//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)=")