Hi!

You have next equation:

e[nT] = u[nT] - H(z)*y[nT]. (1)

You must find discrete TF of (1)

e(z) = u(z) - H(z)*y(z) = u(z) - H(z)*G(z)*u(z) = u(z)*[1 - H(z)*G(z)], (2)

where: u(z) - input, y(z) - output, H(z) - z-TF of feedback, G(z) = y(z)/u(z) - z-TF of closed-loop system.

From (2) we can take z-TF of error:

Ke(z) = e(z)/u(z) = 1 - H(z)*G(z). (3)

Based on the final value propertie, from (3) we can find

es-s[nT] = lim(e[nT]) = lim{(z-1)*Ke(z)*u(z)},

                 n→∞            z→1

where index ''s-s'' mean a steady state.

Best regards.

If there are known rate or magnitude bounds on the system inputs, then the maximum error magnitude can be obtained from l_1 norms of the sensitivity function (see, for example, the books of Dahleh & Diaz-Bobillo, Boyd & Barratt or Whidborne & Liu). 

If there are both rate and magnitude bounds, then less conservative methods exist - there are some pointers in Boyd & Barratt  or (for the continuous time case) see the works of Arunsawatwong.