pole :

Step response with added pole

Dominant and Insignicant Pole

Dominant and Insignicant Pole

Pole-ero !ancellation 

 Assume a three pole system with a zero as shown in equation below. If  the zero at –z is very close to the pole at –p3, then the partial fraction of the equation show that the residue of the eponential decay is much smaller than the amplitude of the second!order response. In other word, the zero and the pole cancel out.

T ( s ) =

 K ( s +  z ) ( s +  p3 )( s

2

+ as + b)

"a#ing second order appro$imation

"a#ing second order appro$imation

%$ample 1 

"ind the step response for:

T 1 ( s) =

T 2 ( s ) =

T 3 ( s ) =

24.542  s

2

+ 4 s + 24.542

245.42

( s + 10) ( s 2 + 4 s + 24.542 ) 73.626

( s + 3) ( s 2 + 4 s + 24.542)

%$ample 1&

%$ample 2 

"ind the step response for: C 1 ( s )

=

C 2 ( s ) 

26.25( s + 4)  s ( s + 3.5)( s + 5)( S  + 6)

=

26.25( s + 4)  s ( s + 4.01)( s + 5)( S  + 6)

The #spansion of those equation: 1 3.5 3.5 1 + − C 1 ( s ) = − ( s + 5) ( s + 6) ( s + 3.5)  s C 2 ( s )

=

0.87  s

−

5.3 ( s + 5)

+

4.4 ( s + 6)

−

0.033 ( s + 4.01)

Review Second 'rder System Specication  s1, 2

= −ζω n ± ω n

ζ 

=

ζ  2 − 1

cos θ 

Review Second 'rder System Specication

ζ  = cos θ  π  T  p = ω d 

−1

ζ  = cos θ 

T  s

T  p

=

π  ω d 

!"# $ !"% Tp% $ Tp# Ts% $ Ts#

4 =

ζω n

Specication on the s-plane

ma&

'verdamped system (within specs)

ma&

*nderdamped system (within specs)

ma&

+-order system (within specs)

ma&

+-order system (o,t of specs)

-order system (within specs)

ma&

-order system (o,t of specs)

ma&

Transient Response Design via Gain  Adjustment

%$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.

%$ample. /hird-order gain design Original Root Locus 6

4

2  s  i  x  A  g0  a  m  I

2

4

6

1 0

9

8

7

6

5

4

Re a l Ax i s

3

2

1

0

1

%$ample. /hird-order gain design -line ln(%OS  / 100) −

ζ  =

π 2 + ln 2 (%OS  / 100)

6

' #.% overshoot corresponds to a damping ratio of .*.

4

2  s  i  x  A  g0  a  m  I

2

4

6

1 0

9

8

7

6

5

4

Re a l Ax i s

3

2

1

0

1

%$ample. /hird-order gain design Closed-loop Poles 6

4

2  s  i  x  A  g0  a  m  I

2

4

"econd+order system appro&imation

6

1 0

9

8

7

6

5

4

Re a l Ax i s

3

2

1

0

1

%$ample. /hird-order gain design Closed-loop Poles & Characteristic of !stem





ase ' and ) yield third poles that are relatively far from the closed!loop zero. "or this two cases there is no pole!zero cancellation, and the second!order system approimation 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!order system approimation can be considered valid.

%$ample. /hird-order gain design