Modern control systems (12th edition) richard c dorf and robert h bishop

© 2011 Pearson Education, Inc., Upper Saddle River, NJ All rights reserved This publication is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458 MODERN CONTROL SYSTEMS SOLUTION MANUAL Richard C Dorf Robert H Bishop University of California, Davis Marquette University A companion to MODERN CONTROL SYSTEMS TWELFTH EDITION Richard C Dorf Robert H Bishop Prentice Hall Upper Saddle River Boston Columbus San Francisco New York Indianapolis London Toronto Sydney Singapore Tokyo Montreal Dubai Madrid Hong Kong Mexico City Munich Paris Amsterdam Cape Town © 2011 Pearson Education, Inc., Upper Saddle River, NJ All rights reserved This publication is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458 P R E F A C E In each chapter, there are five problem types: Exercises Problems Advanced Problems Design Problems/Continuous Design Problem Computer Problems In total, there are over 1000 problems The abundance of problems of increasing complexity gives students confidence in their problem-solving ability as they work their way from the exercises to the design and computer-based problems It is assumed that instructors (and students) have access to MATLAB and the Control System Toolbox or to LabVIEW and the MathScript RT Module All of the computer solutions in this Solution Manual were developed and tested on an Apple MacBook Pro platform using MATLAB 7.6 Release 2008a and the Control System Toolbox Version 8.1 and LabVIEW 2009 It is not possible to verify each solution on all the available computer platforms that are compatible with MATLAB and LabVIEW MathScript RT Module Please forward any incompatibilities you encounter with the scripts to Prof Bishop at the email address given below www.elsolucionario.org The authors and the staff at Prentice Hall would like to establish an open line of communication with the instructors using Modern Control Systems We encourage you to contact Prentice Hall with comments and suggestions for this and future editions Robert H Bishop rhbishop@marquette.edu iii © 2011 Pearson Education, Inc., Upper Saddle River, NJ All rights reserved This publication is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458 T A B L E - O F - C O N T E N T S 10 11 12 13 iv Introduction to Control Systems Mathematical Models of Systems 22 State Variable Models 85 Feedback Control System Characteristics 133 The Performance of Feedback Control Systems 177 The Stability of Linear Feedback Systems 234 The Root Locus Method 277 Frequency Response Methods 382 Stability in the Frequency Domain 445 The Design of Feedback Control Systems 519 The Design of State Variable Feedback Systems 600 Robust Control Systems 659 Digital Control Systems 714 © 2011 Pearson Education, Inc., Upper Saddle River, NJ All rights reserved This publication is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458 C H A P T E R Introduction to Control Systems There are, in general, no unique solutions to the following exercises and problems Other equally valid block diagrams may be submitted by the student Exercises E1.1 A microprocessor controlled laser system: Controller Process www.elsolucionario.org Desired power output Error - Microprocessor Laser Power Sensor power A driver controlled cruise control system: Controller Process Foot pedal Desired speed Power out Measurement Measured E1.2 Current i(t) - Driver Car and Engine Actual auto speed Measurement Visual indication of speed E1.3 Speedometer Although the principle of conservation of momentum explains much of the process of fly-casting, there does not exist a comprehensive scientific explanation of how a fly-fisher uses the small backward and forward motion of the fly rod to cast an almost weightless fly lure long distances (the © 2011 Pearson Education, Inc., Upper Saddle River, NJ All rights reserved This publication is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458 CHAPTER Introduction to Control Systems current world-record is 236 ft) The fly lure is attached to a short invisible leader about 15-ft