I have a simple control system, G(z), with feedback H(z), e[nT]=u[nT]-H(z)*y[nT];
What is the maximum of the magnitude of the e[nT]?
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.
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