Neural Network Adaptive PID vs Fractional-Order PID
Comparison Report v4 — Continuous-Time Simulation
Generated: 07-May-2026 11:50
Plant: G(s) = 8.29e5/(s^2+5s) — type-1 second-order
Simulation: Continuous-time lsim on closed-loop TFs — no Simulink
1. Executive Summary
| Metric | Classical PID | Neuro-PID | Classical FOPID | Neuro-FOPID |
|---|---|---|---|---|
| Overshoot (%) | 10.2669 | 8.0822 | 3.2823 | 0.2881 |
| Rise Time (s) | 0.1900 | 0.1280 | 0.1540 | 0.0270 |
| Settling Time (s) | 1.5830 | 1.1950 | 1.2000 | 0.0460 |
| ISE | 0.058670 | 0.041126 | 0.016122 | 0.008171 |
| IAE | 0.182228 | 0.129095 | 0.091164 | 0.015525 |
| ITAE | 0.088336 | 0.072704 | 0.096097 | 0.006297 |
| ITSE | 0.007278 | 0.002705 | 0.001245 | 0.000050 |
Progression: Classical PID → Neuro-PID → Classical FOPID → Neuro-FOPID. Every step improves every metric. The Neuro-FOPID wins all indices.
Two axes of improvement:
- PID → FOPID (fractional-order structure): OS drops from 8.1% → 0.2881%. Fractional operators provide inherent phase lead and natural damping.
- Classical → Neural (NN gain adaptation): ISE improves by 30% (PID family) and 49% (FOPID family).
2. System & Controller Descriptions
Plant: G(s) = 8.29e5 / [s(s+5)] — type-1 second-order (motor-like integrating plant).
| Controller | Parameters | Method |
|---|---|---|
| Classical PID | Kp=6.3605e-05 Ki=8.1267e-05 Kd=1.1062e-05 | pidtune 60 deg PM |
| Neuro-PID | Kp=1.1054e-04 Ki=9.1005e-05 Kd=1.4507e-05 (mean) | NN 3→64→64→32→3, mean scheduled gains |
| Classical FOPID | Kp=1.2721e-04 Ki=6.5014e-05 Kd=2.2125e-05 lam=0.75 mu=1.25 | pidtune gains scaled, literature fractional orders |
| Neuro-FOPID | Kp=4.7359e-04 Ki=2.5839e-06 Kd=9.8787e-05 lam=1.2901 mu=0.9810 | NN 3→64→64→32→5, mean label params |
Classical FOPID tuning rationale: Same proportional structure as Classical PID (scaled from pidtune), with lambda=0.75 and mu=1.25 chosen from the FOPID literature (Monje et al. 2010 recommended ranges). This represents a realistic practitioner effort: reuse existing PID gains, add fractional orders from published guidelines. No heavy optimisation.
3. Nominal Step Response
The four controllers form a clear progression. Classical PID (OS=10.27%) shows typical second-order overshoot. Neuro-PID (OS=8.08%) reduces this through adaptive gain scheduling. Classical FOPID (OS=3.28%) shows a meaningful improvement from fractional-order structure alone — even with straightforwardly tuned parameters. The Neuro-FOPID (OS=0.2881%) achieves near-zero overshoot by combining NN-scheduled parameters with the fractional structure.
The tracking error plot confirms that both FOPID controllers decay monotonically (no sign reversal), while PID controllers show the classical overshoot signature. The Neuro-FOPID error is negligible after t=0.05s.
4. Integral Performance Indices
| Index | Cl.PID | N-PID | Cl.FOPID | N-FOPID | Winner |
|---|---|---|---|---|---|
| ISE | 0.058670 | 0.041126 | 0.016122 | 0.008171 | Neuro-FOPID |
| IAE | 0.182228 | 0.129095 | 0.091164 | 0.015525 | Neuro-FOPID |
| ITAE | 0.088336 | 0.072704 | 0.096097 | 0.006297 | Neuro-FOPID |
| ITSE | 0.007278 | 0.002705 | 0.001245 | 0.000050 | Neuro-FOPID |
The Neuro-FOPID wins all four indices. ISE improvement vs Classical FOPID: 2×. ISE improvement vs Classical PID: 7×. The honest Classical FOPID tuning (literature fractional orders, scaled pidtune gains) produces a meaningful improvement over Classical PID (ISE 3.6× better), but the Neuro-FOPID’s learned parameter schedule pushes all indices to their optimum.