long, which is in turn attached to a longer and thicker Dacron line The objective is cast the fly lure to a distant spot with deadeye accuracy so that the thicker part of the line touches the water first and then the fly gently settles on the water just as an insect might Fly-fisher Desired position of the fly Controller - Wind disturbance Mind and body of the fly-fisher Process Rod, line, and cast Actual position of the fly Measurement Visual indication of the position of the fly E1.4 Vision of the fly-fisher An autofocus camera control system: One-way trip time for the beam Conversion factor (speed of light or sound) K1 Beam Emitter/ Receiver Beam return Distance to subject Subject Lens focusing motor Lens © 2011 Pearson Education, Inc., Upper Saddle River, NJ All rights reserved This publication is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458 Exercises E1.5 Tacking a sailboat as the wind shifts: Error Desired sailboat direction - Controller Actuators Sailor Rudder and sail adjustment Wind Process Sailboat Actual sailboat direction Measurement Measured sailboat direction Gyro compass E1.6 An automated highway control system merging two lanes of traffic: Controller Error Desired gap - Embedded computer Actuators Brakes, gas or steering Process Active vehicle Actual gap Measurement www.elsolucionario.org Measured gap Radar E1.7 Using the speedometer, the driver calculates the difference between the measured speed and the desired speed The driver throotle knob or the brakes as necessary to adjust the speed If the current speed is not too much over the desired speed, the driver may let friction and gravity slow the motorcycle down Controller Desired speed Error - Driver Actuators Throttle or brakes Measurement Visual indication of speed Speedometer Process Motorcycle Actual motorcycle speed © 2011 Pearson Education, Inc., Upper Saddle River, NJ All rights reserved This publication is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458 CHAPTER E1.8 Introduction to Control Systems Human biofeedback control system: Controller Desired body temp Process Hypothalumus - Message to blood vessels Actual body temp Human body Measurement Visual indication of body temperature E1.9 TV display Body sensor E-enabled aircraft with ground-based flight path control: Corrections to the flight path Desired Flight Path - Controller Aircraft Gc(s) G(s) Flight Path Health Parameters Meteorological data Location and speed Optimal flight path Ground-Based Computer Network Optimal flight path Meteorological data Desired Flight Path E1.10 Specified Flight Trajectory Health Parameters Corrections to the flight path Gc(s) G(s) Controller Aircraft Location and speed Flight Path Unmanned aerial vehicle used for crop monitoring in an autonomous mode: Trajectory error - Controller UAV Gc(s) G(s) Flight Trajectory Sensor Location with respect to the ground Map Correlation Algorithm Ground photo Camera © 2011 Pearson Education, Inc., Upper Saddle River, NJ All rights reserved This publication is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458 Exercises E1.11 An inverted pendulum control system using an optical encoder to measure the angle of the pendulum and a motor producing a control torque: Actuator Voltage Error Desired angle - Controller Process Torque Motor Pendulum Angle Measurement Measured angle E1.12 Optical encoder In the video game, the player can serve as both the controller and the sensor The objective of the game might be to drive a car along a prescribed path The player controls the car trajectory using the joystick using the visual queues from the game displayed on the computer monitor Controller Actuator Process www.elsolucionario.org Desired game objective Error - Player Joystick Measurement Player (eyesight, tactile, etc.) Video game Game objective © 2011 Pearson Education, Inc., Upper Saddle River, NJ All rights reserved This publication is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458 CHAPTER Introduction to Control Systems Problems P1.1 Desired temperature set by the driver An automobile interior cabin temperature