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No part of thi s publication may be reproduced, stored in a retrieval system, or tran smitted, in any form or by any means, electronic, mechanical, photocop ying, recording, or otherwise, without the prior written permission of the publisher. Printed in the United States of America." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 15 "Initializations" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "with(linalg):\nwith(plots): \nwith(plottools):\nwith(inttrans):\nwith(student):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "alias(X=laplace(x(t),t,s), Y=laplace(y(t) ,t,s), F=laplace(f(t),t,s)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 12 "Introduction" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "In the s tudy and design of feedback control systems, engineers make use of the " }{TEXT 261 10 "root-locus" }{TEXT -1 204 " diagram which shows the \+ variation of eigenvalues as a function of a system parameter. The eig envalues are actually found as poles of an appropriate transfer functi on, a function of the complex variable " }{XPPEDIT 18 0 "s" "6#%\"sG" }{TEXT -1 232 " arising from the Laplace transform of the differential equations of the system. For each value of the system parameter, the poles are plotted in the complex plane. The resulting locus of point s in the complex plane is called the " }{TEXT 262 10 "root locus" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 305 "This section will give some mathematical insight into th e construction of a root-locus diagram. We begin with a study of the \+ \"locus of roots,\" and reserve the phrase \"root locus\" specifically for the case pertinent to controls engineering. (For a more complet e engineering perspective, see [1] or [2].)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 13 "Example 36.20 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "In Chapter 12, the mixing-tank problem led to " }{TEXT 259 1 "x" } {TEXT -1 4 "' = " }{TEXT 272 1 "A" }{TEXT 260 1 "x" }{TEXT -1 121 ", t he first-order linear system of ordinary differential equations with c onstant coefficients. The solution is given by " }{XPPEDIT 18 0 "exp( At)" "6#-%$expG6#%#AtG" }{TEXT 273 1 "x" }{XPPEDIT 18 0 "``[0]" "6#&%! G6#\"\"!" }{TEXT -1 8 ", where " }{TEXT 274 1 "x" }{XPPEDIT 18 0 "``[0 ]" "6#&%!G6#\"\"!" }{TEXT -1 3 " = " }{TEXT 275 1 "x" }{TEXT -1 9 "(0) , and " }{XPPEDIT 18 0 "exp(At)" "6#-%$expG6#%#AtG" }{TEXT -1 48 " is \+ the fundamental matrix. The eigenvalues of " }{XPPEDIT 18 0 "A" "6#% \"AG" }{TEXT -1 60 " determine stability of the equilibrium point at t he origin." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "Suppose the matrix " }{TEXT 276 1 "A" }{TEXT -1 24 " contains t he parameter " }{XPPEDIT 18 0 "c" "6#%\"cG" }{TEXT -1 58 ", making the system performance dependent on the value of " }{XPPEDIT 18 0 "c" "6# %\"cG" }{TEXT -1 20 ". For example, let " }{TEXT 277 1 "A" }{TEXT -1 14 " be the matrix" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "A := matrix(2,2,[1,c,3,4]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "The eigenvalues of " }{TEXT 278 1 "A" }{TEXT -1 23 " are then f unctions of " }{XPPEDIT 18 0 "c" "6#%\"cG" }{TEXT -1 19 ", and are giv en by " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "q := eigenvals(A):\nQ := simplify(subs(c=0,[q])):\ni f Q[1]>Q[2] then \nlambda[1] := q[1]:\nlambda[2] := q[2]:\nelse lambda [1]:=q[2]:lambda[2]:=q[1];fi;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 90 "(The additional Maple co mmands are to guarantee the eigenvalues are consistently labeled.)" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "These eig envalues are graphed as functions of " }{XPPEDIT 18 0 "c" "6#%\"cG" } {TEXT -1 16 " in Figure 36.7." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 270 "p1 := plot([lambda[1],lambd a[2]], c=-1..2, color=[black,red], linestyle=[1,2], labels=[c,``], lab elfont=[TIMES,ITALIC,12], xtickmarks=4, ytickmarks=6):\np2 := textplot (\{[.2,5.2,`l`],[1.8,5,`l+`],[.7,.7,`l-`]\}, font=[SYMBOL,12]):\ndispl ay([p||(1..2)], scaling=constrained);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "The solid (bl ack) curve is " }{XPPEDIT 18 0 "lambda[1]=lambda[`+`]" "6#/&%'lambdaG6 #\"\"\"&F%6#%\"+G" }{TEXT -1 36 ", whereas the dotted (red) curve is \+ " }{XPPEDIT 18 0 "lambda[2]=lambda[-``]" "6#/&%'lambdaG6#\"\"#&F%6#,$% !G!\"\"" }{TEXT -1 7 ". For " }{XPPEDIT 18 0 "c<=3/4" "6#1%\"cG*&\"\" $\"\"\"\"\"%!\"\"" }{TEXT -1 112 ", the eigenvalues are complex with p ositive real part, so the equilibrium point at the origin is unstable. For " }{XPPEDIT 18 0 "-3/4<=c" "6#1,$*&\"\"$\"\"\"\"\"%!\"\"F)%\"cG " }{XPPEDIT 18 0 "``<4/3" "6#2%!G*&\"\"%\"\"\"\"\"$!\"\"" }{TEXT -1 98 ", there are two positive real eigenvalues, the origin is a node (o ut) and is again unstable. For " }{XPPEDIT 18 0 "c*`>`" "6#*&%\"cG\" \"\"%\">GF%" }{TEXT -1 1 " " }{XPPEDIT 18 0 "4/3" "6#*&\"\"%\"\"\"\"\" $!\"\"" }{TEXT -1 98 ", one eigenvalue is poistive and the other is ne gative, so the origin is a saddle, again unstable." