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©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e
Chapter 9: Design via Root Locus1
Figure 9.1a. Sample root locus,showing possibledesign point viagain adjustment (A)and desired designpoint that cannot bemet via simple gainadjustment (B);b. responses frompoles at A and B
©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e
Chapter 9: Design via Root Locus2
Figure 9.2Compensationtechniques:a. cascade;b. feedback
©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e
Chapter 9: Design via Root Locus3
Figure 9.3Pole at A is:a. on the rootlocus without compensator;b. not on theroot locus withcompensatorpole added;(figure continues)
©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e
Chapter 9: Design via Root Locus4
Figure 9.3(continued)c. approximately on the root locus withcompensatorpole and zero added
©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e
Chapter 9: Design via Root Locus5
Figure 9.4Closed-loopsystem forExample 9.1:a. beforecompensation;b. after ideal integralcompensation
©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e
Chapter 9: Design via Root Locus6
Figure 9.5Root locus foruncompensatedsystem ofFigure 9.4(a)
©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e
Chapter 9: Design via Root Locus7
Figure 9.6Root locus forcompensatedsystem of Figure 9.4(b)
©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e
Chapter 9: Design via Root Locus8
Figure 9.7Ideal integral compensated system response and theuncompensated systemresponse of Example 9.1
©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e
Chapter 9: Design via Root Locus9
Figure 9.8PI controller
©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e
Chapter 9: Design via Root Locus10
Figure 9.9a. Type 1 uncompensated system;b. Type 1 compensatedsystem;c. compensatorpole-zero plot
©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e
Chapter 9: Design via Root Locus11
Figure 9.10Root locus:a. before lag compensation;b. after lag compensation
©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e
Chapter 9: Design via Root Locus12
Figure 9.11Compensated systemfor Example 9.2
©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e
Chapter 9: Design via Root Locus13
Figure 9.12Root locus forcompensated system of Figure 9.11
©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e
Chapter 9: Design via Root Locus14
Table 9.1Predicted characteristics of uncompensated and lag-compensated systems for Example 9.2
©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e
Chapter 9: Design via Root Locus15
Figure 9.13Step responses ofuncompensated andlag-compensatedsystems forExample 9.2
©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e
Chapter 9: Design via Root Locus16
Figure 9.14Step responses of the system for Example 9.2 using different lagcompensators
©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e
Chapter 9: Design via Root Locus17
Figure 9.15Using ideal derivativecompensation:a. uncompensated;b. compensatorzero at –2;(figure continues)
©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e
Chapter 9: Design via Root Locus18
Figure 9.15(continued)c. compensatorzero at –3;d. compensatorzero at – 4
©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e
Chapter 9: Design via Root Locus19
Figure 9.16Uncompensated system and ideal derivativecompensation solutions from Table 9.2
©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e
Chapter 9: Design via Root Locus20
Table 9.2Predicted characteristics for the systems of Figure 9.15
©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e
Chapter 9: Design via Root Locus21
Figure 9.17Feedbackcontrol systemfor Example 9.3
©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e
Chapter 9: Design via Root Locus22
Figure 9.18Root locus for uncompensatedsystem shown in Figure 9.17
©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e
Chapter 9: Design via Root Locus23
Table 9.3Uncompensated and compensated system characteristics for Example 9.3
©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e
Chapter 9: Design via Root Locus24
Figure 9.19Compensateddominant polesuperimposed over the uncompensatedroot locus forExample 9.3
©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e
Chapter 9: Design via Root Locus25
Figure 9.20Evaluating the location of the compensatingzero for Example 9.3
©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e
Chapter 9: Design via Root Locus26
Figure 9.21Root locus for thecompensated system of Example 9.3
©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e
Chapter 9: Design via Root Locus27
Figure 9.22Uncompensated andcompensated system step responses ofExample 9.3
©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e
Chapter 9: Design via Root Locus28
Figure 9.23PD controller
©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e
Chapter 9: Design via Root Locus29
Figure 9.24Geometry of leadcompensation
©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e
Chapter 9: Design via Root Locus30
Figure 9.25Three of the infinitepossible leadcompensator solutions
©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e
Chapter 9: Design via Root Locus31
Figure 9.26Lead compensatordesign, showingevaluation ofuncompensatedand compensateddominant poles forExample 9.4
©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e
Chapter 9: Design via Root Locus32
Table 9.4Comparison of lead compensation designs for Example 9.4
