f s O L l: T I O N S 1\1 .\ N ll A L to accompany DIGITAL CONTROL I \ OF DYNAMIC SYSTEMS Third Edition Solutions Manual to accompany DIGITAL CONTROL OF DYNAMIC SYSTEMS Solutions Manual to accompany DIGITAL CONTROL OF DYNAMIC SYSTEMS Third Edition by Gene F. Franklin Stanford University J. David Powell Stanford University Michael L. Workman IBM Corporation Prepared by Chen-Fang Chang Carlos Coelho Gene F. Franklin Deirdre Meldrum J. David Powell Micheal L. Work.man 'lff~ ADDISON-WESLEY An imprint of Addison Wesley Longman. Inc. Menlo Park, Califomia • Rcatling, Masschusctts • New York• Harlow, E11{:lt11ul Bl!rkelc·~·. California• Dun Mills, Ontario• Sydney• Bon11 • Amsterdam• Tokyo• Mexico Cir~ Solutions Manual to accompany Fraklin/Powell's DIGITAL CONTROL OF DYNAMIC SYSTEMS, Third Edition Copyright© 1998 by Addison Wesley Longman, Inc. All riglits reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or any other media embodiments now known of hereafter to become known, without the prior written permission of the publisher. Manufactured in the United States of America. ISBN 0-201-82056-0 123 4 5 6 7 8 9 10-VG-Ol 00 99 98 Addison Wesley Longman, Inc. 2725 Sand Hill Road Menlo Park, CA 94025 SOLUTI ON.5 TO CHAPTER l PROBLEM.S 1.1 Suppose a r:idar search antenna at the San Francisco airport rotates at 6 rev/min. and JJta points corresponding to the position of flight 108 l :ire plotted on the controller's scre:.:n once per antenn:i revolution. Flight 1081 is tr.ivcling directly tow:ird the airport at 540 mi/hr. A feedback control system is est.iblished through the controller who gives course corrections to the pilot. He wishes to do so each 9 mi of tr.ivel of the aircraft. and his instructions consist of course headings in integral degree values. (:1) What is the sampling rate, in seconds, of the r,mge sign:il plotted on the r.id.ir screen? (b) Wh:it is the s.impling rote, in seconds, of the controller's instructions? (c) Identify the following sign.als as continuous, discrete, or digit:il: i. the .iircr:ift's r.inge from the airport, ii. the range d:ita as plotted on the rad:ir screen, iii. the controller's instructioas to the pilot, iv. the pilot's actions on the aircr:ift control surfaces. (d) Is this a continuous, sampled-data. or digital control system? (e) Show that it is possible for the pilot of flight 1081 to fly .i zigzag course which would show up as a straight line on the controller's screen. What is the (lowest) frequency of a sinusoidal zigzag course which will be hidden from the controller's r:id:ir? o. ( 0 s e'-• = i I{ 2: SO..Y\pl~ ,ose«. b) IAstruc.tton S<1mplt."j-: 9 mi. . 1 hr. . G,Orn~:\ •• ~OsH. - 1 SOl'f\ pl t 5 4 0 mi. l hr l mi1 •. - sam,I «. = c.01 rc1 1-\~ <.) L) • cotrti ·" yous. l~ of ra,,g e. s 1 ·, v) Pi lot's I .S °'- cof\trol syste.f't1. -1- e) i ! T-:;. 10 S-2<.. f . I . t "-' = - = o.o5 Hi: nj ~ t Z. T .. sin os)-t) (Zlff'"(O· 1.2. If a signal varies between O and 10 volts (called the dynamic range) and it is required th11t the signal must be represented in the digital computer to the nearest 5 millivolts, that Is, if the resolution must be S mv, determine how m.iny bits the analog-to-digital convener must have. A r~ oL u ti o" of 5 rn V -= • 0 u 5 V fer -the. j . of d ~ o ~.n·, <... 