ch10_2010

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PROF. DR. JOÃO EDGAR CHAVESFILHO

DEPARTAMENTO DE ELETRICIDADE

Control Systems Engineering, Fourth Edition by

Norman S. Nise

reserved.

SISTEMAS DE CONTROL

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Objetivos

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reserved.

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Introdução

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Os métodos de resposta em freqüênciadesenvolvidos por Nyquist e Bode (anos 30), sãomais antigos do que o método do lugar dasraízes que foi descoberto por Evans 1948.

Esta técnica tem vantagens distintas nasseguintes situações:

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Figure 10.1 The HP 35670A DynamicSignal Analyzer obtains

frequency responsedata from a physicalsystem. Thedisplayed data can be used toanalyze, design, or determinea mathematical model

for the system.

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reserved.

Courtesy of Hewlett-Packard.

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O Conceito de Resposta emFreqüência

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Figure 10.2 Sinusoidalfrequencyresponse:

a. system;b. transfer function; c. input andoutputwaveforms

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Figure 10.3 System with

sinusoidal input

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As equações abaixo formam nossa definição de resposta em

Freqüência

M (ω) e Ф(ω) são a magnitude e a fase da resposta em

Freqüênncia.

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Figure 10.4 Frequency responseplots for G(s) = 1/(s + 2):separate magnitudeand phase

Norman S. Nise

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Figure 10.5 Frequency response

plots for G(s)= 1/(s + 2): polar plot

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Figure 10.6

Bode plots of (s + a):a. magnitude plot;b. phase plot.

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Table 10.1

Asymptotic and actualnormalized and scaledfrequency response datafor (s + a)

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Figure 10.7 Asymptotic and actual normalized and scaledmagnitude response of (s + a)

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Figure 10.8 Asymptotic and actual normalized and scaled phaseresponse of (s + a)

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Figure 10.9Normalized and scaledBode plots for a. G(s) = s;b. G(s) = 1/s;

c. G(s) = (s + a);d. G(s) = 1/(s + a)

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Closed-loopunity

feedbacksystem

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Figure

10.11 Bodelog-magnitudeplot for Example 10.2:a. components;b. composite

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Figure

10.12 Bode phaseplot for Example 10.2:a. components;b. composite

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Figure 10.13Bode asymptotesfor normalizedand scaled G(s) =

a. magnitude;b. phase

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Table 10.4Data for normalized andscaled log-magnitude andphase plots for (s2 + 2 ns +

).Mag = 20 log(M / n2)

(table continues)

Norman S. Nise

reserved.

Freq. / n

(dB) = 0.1

(deg) = 0.1

(dB) = 0.2

(deg) = 0.2

(dB) = 0.3

(deg) = 0.3

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Table 10.4

(continued)

Norman S. Nise

Freq. / n

(dB) = 0.5

(deg) = 0.5

(dB) = 0.7

(deg) = 0.7

(dB) = 1

(deg) = 1

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Figure 10.14Normalized and scaled

log-magnitude response for

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Figure 10.15 Scaled phase response for

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Table 10.5Data for normalized andscaled log-magnitude andphase plots for

+ 2 ns + n2

).Mag = 20 log(M / n2)

(table continues)

Norman S. Nise

Freq. / n

(dB) = 0.1

(deg) = 0.1

(dB) = 0.2

(deg) = 0.2

(dB) = 0.3

(deg) = 0.3

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Table 10.5 (continued)

Norman S. Nise

Freq. / n

(dB) = 0.5

(deg) = 0.5

(dB) = 0.7

(deg) = 0.7

(dB) = 1

(deg) = 1

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Figure 10.16Normalized and scaled log magnituderesponse for

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Figure 10.17 Scaled phase response for

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Figure

10.18Bode magnitudeplot for G(s) =(s + 3)/[(s + 2)(s2 + 2s + 25)]:a. components;b. composite

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Table 10.7

Phase diagram slopes for Example 10.3

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(s + 3)/[(s +2)

(s2 + 2s +25)]:a.

components;

b.compositeControl Systems Engineering, Fourth Edition by

Norman S. Nise

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Figure 10.20 Closed-loop controlsystem

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. Mapping contour A through function

F (s)to contour B

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Figure 10.22 Examples of contour mapping

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Figure 10.23

Vector representationof mapping

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Figure10.24 Contour enclosingright half-plane

to determine stability

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Figure 10.25 Mapping examples:a. contour doesnot encloseclosed-loop poles;b. contour doesencloseclosed-loop poles

