In a chemical process, 30 successive values of viscosity ???????? that occurred during a period when the control variable (gas rate) ???????? was held fixed at its standard reference origin were recorded as follows:

Time Viscosities 1--10 92 92 96 96 96 98 98 100 100 94 11--20 98 88 88 88 96 96 92 92 90 90 21--30 90 94 90 90 94 94 96 96 96 96 Reconstruct and plot the error sequence (deviations from target) ???????? and adjustments

????????, which would have occurred if the optimal feedback control scheme

???????? = −10???????? + 5????????−1 (1)

had been applied during this period. It is given that the dynamic model is

???????? = 0.5????????−1 + 0.10????????−1 (2)

and that the error signal may be obtained from

???????? = ????????−1 + ∇???????? + ???????? (3)

Your calculation sequence should proceed in the order (2), (3), and (1) and initially you should assume that ????1 = 0, ????1 = 0, ????1 = 0. Can you devise a more direct way to compute ???????? from ?????????