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "We have just seen that wh en " }{XPPEDIT 18 0 "c" "6#%\"cG" }{TEXT -1 141 " causes both eigenval ues to fall into the left-half of the complex plane, the origin is an \+ asymptotically stable equilibrium point, but when " }{XPPEDIT 18 0 "c " "6#%\"cG" }{TEXT -1 337 " causes at least one eigenvalue to fall int o the right-half of the complex plane, the origin is an unstable equil ibrium point. A plot of the eigenvalues as points in the complex plan e is an important tool in the design of feedback control systems which also can be modeled with systems of linear differential equations. I n the complex " }{XPPEDIT 18 0 "s" "6#%\"sG" }{TEXT -1 114 "-plane, th e curve traced by the eigenvalues as they vary with an appropriate par ameter of the system, is called a " }{TEXT 279 10 "root-locus" }{TEXT -1 62 ". Figure 36.8 shows, for this example, the locus of roots as \+ " }{XPPEDIT 18 0 "c" "6#%\"cG" }{TEXT -1 24 " varies in the interval \+ " }{XPPEDIT 18 0 "[-2,4]" "6#7$,$\"\"#!\"\"\"\"%" }{TEXT -1 18 ". (Ou r parameter " }{XPPEDIT 18 0 "c" "6#%\"cG" }{TEXT -1 79 " is not one w hich would be found in a typical problem in controls engineering.)" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 388 "p1 := complexplot(lambda[1], c=-2..4, color=black, thickness=3) :\np2 := complexplot(lambda[2], c=-2..4, color=cyan, thickness=3):\np3 := textplot(\{[2.5,2,`c = -2`], [2.5,-2.1,c = -2], [-1,.3,`c = 2`], [ 6,.3,`c = 2`]\}):\np4 := textplot(\{[2.7,1.4,`l+`],[2.7,-1.4,`l-`]\},f ont=[SYMBOL,12], align=RIGHT):\ndisplay([p||(1..4)],scaling=constraine d,xtickmarks=5,ytickmarks=4, labels=[` Re`,`Im `]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 107 "The initial and terminal points on each locus are indicated on the graph. The black curve is the locus of " }{XPPEDIT 18 0 "lambda[ 1]=lambda[`+`]" "6#/&%'lambdaG6#\"\"\"&F%6#%\"+G" }{TEXT -1 41 ", wher eas the cyan curve is the locus of " }{XPPEDIT 18 0 "lambda[2]=lambda[ -``]" "6#/&%'lambdaG6#\"\"#&F%6#,$%!G!\"\"" }{TEXT -1 8 ". When " } {XPPEDIT 18 0 "c=-2" "6#/%\"cG,$\"\"#!\"\"" }{TEXT -1 75 ", both eigen values are complex. In fact, they are complex conjugates. As " } {XPPEDIT 18 0 "c" "6#%\"cG" }{TEXT -1 19 " increases towards " } {XPPEDIT 18 0 "c=-3/4" "6#/%\"cG,$*&\"\"$\"\"\"\"\"%!\"\"F*" }{TEXT -1 51 ", the two eigenvalues approach the real axis. For " }{XPPEDIT 18 0 "-3/4`" "6# *&%\"cG\"\"\"%\">GF%" }{TEXT -1 1 " " }{XPPEDIT 18 0 "4/3" "6#*&\"\"% \"\"\"\"\"$!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "lambda[1]=lambda[`+ `]" "6#/&%'lambdaG6#\"\"\"&F%6#%\"+G" }{TEXT -1 40 " remains on the po sitive real axis, but " }{XPPEDIT 18 0 "lambda[2]=lambda[-``]" "6#/&%' lambdaG6#\"\"#&F%6#,$%!G!\"\"" }{TEXT -1 40 " is now found on the nega tive real axis." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "The following animation shows the dynamic development of \+ these loci as " }{XPPEDIT 18 0 "c" "6#%\"cG" }{TEXT -1 11 " increases. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 496 "f1 := t -> plot([Re(lambda[1]), Im(lambda[1]), c=-2. .t], color=black, thickness=3):\nf2 := t -> plot([Re(lambda[2]), Im(la mbda[2]), c=-2..t], color=cyan, thickness=3):\np4 := display([seq(f1(k /5),k=-10..10)],insequence=true):\np5 := display([seq(f2(k/5),k=-10..1 0)],insequence=true):\np6 := disk([2.5,2],.1,color=black):\np7 := disk ([2.5,-2],.1,color=cyan):\np8 := textplot(\{[2.9,2,`c = -2`],[2.9,-2,` c = -2`]\}):\ndisplay([p||(4..8)],scaling=constrained, labels=[`Re`,`I m`], xtickmarks=6, ytickmarks=5);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 13 "Example 36.21" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "The first-order linear s ystem " }{TEXT 263 1 "x" }{TEXT -1 4 "' = " }{TEXT 280 1 "A" }{TEXT 264 1 "x" }{TEXT -1 22 " for which the matrix " }{TEXT 281 1 "A" } {TEXT -1 12 " is given by" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "A := matrix(2,2,[c,-5,3,4]);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "has eigenvalues" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "q := eigenvals(A):\nQ:=simp lify(subs(c=20,[q])):\nif Q[1]>Q[2] then \nlambda[1] := q[1]:\nlambda[ 2] := q[2]:\nelse lambda[1]:=q[2]:lambda[2]:=q[1];fi;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "Figure 36.9 contains a graph of these eigenvalues as functions \+ of the parameter " }{XPPEDIT 18 0 "c" "6#%\"cG" }{TEXT -1 6 " with " } {XPPEDIT 18 0 "lambda[1]=lambda[`+`]" "6#/&%'lambdaG6#\"\"\"&F%6#%\"+G " }{TEXT -1 21 " shown in black, and " }{XPPEDIT 18 0 "lambda[2]=lambd a[-``]" "6#/&%'lambdaG6#\"\"#&F%6#,$%!G!\"\"" }{TEXT -1 15 " shown in \+ cyan." