c) L Û µ Ä d) Feedback ratio = L Û -...

Automatic Control Systems, 9th Edition Chapter 3 Solutions Golnaraghi, Kuo Chapter 3__________________________________________________________________________

3-1) a)

b)

c)

d) Feedback ratio = G H

e)

3-2)

Characteristic equation: 1 2 1 0

3 2 1 0

3‐1

Automatic Control Systems, 9th Edition Chapter 3 Solutions Golnaraghi, Kuo 3-3)

11

1

1 HGG−

11

1

1 HGG

−

2

2

GH

11

21

1 HGGG

− 2

23 G

HG +

1

1 1

1

3‐2


332

2

1 HGGG

+

G1 G3

H1

X+

- 22332

2

1 HGHGGG

++ Y

3‐3

, 9th Edition A Automatic Control Systems Chapter 3 Solutionns Golnarraghi, Kuo

3

X

3-5)

+

-

1

332

21

1 HGGGG

+G

3‐4

3

1

GH

G

22

3

HGY

+

Automatic Control Systems, 9th Edition Chapter 3 Solutions Golnaraghi, Kuo 3-6) MATLAB

syms s G=[2/(s*(s+2)),10;5/s,1/(s+1)] H=[1,0;0,1] A=eye(2)+G*H B=inv(A) Clp=simplify(B*G) G = [ 2/s/(s+2), 10] [ 5/s, 1/(s+1)] H = 1 0 0 1 A = [ 1+2/s/(s+2), 10] [ 5/s, 1+1/(s+1)] B = [ s*(s+2)/(s^2-48*s-48), -10/(s^2-48*s-48)*(s+1)*s] [ -5/(s^2-48*s-48)*(s+1), (s^2+2*s+2)*(s+1)/(s+2)/(s^2-48*s-48)] Clp = [ -2*(24+25*s)/(s^2-48*s-48), 10/(s^2-48*s-48)*(s+1)*s] [ 5/(s^2-48*s-48)*(s+1), -(49*s^2+148*s+98)/(s+2)/(s^2-48*s-48)]

3‐5


33-7)

3

3-8)

3‐6


33-9)

3

3

3-10)

3-11)

3‐7


33-12)

33-13)

ssyms t ff=100*(1‐0.33*exp(‐6*t)‐00.7*exp(‐10*t)) FF=laplace(f) ssyms s

FF=eval(F) GGc=F*s MM=30000 ssyms K OOlp=simplify(KK*Gc/M/s) KKt=0.15 CClp= simplify(Olp/(1+Olp*KKt)) ss=0 EEss=eval(Clp) ff = 1100‐30*exp(‐66*t)‐70*exp(‐‐10*t) FF = 880*(11*s+75)/s/(s+6)/(s+110) aans =

3‐8


/s/(s+6)/(s+110) (880*s+6000)

GGc = (880*s+6000)/(s+6)/(s+100) MM = 30000 OOlp = 11/375*K*(11**s+75)/s/(s+66)/(s+10) KKt = 0.1500 CClp = 2 s E2 3

20/3*K*(11*s

s = 0

Ess = 20/3

3-14)

s+75)/(2500*s^3+40000*ss^2+150000*ss+11*K*s+75*K)

3‐9

Automatic Control Systems, 9th Edition Chapter 3 Solutions Golnaraghi, Kuo

3-15)

Note: If ‐1G(s) = g(t), then ‐1{e‐asG(s)} = u(t ‐ a) • g(t ‐ a)

syms t s f=100*(1‐0.3*exp(‐6*(t‐0.5))) F=laplace(f)*exp(‐0.5*s)

F=eval(F) Gc=F*s M=30000 syms K Olp=simplify(K*Gc/M/s) Kt=0.15 Clp= simplify(Olp/(1+Olp*Kt)) s=0 Ess=eval(Clp) digits (2) Fsimp=simplify(expand(vpa(F))) Gcsimp=simplify(expand(vpa(Gc))) Olpsimp=simplify(expand(vpa(Olp))) Clpsimp=simplify(expand(vpa(Clp))) f = 100‐30*exp(‐6*t+3) F = (100/s‐30*exp(3)/(s+6))*exp(‐1/2*s) F = (100/s‐2650113767660283/4398046511104/(s+6))*exp(‐1/2*s) Gc = (100/s‐2650113767660283/4398046511104/(s+6))*exp(‐1/2*s)*s M = 30000 Olp = ‐1/131941395333120000*K*(2210309116549883*s‐2638827906662400)/s/(s+6)*exp(‐1/2*s) Kt = 0.1500 Clp =

