MIACE and Monotone Tracking: Exact Tuning for First-Order Plants and Positive Cascades with Pure Delay
Year: 2026
Type: preprint
Abstract: The Module of Integral Absolute Complex Error (MIACE) was created by me 23 years ago. It gave very nice behaviour to the transient response of the controlled systems I tested. It combines absolute tracking-error and equilibrium-centered command areas. For a stable unit-gain plant with known pure delay, an exact identity separates squared MIACE excess into a negative-area penalty and a weighted area imbalance. The resulting lower bound depends on the plant's first low-frequency moment and the delay. Every attaining response of a positive cascade is monotone, independently of controller structure. For a first-order plant, output variation and peak overshoot also admit quantitative bounds. Exact conditions identify the unique positive PI controller attaining the delayed bound; filtered PID extends attainment beyond this PI range. For cascades of arbitrary order, Hermite interpolation and a finite-support delay remainder construct proper rational controllers of the same order, with internal exponential stability proved by positive convolution representations. An exponential-moment characterization then closes the attainment question when internal delay blocks are allowed: an ordinary command attains the bound if and only if $α>μ+δ-1$. An explicit internal-model controller realizes every feasible weight and delay. A sharp amplitude-cap condition determines the least command peak among all attaining responses and the complete capped sign-preserving area front. A small-gain estimate supplies an explicit MIACE deviation bound under delay error. A storage balance also extends the exact MIACE identity and monotone attainment to a nonlinear first-order class without delay. Independent physical integrations retain delays exactly; quadratic comparisons provide complementary benchmarks.
Source: arXiv (Cornell University)
Author: Tomov, Latchezar
Institution: New Bulgarian University
Language: English
Subfield: Control and Systems Engineering
Field: Engineering
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7223659780