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 314 "pp1 := plot([lambda[1],lambda[2]], c=-10..20, color= [black,cyan], linestyle=[1,2], numpoints=500):\npp2 := textplot(\{[1,1 7,`l`],[18,14.5,`l+`],[18,4,`l-`], [-8,4,`l+`],[-8,-9,`l-`]\},font=[SY MBOL,12]):\ndisplay([pp||(1..2)], scaling=constrained,labels=[c,``], l abelfont=[TIMES,ITALIC,12], xtickmarks=7, ytickmarks=5);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "There are definitely values of " }{XPPEDIT 18 0 "c" "6#%\"cG" } {TEXT -1 75 " for which the eigenvalues are of opposite sign. There a re also values of " }{XPPEDIT 18 0 "c" "6#%\"cG" }{TEXT -1 84 " for wh ich the eigenvalues are complex. It is not yet clear if there are val ues of " }{XPPEDIT 18 0 "c" "6#%\"cG" }{TEXT -1 62 " for which both ei genvalues remain positive. Thus, we compute" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 265 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "limit(lambd a[-``],c=infinity)=4" "6#/-%&limitG6$&%'lambdaG6#,$%!G!\"\"/%\"cG%)inf inityG\"\"%" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 8 "that is," } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "Limit(lambda[2],c=infinity) = limit(lambda[2],c=infinity);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "to determine if, for large " }{XPPEDIT 18 0 "c" "6#%\"cG " }{TEXT -1 17 ", the eigenvalue " }{XPPEDIT 18 0 "lambda[2]=lambda[-` `]" "6#/&%'lambdaG6#\"\"#&F%6#,$%!G!\"\"" }{TEXT -1 31 " remains posit ive. Hence, for " }{XPPEDIT 18 0 "c" "6#%\"cG" }{TEXT -1 45 " large e nough, both eigenvalues are positive." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 36 "To determine the specific ranges of \+ " }{XPPEDIT 18 0 "c" "6#%\"cG" }{TEXT -1 73 " for which each outcome o ccurs, obtain the characteristic polynomial for " }{TEXT 282 1 "A" } {TEXT -1 15 ". Thus, obtain" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "char_poly := charpoly(A,s); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "for the characteristic polynomial, and" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "disc riminant := discrim(char_poly,s);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 129 "for its disc riminant. The transition between real and complex takes place where t he discriminant vanishes, so solve the equation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "q1 := discr iminant = 0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "for " }{XPPEDIT 18 0 "c=4" "6#/%\"cG\"\"% " }{TEXT -1 1 " " }{TEXT 283 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "2*s qrt(15)" "6#*&\"\"#\"\"\"-%%sqrtG6#\"#:F%" }{TEXT -1 67 " as the value s for transition between real and complex eigenvalues." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "Using Maple, these v alues are found in exact and floating-point form by" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "solve(q1, c);\nfsolve(q1,c);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "Figure 36.10 shows the locus of r oots for " }{XPPEDIT 18 0 "-5<=c" "6#1,$\"\"&!\"\"%\"cG" }{XPPEDIT 18 0 "``<=15" "6#1%!G\"#:" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 410 "p1 := complexplot(lamb da[1], c=-5..15, color=black, thickness=3):\np2 := complexplot(lambda[ 2], c=-5..15, color=cyan, thickness=3):\np3:=textplot(\{[-3,.3,`c = -5 `], [2,.3,c = -5], [6,.3,`c = 15`], [13.5,.3,`c = 15`]\}):\np4:=textpl ot(\{[7,3.1,`l+`],[7,-3.1,`l-`]\},font=[SYMBOL,12],align=RIGHT):\ndisp lay([p||(1..4)], scaling=constrained, xtickmarks=5, ytickmarks=4, labe ls=[` Re`,`Im `]);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "For " } {XPPEDIT 18 0 "c" "6#%\"cG" }{TEXT -1 14 " in the range " }{XPPEDIT 18 0 "-5<=c" "6#1,$\"\"&!\"\"%\"cG" }{XPPEDIT 18 0 "``<=4-2*sqrt(15)" "6#1%!G,&\"\"%\"\"\"*&\"\"#F'-%%sqrtG6#\"#:F'!\"\"" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "lambda[1]=lambda[`+`]" "6#/&%'lambdaG6#\"\"\"&F%6#%\"+G " }{TEXT -1 49 " (in black) is along the positive real axis, but " } {XPPEDIT 18 0 "lambda[2]=lambda[-``]" "6#/&%'lambdaG6#\"\"#&F%6#,$%!G! \"\"" }{TEXT -1 49 " (in cyan) is along the negative real axis. For \+ " }{XPPEDIT 18 0 "c" "6#%\"cG" }{TEXT -1 14 " in the range " } {XPPEDIT 18 0 "4-2*sqrt(15)<=c" "6#1,&\"\"%\"\"\"*&\"\"#F&-%%sqrtG6#\" #:F&!\"\"%\"cG" }{XPPEDIT 18 0 "``<=4+2*sqrt(15)" "6#1%!G,&\"\"%\"\"\" *&\"\"#F'-%%sqrtG6#\"#:F'F'" }{TEXT -1 141 ", the eigenvalues are comp lex conjugates. The locus of roots consists of the upper and lower ha lves of a circle with radius 4 and center at " }{XPPEDIT 18 0 "``(4,0) " "6#-%!G6$\"\"%\"\"!" }{TEXT -1 7 ". For " }{XPPEDIT 18 0 "c*`>`" "6 #*&%\"cG\"\"\"%\">GF%" }{TEXT -1 1 " " }{XPPEDIT 18 0 "4+2*sqrt(15)" " 6#,&\"\"%\"\"\"*&\"\"#F%-%%sqrtG6#\"#:F%F%" }{TEXT -1 77 ", both eigen values are on the positive real axis. We have already seen that " } {XPPEDIT 18 0 "lambda[2]=lambda[-``]" "6#/&%'lambdaG6#\"\"#&F%6#,$%!G! \"\"" }{TEXT -1 25 " tends towards the point " }{XPPEDIT 18 0 "``(4,0) " "6#-%!G6$\"\"%\"\"!" }{TEXT -1 40 ". The locus of roots indicates t hat as " }{XPPEDIT 18 0 "c" "6#%\"cG" }{TEXT -1 1 " " }{XPPEDIT 18 0 " -``" "6#,$%!G!\"\"" }{XPPEDIT 18 0 "`>`*infinity" "6#*&%\">G\"\"\"%)in finityGF%" }{TEXT -1 17 ", the eigenvalue " }{XPPEDIT 18 0 "lambda[1]= lambda[`+`]" "6#/&%'lambdaG6#\"\"\"&F%6#%\"+G" }{TEXT -1 53 " also ten ds to infinity along the positive real axis." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "The following animation shows t he dynamic development of this locus of roots." