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+//caption:root_locus_and_value_of_K
+//example 12.46
+//page 570
+s=%s;
+syms K;
+GH=K/(s*((s+6)^2))
+disp("the characterstics eq. is determined as:")
+CH=(s*((s+6)^2))+K
+CH=sym('(s*((s+6)^2))+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*((s+6)^2))')
+d=diff(K,s)
+e=3*s^2+24*s+36 
+r1=roots(e)
+disp(r1,"roots=")
+disp("-2 is break away point sinc it lies on root locus")
+disp(routh,"routh=")
+disp("for given system to be marginally stable:");
+disp("(-(K-432)/12)=0 ");
+disp("which gives:");
+disp("K=432");
+K=432;//considering positive value
+k=432
+a=12*s^2+k//intersection of root locus with imaginary axis plane
+r=roots(a)
+g=k/(s*((s+6)^2))
+G=syslin('c',g)
+clf();
+evans(g,5)
+xgrid(2)
+disp(r,"the point of intersection of root locus with imaginary axis =")