3‐10


2520/3*K*(221052776558133

03091165498324800000*s+

883*s‐263882+2210309116

279066624006549883*K*e

0)*exp(‐1/2*sexp(‐1/2*s)*s‐

)/(‐87960930‐2638827906

022208000006662400*K*ex

0*s^2‐xp(‐1/2*s))

ss = 0 EEss = 220/3 FFsimp = ‐.10e3*exp(‐.550*s)*(5.*s‐66.)/s/(s+6.) GGcsimp = .10e3*exp(‐.5‐ 50*s)*(5.*s‐66.)/(s+6.) OOlpsimp = .10e‐2*K*exp‐ p(‐.50*s)*(177.*s‐20.)/s/(s++6.) CClpsimp = 5 5.*K*exp(‐.500*s)*(15.*s‐117.)/(‐.44e4*ss^2‐.26e5*s+111.*K*exp(‐.550*s)*s‐13.*KK*exp(‐.50*s)))

33-16)

3‐11


15 6 3 0.5

1 0.5

10.5 5 0.5

3-18)

1.5 0.5 0.

0.5 0.50.5 0.5

u1

u2

0.5

0.5

0.5

1/s

0.5 x3

0.5 1.5

-1 1/s

3

-6

x2-5

0.5

1/s

0.5

0.5

0.5

x1

1

1 z1

z2-0.5

1

3‐12


uB0 1/s

x

-A0

-A1

1/sy

B1

1

3-20)

3‐13


3

3

3-21)

3-22)

3‐14


3

3-23)

3‐15


33-24)

3

3

3

3-25)

3-26)

3-27)

3‐16


3

3-28)

3‐17

, 9th Edition A Automatic Co

3

3-29)

ntrol Systems Chapter 3 Solutionns Golnarraghi, Kuo

3‐18


3-30) UUse Mason’ss formula:

3-31) MMATLAB syms s K G=100//(s+1)/(s+5) g=ilaplace(G/s) H=K/s YN=simmplify(G/(1+G*H)) Yn=ilapplace(YN/s) G = 100/(ss+1)/(s+5) g = ‐25*exxp(‐t)+5*expp(‐5*t)+20 H = K/s

3‐19


YN = 100*s/(s^3+6*s^2+5*s+100*K) Apply Routh‐Hurwitz within Symbolic tool of ACSYS (see chapter 3)

RH = [ 1, 5] [ 6, 100*k] [ ‐50/3*k+5, 0] [ 100*k, 0] Stability requires: 0<k<3/10.

3‐20

, 9th Edition A Automatic Co

3‐32)

ntrol Systems Chapter 3 Solutionns Golnarraghi, Kuo

3‐21


3‐33) MMATLAB soluutions are inn 3‐34.

3‐34) Mclear p = [1roots(G=tf(1step(G

MATLAB all 1 5 6 10] (p) 1,p) G)

% Define p

polynomial

3‐22

s^3+5*s^22+6*s+10=0


p = 1 5 6 10 ans = ‐4.1337 ‐0.4331 + 1.4938i ‐0.4331 ‐ 1.4938i Transfer function: 1 ‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐ s^3 + 5 s^2 + 6 s + 10

3‐23


Alternatively: clear all syms s G=1/( s^3 + 5*s^2 + 6*s + 10) y=ilaplace(G/s) s=0 yfv=eval(G) G = 1/(s^3+5*s^2+6*s+10) y = 1/10+1/5660*sum((39*_alpha^2-91+160*_alpha)*exp(_alpha*t),_alpha = RootOf(_Z^3+5*_Z^2+6*_Z+10)) s = 0 yfv = 0.1000

Problem finding the inverse Laplace. Use Toolbox 2‐5‐1 to find the partial fractions to better find inverse Laplace clear all B=[1] A = [1 5 6 10 0] % Define polynomial s*(s^3+5*s^2+6*s+10)=0 [r,p,k]=residue(B,A) B = 1 A = 1 5 6 10 0 r = -0.0152 -0.0424 + 0.0333i -0.0424 - 0.0333i 0.1000 p = -4.1337 -0.4331 + 1.4938i -0.4331 - 1.4938i 0 k = []

So partial fraction of Y is:1 0.0152 0.0424 0.0333i 0.0424 - 0.0333i

4.1337 0.4331 + 1.4938i 0.4331 - 1.4938is s s s− − + −

+ + +− − −

3‐24


3‐35) MMATLAB soluutions are inn 3‐36.

