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//caption:root_locus
//example 7.24.4
//page 301
clc;
s=%s;
syms K;
GH=K/(s*(s+4)*(s^2+4*s+13))
disp("the characterstics eq. is determined as:")
CH=(s*(s+4)*(s^2+4*s+13))+K
CH=sym('(s*(s+4)*(s^2+4*s+13))+K');
disp('=0',CH,"characterstics_eq,CH=")
c0=coeffs(CH,'s',0);
c1=coeffs(CH,'s',1);
c2=coeffs(CH,'s',2);
c3=coeffs(CH,'s',3);
c4=coeffs(CH,'s',4);
b=[c0 c1 c2 c3 c4 ]
routh=[b([5,3,1]);b([4,2]),0]
routh=[routh;-det(routh(1:2,1:2))/routh(2,1),routh(1,3),0]
routh(3,1)=simple(routh(3,1))
t=routh(2:3,1:2)
l=simple(-det(t)/t(2,1))
routh=[routh;l,0,0]
routh=[routh;K,0,0]
K=sym('(s*(s+4)*(s^2+4*s+13))')
d=diff(K,s)
e=(-4*s^3+24*s^2+58*s+52)
r=roots(e)
disp("since -2 lies on root locus so complex breakaway point is -2+i1.58 and -2-i1.58")
disp(routh,"routh=")
disp("for given system to be marginally stable:");
disp("((20-4K)/5)=0 ");
disp("which gives:");
disp("K=5");
K=5;
k=5
a=5*s^2+5//intersection of root locus with s plane
r=roots(a)
g=k/(s*(s+2)*(s^2+2*s+2))
G=syslin('c',g)
evans(g,200)
xgrid(2)
eq=(s*(s+4)*(s^2+4*s+13))
p=roots(eq)
disp(p,"open loop poles are:")
phi1=180-(atan(3/2)*180/%pi)
phi2=atan(3/2)*180/%pi
phi3=90
phi_p2=180-(phi1+phi2+phi3)
phi_p3=-phi_p2
disp(phi_p2,"angle of departure for -2+3i=")
disp(phi_p3,"angle of departure for -2-3i=")
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