Exact MIACE Optimization for Constrained 2DOF PIDF Reference Shaping with Pure Delay

This paper studies theModule of Integral Absolute Complex Error (MIACE) in a constrained two-degree-of-freedomproportional–integral–derivative controller with a filtered derivative (PIDF). The criterion that was developed by the author in 2003, balances integral absolute tracking error against the accumulated gain-normalised commandexcursion about equilibrium. For a strictly proper rational plant with pure delay and internallyexponentially stable feedback, we characterise the all-time monotone feasible set and provethat minimum integral absolute tracking error is attained at its largest feasible proportionalweight. When the centred command is also nonnegative, MIACE minimisation reducesexactly to projection onto a feasible weight interval. An explicit delayed second-order familyattains the universal MIACE lower bound using only the two reference weights. Along thismonotone family, moving from one MIACE optimum to the minimum-error endpoint reducestracking-error area by 10 percent while increasing centred-command area by 50 percent. Thedisturbance and noise channels remain identical. A complementary modal design uses exactfilter-placement regions, command-jump limits and mismatch bounds to construct constrainedconic optimisations for an uncertain oscillatory benchmark. The transport delay is retainedwithout rational approximation, and analytic all-time guarantees are distinguished from finitenumerical checks.