control system block diagram: Error - Controller Process Thermostat and air conditioning unit Automobile cabin Automobile cabin temperature Measurement Measured temperature P1.2 Temperature sensor A human operator controlled valve system: Controller Process Error * Desired fluid output * - Tank Valve Fluid output Measurement Visual indication of fluid output * Meter * = operator functions P1.3 A chemical composition control block diagram: Controller Process Error Desired chemical composition - Mixer tube Valve Measurement Measured chemical composition Infrared analyzer Chemical composition © 2011 Pearson Education, Inc., Upper Saddle River, NJ All rights reserved This publication is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458 735 Advanced Problems 0.9 0.8 Amplitude 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.5 1.5 2.5 3.5 No of Samples FIGURE AP13.4 Step response with T = 0.1s AP13.5 The maximum gain for stability is Kmax = 63.15 www.elsolucionario.org Root Locus Unit circle (dashed line) 1.5 Imaginary Axis 0.5 System: sysz Gain: 63.2 Pole: 0.725 − 0.686i Damping: 0.00308 Overshoot (%): 99 Frequency (rad/sec): 7.58 −0.5 −1 −1.5 −2 −3 −2.5 −2 −1.5 −1 −0.5 Real Axis FIGURE AP13.5 Root locus for + K 0.004535z+0.004104 z −1.741z+0.7408 = 0 0.5 1.5 © 2011 Pearson Education, Inc., Upper Saddle River, NJ All rights reserved This publication is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458 736 CHAPTER 13 Digital Control Systems Design Problems The plant model with parameters given in Table CDP2.1 in Dorf and Bishop is given by: 26.035 , s(s + 33.142) Gp (s) = where we neglect the motor inductance Lm and where we switch off the tachometer feedback (see Figure CDP4.1 in Dorf and Bishop) Letting G(z) = Z G≀ (∫ )G√ (∫ ) we obtain G(z) = 1.2875e − 05(z + 0.989) (z − 1)(z − 0.9674) A suitable controller is D(z) = 20(z − 0.5) z + 0.25 The step response is shown below The settling time is under 250 samples With each sample being ms this means that Ts < 250 ms, as desired Also, the percent overshoot is P.O < 5% 1.2 0.8 Amplitude CDP13.1 0.6 0.4 0.2 0 50 100 150 No of Samples 200 250 300 © 2011 Pearson Education, Inc., Upper Saddle River, NJ All rights reserved This publication is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458 737 Design Problems DP13.1 (a) Given the sample and hold with Gp (s), we determine that KG(z) = K 0.1228 z − 0.8465 The root locus is shown in Figure DP13.1a For stablity: ≤ K < 15 Unit circle (dashed line) 1.5 Imag Axis 0.5 x -0.5 -1 -1.5 www.elsolucionario.org -1.5 -1 -0.5 0.5 1.5 Real Axis FIGURE DP13.1 0.1228 (a) Root locus for + K z−0.8465 = with unit circle (dashed line) (b) A suitable compensator is Gc (s) = 15(s + 0.5) s+5 Utilizing the Gc (s)-to-D(z) method (with T = 0.5 second), we determine D(z) = C z−A z − 0.7788 = 6.22 z−B z − 0.0821 We use the relationships A = e−aT , B = e−bT , and C 1−A a =K , 1−B b to compute A = e−0.5(0.5) = 0.7788 , B = e−0.5(5) = 0.0821 , and C = 6.22 © 2011 Pearson Education, Inc., Upper Saddle River, NJ All rights reserved This publication is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458 738 CHAPTER 13 Digital Control Systems (c) The step response is shown in Figure DP13.1b 0.8 0.7 0.6 Amplitude 0.5 0.4 0.3 0.2 0.1 0 10 12 14 16 18 20 No of Samples FIGURE DP13.1 CONTINUED: (b) Closed-loop system step response DP13.2 With the sample and hold (T=10ms), we have G(z) = 0.00044579z + 0.00044453 z − 1.9136z + 0.99154 A suitable compensator is D(z) = K z − 0.75 , z + 0.5 √ where K is determined so that ζ of the system is 1/ The root locus is shown in Figure DP13.2 We choose K = 1400 © 2011 Pearson Education, Inc., Upper Saddle River, NJ All rights reserved This publication is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458 739 Design Problems Root Locus 1.5 Curve of constant zeta=0.707 (dashed line) Imaginary Axis 0.5 −0.5 −1 −1.5 −5 −4 −3 −2 −1 Real Axis FIGURE DP13.2 0.00044579z+0.00044453 Root locus for + K z−0.75 = z+0.5 z −1.9136z+0.99154 www.elsolucionario.org The root locus is shown in Figure DP13.3a Curve of constant zeta=0.707 (dashed line) 1.5 0.5 Imag Axis DP13.3 o x x 0.5 -0.5 -1 -1.5 -2 -2 -1.5 -1 -0.5 1.5 Real Axis FIGURE DP13.3 z+1 (a) Root locus for + K (z−1)(z−0.5) = The gain for ζ = 0.707 is K = 0.0627 The step response is shown in Figure DP13.3b The settling time is Ts = 14T = 1.4s and P.O = 5% © 2011 Pearson Education, Inc., Upper Saddle River, NJ All rights reserved This publication is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458 740 CHAPTER 13 Digital Control Systems 1.2 Amplitude 0.8 0.6 0.4 0.2 0 10 12 14 16 18 No of Samples FIGURE DP13.3 CONTINUED: (b) Step response with K = 0.0627 With the sample and hold (T=1s), we have G(z) = 0.484(z + 0.9672) (z − 1)(z − 0.9048) Curve of constant zeta=0.5 (dashed line) 1.5 0.5 Imag Axis DP13.4 o x ox x -0.5 -1 -1.5 -2 -2 -1.5 -1 -0.5 0.5 Real Axis FIGURE DP13.4 0.484(z+0.9672) (a) Root locus for + K z−0.88 z+0.5 (z−1)(z−0.9048) = 1.5 20 © 2011 Pearson Education, Inc., Upper Saddle River, NJ All rights reserved This publication is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458 741 Design Problems 1.2 Amplitude 0.8 0.6 0.4 0.2 0 10 12 14 16 No of Samples FIGURE DP13.4 CONTINUED: (b) Step response for K = 12.5 A suitable compensator is www.elsolucionario.org D(z) = K z − 0.88 , z + 0.5 where K is determined so that ζ of the system is 0.5 The root locus is shown in Figure DP13.4a We choose K = 12.5 The step response is shown in Figure DP13.4b Also, Kv = 1, so the steady-state error specification is satisfied DP13.5 Select T = second With the sample and hold, we have G(z) = 0.2838z + 0.1485 − 1.135z + 0.1353 z2 The root locus is shown in Figure DP13.5 To meet the percent overshoot specification, we choose K so that ζ of the system is 0.7 This results in K = The step response has an overshoot of P.O = 4.6% Also, from Figure 13.21 in Dorf and Bishop, we determine that the steady-state error to a ramp input is ess = (since T /τ = 2, and Kτ = 0.3) © 2011 Pearson Education, Inc., Upper Saddle River, NJ All rights reserved This publication is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458 742 CHAPTER 13 Digital Control Systems Curve of constant zeta=0.7 (dashed line) 1.5 Imag Axis 0.5 o x x -0.5 -1 -1.5 -2 -2 -1.5 -1 -0.5 0.5 1.5 Real Axis FIGURE DP13.5 Root locus for + K z 20.2838z+0.1485 −1.135z+0.1353 = DP13.6 With the sample and hold at T = , we have G(z) = z2 Consider the digital controller 0.298z + 0.296 − 1.98z + 0.9802 Dz) = K z − 0.9 z + 0.6 The root locus is shown in Figure DP13.6 To meet the percent overshoot specification, we choose K so that ζ of the system is greater than 0.52 We select K = The step response has an overshoot of P.O = 11.9% and the settling time is Ts = 17.8s © 2011 Pearson Education, Inc., Upper Saddle River, NJ All rights reserved This publication is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458 743 Design Problems Root Locus 0.8 0.6 Imaginary Axis 0.4 0.2 −0.2 −0.4 −0.6 −0.8 −1 −5 −4 −3 −2 −1 Real Axis FIGURE DP13.6 0.298z+0.296 Root locus for + K z−0.9 z+0.6 z −1.98z+0.9802 = www.elsolucionario.org Step Response 1.4 System: syscl Peak amplitude: 1.12 Overshoot (%): 11.9 At time (sec): 1.2 System: syscl Settling Time (sec): 17.8 Amplitude 0.8 0.6 0.4 0.2 0 10 15 20 Time (sec) FIGURE DP13.6 CONTINUED: (b) Step response for K = 25 30 35 © 2011 Pearson Education, Inc., Upper Saddle River, NJ All rights reserved This publication is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458 744 CHAPTER 13 Digital Control Systems Computer Problems CP13.1 The m-file script and unit step response are shown in Figure CP13.1 num=[0.2145 0.1609]; den=[1 -0.75 0.125]; sysd = tf(num,den,1); step(sysd,0:1:50) 1.2 Amplitude 0.8 0.6 0.4 0.2 0 10 15 20 25 30 35 40 45 50 No of Samples FIGURE CP13.1 Step response CP13.2 The m-file script utilizing the c2d function is shown in Figure CP13.2 % Part (a) num = [1]; den = [1 0]; T = 1; sys = tf(num,den); sys_d = c2d(sys,T,'zoh') % % Part (b) num = [1 0]; den = [1 2]; T = 1; sys = tf(num,den); sys_d=c2d(sys,T,'zoh') FIGURE CP13.2 Script utilizing the c2d function for (a) and (b) Transfer function: z-1 Transfer