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 471 "f1 := t -> plot([Re(lambda[1]),Im(lambda[1]),c=-5..t], color=black, thickness=3) :\nf2 := t -> plot([Re(lambda[2]),Im(lambda[2]),c=-5..t], color=cyan, \+ thickness=3):\np4 := display([seq(f1(k/2),k=-10..30)],insequence=true) :\np5 := display([seq(f2(k/2),k=-10..30)],insequence=true):\nP1 := sub s(c=-5,lambda[1]):\nP2 := subs(c=-5,lambda[2]):\np6 := disk([P1,0],.2, color=black):\np7 := disk([P2,0],.2,color=cyan):\ndisplay([p||(4..7)], scaling=constrained, xtickmarks=8, ytickmarks=5);" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 37 "Transf er Functions and Block Diagrams" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "Recall that for a differential equ ation of the form" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 266 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "`y''`+2*`y'`+10*y=f(t)" "6#/,(%$y''G \"\"\"*&\"\"#F&%#y'GF&F&*&\"#5F&%\"yGF&F&-%\"fG6#%\"tG" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 8 "that is," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "q := diff(y(t),t,t)+ 2*diff(y(t),t) + 10*y(t) = f(t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "with the iner t initial conditions " }{XPPEDIT 18 0 "y(0)" "6#-%\"yG6#\"\"!" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "y" "6#%\"yG" }{TEXT -1 1 "'" }{XPPEDIT 18 0 "``(0)=0" "6#/-%!G6#\"\"!F'" }{TEXT -1 32 ", the Laplace transform l eads to" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 267 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "s^2*Y+2*s*Y+10*Y=F" "6#/,(*&%\"sG\"\"#%\"YG\"\" \"F)*(F'F)F&F)F(F)F)*&\"#5F)F(F)F)%\"FG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 8 "that is," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "q1 := subs(y(0)=0,D(y)(0)=0,laplace (q,t,s));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "and " }}{PARA 268 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Y(s)=F(s)/(s^2+2*s+10)" "6#/-%\"YG6#%\"sG*&-%\"FG6#F'\" \"\",(*$F'\"\"#F,*&F/F,F'F,F,\"#5F,!\"\"" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 7 "that is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "q2 := isolate(q1,Y);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "The function " }{XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" }{TEXT -1 110 " is an \"input\" to the system, providing the system with a dr iving force or excitation function. The solution " }{XPPEDIT 18 0 "y( t)" "6#-%\"yG6#%\"tG" }{TEXT -1 40 " is the system \"output,\" so the \+ function" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 269 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "T(s)=1/(s^2+2*s+10" "6#/-%\"TG6#%\"sG*&\"\"\" F),(*$F'\"\"#F)*&F,F)F'F)F)\"#5F)!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 8 "that is," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "T = coeff(rhs(q2),F);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "is called the " }{TEXT 265 17 "transfer function" }{TEXT -1 39 " for the differential equation. Since " }{XPPEDIT 18 0 "Y/F=T" "6#/* &%\"YG\"\"\"%\"FG!\"\"%\"TG" }{TEXT -1 235 ", the transfer function is characterized as the ratio of Laplace transforms of the system output over the system input. (See Section 6.11 for a determination of the \+ transfer function by means of the Dirac delta function.) Written as \+ " }{XPPEDIT 18 0 "Y(s)=T(s)*F(s)" "6#/-%\"YG6#%\"sG*&-%\"TG6#F'\"\"\"- %\"FG6#F'F," }{TEXT -1 24 ", the transfer function " }{XPPEDIT 18 0 "T (s)" "6#-%\"TG6#%\"sG" }{TEXT -1 41 " determines how the action of the input, " }{XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" }{TEXT -1 10 ", will be " }{TEXT 284 11 "transferred" }{TEXT -1 18 " into the output, " } {XPPEDIT 18 0 "y(t)" "6#-%\"yG6#%\"tG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 217 "The engineering appro ach to feedback control systems is couched in the language of transfer functions and block diagrams. To illustrate this approach, consider \+ the first-order linear system on the left in Table 36.2." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " `x'`=x+2*y" "6#/%#x'G,&%\"xG\"\"\"*&\"\"#F'%\"yGF'F'" }{TEXT -1 30 " \+ " }{XPPEDIT 18 0 "s*X=X+2*Y" "6#/*&%\"sG\" \"\"%\"XGF&,&F'F&*&\"\"#F&%\"YGF&F&" }{TEXT -1 32 " \+ " }{XPPEDIT 18 0 "X=G[1]*Y" "6#/%\"XG*&&%\"GG6#\"\"\"F)% \"YGF)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`y'`=-K*x+4*y+3*f(t)" "6#/%#y'G,(*&%\"KG\"\"\"%\"xGF(!\"\"*&\"\"%F( %\"yGF(F(*&\"\"$F(-%\"fG6#%\"tGF(F(" }{TEXT -1 11 " " } {XPPEDIT 18 0 "s*Y=-K*X+4*Y+3*F" "6#/*&%\"sG\"\"\"%\"YGF&,(*&%\"KGF&% \"XGF&!\"\"*&\"\"%F&F'F&F&*&\"\"$F&%\"FGF&F&" }{TEXT -1 17 " \+ " }{XPPEDIT 18 0 "Y=-K*G[2]*X+G[3]*F" "6#/%\"YG,&*(%\"KG\"\"\"& %\"GG6#\"\"#F(%\"XGF(!\"\"*&&F*6#\"\"$F(%\"FGF(F(" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 70 "_________________________________________ _____________________________" }}{PARA 0 "" 0 "" {TEXT -1 71 " \+ Table 36.2 " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "The syste m parameter " }{XPPEDIT 18 0 "K" "6#%\"KG" }{TEXT -1 19 " is often cal led a " }{TEXT 287 4 "gain" }{TEXT -1 115 " in the controls literature . The Laplace transform of each equation, in concert with the inert i nitial conditions " }{XPPEDIT 18 0 "x(0) = y(0);" "6#/-%\"xG6#\"\"!