©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e
Chapter 9: Design via Root Locus33
Figure 9.27s-plane pictureused to calculatethe location ofthe compensatorpole for Example 9.4
©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e
Chapter 9: Design via Root Locus34
Figure 9.28Compensated systemroot locus
©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e
Chapter 9: Design via Root Locus35
Figure 9.29Uncompensatedsystem and leadcompensationresponses forExample 9.4
©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e
Chapter 9: Design via Root Locus36
Figure 9.30PID controller
©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e
Chapter 9: Design via Root Locus37
Figure 9.31Uncompensated feedback control system for Example 9.5
©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e
Chapter 9: Design via Root Locus38
Figure 9.32Root locus for theuncompensatedsystem ofExample 9.5
©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e
Chapter 9: Design via Root Locus39
Table 9.5Predicted characteristics of uncompensated, PD- , and PID- compensated systems of Example 9.5
©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e
Chapter 9: Design via Root Locus40
Figure 9.33Calculating thePD compensatorzero for Example 9.5
©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e
Chapter 9: Design via Root Locus41
Figure 9.34Root locus forPD-compensatedsystem ofExample 9.5
©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e
Chapter 9: Design via Root Locus42
Figure 9.35Step responses foruncompensated,PD-compensated, andPID-compensatedsystems ofExample 9.5
©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e
Chapter 9: Design via Root Locus43
Figure 9.36Root locus for PID-compensatedsystemof Example 9.5
©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e
Chapter 9: Design via Root Locus44
Figure 9.37Uncompensatedsystem forExample 9.6
©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e
Chapter 9: Design via Root Locus45
Figure 9.38Root locus for uncompensatedsystem of Example 9.6
©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e
Chapter 9: Design via Root Locus46
Table 9.6Predicted characteristics of uncompensated, lead-compensated, and lag-lead-compensated systems of Example 9.6
©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e
Chapter 9: Design via Root Locus47
Figure 9.39Evaluating thecompensator pole forExample 9.6
©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e
Chapter 9: Design via Root Locus48
Figure 9.40Root locus for lead-compensated system of Example 9.6
©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e
Chapter 9: Design via Root Locus49
Figure 9.41Root locus for lag-lead-compensated systemof Example 9.6
©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e
Chapter 9: Design via Root Locus50
Figure 9.42Improvement in stepresponse forlag-lead-compensatedsystem ofExample 9.6
©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e
Chapter 9: Design via Root Locus51
Figure 9.43Improvement inramp response error for the system ofExample 9.6:a. lead-compensated;b. lag-lead-compensated
©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e
Chapter 9: Design via Root Locus52
Figure 9.44a. Root locusbefore cascading notch filter;b. typical closed-loopstep response before cascading notch filter;(figure continues)
©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e
Chapter 9: Design via Root Locus53
Figure 9.44(continued)c. pole-zero plot of a notch filter;d. root locus aftercascading notch filter;e. closed-loop step response after cascading notch filter.
©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e
Chapter 9: Design via Root Locus54
Table 9.7Types of cascade compensators (slide 1 of 2)
(continued next slide)
©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e
Chapter 9: Design via Root Locus55
Table 9.7Types of cascade compensators (slide 2 of 2)
©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e
Chapter 9: Design via Root Locus56
Figure 9.45Generic controlsystem with feedbackcompensation
©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e
Chapter 9: Design via Root Locus57
Figure 9.46A position control system that uses a tachometer as adifferentiator in the feedback path. Can you see the similarity between this system and the schematic onthe front end papers?
Photo by Mark E. Van Dusen.
©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e
Chapter 9: Design via Root Locus58
Figure 9.47a. Transfer function of a tachometer;b. tachometer feedback compensation
©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e
Chapter 9: Design via Root Locus59
Figure 9.48Equivalentblock diagramof Figure 9.45
©2000, John Wiley & Sons, Inc.Nise/Control Systems Engineering, 3/e
Chapter 9: Design via Root Locus60
Figure 9.49a. System forExample 9.7;b. system withrate feedbackcompensation;c. equivalentcompensatedsystem;d. equivalentcompensatedsystem, showingunity feedback
©2000, John Wiley & Sons, Inc.
Nise/Control Systems Engineering, 3/e
Chapter 13: Digital Control Systems29
Figure 13.25a. Digital controlsystem showingthe digital computerperformingcompensation;b. continuous systemused for design;c. transformed digitalsystem
©2000, John Wiley & Sons, Inc.
Nise/Control Systems Engineering, 3/e
Chapter 13: Digital Control Systems30
Figure 13.26Closed-loop responsefor the compensatedsystem of Example13.12 showing effectof three differentsampling frequencies