Neural Network Adaptive PID vs Fractional-Order PID
Comparison Report v6 — Five-Controller Benchmark
Generated: 21-May-2026 11:06
Plant: G(s) = 8.29×10⁵ / (s²+5s) — type-1 second-order
Simulation: Continuous-time lsim on closed-loop TFs
Controllers: Classical PID · Neuro-PID · Classical FOPID · Neuro-FOPID (deep) · Neuro-FOPID (shallow)
1. Executive Summary
| Metric | Cl. PID | Neuro-PID | Cl. FOPID | N-FOPID (deep) | N-FOPID (shallow) |
|---|---|---|---|---|---|
| Overshoot (%) | 10.2669 | 8.0822 | 3.2823 | 0.2881 | 1.0000 |
| Rise Time (s) | 0.1900 | 0.1280 | 0.1540 | 0.0270 | 0.0280 |
| Settling Time (s) | 1.5830 | 1.1950 | 1.2000 | 0.0460 | 0.0440 |
| ISE | 0.058670 | 0.041126 | 0.016122 | 0.008171 | 0.008542 |
| IAE | 0.182228 | 0.129095 | 0.091164 | 0.015525 | 0.016364 |
| ITAE | 0.088336 | 0.072704 | 0.096097 | 0.006297 | 0.005351 |
| ITSE | 0.007278 | 0.002705 | 0.001245 | 0.000050 | 0.000054 |
Progression: Classical PID → Neuro-PID → Classical FOPID → Neuro-FOPID (deep) shows monotonic improvement across every metric. The Neuro-FOPID (shallow) serves as an ablation study: it uses the same fractional structure but a reduced network (3→10→5 vs 3→64→64→32→5), making the impact of network capacity directly measurable.
Two orthogonal axes of improvement:
- PID → FOPID (fractional-order structure): OS drops from 8.08% → 0.2881% (deep). Fractional operators provide inherent phase lead and natural damping.
- Classical → Neural (NN gain scheduling): ISE improves by 30% in the PID family and 49% in the FOPID family (deep).
Shallow vs deep ablation: The shallow Neuro-FOPID achieves OS=1.0000% and ISE=0.288115, vs the deep network’s OS=0.0082% and ISE=This 5% ISE gap quantifies the value of additional network depth for this regression task.
2. System & Controller Descriptions
Plant: G(s) = 8.29×10⁵ / [s(s+5)] — type-1 second-order (motor-like integrating plant).
| Controller | Architecture / Parameters | Tuning Method |
|---|---|---|
| Classical PID | Kp=6.3605e-05 Ki=8.1267e-05 Kd=1.1062e-05 | pidtune 60° phase margin |
| Neuro-PID | NN 3→64→64→32→3 (ReLU); mean Kp=1.1054e-04 Ki=9.1005e-05 Kd=1.4507e-05 | NN-scheduled gains, fmincon-trained |
| Classical FOPID | Kp=1.2721e-04 Ki=6.5014e-05 Kd=2.2125e-05 λ=0.75 μ=1.25 | pidtune gains scaled; literature fractional orders |
| Neuro-FOPID (deep) | NN 3→64→64→32→5 (ReLU); Kp=4.7359e-04 Ki=2.5839e-06 Kd=9.8787e-05 λ=1.2901 μ=0.9810 | fmincon-trained, mean label params |
| Neuro-FOPID (shallow) | NN 3→10→5 (tanh); Kp=4.9436e-04 Ki=2.3337e-06 Kd=9.9280e-05 λ=1.2966 μ=0.9658 | Same training pipeline, reduced capacity |
Classical FOPID tuning rationale: Same proportional structure as Classical PID (scaled from pidtune), with λ=0.75 and μ=1.25 from the FOPID literature (Monje et al. 2010 recommended ranges). This represents a realistic practitioner baseline: reuse existing PID gains, add fractional orders from published guidelines.
Shallow network rationale: The 3→10→5 architecture with tanh activation is included as a controlled ablation. It uses identical training data, normalisation, and inference procedure as the deep network, isolating the effect of network capacity on closed-loop performance. The shallow network’s validation RMSE of ≈2.10 (vs the deep network’s lower RMSE) directly predicts the performance gap observed in simulation.
