Necessary and Sufficient Criteria for Global PID Regulation of Nonlinear Uncertain System

@misc{zhao2026necessary,
  title = {{Necessary and Sufficient Criteria for Global PID Regulation of Nonlinear Uncertain System}},
  author = {Cheng Zhao and Jingru Zhu and Lei Guo},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.32703},
  eprint = {2610.32703},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.32703},
}

TY  - UNPB
TI  - Necessary and Sufficient Criteria for Global PID Regulation of Nonlinear Uncertain System
AU  - Cheng Zhao
AU  - Jingru Zhu
AU  - Lei Guo
PY  - 2026
DA  - 2026/10/23
PB  - arXiv
DO  - 10.48550/arxiv.2610.32703
UR  - https://arxiv.org/abs/2610.32703
AB  - Although classical PID control is widely used in engineering practice, identifying all PID gains that guarantee global regulation of nonlinear uncertain dynamics remains challenging. For a class of second-order nonaffine systems under one-sided derivative bounds, we establish necessary and sufficient conditions for global PID regulation: a fixed PID controller globally regulates the entire infinite-dimensional uncertainty class if and only if it regulates a single boundary linear system. Applying the Routh--Hurwitz criterion to this boundary linear system yields the exact admissible PID gain region. To establish this equivalence, we prove a structured Markus--Yamabe theorem for third-order integrator chains under uniform derivative-ratio conditions. Its proof combines a two-dimensional storage function constructed via viscosity solutions of a Hamilton--Jacobi equation with a scalar comparison argument. We further show that the same PID gain region remains exact for multidimensional affine systems under symmetry assumptions on the state partial Jacobians and the input matrix.
ER  -