MIACE and Monotone Tracking: Exact Tuning for First-Order Plants and Positive Cascades with Pure Delay
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.
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BibTeX
@misc{tomov2026miace,
title = {{MIACE and Monotone Tracking: Exact Tuning for First-Order Plants and Positive Cascades with Pure Delay}},
author = {Latchezar Tomov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.32401},
eprint = {2610.32401},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.32401},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - MIACE and Monotone Tracking: Exact Tuning for First-Order Plants and Positive Cascades with Pure Delay
AU - Latchezar Tomov
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.32401
UR - https://arxiv.org/abs/2610.32401
AB - 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.
ER -