Consider (Box and MacGregor, 1976) a system for which the process transfer function is gB and the noise model is (1 − ????)???????? = (1 − ????????)???????? so that the error ???????? at the output satisfies

(1 − ????)???????? = ????(1 − ????)????????−1+ + (1 − ????????)????????

Suppose that the system is controlled by a known discrete ‘‘integral’’

controller

(1 − ????)????????+ = −????????????

(a) Show that the errors ???????? at the output will follow the ARMA(1, 1) process

(1 − ????????)???????? = (1 − ????????)???????? ???? =1− ????????

and hence that the problem of estimating ???? and ???? using data from a pilot control scheme is equivalent to that of estimating the parameters in this ARMA(1, 1)

model.

(b) Show also that the optimal control scheme is such that ???? = ????0 = (1 − ????)∕???? and hence that if the pilot scheme used in collecting the data happens to be optimal already, then 1 − ???? and ???? cannot be separately estimated.