-% \"yG6#F'" }{TEXT -1 99 " = 0 leads to the equations in the center of T able 36.2. Solving the first of these equations for " }{XPPEDIT 18 0 "X" "6#%\"XG" }{TEXT -1 21 ", and the second for " }{XPPEDIT 18 0 "Y" "6#%\"YG" }{TEXT -1 54 " gives the equations on the right in Table 36. 2, where" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 270 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "G[1]=2/(s-1)" "6#/&%\"GG6#\"\"\"*&\"\"#F',&% \"sGF'F'!\"\"F," }{TEXT -1 1 " " }}{PARA 271 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "G[2]=1/(s-4)" "6#/&%\"GG6#\"\"#*&\"\"\"F),&%\"sGF)\"\"% !\"\"F-" }{TEXT -1 1 " " }}{PARA 272 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "G[3]=3/(s-4)" "6#/&%\"GG6#\"\"$*&F'\"\"\",&%\"sGF)\"\"%!\"\"F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 23 "are transfer functions. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "This \+ form of the differential equations, namely," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "q3 := diff(x(t),t ) = x(t)+2*y(t);\nq4 := diff(y(t),t) = -K*x(t)+4*y(t)+3*f(t);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 " is modeled by the block diagram in Figure 36.11." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 739 "p1 := arrow([0,0],[1,0],.1,.3,.2,color=black):\np2 := rectangle ([1,.5],[2,-.5]):\np3 := arrow([2,0],[3.25,0],.1,.3,.2,color=black):\n p4 := circle([3.5,0],.25):\np5 := arrow([3.75,0],[5,0],.1,.3,.2,color= black):\np6 := rectangle([5,.5],[6,-.5]):\np7 := arrow([6,0],[8,0],.1, .3,.2,color=black):\np8 := rectangle([5,-1.5],[6,-2.5]):\np9 := arrow( [7,0],[7,-2],.1,.3,.2,color=black):\np10 := arrow([7,-2],[6,-2],.1,.3, .2,color=black):\np11 := arrow([5,-2],[3.5,-2],.1,.3,.2,color=black): \np12 := arrow([3.5,-2],[3.5,-.25],.1,.3,.2,color=black):\np13 := text plot(\{[1.5,0,`G3`],[5.5,0,`G1`],[5.5,-2,`K G2`], [.2,.2,`F`], [4.25,. 2,`Y`], [7.3,.2,X], [3.25,-.45,`-`], [3.15,.3,`+`]\}, font=[TIMES,BOLD ,10]):\ndisplay([p||(1..13)], scaling=constrained, axes=none);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 150 "The circle is a junction node, the squares are \"blocks \" representing the transfer functions, and the secret to reading the \+ diagram is to realize that " }{XPPEDIT 18 0 "F" "6#%\"FG" }{TEXT -1 21 " represents a system " }{TEXT 285 5 "input" }{TEXT -1 6 ", and " } {XPPEDIT 18 0 "X" "6#%\"XG" }{TEXT -1 11 ", a system " }{TEXT 286 6 "o utput" }{TEXT -1 47 ". The loop containing the blocks representing " }{XPPEDIT 18 0 "G[1]" "6#&%\"GG6#\"\"\"" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "G[2]" "6#&%\"GG6#\"\"#" }{TEXT -1 323 " is the feedback loop. The surprise is that inhomogeneous first-order systems arising from mixing-tank problems, and studied in Chapter 12, can be interpre ted as feedback systems. Hence, feedback is inherently found in the c oupling of the differential equations, and not necessarily in an array of sensors and amplifiers." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 85 "The Laplace transform of each equation, in conc ert with the inert initial conditions " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "inits := \{x(0)=0,y(0)= 0\};" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "leads to" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "q5 := subs(inits,laplace(q3, t,s));\nq6 := subs(inits,laplace(q4,t,s));" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Solve the f irst equation for " }{XPPEDIT 18 0 "X(s)" "6#-%\"XG6#%\"sG" }{TEXT -1 22 ", and the second, for " }{XPPEDIT 18 0 "Y(s)" "6#-%\"YG6#%\"sG" } {TEXT -1 19 ", thereby obtaining" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "q7 := expand(isolate(q5,X)); \nq8 := isolate(q6,Y);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Define the transfer functions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "G[1]=2/(s-1)" "6#/&%\"GG6#\"\"\"*&\"\"#F',&%\"sGF'F'!\" \"F," }}{PARA 262 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "G[2]=1/(s-4) " "6#/&%\"GG6#\"\"#*&\"\"\"F),&%\"sGF)\"\"%!\"\"F-" }}{PARA 263 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "G[3]=3/(s-4)" "6#/&%\"GG6#\"\"$*&F'\" \"\",&%\"sGF)\"\"%!\"\"F-" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 99 "then rewrite the transformed differential equation s in terms of these transfer functions, obtaining" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "q9 := X = \+ G[1]*Y;\nq10 := Y = -K*G[2]*X+G[3]*F;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Finally, solv e for " }{XPPEDIT 18 0 "X(s)" "6#-%\"XG6#%\"sG" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "Y(s)" "6#-%\"YG6#%\"sG" }{TEXT -1 46 " in terms of the \+ transfer functions, obtaining" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "q11 := solve(\{q9,q10\},\{X, Y\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "The transfer function connecting the input " } {XPPEDIT 18 0 "f(t)" "6#-%\"fG6#%\"tG" }{TEXT -1 15 " to the output " }{XPPEDIT 18 0 "x(t)" "6#-%\"xG6#%\"tG" }{TEXT -1 8 " is then" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 264 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "T=G[3]*G[1]/(1+K*G[1]*G[2])" "6#/%\"TG*(&%\"GG6#\"\"$\" \"\"&F'6#F*F*,&F*F**(%\"KGF*&F'6#F*F*&F'6#\"\"#F*F*!