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+//caption:root_locus
+//example 7.24.8
+//page 308
+clc;
+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)
+disp(r,"intersection point of root locus with imaginary axis=")
+g=k/(s*(s+1)*(s+3))
+G=syslin('c',g)
+evans(g,8)
+xgrid(2)
+disp("for K=6")
+k=6;
+GH=k/(s*(s+1)*(s+3))
+gm=K/k  //gm=K(marginal_stability)/K(desired)
+disp(gm,"gain_margin=")
+disp("the point where K=6 is s=1.2")
+pm=180+(-90-(atan(1.2/1)*180/%pi)-(atan(1.2/3)*180/%pi))
+disp(pm,"phase margin=")
+\ No newline at end of file