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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=");