Chapter 8: Root Locus Techniques 1 ©2000, John Wiley & Sons, Inc. Nise/Control Systems Engineering,...

©2000, John Wiley & Sons, Inc.

Nise/Control Systems Engineering, 3/e

Chapter 8: Root Locus Techniques1

Chapter 8

Root Locus Techniques




Figure 8.1a. Closed-loop system;b. equivalent transferfunction



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




Figure 8.3Vector representationof Eq. (8.7)




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.




Table 8.1Pole location as a function of gain for the system of Figure 8.4




Figure 8.5a. Pole plot fromTable 8.1;b. root locus




Figure 8.6a. Example system;b. pole-zero plot of G (s)




Figure 8.7Vector representationof G(s) from Figure8.6(a) at -2+ j 3




Figure 8.8Poles and zeros of a general open-loopsystem withtest points, Pi, on the real axis




Figure 8.9Real-axis segments ofthe root locus for thesystem of Figure 8.6




Figure 8.10Complete root locus for the system ofFigure 8.6




Figure 8.11System forExample 8.2




Figure 8.12Root locus andasymptotes for the system of Figure 8.11




Figure 8.13Root locus exampleshowing real- axisbreakaway (-1) andbreak-in points (2)




Figure 8.14Variation of gain along the real axis for the root locus ofFigure 8.13




Table 8.2Data for breakaway and break-in points for the root locus of Figure 8.13




Table 8.3Routh table for Eq. (8.40)




Figure 8.15Open-loop polesand zeros andcalculation of:a. angle of departure;b. angle ofarrival




Figure 8.16Unity feedback systemwith complex poles




Figure 8.17Root locus for systemof Figure 8.16showing angle ofdeparture




Figure 8.18Finding and calibrating exact points on the root locus ofFigure 8.12




Figure 8.19a. System forExample 8.7;b. root locus sketch




Figure 8.20Making second-orderapproximations




Figure 8.21System forExample 8.8




Figure 8.22Root locus forExample 8.8




Table 8.4Characteristics of the system of Example 8.8




Figure 8.23Second- and third-orderresponses forExample 8.8:a. Case 2;b. Case 3




Figure 8.24System requiring aroot locus calibratedwith p1 as aparameter




Figure 8.25Root locus for the system of Figure 8.24, with p1 as aparameter




Figure 8.26Positive-feedbacksystem




Figure 8.27a. Equivalent positive-feedback system forExample 8.9;b. root locus




Figure 8.28Portion of the root locus for the antennacontrol system




Figure 8.29Step response of thegain-adjusted antennacontrol system




Figure 8.30Root locus of pitch control loop withoutrate feedback, UFSS vehicle




Figure 8.31 Computer simulation of step response of pitch control loopwithout rate feedback,UFSS vehicle




Figure 8.32Root locus of pitch control loop with rate feedback, UFSS vehicle




Figure 8.33Computer simulation of step response of pitch control loop withrate feedback, UFSS vehicle




Figure P8.1




Figure P8.2




Figure P8.3




Figure P8.4




Figure P8.5




Figure P8.6




Figure P8.7




Figure P8.8




Figure P8.9




Figure P8.10




Figure P8.11




Figure P8.12




Figure P8.13a. Robot equipped to perform arc welding;(figure continues)

Courtesy of FANUC Robotics.




Figure P8.13(continued)b. block diagram for swing motion system

© 1967 H.L. Hardy.




Figure P8.14Block diagram ofsmoother

© 1985 Rockwell International.




Figure P8.15a. Active vibration absorber((c)1992 AIAA);b. control systemblock diagram




Figure P8.16Floppy disk drive:a. physicalrepresentation;b. block diagram




Figure P8.17Simplified blockdiagram of pupilservomechanism




Figure P8.18Active suspensionsystem

© 1985 ASME.




Figure P8.19F4-E pitchstabilization loop




Figure P8.20Pitch axis attitude control system utilizing momentum wheel




Figure P8.21a. Combustor with microphone and loud speaker((c)1995 IEEE);

b. block diagram((c)1995 IEEE)




Figure P8.22a. Wind turbinesgenerating electricitynear Palm Springs, California;(figure continues)

© Jim Corwin/ Photo Researchers




Figure P8.22(continued)b. Control loop for a constant-speedpitch-controlledwind turbine((c)1998 IEEE);c. Drivetrain((c)1998 IEEE)




Table 8.1Pole location as a function of gain for the system of Figure 8.4




Table 8.2Data for breakaway and break-in points for the root locus of Figure 8.13




Table 8.3Routh table for Eq. (8.40)




Table 8.4Characteristics of the system of Example 8.8

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