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©2000, John Wiley & Sons, Inc.
Nise/Control Systems Engineering, 3/e
Chapter 8: Root Locus Techniques1
Chapter 8
Root Locus Techniques
©2000, John Wiley & Sons, Inc.
Nise/Control Systems Engineering, 3/e
Chapter 8: Root Locus Techniques2
Figure 8.1a. Closed-loop system;b. equivalent transferfunction
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Chapter 8: Root Locus Techniques3Figure 8.2Vector representationof complex numbers:a. s = + j;b. (s + a);c. alternaterepresentationof (s + a);d. (s + 7)|s5 + j2
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Chapter 8: Root Locus Techniques4
Figure 8.3Vector representationof Eq. (8.7)
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Chapter 8: Root Locus Techniques5
Figure 8.4a. CameraMan®Presenter Camera System automatically follows a subject who wears infrared sensors on their front and back (the front sensor is also a microphone); trackingcommands and audio are relayed to CameraMan via a radio frequency link from a unit worn by the subject. b. block diagram.c. closed-loop transfer function.
Courtesy of ParkerVision.
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Chapter 8: Root Locus Techniques6
Table 8.1Pole location as a function of gain for the system of Figure 8.4
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Chapter 8: Root Locus Techniques7
Figure 8.5a. Pole plot fromTable 8.1;b. root locus
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Chapter 8: Root Locus Techniques8
Figure 8.6a. Example system;b. pole-zero plot of G (s)
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Chapter 8: Root Locus Techniques9
Figure 8.7Vector representationof G(s) from Figure8.6(a) at -2+ j 3
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Chapter 8: Root Locus Techniques10
Figure 8.8Poles and zeros of a general open-loopsystem withtest points, Pi, on the real axis
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Chapter 8: Root Locus Techniques11
Figure 8.9Real-axis segments ofthe root locus for thesystem of Figure 8.6
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Figure 8.10Complete root locus for the system ofFigure 8.6
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Figure 8.11System forExample 8.2
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Chapter 8: Root Locus Techniques14
Figure 8.12Root locus andasymptotes for the system of Figure 8.11
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Figure 8.13Root locus exampleshowing real- axisbreakaway (-1) andbreak-in points (2)
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Chapter 8: Root Locus Techniques16
Figure 8.14Variation of gain along the real axis for the root locus ofFigure 8.13
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Chapter 8: Root Locus Techniques17
Table 8.2Data for breakaway and break-in points for the root locus of Figure 8.13
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Chapter 8: Root Locus Techniques18
Table 8.3Routh table for Eq. (8.40)
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Chapter 8: Root Locus Techniques19
Figure 8.15Open-loop polesand zeros andcalculation of:a. angle of departure;b. angle ofarrival
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Chapter 8: Root Locus Techniques20
Figure 8.16Unity feedback systemwith complex poles
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Chapter 8: Root Locus Techniques21
Figure 8.17Root locus for systemof Figure 8.16showing angle ofdeparture
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Chapter 8: Root Locus Techniques22
Figure 8.18Finding and calibrating exact points on the root locus ofFigure 8.12
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Chapter 8: Root Locus Techniques23
Figure 8.19a. System forExample 8.7;b. root locus sketch
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Figure 8.20Making second-orderapproximations
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Chapter 8: Root Locus Techniques25
Figure 8.21System forExample 8.8
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Figure 8.22Root locus forExample 8.8
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Chapter 8: Root Locus Techniques27
Table 8.4Characteristics of the system of Example 8.8
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Figure 8.23Second- and third-orderresponses forExample 8.8:a. Case 2;b. Case 3
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Figure 8.24System requiring aroot locus calibratedwith p1 as aparameter
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Chapter 8: Root Locus Techniques30
Figure 8.25Root locus for the system of Figure 8.24, with p1 as aparameter
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Figure 8.26Positive-feedbacksystem
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Figure 8.27a. Equivalent positive-feedback system forExample 8.9;b. root locus
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Figure 8.28Portion of the root locus for the antennacontrol system
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Figure 8.29Step response of thegain-adjusted antennacontrol system
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Chapter 8: Root Locus Techniques35
Figure 8.30Root locus of pitch control loop withoutrate feedback, UFSS vehicle
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Chapter 8: Root Locus Techniques36
Figure 8.31 Computer simulation of step response of pitch control loopwithout rate feedback,UFSS vehicle
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Figure 8.32Root locus of pitch control loop with rate feedback, UFSS vehicle
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Figure 8.33Computer simulation of step response of pitch control loop withrate feedback, UFSS vehicle
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Figure P8.1
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Chapter 8: Root Locus Techniques40
Figure P8.2
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Chapter 8: Root Locus Techniques41
Figure P8.3
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Figure P8.4
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Figure P8.5
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Figure P8.6
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Figure P8.7
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Chapter 8: Root Locus Techniques46
Figure P8.8
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Chapter 8: Root Locus Techniques47
Figure P8.9
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Chapter 8: Root Locus Techniques48
Figure P8.10
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Chapter 8: Root Locus Techniques49
Figure P8.11
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Figure P8.12
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Chapter 8: Root Locus Techniques51
Figure P8.13a. Robot equipped to perform arc welding;(figure continues)
Courtesy of FANUC Robotics.
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Chapter 8: Root Locus Techniques52
Figure P8.13(continued)b. block diagram for swing motion system
© 1967 H.L. Hardy.
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Chapter 8: Root Locus Techniques53
Figure P8.14Block diagram ofsmoother
© 1985 Rockwell International.
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Figure P8.15a. Active vibration absorber((c)1992 AIAA);b. control systemblock diagram
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Figure P8.16Floppy disk drive:a. physicalrepresentation;b. block diagram
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Figure P8.17Simplified blockdiagram of pupilservomechanism
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Figure P8.18Active suspensionsystem
© 1985 ASME.
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Figure P8.19F4-E pitchstabilization loop
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Figure P8.20Pitch axis attitude control system utilizing momentum wheel
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Figure P8.21a. Combustor with microphone and loud speaker((c)1995 IEEE);
b. block diagram((c)1995 IEEE)
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Chapter 8: Root Locus Techniques61
Figure P8.22a. Wind turbinesgenerating electricitynear Palm Springs, California;(figure continues)
© Jim Corwin/ Photo Researchers
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Chapter 8: Root Locus Techniques62
Figure P8.22(continued)b. Control loop for a constant-speedpitch-controlledwind turbine((c)1998 IEEE);c. Drivetrain((c)1998 IEEE)
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Table 8.1Pole location as a function of gain for the system of Figure 8.4
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Table 8.2Data for breakaway and break-in points for the root locus of Figure 8.13
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Table 8.3Routh table for Eq. (8.40)
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Chapter 8: Root Locus Techniques66
Table 8.4Characteristics of the system of Example 8.8