3‐25


3‐36) clear all p = [1 4 3 5 1] % Define polynomial s^4+4*s^3+3*s^2+5*s+1=0 roots(p) G=tf(1,p) step(G) p = 1 4 3 5 1 ans = -3.5286 -0.1251 + 1.1250i -0.1251 - 1.1250i -0.2212 Transfer function: 1 ----------------------------- s^4 + 4 s^3 + 3 s^2 + 5 s + 1

3‐26


Alternatively: clear all syms s t G=1/(s^4+4*s^3+3*s^2+5*s+1) y=ilaplace(G/s) s=0 yfv=eval(G) G = 1/(s^4+4*s^3+3*s^2+5*s+1) y = 1-1/14863*sum((3955*_alpha^3+16873+14656*_alpha^2+7281*_alpha)*exp(_alpha*t),_alpha = RootOf(_Z^4+4*_Z^3+3*_Z^2+5*_Z+1)) s = 0 yfv = 1

Problem finding the inverse Laplace. Use Toolbox 2‐5‐1 to find the partial fractions to better find inverse Laplace clear all B=[1] A = [1 4 3 5 1] % Define polynomial s^4+4*s^3+3*s^2+5*s+1=0 [r,p,k]=residue(B,A) B = 1 A = 1 4 3 5 1 r = -0.0235 -0.1068 + 0.0255i -0.1068 - 0.0255i 0.2372 p = -3.5286 -0.1251 + 1.1250i -0.1251 - 1.1250i -0.2212 k = []

3‐27


3-37)

3-38) MUse TFWindoTFcal

MATLAB Fcal in ACSYow).

YS (go to AACSYS folde

3‐28

er and type inn TFcal in thhe MATLAB

B Commandd


Alternatively use toolboxes 3-3-1 and 3-3-2 clear all syms s G1=100 G2=(s+1)/(s+2) G3=10/s/(s+20) G4=(101*s^3+2122*s^2+2040*s)/10/(s+1) H1=1 H2=1 simplify(G1*G2*G3/(1+G1*G2*H1+G1*G2*H2+G1*G2*G3)) G1 = 100 G2 = (s+1)/(s+2) G3 = 10/s/(s+20) G4 = (101/10*s^3+1061/5*s^2+204*s)/(s+1) H1 = 1 H2 = 1 ans = 1000*(s+1)/(201*s^3+4222*s^2+5040*s+1000) clear all TF=tf([1000 1000],[201 4222 5040 1000]) step(TF)

3‐29


3-39)

clear all syms s P1 = 2*s^6+9*s^5+15*s^4+25*s^3+25*s^2+14*s+6 % Define polynomial P2 = s^6+8*s^5+23*s^4+36*s^3+38*s^2+28*s+16 % Define polynomial solve(P1, s) solve(P2, s) collect(P2-P1) collect(P2+P1) collect((P1-P2)*P1) P1 = 2*s^6+9*s^5+15*s^4+25*s^3+25*s^2+14*s+6 P2 = s^6+8*s^5+23*s^4+36*s^3+38*s^2+28*s+16 ans =

3‐30


-1 -3 i*2^(1/2) -i*2^(1/2) -1/4+1/4*i*7^(1/2) -1/4-1/4*i*7^(1/2) ans = -2 -4 i -i -1+i -1-i ans = -s^6-s^5+8*s^4+11*s^3+13*s^2+14*s+10 ans = 3*s^6+17*s^5+38*s^4+61*s^3+63*s^2+42*s+22 ans = -60+2*s^12+11*s^11+8*s^10-54*s^9-195*s^8-471*s^7-796*s^6-1006*s^5-1027*s^4-848*s^3-524*s^2-224*s Alternative: clear all P1 = [2 9 15 25 25 14 6] % Define polynomial roots(P1) P2 = [1 8 23 36 38 28 16] % Define polynomial roots(P2) P1 = 2 9 15 25 25 14 6 ans = -3.0000 -0.0000 + 1.4142i -0.0000 - 1.4142i -1.0000 -0.2500 + 0.6614i -0.2500 - 0.6614i

