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+//caption:root_locus

+//example 7.24.7

+//page 306

+s=%s;

+syms K;

+GH=K/(s*(s+2)*(s+4))

+//since Mp=40%, so .4=exp((-zeta*%pi)/(sqrt(1-zeta^2))

+zeta=0.3

+//from given data 

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

+CH=(s*(s+2)*(s+4))+K

+K=sym('-(s^3+6*s^2+8*s)')

+d=diff(K,s)

+e=-3*s^2-12*s-8 

+r1=roots(e)

+disp(r1,"roots=")

+disp("-0.842 is break away point sinc it lies on root locus")

+CH=sym('s^3+6*s^2+8*s+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]

+disp(routh,"routh=")

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

+disp("(48-K)=0 ");

+disp("which gives:");

+disp("K=48");

+K=48;

+k=48

+a=6*s^2+48//intersection of root locus with imaginary plane

+r=roots(a)

+g=k/(s*(s+2)*(s+4))

+G=syslin('c',g)

+evans(g,8)

+xgrid(2)

+disp("the line theta=acos(zeta)=72.5 intersects root locus at sa=(-0.5+i1.65)")

+disp("the value of K at s=sa is find to be 14.87 for Mp=40%")

+K=14.87

+ts=4/0.5     //ts=4/(zeta*wn)

+Kv=limit(s*GH,s,0)

+Kv=eval(Kv)

+Kv=float(Kv)

+disp(Kv,"Kv=");