EML5311_Chapter6_p3

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Control Systems Engineering, Fourth Edition by Norman S. Nise

Copyright © 2004 by John Wiley & Sons. All rights reserved.

Pole at A is:

a. on the root locus withoutcompensator; (P-controller )

b. not on the root locus

with compensator poleadded (I-controller )

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Figure (continued)

c. approximately onthe root locus with

compensator poleand zero added

PI Compensator

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a. Before compensation

b. after ideal integral compensation: zero (a=0.1) is small

Closed-loop system for Example 9.1:

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Root locus for uncompensated

system of Figure (a)

Root locus for compensated

system of Figure (b)

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Ideal integral compensated system response and the

uncompensated system response of Example 9.1

• Ideal integral effect was reduce in steady-state error without changing the transient response,

• and yield approximately the same original root locus

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PI controller Implementation:

Operational Amplifiers of Chapter 2Or, Virtual Instruments of LabView, Matlab, DSpace

or other controller software

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a. Type 1uncompensatedsystem;

b. Type 1

compensatedsystem;

c. LAG compensator pole-zero plot|zc| > |pc|

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a. Type 1uncompensatedsystem;

b. Type 1compensated

system;

c. LAGcompensator pole-zero plot|zc| > |pc|

)()zz(z

zz

z)z()(

21c

21c

21

21

c

c

cc

p p p K K

rCompensatoCascade

p p

K K

K

Ap

p K

A

K

Ae

vn

vo

vovov

=

=

===∞

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Root locus:a. before lag compensation;b. after lag compensation

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Root locus for Lagcompensated systemExample 9.2

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Predicted characteristics of uncompensated andLag-compensated systems for Example 9.2

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Step responses ofuncompensated andlag-compensated systems forExample 9.2

Step responses of thesystem for Example 9.2using different lagcompensators

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Using ideal derivativecompensation:

a. uncompensated

b. compensator zero at –2

c. compensator zero at –3;

d. compensator zero at – 4

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Table 9.2

Predicted characteristics for the systems of previous Figuresideal derivative compensations

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Uncompensated system and ideal derivative

compensation solutions from Table 9.2

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Root locus foruncompensated

system shown

Root locus for thePD compensatedsystem of Example 9.3

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Table 9.3

Uncompensated and compensated system characteristicsfor Example 9.3

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Compensateddominant polesuperimposed over theuncompensatedroot locus for Example 9.3

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Evaluating the

location of thecompensatingzero for Example9.3

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Uncompensated andcompensated systemstep responses of Example 9.3

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PD controller

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Geometry of leadcompensation

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Three of the infinitepossible lead

compensator solutions)(

)zz(z

zz)z(

)(

21c

21c

21

21

cc

p p p

K K

rCompensatoCascade

p p

K K

p K

A

K

Ae

pn

po

po p

=

=

+

=

+

=∞

11

Which p c

and

z c to choose?

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Lead compensator design, showingevaluation of uncompensated

and compensateddominant poles for Example 9.4

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s-plane pictureused to calculatethe location of

the compensator polefor Example 9.4

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Table 9.4

Comparison of lead compensation designs for Example 9.4

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Compensated systemroot locus

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Uncompensatedsystem and leadcompensationresponses for

Example 9.4

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PID controller

Procedure for Design

• Design PD to improve transient response

• Add Integrator and Design to reduce steady-state error

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Uncompensated feedback control

system for Example 9.5

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Root locus for the

uncompensatedsystem of Example 9.5

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Table 9.5

Predicted characteristics of uncompensated, PD- , and PID-compensated systems of Example 9.5

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Calculating the

PD compensator zero forExample 9.5

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Root locus for

PD-compensatedsystem of Example 9.5

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Step responses for uncompensated, PD-compensated,

and PID-compensated systems of Example 9.5

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Root locus for PID-compensatedsystem of Example 9.5

What are the resulting

PID controller gains for

parallel PID structure

block diagram given in slide 30?

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Uncompensated system for Example 9.6

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Control Systems Engineering, Fourth Edition by Norman S. NiseCopyright © 2004 by John Wiley & Sons. All rights reserved.

Root locus for uncompensatedsystem of Example 9.6

T bl 9 6

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Note: Pole-zero

cancellation

Table 9.6

Predicted characteristics of uncompensated, lead-compensated,

and lag-lead- compensated systems of Example 9.6

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Evaluating the compensator pole for Example 9.6

Note: Chose Lead compensator zero to achieve pole-zero cancellation of -6 pole of the uncompensated system.

Next, determine compensator pole for desired closed-loop

system dominant pole location (-3.588 + j 7.003).

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Root locus for lead-compensated system of

Example 9.6

Note: Pole-zero cancellation with zc

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Root locus for lag-lead compensated

system of Example 9.6

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Improvement in step response for

lag-lead- compensated system of Example 9.6

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Improvement in rampresponse error for thesystem of Example 9.6:

a. lead-compensated;b. lag-lead-compensated

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a. Root locus beforecascading notch filter;b. typical closed-loop

step response beforecascading notch filter;(figure continues)

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(continued)

c. pole-zero plotof a notch filter;d. root locus after cascading notchfilter;

e. closed-loopstep responseafter cascading

notch filter.

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Figure 9.45Generic control system withfeedback compensation

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A position control system that uses a tachometer as

a differentiator in the feedback path. Can you see thesimilarity between this system and the schematic onthe front end papers?

Photo by Mark E. Van Dusen.

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a. Transfer function of a tachometer;

b. tachometer feedback compensation

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Equivalent block diagram of Figure 9.45

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a. System for Example 9.7;

b. system with rate feedbackcompensation

c. Equivalent compensated

system;

d. Equivalent compensatedsystem, showing unityfeedback

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EML5311_Chapter6_p3