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+//caption:root_locus_and_value_of_K
+//example 12.43
+//page 564
+s=%s;
+syms K;
+GH=K/(s*(s+2)*(s^2+2*s+2))
+disp("the characterstics eq. is determined as:")
+CH=(s*(s+2)*(s^2+2*s+2))+K
+CH=sym('s^4+4*s^3+6*s^2+4*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);
+c4=coeffs(CH,'s',4);
+b=[c0 c1 c2 c3 c4 ]
+routh=[b([5,3,1]);b([4,2]),0]
+routh=[routh;-det(routh(1:2,1:2))/routh(2,1),routh(1,3),0]
+routh(3,1)=simple(routh(3,1))
+t=routh(2:3,1:2)
+l=simple(-det(t)/t(2,1))
+routh=[routh;l,0,0]
+routh=[routh;K,0,0]
+K=sym('s^4+4*s^3+6*s^2+4*s')
+d=diff(K,s)
+e=-(4*s^3+12*s^2+12*s+4)
+r=roots(e)
+disp(routh,"routh=")
+disp("for given system to be marginally stable:");
+disp("((20-4K)/5)=0 ");
+disp("which gives:");
+disp("K=5");
+K=5;
+k=5
+a=5*s^2+5//intersection of root locus with s plane
+r=roots(a)
+disp(r,"intersection point with imaginary axis=")
+g=k/(s*(s+2)*(s^2+2*s+2))
+G=syslin('c',g)
+evans(g,200)
+xgrid(2)
+disp("angle of departure=-90 and +90")
+disp("breakaway point is -1 and +j and -j")
+disp("on solving we find K=1")