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+//caption:root_locus_description

+//example 7.24.2

+//page 296

+s=%s;

+syms K;

+GH=K/(s*(s+1)*(s+3))

+zeta=0.5

+//from given data 

+disp("the characterstics eq. is determined as:")

+CH=(s*(s+1)*(s+3))+K

+CH=sym('(s*(s+1)*(s+3))+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);

+b=[c0 c1 c2 c3]

+n=4;

+routh=[b([4,2]);b([3,1])];

+routh=[routh;-det(routh)/routh(2,1),0]

+t=routh(2:3,1:2)

+routh=[routh;-det(t)/t(2,1),0]

+K=sym('-(s^3+4*s^2+3*s)')

+d=diff(K,s)

+e=-3*s^2-8*s-3 

+r1=roots(e)

+disp(r1,"roots=")

+disp("-0.45 is break away point since it lies on root locus")

+disp(routh,"routh=")

+disp("for given system to be marginally stable:");

+disp("(12-K)=0 ");

+disp("which gives:");

+disp("K=12, for margianl stability");

+K=12;

+k=12

+a=4*s^2+k//intersection of root locus with imaginary plane

+r=roots(a)

+g=k/(s*(s+1)*(s+3))

+G=syslin('c',g)

+evans(g,8)

+xgrid(2)

+disp("the line theta=acos(zeta)=60 intersects root locus at sa=(-0.35+i0.6)")

+disp("the value of K at s=sa is find to be 1.66 ")

+disp("the value of K at s=-4")

+disp("K=12")

+disp("at K=1.66")

+k=1.66

+H=1

+G=k/(s*(s+1)*(s+3))

+CL=G/(1+G*H)

+disp(CL,"C(s)/R(s)=")
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