3‐31


P2 = 1 8 23 36 38 28 16 ans = -4.0000 -2.0000 -1.0000 + 1.0000i -1.0000 - 1.0000i 0.0000 + 1.0000i 0.0000 - 1.0000i

3-40) clear all syms s P6 = (s+1)*(s^2+2)*(s+3)*(2*s^2+s+1) % Define polynomial P7 = (s^2+1)*(s+2)*(s+4)*(s^2+s+1) % Define polynomial digits(2) vpa(solve(P6, s)) vpa(solve(P7, s)) collect(P6) collect(P7)

P6 = (s+3)*(s+1)*(2*s^2+s+1)*(s^2+2) P7 = (s^2+1)*(s+2)*(s+4)*(s^2+s+1) ans = -1. -3. 1.4*i -1.4*i -.25+.65*i -.25-.65*i ans = -2. -4. i -1.*i -.50+.85*i -.50-.85*i ans = 2*s^6+9*s^5+15*s^4+25*s^3+25*s^2+14*s+6 ans = 8+s^6+7*s^5+16*s^4+21*s^3+23*s^2+14*s

3‐32

Automatic Control Systems, 9th Edition Chapter 3 Solutions Golnaraghi, Kuo 3‐41)

Use Toolbox 2‐5‐1 to find the partial fractions

clear all B= conv(conv(conv([1 1],[1 0 2]),[1 4]),[1 10]) A= conv(conv(conv([1 0],[1 2]),[1 2 5]),[2 1 4]) [r,p,k]=residue(B,A) B = 1 15 56 70 108 80 A = 2 9 26 45 46 40 0 r = -1.0600 - 1.7467i -1.0600 + 1.7467i 0.9600 -0.1700 + 0.7262i -0.1700 - 0.7262i 2.0000 p = -1.0000 + 2.0000i -1.0000 - 2.0000i -2.0000 -0.2500 + 1.3919i -0.2500 - 1.3919i 0 k = []

3‐33


3‐42) Use toolbox 3‐3‐2 clear all B= conv(conv(conv([1 1],[1 0 2]),[1 4]),[1 10]) A= conv(conv(conv([1 0],[1 2]),[1 2 5]),[2 1 4]) G1=tf(B,A) YR1=G1/(1+G1) pole(YR1)

B = 1 15 56 70 108 80 A = 2 9 26 45 46 40 0 Transfer function: s^5 + 15 s^4 + 56 s^3 + 70 s^2 + 108 s + 80 ‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐ 2 s^6 + 9 s^5 + 26 s^4 + 45 s^3 + 46 s^2 + 40 s Transfer function: 2 s^11 + 39 s^10 + 273 s^9 + 1079 s^8 + 3023 s^7 + 6202 s^6 + 9854 s^5 + 12400 s^4 + 11368 s^3 + 8000 s^2 + 3200 s ‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐ 4 s^12 + 38 s^11 + 224 s^10 + 921 s^9 + 2749 s^8 + 6351 s^7 + 11339 s^6 + 16074 s^5 + 18116 s^4 + 15048 s^3 + 9600 s^2 + 3200 s ans = 0 ‐0.7852 + 3.2346i ‐0.7852 ‐ 3.2346i ‐2.5822 ‐1.0000 + 2.0000i ‐1.0000 ‐ 2.0000i ‐2.0000 ‐0.0340 + 1.3390i ‐0.0340 ‐ 1.3390i ‐0.2500 + 1.3919i ‐0.2500 ‐ 1.3919i ‐0.7794

C= [1 12 47 60] D= [4 28 83 135 126 62 12] G2=tf(D,C) YR2=G2/(1+G2) pole(YR2) C = 1 12 47 60