\"\"" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "``(3/(s-4))*``(2/(s-1))/(1+K*(2/(s-1))*(1/(s-4)) )" "6#*(-%!G6#*&\"\"$\"\"\",&%\"sGF)\"\"%!\"\"F-F)-F%6#*&\"\"#F),&F+F) F)F-F-F),&F)F)*(%\"KGF)*&F1F),&F+F)F)F-F-F)*&F)F),&F+F)F,F-F-F)F)F-" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "The poles of " }{XPPEDIT 18 0 "T(s)" "6#-%\"TG6#%\"sG" } {TEXT -1 15 " are values of " }{XPPEDIT 18 0 "s" "6#%\"sG" }{TEXT -1 30 " for which the denominator of " }{XPPEDIT 18 0 "T(s)" "6#-%\"TG6#% \"sG" }{TEXT -1 54 " vanishes. The equation whose roots are the poles of " }{XPPEDIT 18 0 "T(s)" "6#-%\"TG6#%\"sG" }{TEXT -1 53 " is the ch aracteristic equation for the system matrix" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "A := genmatrix([rhs(q3),rhs(q4)],[x(t),y( t)]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "In fact, the characteristic polynomial for this matrix is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 14 "charpoly(A,s);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "and the eigen values of " }{TEXT 290 1 "A" }{TEXT -1 4 " are" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "eigenvals(A );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 19 "The denominator of " }{XPPEDIT 18 0 "T(s)" "6#-%\" TG6#%\"sG" }{TEXT -1 3 " is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "q12 := 1+2*K/(s-1)/(s-4);" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 47 "which, if written with a common denominator, is" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "normal(q12);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "The denominator of " }{XPPEDIT 18 0 "T(s)" "6#-%\"TG6#%\"sG" }{TEXT -1 48 " vanishes when the characteri stic polynomial of " }{TEXT 289 1 "A" }{TEXT -1 53 " vanishes! Thus, \+ the poles of the transfer function " }{XPPEDIT 18 0 "T(s)" "6#-%\"TG6# %\"sG" }{TEXT -1 4 " are" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "solve(q12 = 0,s);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "w hich are precisely the eigenvalues of " }{TEXT 288 1 "A" }{TEXT -1 166 ". Thus, the stability of the system is determined by the locatio n of the poles of the transfer function because these poles are the ei genvalues of the system matrix." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 27 "The prevalence of the form " }{XPPEDIT 18 0 "1+K*H(s)" "6#,&\"\"\"F$*&%\"KGF$-%\"HG6#%\"sGF$F$" }{TEXT -1 45 " in the denominator of the transfer function " }{XPPEDIT 18 0 "T(s)" "6#-%\"TG6#%\"sG" }{TEXT -1 20 " leads to the Maple " }{TEXT 266 9 "ro otlocus" }{TEXT -1 50 " command which draws the root locus for solutio ns " }{XPPEDIT 18 0 "s=s(K)" "6#/%\"sG-F$6#%\"KG" }{TEXT -1 17 " of th e equation " }{XPPEDIT 18 0 "1+K*h(s)=0" "6#/,&\"\"\"F%*&%\"KGF%-%\"hG 6#%\"sGF%F%\"\"!" }{TEXT -1 33 ". The inputs to the command are " } {XPPEDIT 18 0 "H(s)" "6#-%\"HG6#%\"sG" }{TEXT -1 17 " and a range for \+ " }{XPPEDIT 18 0 "K" "6#%\"KG" }{TEXT -1 124 ". The use of this comma nd, and the mathematics it takes to draw the root locus, will be explo red in the following examples." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 " " {TEXT -1 13 "Example 36.22" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "Suppose the denominator of a trans fer function for a feedback system is " }{XPPEDIT 18 0 "1+k*h(s)" "6#, &\"\"\"F$*&%\"kGF$-%\"hG6#%\"sGF$F$" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "h(s)" "6#-%\"hG6#%\"sG" }{TEXT -1 12 " is given by" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "h1 : = (s+1)/s^2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Figure 36.12, created by Maple's " } {TEXT 267 9 "rootlocus" }{TEXT -1 107 " command, provides a graph of t hat part of the root locus corresponding to the range provided for the gain " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 26 ". Thus, we have th e graph" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "rootlocus(h1, s, 0..5, scaling=constrained,thickness =3, xtickmarks=4, ytickmarks=3, labels=[` Re`,`Im `]);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 166 "In any design process based on this root-locus diagram, \+ it would be important to know where in the complex s-plane the roots ( eigenvalues) were for various values of " }{XPPEDIT 18 0 "k" "6#%\"kG " }{TEXT -1 58 ". There are a variety of ways to obtain this informat ion." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 88 "F irst, form the denominator of the transfer function and set it equal t o zero, obtaining" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 33 "unassign('k'):\nq := 1 + k*h1 = 0;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Adding fractions leads to" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "normal(q);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "so the equation is equivalent to the vanishing of the numerator . Thus," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "q1 := numer(normal(lhs(q))) = 0;" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "is a ctually the characteristic equation for the system." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The roots as a function of " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 11 ", that is, " } {XPPEDIT 18 0 "s=s(k)" "6#/%\"sG-F$6#%\"kG" }{TEXT -1 24 ", would then be given by" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "q2 := solve(q,s);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "The character istic equation is quadratic in " }{TEXT 256 1 "s" }{TEXT -1 56 " so th ere are two roots. However, each root depends on " }{XPPEDIT 18 0 "k " "6#%\"kG" }{TEXT -1 31 " so each root is a function of " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 104 ". As we did earlier, we could anima te the tracing of the root locus, obtaining the following animation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 482 "f1 := t -> plot([Re(q2[1]), Im(q2[1]), k=-1..t], color=black, thickness=3):\nf2 := t -> plot([Re(q2[2]), Im(q2[2]), k=-1..t], color =cyan, thickness=3):\np4 := display([seq(f1(n/2),n=-2..10)],insequence =true):\np5 := display([seq(f2(n/2),n=-2..10)],insequence=true):\np6 : = disk([-.62,0],.1,color=cyan):\np7 := disk([1.62,0],.1,color=black): \np8 := textplot(\{[-.62,.2,`k = -1`],[1.62,.2,`k = -1`]\}):\ndisplay( [p||(4..8)],scaling=constrained, labels=[`Re`,`Im`], xtickmarks=5, yti ckmarks=3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "Another approach to studying the dependen ce of the characteristic roots on the gain " }{XPPEDIT 18 0 "k" "6#%\" kG" }{TEXT -1 26 " is to solve the equation " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "q;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "for " }{XPPEDIT 18 0 "k=k(s)" "6#/%\"kG-F$6#%\"sG" }{TEXT -1 13 " rather than " }{XPPEDIT 18 0 "s=s(k)" "6#/%\"sG-F$6#%\"kG" }{TEXT -1 13 ". This gives" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "K := solve(q,k);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "But in t he complex plane, " }{XPPEDIT 18 0 "s" "6#%\"sG" }{TEXT -1 25 " is the complex variable " }{XPPEDIT 18 0 "s=x+i*y" "6#/%\"sG,&%\"xG\"\"\"*&% \"iGF'%\"yGF'F'" }{TEXT -1 14 ". What about " }{XPPEDIT 18 0 "k=k(x+i *y)" "6#/%\"kG-F$6#,&%\"xG\"\"\"*&%\"iGF)%\"yGF)F)" }{TEXT -1 19 "? F or this, we get" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 25 "q3 := subs(s = x+I*y, K);" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "What do we know about " }{XPPEDIT 18 0 "k(x+i*y)" "6#-%\"kG6#,&%\"xG\"\"\" *&%\"iGF(%\"yGF(F(" }{TEXT -1 13 "? Only that " }{XPPEDIT 18 0 "k" "6 #%\"kG" }{TEXT -1 24 " is real. Hence, split " }{XPPEDIT 18 0 "k(x+i* y)" "6#-%\"kG6#,&%\"xG\"\"\"*&%\"iGF(%\"yGF(F(" }{TEXT -1 76 " into it s real and imaginary parts and set the imaginary part equal to zero." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "q4 := simplify(evalc(Re(q3)));\nq5 := simplify(evalc(Im(q3))); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "The equation " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Im(k) = -y*(x^2+y^2+2 *x)/(x^2+2*x+1+y^2)" "6#/-%#ImG6#%\"kG,$*(%\"yG\"\"\",(*$%\"xG\"\"#F+* $F*F/F+*&F/F+F.F+F+F+,**$F.F/F+*&F/F+F.F+F+F+F+*$F*F/F+!\"\"F6" } {TEXT -1 4 " = 0" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "implicitly defines a curve y = y(x). In this example we \+ can obtain this curve explicitly. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "solve(q5,y);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "However, the vanishing of the numerator leads to the equation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "completesquare(numer(q5)/(-y),x) = 0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "which is the implicit form of a circle with radius 1 and center at " } {XPPEDIT 18 0 "``(-1,0)" "6#-%!G6$,$\"\"\"!\"\"\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 94 "The sta ndard approach to constructing a root locus is numeric. For a fixed v alue of the gain " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 14 ", the eq uation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1+k*(s+1)/s^2 = 0" "6#/,&\"\"\"F%*(%\"kGF%,&%\"s GF%F%F%F%*$F)\"\"#!\"\"F%\"\"!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 69 "is solved numerically for the two corresp onding characteristic roots " }{XPPEDIT 18 0 "s(k)" "6#-%\"sG6#%\"kG" }{TEXT -1 19 ". For example, if " }{XPPEDIT 18 0 "k=1" "6#/%\"kG\"\" \"" }{TEXT -1 29 ", the corresponding roots are" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "fsolve(subs (k=1,q1),s,complex);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "If this process is repeated for a sequence of values of " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 40 ", \+ a collection of points in the complex " }{XPPEDIT 18 0 "s" "6#%\"sG" } {TEXT -1 38 "-plane will result. For example, for " }{XPPEDIT 18 0 "k [n]=n/10,n=0,1,`...`,50" "6'/&%\"kG6#%\"nG*&F'\"\"\"\"#5!