PID-GM: PID Control with Gain Mapping
Bibliographic record
Abstract
Proportional-Integral-Differential (PID) control is widely used in industrial control systems. However, up to now there are at least two open problems related with PID control. One is to have a comprehensive understanding of its robustness with respect to model uncertainties and disturbances. The other is to build intuitive, explicit and mathematically provable guidelines for PID gain tuning. In this paper, we introduce a simple nonlinear mapping to determine PID gains from three auxiliary parameters. By the mapping, PID control is shown to be equivalent to a new PD control (serving as a nominal control) plus an uncertainty and disturbance compensator (to recover the nominal performance). Then PID control can be understood, designed and tuned in a Two-Degree-of-Freedom (2-DoF) control framework. We discuss some basic properties of the mapping, including the existence, uniqueness and invertibility. Taking as an example the PID control applied to a general uncertain second-order plant, we prove by the singular perturbation theory that the closed-loop steady-state and transient performance depends explicitly on one auxiliary parameter which can be viewed as the virtual singular perturbation parameter (SPP) of PID control. All the three PID gains are monotonically decreasing functions of the SPP, indicating that the smaller the SPP is, the higher the PID gains are, and the better the robustness of PID control is. Simulation and experimental examples are provided to demonstrate the properties of the mapping as well as the effectiveness of the mapping based PID gain turning.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".