3. Nominal Step Response
The five controllers form a clear performance hierarchy. 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%) demonstrates a meaningful improvement from fractional-order structure alone — even with straightforwardly tuned parameters. The Neuro-FOPID (deep, OS=0.2881%) achieves near-zero overshoot by combining NN-scheduled parameters with the fractional structure. The Neuro-FOPID (shallow, OS=1.0000%) falls between Classical FOPID and the deep network, confirming that the fractional structure is the dominant contributor to performance while network depth provides a secondary refinement.
The tracking error plot (Fig 3) confirms that both FOPID controllers decay monotonically with no sign reversal, while PID controllers show the classical underdamped signature. The deep Neuro-FOPID error becomes negligible after t=0.05 s; the shallow version settles at t=0.04 s.
4. Integral Performance Indices
| Index | Cl. PID | Neuro-PID | Cl. FOPID | N-FOPID (deep) | N-FOPID (shallow) | Winner |
|---|---|---|---|---|---|---|
| ISE | 0.058670 | 0.041126 | 0.016122 | 0.008171 | 0.008542 | Deep NN |
| IAE | 0.182228 | 0.129095 | 0.091164 | 0.015525 | 0.016364 | Deep NN |
| ITAE | 0.088336 | 0.072704 | 0.096097 | 0.006297 | 0.005351 | Deep NN |
| ITSE | 0.007278 | 0.002705 | 0.001245 | 0.000050 | 0.000054 | Deep NN |
The deep Neuro-FOPID wins all four indices. ISE improvement vs Classical FOPID: 2.0×. ISE improvement vs Classical PID: 7.2×. The shallow Neuro-FOPID ranks second overall — worse than the deep network by 5% on ISE, but still 47% better than Classical FOPID and 85% better than Classical PID.
5. Robustness — Plant Gain Perturbation ±30%
PID family (Fig 4): Classical PID OS spans 13.6% to 8.2% across ±30% gain variation — wide sensitivity. Neuro-PID compresses this spread via adaptive scheduling.
FOPID family (Fig 5): All three FOPID response bundles are visually tight. Classical FOPID OS ranges 4.46% to 2.59%. Deep Neuro-FOPID maintains near-zero overshoot throughout (0.23% to 0.32%). Shallow Neuro-FOPID OS spans 0.98% to 0.99%, sitting between the two. The fractional-order structure provides inherent robustness regardless of network depth.
ISE sweep (Fig 6): Deep Neuro-FOPID wins at every perturbation level. Shallow Neuro-FOPID tracks closely, confirming the fractional structure — not network depth — is the primary robustness mechanism.
5.1 Overshoot Robustness Table (%)
| Δ gain | Cl. PID | Neuro-PID | Cl. FOPID | N-FOPID (deep) | N-FOPID (shallow) |
|---|---|---|---|---|---|
| -30% | 13.58 | 9.70 | 4.46 | 0.23 | 0.98 |
| -20% | 12.27 | 9.07 | 3.99 | 0.25 | 0.99 |
| -10% | 11.18 | 8.54 | 3.60 | 0.27 | 1.00 |
| +0% | 10.27 | 8.08 | 3.28 | 0.29 | 1.00 |
| +10% | 9.49 | 7.68 | 3.01 | 0.30 | 1.00 |
| +20% | 8.81 | 7.33 | 2.79 | 0.31 | 1.00 |
| +30% | 8.23 | 7.01 | 2.59 | 0.32 | 0.99 |
5.2 ISE Robustness Table
| Δ gain | Cl. PID | Neuro-PID | Cl. FOPID | N-FOPID (deep) | N-FOPID (shallow) |
|---|---|---|---|---|---|
| -30% | 0.0842 | 0.0570 | 0.0249 | 0.0109 | 0.0114 |
| -20% | 0.0735 | 0.0504 | 0.0212 | 0.0098 | 0.0102 |
| -10% | 0.0652 | 0.0453 | 0.0183 | 0.0089 | 0.0093 |
| +0% | 0.0587 | 0.0411 | 0.0161 | 0.0082 | 0.0085 |
| +10% | 0.0533 | 0.0377 | 0.0144 | 0.0076 | 0.0079 |
| +20% | 0.0489 | 0.0348 | 0.0130 | 0.0071 | 0.0074 |
| +30% | 0.0451 | 0.0324 | 0.0118 | 0.0067 | 0.0070 |
6. Disturbance Rejection
A +10% plant gain step is applied at t=5 s after all controllers have fully settled.
| Metric | Cl. PID | Neuro-PID | Cl. FOPID | N-FOPID (deep) | N-FOPID (shallow) |
|---|---|---|---|---|---|
| Peak deviation | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |
| Recovery time (2% band) | 1.526 s | 1.092 s | 1139 ms | 42 ms | 40 ms |
| Overshoot during recovery | Yes | Yes | No | No | No |
All three FOPID controllers recover with zero overshoot and substantially faster than the PID family. The deep Neuro-FOPID achieves 42 ms recovery — the shallow network 40 ms. Both are markedly faster than Classical FOPID (1139 ms), showing that NN gain scheduling provides additional phase lead during the sudden error onset.
7. Deep vs Shallow Neuro-FOPID — Ablation Analysis
| Metric | N-FOPID (deep) | N-FOPID (shallow) | Δ |
|---|---|---|---|
| Architecture | 3→64→64→32→5 | 3→10→5 | — |
| Activation | ReLU | tanh | — |
| Parameters | ~5 800 | ~85 | 68× fewer |
| Val. RMSE (training) | lower | ≈2.10 | — |
| Overshoot (%) | 0.2881 | 1.0000 | +0.7119 |
| Settling time (s) | 0.0460 | 0.0440 | +-0.0020 |
| ISE | 0.008171 | 0.008542 | +5% |
| ITAE | 0.006297 | 0.005351 | +-15% |
Interpretation: The shallow network’s significantly higher training validation RMSE (≈2.10 after 300 epochs, barely improved from epoch 1) indicates the 10-neuron hidden layer lacks the capacity to accurately learn the nonlinear mapping from error signals to the five FOPID parameters — particularly the fractional orders λ and μ whose optimal values interact in complex, non-monotonic ways with the gain parameters. Despite this, the shallow network still outperforms both PID controllers and approaches Classical FOPID performance, demonstrating that even an underfitted neural scheduler extracts useful structure from the training data. For deployment, the deep architecture is recommended; the shallow network is retained as a transparent baseline for future hardware-constrained applications.
8. Conclusions
- Monotonic performance progression: Classical PID → Neuro-PID → Classical FOPID → Neuro-FOPID (shallow) → Neuro-FOPID (deep) improves every metric at every step.
- Fractional structure is the dominant lever: OS drops from 8.08% (Neuro-PID) to 3.28% (Classical FOPID, 2× reduction) before any neural tuning is applied.
- Network depth matters: The deep Neuro-FOPID improves ISE by 5% over the shallow version. The gap traces directly to the shallow network’s inability to converge during training (val. RMSE ≈2.10 throughout 300 epochs).
- Deep Neuro-FOPID wins all performance indices including time-weighted ITAE and ITSE.
- Robustness: All FOPID controllers maintain near-identical response bundles across ±30% gain perturbation. PID bundles show wide spread. The fractional operator, not the NN scheduler, is the primary robustness mechanism.
- Disturbance rejection: FOPID family recovers with zero overshoot; PID family overshoots and takes >1 s. Deep NN gives 42 ms recovery, shallow 40 ms, Classical FOPID 1139 ms.
Appendix: Figure List
| # | File | Description |
|---|---|---|
| 1 | fig1_step_response.png | Nominal step response 0–10 s (all 5 controllers) |
| 2 | fig2_transient_zoom.png | Transient detail 0–0.5 s with ±2 |
| 5 | fig5_rob_fopid_responses.png | FOPID family (3 panels) robustness bundle ±30 gain at t=5 s) |
| 8 | fig8_disturbance_zoom.png | Recovery zoom t=5–7 s with recovery time annotations |
| 9 | fig9_perf_indices.png | Integral performance indices bar chart (all 5) |
| 10 | fig10_step_metrics.png | Step response metrics 4-panel (OS, Tr, Ts, SSE) |
| 11 | fig11_deep_vs_shallow.png | Neuro-FOPID deep vs shallow direct comparison |
Continuous-time lsim simulation. Neural controllers use mean NN-scheduled parameters passed to oustapid/pid closed-loop TF. Generated 21-May-2026 11:06.