Exact Schur-Stability Regions and Computational Controller Selection for Delayed Scalar Discrete-Time Systems

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Exact Schur-Stability Regions and Computational Controller Selection for Delayed Scalar Discrete-Time Systems
by Ramy-Badr Ahmed
[ORCID]
Institute for Thermal Energy Technology and Safety (ITES), Resilient and Smart Infrastructure Systems (RESIS), Karlsruhe Institute of Technology (KIT), Hermann-von-Helmholtz-Platz 1, 76344 Eggenstein-Leopoldshafen, Germany
Algorithms 2026, 19(10), 823; https://doi.org/10.3390/a19100823
Submission received: 17 August 2026 / Revised: 21 September 2026 / Accepted: 22 September 2026 / Published: 24 September 2026
(This article belongs to the Section Algorithms for Multidisciplinary Applications)
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Abstract
Delayed feedback converts local stabilization of a scalar map with delay 𝜏
into a degree-(𝜏+1) Schur-stability problem. Using the classical Kuruklis criterion as the analytical starting point, we derive exact Schur-stability regions in the multiplier–gain plane for fixed-point feedback control (FPFC), time-delayed feedback control (TDFC), and nonlinear predictive feedback control (NFC) as pullbacks of a universal Schur region. ... mehrA direct unit-circle parametrization yields the curved boundary and supports exact results on same-parity nesting, large-delay intersections, stabilizable-multiplier projections, delay budgets, and necessary-and-sufficient common-gain conditions under interval uncertainty. These results lead to reproducible algorithms for certified pointwise classification, 𝑂(𝑁) boundary generation, endpoint-based robust-gain testing, and gain-constrained controller selection from complete feasible-gain intervals. Non-explicit calculations reduce to monotone scalar bisection, avoiding repeated characteristic-root calculations in routine region construction. Independent root computations at 114,000 validation points, including delays up to 𝜏=100, show no classification mismatch. Logistic-map examples and a non-polynomial tumour-growth map illustrate the local stabilization and controller-selection implications. The resulting framework provides an exact analytical and algorithmic treatment of arbitrary-delay local stabilization in scalar discrete-time systems.