\"\"/F'\"\"!F )%$...G\"#]" }{TEXT -1 30 ", the collection of solutions " }{XPPEDIT 18 0 "s(k)" "6#-%\"sG6#%\"kG" }{TEXT -1 9 " would be" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "q6 := [se q(fsolve(subs(k=n/10,q1),s,complex),n=0..50)];" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "A plot of these points would then give an outline of the root locus. Here, \+ Maple's " }{TEXT 257 11 "complexplot" }{TEXT -1 18 " command from the \+ " }{TEXT 258 5 "plots" }{TEXT -1 104 " package will plot a list of com plex numbers as points in the complex plane, giving the following grap h." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "complexplot(q6, style=point, scaling=constrained);" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 48 "This is, in fact, the algorithm used by Maple's " } {TEXT 271 9 "rootlocus" }{TEXT -1 114 " command, and by the equivalent functionality in other commercial packages which provide the root loc us. Maple's " }{TEXT 270 9 "rootlocus" }{TEXT -1 78 " command also co ntains heuristics for joining the points together into curves." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 13 "Example 36.23" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "Le t the function " }{XPPEDIT 18 0 "h(s)" "6#-%\"hG6#%\"sG" }{TEXT -1 3 " be" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "h2 := (s+1)/s^3;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "The root locu s (Figure 36.13), that is, the curve " }{XPPEDIT 18 0 "s=s(k)" "6#/%\" sG-F$6#%\"kG" }{TEXT -1 35 " defined implicitly by the equation" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "1+k*h(s)=0" "6#/,&\"\"\"F%*&%\"kGF%-%\"hG6#%\"sGF%F%\" \"!" }}{PARA 0 "" 0 "" {TEXT -1 18 "can be obtained by" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "rootl ocus(h2, s, 0..5, xtickmarks=2, scaling=constrained, labels=[` \+ Re`,`Im `], thickness=3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "Maple's " } {TEXT 268 9 "rootlocus" }{TEXT -1 65 " command draws the root locus by numerically computing solutions " }{XPPEDIT 18 0 "s(k)" "6#-%\"sG6#% \"kG" }{TEXT -1 26 " for a range of values of " }{XPPEDIT 18 0 "k" "6# %\"kG" }{TEXT -1 105 ". To see this, we first obtain the characterist ic equation. The denominator of the transfer function is" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "q7 : = 1+k*h2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "and the characteristic equation is" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "q8 := numer(normal(q7)) = 0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "Points on the root locus are obtained by fixing " }{XPPEDIT 18 0 "k" "6#%\"kG" } {TEXT -1 40 " at some value, and computing the roots " }{XPPEDIT 18 0 "s(k)" "6#-%\"sG6#%\"kG" }{TEXT -1 19 ". For example, if " }{XPPEDIT 18 0 "k=1/10" "6#/%\"kG*&\"\"\"F&\"#5!\"\"" }{TEXT -1 62 ", then the t hree roots of the characteristic equation would be" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "fsolve(subs (k=.1,q8),s,complex);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "Computing and plotting in the co mplex " }{XPPEDIT 18 0 "s" "6#%\"sG" }{TEXT -1 69 "-plane a large numb er of such points leads to the root-locus diagram." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 "Alternatively, since the \+ characteristic equation is just a cubic in " }{XPPEDIT 18 0 "s" "6#%\" sG" }{TEXT -1 38 ", we can obtain the analytic solution " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "q9 : = solve(q8,s);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 88 "There are three branches, each of \+ which can be plotted in the complex plane via Maple's " }{TEXT 269 11 "complexplot" }{TEXT -1 126 " command. The result is the following ve rsion of the root locus, where the three branches are shown in black, \+ red, and green." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 331 "S:=[seq(n*.0011,n=1..100),seq(n*.1,n=1..50)]: \np1 := complexplot(q9[1],k=0..5,color=black, thickness=3, sample=S): \np2 := complexplot(q9[2],k=0..5,color=red, thickness=3, sample=S):\np 3 := complexplot(q9[3],k=0..5,color=green, thickness=3, sample=S):\ndi splay([p||(1..3)], scaling=constrained, xtickmarks=[-1,0,1], view=[-1. .1,-3..3]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "For " }{XPPEDIT 18 0 "k=0" "6#/%\"kG\"\"! " }{TEXT -1 20 ", the roots are all " }{XPPEDIT 18 0 "s=0" "6#/%\"sG\" \"!" }{TEXT -1 87 ", so the three branches of the root locus start at \+ the origin, and become unbounded as " }{XPPEDIT 18 0 "k" "6#%\"kG" } {TEXT -1 11 " increases." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "References" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 129 "[1] Charles L. Phillips and Royce D. Harbor, Basic Feedb ack Control Systems, Alternate Second Edition, Prentice-Hall, Inc., 19 91." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 139 "[ 2] Gene F. Franklin, J. David Powell and Abbas Emami-Naeini, Feedback \+ Control of Dynamic Systems, Addison-Wesley Publishing Company, 1986." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{MARK "1" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }