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7/31/2019 Notas Sistemas Controle

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Effects of Addition of Poles and Zeros

Figure: Effect of Adding Poles

Figure: Effect of Adding Zero

1. Addition of poles pulls the root locus to the right

2. Additional zero pulls the root-locus to the left

Controller Design

A compensator or controller placed in the forward path of a control system will modify the shape

of the loci if it contains additional poles and zeros.

In compensator design, hand calculation is cumbersome, and a suitable computer pakage is

generally used.

Compensator Characteristics

PD One additional zero

PI One additional zero and a pole at origin

PID Two additional zeros and a additional pole at origin

Example

A control system has the open-loop transfer function, ( ) ; 1.( 2)( 5)

f

KG s k

s s s= =

+ +A PD compensator of the form 1( ) ( )cG s K s a= + is to be introduced in the forward path toachieve the performance specification:

Overshoot less than 5%, settling time less than 2seconds

Determine the values of 1K and a to meet the specification.


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Original Controller

For 0.5, = . . 16.3%P O =For 0.7, = . . 4.6%P O =

0.7, = corresponds to thecontroller gain of 7.13K = .

Settling time condition is not

met.

PD compensator design

When PD compensator is used, we actually add a zero in the open-loop transfer function. Potential

locations of zero include:

1. At 1s = , 2. At 2s = , 3. At 3s =

1( 1)

( ) ( )( 2)( 5)

K sG s H s

s s s

+=

+ +

10.7, 15, 3.9

sK t = = = sec,

Here, the closed-loop pole in the

root locus branch between 0 and -1

dominate the time response with

oscillatory 2nd-order response

superimposed.

1( 2)

( ) ( )( 2)( 5)

K sG s H s

s s s

+=

+ +

10.7, 12.8, 1.7

sK t = = = sec,

. . 4.1%P O =


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1( 3)

( ) ( )( 2)( 5)

K sG s H s

s s s

+=

+ +

10.7, 5.3, 3.1sK t = = = sec,. . 5.3%P O =

Summary

Of the three compensators considered, only option 2 meets the performance specifications. The

recommended compensator is therefore, ( ) 12.8( 2)cG s s= + .

The time-domain responses for the four conditions are shown in Figure below.


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Realization of Compensator using Passive Components

It is not possible to design isolated zero or pole at origin using passive components. In that case a

pair of pole and zero is produced. Compensators may be of four types: Lead compensator, Lag

compensator, Lag-lead compensator, and feedback compensator. [Cascade compensator]*

A. Lead Compensator

where 1R C =

and2

1 2

1R

R R =

2 2

1 2 1 2

2 2 2

1 2 1 2 22 2

2 1 2

1/ 1( )1/ 1 ( )

1/ 1/

1/ /

1 1/

1/

c R Cs R CsG sR R Cs R R Cs

s R C R s R C

R R R R RR C s s R CR R R

s

s

+ += =+ + + ++ +

= =+ + + + +

+=

+


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C. Lag-lead Compensator

where, 1, 1. >

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