Here's my take on your problem: I've invented an arbitrary continuous TF (since you neglected to tell us what you are dealing with), and then discretised it in a variety of ways, including your "AA" method. Results are relatively poor, except at small sample times.
tau = 2; zeta = 0.5;
Gc = tf([5 2],conv([6 5 4],[tau^2 2*tau*zeta 1])) % continuous TF
Ts = 1; % sampling time [s]
B = cell2mat(Gc.Numerator);
A = cell2mat(Gc.Denominator);
G = poly2sym(B,s)/poly2sym(A,s);
syms s z T
Gd = subs(G,s,8*(1-1/z)/(7*T*(1+1/7/z)))
Gd = subs(Gd,T,Ts); [N,D] = numden(Gd); Gd1 = tf(sym2poly(N), sym2poly(D),Ts);
%Gd = subs(Gd2,T,Ts); [N,D] = numden(Gd); Gd2 = tf(sym2poly(N), sym2poly(D),Ts);
Gd2 = c2d(Gc,Ts,'zoh');
Gd3 = c2d(Gc,Ts,'foh');
Gd4 = c2d(Gc,Ts,'tustin');
step(Gc, Gd1, Gd2, Gd3, Gd4)
legend('G_c','AA')