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Figure

10.26a. Turbine andgenerator;b. block diagramof speed control systemfor

Example 10.4

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Figure 10.27 Vector evaluation of the Nyquist diagramfor Example 10.4:a. vectors on contour at lowfrequency;b. vectors on contour aroundinfinity;c. Nyquist diagram

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Figure 10.28Detouring around open-loop poles:a. poles on contour;b. detour right;c. detour left

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Figure 10.29a. Contour for Example 10.5;b. Nyquist diagram for Example 10.5

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Figure 10.30Demonstrating

Nyquist stability:a. system;b. contour;c. Nyquist diagram

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Figure 10.31 a. Contour for Example 10.6;b. Nyquist diagram

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Figure 10.32a. Contour and root locus of system thatis stable for small gain and unstable for large gain;b. Nyquist diagram

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Figure 10.33a. Contour and root locus of system that is unstable for small gain and stable for large gain;b. Nyquist diagram

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Figure 10.34 a. Portion of contour to be mapped for Example 10.7;b. Nyquist diagram of mapping of positive imaginary axis

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Figure10.35Nyquist diagramshowing gainand phasemargins

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Figure

10.36 Bodelog-magnitudeand phase diagramsfor the system

of Example 10.9

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10.37 Gain and

phasemargins on

the Bodediagrams

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Figure 10.38 Second-order

closed-loopsystem

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Figure

10.39 Representative log-magnitude plot of Eq. (10.51)

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Figure 10.40Closed-loop frequency percent

overshoot for a two-pole system

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vs.damping

ratiofor:a. settling

time;b. peak

time;c. rise time

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Figure 10.42Constant M circles

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Figure 10.43 Constant N

circles

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Figure

10.44 Nyquist diagramfor Example 10.11 andconstantM and N circles

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10.45 Closed-

loopfrequencyresponsefor Example

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.Ni h l

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. Nicholschart

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Figure 10.47Nichols chart with frequency response for G(s) = K /[s(s + 1)(s + 2)] superimposed.Values for K = 1 and K = 3.16 are shown.

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Figure 10.48 Phase margin vs.damping ratio

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Figure 10.49Open-loop gain vs. open-loop phase angle for –3 dB closed-loopgain

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Figure 10.50

a. Block diagram(figure continues)

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ed) b. Bode

diagrams for systemof Example

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Bodelog-

magnitudeplots for Example

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Bode log-magnitude plot

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Bode log magnitude plotfor

Skill-Assessment Exercise10.10

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Effect of

delayuponfrequencyresponse

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Figure

10.55Frequency responseplots for G(s) =K/ [s (s + 1)(s + 10)]with a delay of 1 second and

K = 1:a. magnitude plot;b. phase plot

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Figure 10.56Step response for closed-loop systemwith G(s) =5/[s(s +1)(s + 10)]:a. with a 1 second

delay;b. without delay

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Figure

10.57 Bode plots for subsystem withundeterminedtransfer function

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Figure 10.58 Original Bode plots

minus response of G1(s) =25/(s2 + 2.4s + 25)

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Figure 10.59 Original Bode plotminus responseof G1(s)G2(s) =[25/(s2 + 2.4s +25)]

· [90/(s + 90)]

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Figure

10.60 Bode plots for Skill-AssessmentExercise 10.12

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response

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pplots

for theantennacontrol

system(K = 1)

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Figure P10-1 (p. 675) Control Systems Engineering, Fourth Edition by

Norman S. Nise

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Norman S. Nise

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Figure

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Figure P10-4 (p. 677) Control Systems Engineering, Fourth Edition by

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Figure P10.5

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Fi P10 6

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Figure P10.6

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Figure P10-9 (p. 681)

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Figure P10.10

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Figure P10.11

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Figure P10 12

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Figure P10.12 Soft Arm position

control systemblock diagram

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Figure P10.13

Floppy disk driveblock diagram

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Figure

P10.14 AdeptOne, a four- or five-axisindustrial robot, is used for assembly, packaging,and other manufacturing tasks.

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Figure

P10.15 a. A cutaway viewof a Nikon 35-mm camerashowing parts of the CCDautomatic focusing

system;b. functional blockdiagram;c. block diagram

Norman S. Nise

Courtesy of Nikon, Inc.

oc agram oa

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aship’s roll

stabilizingsystem

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Table 10.2Bode magnitude plot: slop contribution from each pole and zero in Example 10.2

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Table 10.3Bode phase plot: slop contribution from each pole and zero in Example 10.2

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Table 10.6Magnitude diagram slopes for Example 10.3

Control Systems Engineering Fourth Edition by

ch10_2010

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