3‐34


D = 4 28 83 135 126 62 12 Transfer function: 4 s^6 + 28 s^5 + 83 s^4 + 135 s^3 + 126 s^2 + 62 s + 12 ------------------------------------------------------- s^3 + 12 s^2 + 47 s + 60 Transfer function: 4 s^9+76 s^8+607 s^7+2687 s^6+7327 s^5+12899 s^4+14778 s^3+10618 s^2 + 4284 s + 720 ------------------------------------------------------------------------------- 4 s^9+76 s^8+607 s^7+2688 s^6+7351 s^5+13137 s^4+16026 s^3+14267 s^2+9924 s+4320 ans = -5.0000 -4.0000 0.0716 + 0.9974i 0.0716 - 0.9974i -1.4265 + 1.3355i -1.4265 - 1.3355i -3.0000 -2.1451 + 0.3366i -2.1451 - 0.3366i

3-43) Use Toolbox 3-3-1

G3=G1+G2 G4=G1-G2 G5=G4/G3 G6=G4/(G1*G2) G3=G1+G2 G4=G1-G2 G5=G4/G3 G6=G4/(G1*G2) Transfer function: 8 s^12 + 92 s^11 + 522 s^10 + 1925 s^9 + 5070 s^8 + 9978 s^7 + 15154 s^6 + 18427 s^5 + 18778 s^4 + 16458 s^3 + 13268 s^2 + 10720 s + 4800

3‐35


------------------------------------------------------------------------------- 2 s^9 + 33 s^8 + 228 s^7 + 900 s^6 + 2348 s^5 + 4267 s^4 + 5342 s^3 + 4640 s^2 + 2400 s Transfer function: -8 s^12 - 92 s^11 - 522 s^10 - 1925 s^9 - 5068 s^8 - 9924 s^7 - 14588 s^6 - 15413 s^5 - 9818 s^4 - 406 s^3 + 7204 s^2 + 9760 s + 4800 ------------------------------------------------------------------------------- 2 s^9 + 33 s^8 + 228 s^7 + 900 s^6 + 2348 s^5 + 4267 s^4 + 5342 s^3 + 4640 s^2 + 2400 s Transfer function: -16 s^21 - 448 s^20 - 5904 s^19 - 49252 s^18 - 294261 s^17 - 1.346e006 s^16 - 4.906e006 s^15 - 1.461e007 s^14 - 3.613e007 s^13 - 7.482e007 s^12 - 1.3e008 s^11 - 1.883e008 s^10 - 2.234e008 s^9 - 2.078e008 s^8 - 1.339e008 s^7 - 2.674e007 s^6 + 6.595e007 s^5 + 1.051e008 s^4 + 8.822e007 s^3 + 4.57e007 s^2 + 1.152e007 s ------------------------------------------------------------------------------- 16 s^21 + 448 s^20 + 5904 s^19 + 49252 s^18 + 294265 s^17 + 1.346e006 s^16 + 4.909e006 s^15 + 1.465e007 s^14 + 3.643e007 s^13 + 7.648e007 s^12 + 1.369e008 s^11 + 2.105e008 s^10 + 2.803e008 s^9 + 3.26e008 s^8 + 3.343e008 s^7 + 3.054e008 s^6 + 2.493e008 s^5 + 1.788e008 s^4 + 1.072e008 s^3 + 4.8e007 s^2

3‐36


3‐37

+ 1.152e007 s Transfer function: -16 s^21 - 448 s^20 - 5904 s^19 - 49252 s^18 - 294261 s^17 - 1.346e006 s^16 - 4.906e006 s^15 - 1.461e007 s^14 - 3.613e007 s^13 - 7.482e007 s^12 - 1.3e008 s^11 - 1.883e008 s^10 - 2.234e008 s^9 - 2.078e008 s^8 - 1.339e008 s^7 - 2.674e007 s^6 + 6.595e007 s^5 + 1.051e008 s^4 + 8.822e007 s^3 + 4.57e007 s^2 + 1.152e007 s ------------------------------------------------------------------------------- 8 s^20 + 308 s^19 + 5270 s^18 + 54111 s^17 + 379254 s^16 + 1.955e006 s^15 + 7.778e006 s^14 + 2.471e007 s^13 + 6.416e007 s^12 + 1.383e008 s^11 + 2.504e008 s^10 + 3.822e008 s^9 + 4.919e008 s^8 + 5.305e008 s^7 + 4.73e008 s^6 + 3.404e008 s^5 + 1.899e008 s^4 + 7.643e007 s^3 + 1.947e007 s^2 + 2.304e006 s

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c) L Û µ Ä d) Feedback ratio = L Û -...

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