function: 0.6985 z - 0.6985 -z^2 - 0.3119 z + © 2011 Pearson Education, Inc., Upper Saddle River, NJ All rights reserved This publication is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458 745 Computer Problems % Part (c) num = [1 4]; den = [1 3]; T = 1; sys = tf(num,den); sys_d = c2d(sys,T,'zoh') % % Part (d) num = [1]; den = [1 0]; T = 1; sys = tf(num,den); sys_d = c2d(sys,T,'zoh') Transfer function: z + 0.267 z - 0.04979 Transfer function: 0.1094 z + 0.01558 z^2 - z + 0.0003355 FIGURE CP13.2 CONTINUED: Script utilizing the c2d function for (c) and (d) CP13.3 The continuous system transfer function (with T = 0.1 sec) is T (s) = s2 13.37s + 563.1 + 6.931s + 567.2 The step response using the dstep function is shown in Figure CP13.3a The contrinuous system step response is shown in Figure CP13.3b www.elsolucionario.org 1.8 1.6 1.4 Amplitude 1.2 0.8 0.6 0.4 0.2 0 No of Samples FIGURE CP13.3 (a) Unit step response using the dstep function 10 12 14 © 2011 Pearson Education, Inc., Upper Saddle River, NJ All rights reserved This publication is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458 746 CHAPTER 13 Digital Control Systems 1.8 * 1.6 1.4 * 1.2 * * * * * * * * * * 0.8 * * 0.6 0.4 0.2 0* 0.2 0.4 0.6 0.8 1.2 1.4 FIGURE CP13.3 CONTINUED: (b) Continuous system step response (* denote sampled-data step response) The root locus in shown in Figure CP13.4 For stability: < K < 2.45 Root Locus 1.5 Imaginary Axis CP13.4 0.5 −0.5 −1 −1.5 −2 −2 −1.5 −1 FIGURE CP13.4 z Root locus for + K z −z+0.45 = −0.5 Real Axis 0.5 1.5 © 2011 Pearson Education, Inc., Upper Saddle River, NJ All rights reserved This publication is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458 747 Computer Problems CP13.5 The root locus in shown in Figure CP13.5 For stability: < K < ∞ 0.8 0.6 Imag Axis 0.4 0.2 -0.2 -0.4 -0.6 -0.8 -1 -1 -0.8 -0.6 -0.4 -0.2 0.2 Real Axis 0.4 0.6 0.8 FIGURE CP13.5 (z−0.2)(z+1) Root locus for + K (z−0.08)(z−1) = www.elsolucionario.org The root locus is shown in Figure CP13.6 Root Locus 1.5 Imaginary Axis CP13.6 0.5 0.5 1.5 1.5 0.5 Real Axis FIGURE CP13.6 Root locus 0.5 1.5 © 2011 Pearson Education, Inc., Upper Saddle River, NJ All rights reserved This publication is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458 748 CHAPTER 13 Digital Control Systems We determine the range of K for stability is 0.4 < K < 1.06 % Part (a) num=[1 4.25 ]; den=[1 -0.1 -1.5]; sys = tf(num,den); rlocus(sys), hold on xc=[-1:0.1:1];c=sqrt(1-xc.^2); plot(xc,c,':',xc,-c,':') hold off % % Part (b) rlocfind(sys) rlocfind(sys) ÈSelect a point in the graphics window selected_point = -0.8278 + 0.5202i ans = 0.7444 Kmax Select a point in the graphics window selected_point = -0.9745 - 0.0072i ans = 0.3481 Kmin FIGURE CP13.6 CONTINUED: Using the rlocus and rlocfind functions Using root locus methods, we determine that an acceptable compensator is Gc (s) = 11.7 s+6 s + 20 With a zero-order hold and T = 0.02 sec, we find that 1.2 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 0.8 Amplitude CP13.7 * * * 0.6 * * 0.4 * * 0.2 * * 0* * 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Time (sec) FIGURE CP13.7 System step response (* denotes sampled-data response) 0.8 0.9 © 2011 Pearson Education, Inc., Upper Saddle River, NJ All rights reserved This publication is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458 749 Computer Problems D(z) = 11.7z − 10.54 z − 0.6703 The closed-loop step response is shown in Figure CP13.7 www.elsolucionario.org ... long, which is in turn attached to a longer and thicker Dacron line The objective is cast the fly lure to a distant spot with deadeye accuracy so that the thicker part of the line touches the water... communication with the instructors using Modern Control Systems We encourage you to contact Prentice Hall with comments and suggestions for this and future editions Robert H Bishop rhbishop@marquette.edu... distance is to make contact with the ball with a high bat velocity This is more important than the bat’s weight, which is usually around 33 ounces (compared to Ty Cobb’s bat which was 41 ounces!)