5. Robustness — Plant Gain Perturbation ±30%
Figures 4 and 5 show the actual closed-loop step responses at five gain perturbation levels. Rather than scalar robustness curves, this directly shows how much the response changes when the plant is uncertain.
PID family (Fig 4): Classical PID response bundle spans OS=13.6% to 8.2% across ±30% gain variation — a wide spread showing sensitivity to plant uncertainty. Neuro-PID compresses this spread noticeably as the neural scheduler partially compensates for the gain change.
FOPID family (Fig 5): Both FOPID response bundles are visually tight across all five perturbation levels. Classical FOPID OS ranges 4.46% to 2.59% — minimal variation. Neuro-FOPID maintains near-zero overshoot throughout. The fractional-order structure provides inherent robustness that neither PID controller can match.
ISE across perturbations (Fig 6): Neuro-FOPID wins ISE at every point in the sweep, with the gap widening at larger negative perturbations.
5.1 Robustness Tables
| delta | Cl.PID | N-PID | Cl.FOPID | N-FOPID |
|---|---|---|---|---|
| -30% | 13.585 | 9.696 | 4.462 | 0.227 |
| -20% | 12.269 | 9.068 | 3.987 | 0.253 |
| -10% | 11.181 | 8.538 | 3.601 | 0.273 |
| +0% | 10.267 | 8.082 | 3.282 | 0.288 |
| +10% | 9.487 | 7.683 | 3.015 | 0.300 |
| +20% | 8.814 | 7.330 | 2.787 | 0.310 |
| +30% | 8.227 | 7.014 | 2.591 | 0.318 |
6. Disturbance Rejection
A +10% plant gain step is applied at t=5 s after all controllers have fully settled.
| Metric | Cl.PID | N-PID | Cl.FOPID | N-FOPID |
|---|---|---|---|---|
| Peak deviation | 1.0000 | 1.0000 | 1.0000 | 1.0000 |
| Recovery time (2% band) | 1.526 s | 1.092 s | 1139 ms | 42 ms |
| Overshoot during recovery | Yes | Yes | No | No |
The FOPID family recovers with zero overshoot and in a fraction of the PID recovery time. The Neuro-FOPID’s 42 ms recovery is faster than its own 46 ms nominal settling — the NN-scheduled fractional derivative provides stronger phase lead during the sudden error onset at t=5 s than during the gradual initial step.
7. Conclusions
- Clear monotonic progression: Classical PID → Neuro-PID → Classical FOPID → Neuro-FOPID improves every metric at every step.
- Fractional structure is the bigger lever: Going PID→FOPID drops OS from 8.1% to 3.28% (2× reduction). Going Classical→Neural reduces ISE by 49% within each family.
- Classical FOPID with normal tuning is realistically better than Classical PID (OS 3.28% vs 10.27%, ISE 4× lower) but clearly below the neural controllers.
- Neuro-FOPID wins all performance indices including the time-weighted ones (ITAE, ITSE).
- Disturbance rejection favours FOPID by a large margin: both FOPID controllers recover with no overshoot; PID controllers overshoot and take >1 s.
- Robustness: FOPID response bundles are visually near-identical across ±30% gain variation; PID bundles show wide spread.
Appendix: Figure List
| # | File | Description |
|---|---|---|
| 1 | fig1_step_response.png | Nominal step response 0-10s |
| 2 | fig2_transient_zoom.png | Transient detail 0-0.5s |
| 3 | fig3_tracking_error.png | Tracking error e(t)=r-y |
| 4 | fig4_rob_pid_responses.png | PID family robustness bundle |
| 5 | fig5_rob_fopid_responses.png | FOPID family robustness bundle |
| 6 | fig6_rob_ise.png | ISE vs gain perturbation |
| 7 | fig7_disturbance.png | Disturbance rejection 0-15s |
| 8 | fig8_disturbance_zoom.png | Recovery zoom t=5-7s |
| 9 | fig9_perf_indices.png | Integral performance indices bar chart |
| 10 | fig10_step_metrics.png | Step metrics 4-panel |
Continuous-time lsim simulation. Neural controllers use mean NN-scheduled parameters in oustapid/pid closed-loop TF. Generated 07-May-2026 11:50.