1-0.n 3 <.. O ic IO v Is l par f- .. oos II f\ 2-000 ' - _J_' - Zt<.10 ) • (; To repre.~ent 2oc.,o bif\&trj rtt,Uires a+ 1ri leo.51- \ti b;t.s.} (b:,..(Jf'j d.~~;+j) -- SI f\ (. e. c.,..j. 11 io+5. -2- l.J Describe five digita~ ~on~I systems that you are familiar with. State what you think the advantages of the d1g1tal implementation are over an analog implementation. s· -h t a u. o V\. .. ec..) A"" ~. c.-t-°f+ c-l~a 'ria..l ~--kp ,l 01-. ~a~ ,'er -+o ~Ov\.d e. ~oJ~ S Ch.0 <jCUII\.£. b) A cL s k ~;:( r, ve 5 e r v o . Lo S t <'°- -{!o r ft.a c k o-.f -f ju.1-'Ar\lAA ~a.....+ e-ed. L:,.qc k- co~+r-otl.s?-r > l.t.Aor~ a. c::c u ro..=t'~. +u ""a c.) € a'IM.- e L l ~ ec f,o I.,\.. C>V\e,_ Se;- ~QA.d ~C+l re _ 6-c pro~av.,..IM.Q.d &L~ffe.re.v..-+l, -fu ~· Cot.-.. o. \.AA a. \J\ 1 lAA. C) e l s . =r (.,"' ( cl ) AV\ t->- s ~' J b ra.~"'s. "8 k {C, (c. Wv.C- ea.sl~ w~i~ a.. cLl:s1~ 57!delAA.. +u ("" t-k~o {o ..f. ~'a e ') \I\C( ~ ~ ,{c:,-Q. VV'-""-~e .s ~ 1- ea s1 -(b ~ ~ l7 f royto..""""-S, 1.4 Historically, house heating system thermostats were a bimetallic strip ihat would make or b~ak the contact. depending on temperature. Today, most thermostats are digital Dcscnbe how you think they work and list some of their benefits. · 5 °t\ c) ( LL 0 \A.. ', A Sa..wy,kJ ~7r·~i.,._ A/b t-~er~s~ev- ,~ +o {u...-11\..a~ Co1A\J..el\.{eV"" a..~ lo~'c:... i..J. -~ = ~ o r ,'s. ( ocl. e) I -t-L_ ,)-< f, r, D ,[{ LA.. SS 2CIS( -h, i111. clv..de Jci,;r\'r7o t\-cue.s-)... wee~ Je':>-1,~J ""tefro..~ -3- ht fl t.S Use MAll.AB (obtain a copy of the Student Edition or use what's available to you) and = = plot y vs .r ior .t I to IO where y .r:. Label each axis and put :i. title on it » x=l:10; » y=x."2; » plot(x,y) » xlabel('x') » y 1 abe 1 ( ' y ' ) » title ( 'Problem 1. 5' ) Problem 1.5 /. 90 ~ I ~ 80 I I 70 1 ~ ~ 60 I j >- 50 I I ...'. ~ I I I I -j I J I I I a 7 9 10 x -4- 1.6 Use MATLAB (obtain a copy of the Student Edition or use what's available to you) and make two plots (use MATLAB='s . s;u;b plot) of y vs .t for .t = I to 10. Put a plot of y = .t2 on the top of the page and y on the bottom. » x=l:10; » y=x."2; » subplot(2,1,1) » plot(x,y) » » y=sqrt (x) ; 100 ,-------.- » subplot(2,1,2) » plot(x,y) 80 ' t I 60 40 ..; i I 20 oL.=:=:::=:r=.:...~---L~~......1...~~_J,_~~..l-~~J...._~~l--~---1.~~_J 1 2 3 4 5 6 7 8 9 10 3.5 3 2.5 i i 1 2 I 1.5 1 1 1 2 3 4 5 6 7 8 9 10 -s-- • 2.1 Design feedb3Ck with lc:id compcns:ition for the open-loop system =~ JO- G(s) s· The rise time should be I sec or less and the overshoot should be less th.in 10%. :::>o l~ 1\o"'- : Iie,c...·~ e ·--tx-s- -~sl \ / = ~r 'b(S) % Prob 2.1 3 I clear elf 2 root5 a::.:. 5; I~-: b::.:100; 1 --e- sys=tf(lO, [1 0 OJ) *tf( [1 a), [l b)) !(I)o K=0:.5:100; subplot(2,l,l) f--•.:.. .. rlocus(sys,K) ~ axis([-10 1 -3 3)) -1 [k,p)=rlocfind(sys) damp(p) -2 sysCL=feedhack(k*sys,l) subplot(2,l,2) step(sysCL) -3 y=step(sysCL); -4 -2 0 ymax::.max(y) Real Axis Step Response 1.s ---r-----,,----,-------,----.---~ 1 1 . ... . 0.5 oo;-~~~~1~~~~2-;;-~~~~3:--~~~L4~~~~sL--~~__js Time (sec.) s s +C5, 40 5+1Cb - I'\,- 6. to ,J.t;. ' (8%J I, 06 -{f; Kf <'. I 0% 15 cl,. C«. l-\-- . -~-