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+//caption:root_locus
+//example 7.24.4
+//page 301
+clc;
+s=%s;
+syms K;
+GH=K/(s*(s+4)*(s^2+4*s+13))
+disp("the characterstics eq. is determined as:")
+CH=(s*(s+4)*(s^2+4*s+13))+K
+CH=sym('(s*(s+4)*(s^2+4*s+13))+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*(s+4)*(s^2+4*s+13))')
+d=diff(K,s)
+e=(-4*s^3+24*s^2+58*s+52)
+r=roots(e)
+disp("since -2 lies on root locus so complex breakaway point is -2+i1.58 and -2-i1.58")
+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)
+g=k/(s*(s+2)*(s^2+2*s+2))
+G=syslin('c',g)
+evans(g,200)
+xgrid(2)
+eq=(s*(s+4)*(s^2+4*s+13))
+p=roots(eq)
+disp(p,"open loop poles are:")
+phi1=180-(atan(3/2)*180/%pi)
+phi2=atan(3/2)*180/%pi
+phi3=90
+phi_p2=180-(phi1+phi2+phi3)
+phi_p3=-phi_p2
+disp(phi_p2,"angle of departure for -2+3i=")
+disp(phi_p3,"angle of departure for -2-3i=")