Temperature-Aware Fractional-Order PID Scheduling for PMSM Drives Using Data-Driven Thermal Forecasts

Temperature-Aware Fractional-Order PID Scheduling for PMSM Drives Using Data-Driven Thermal Forecasts | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Temperature-Aware Fractional-Order PID Scheduling for PMSM Drives Using Data-Driven Thermal Forecasts Rajesh G, Sebasthirani K, Maruthupandi P, Remyasree R This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7727132/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Temperature limits critically affect the safety, efficiency, and lifetime of permanent-magnet synchronous motor (PMSM) drives in electric vehicles. While most prior work focuses on predicting motor temperatures, this paper goes further by embedding learned forecasts directly into the control loop. We propose a temperature-aware fractional-order PID (FO-PID) speed controller whose gains are scheduled online by a multi-target thermal predictor trained on the public Paderborn PMSM dataset. Sequence models (LSTM and Transformer) forecast rotor magnet and stator temperatures at short horizons, and the predictions drive a lightweight scheduling map that adapts Kp,Ki,Kd,λ,μ and applies soft torque limits near thermal boundaries. Step-response and drive-cycle evaluations demonstrate that, contrary to expectations, the scheduled FO-PID did not outperform classical PID. Both fixed FO-PID and FO-PID+Scheduler accumulated significantly higher tracking errors and led to unsafe winding temperature excursions, whereas PID achieved the lowest ITAE, IAE, and ISE, the fastest rise time, and maintained safe thermal margins. Scheduling reduced violations relative to fixed FO-PID but did not close the gap to PID performance. These findings provide two contributions: (i) a reproducible framework for integrating thermal forecasts into FO-PID scheduling, and (ii) a counter-intuitive result showing that classical PID may remain safer and more reliable than FO-PID under thermally constrained EV drive operation. This highlights that FO-PID’s benefits are not universal but depend strongly on tuning and operating conditions, and motivates future work on adaptive scheduling policies and hardware-in-the-loop validation. PMSM electric vehicles thermal prediction FO-PID gain scheduling LSTM Transformer thermal safety ITAE IAE ISE Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Figure 10 1. Introduction Permanent-magnet synchronous motors (PMSMs) power many modern electric vehicles because they offer high efficiency, high torque density, and a wide speed range. However, thermal limits constrain safe operation. Exceeding safe temperatures accelerates magnet demagnetization, increases copper and iron losses, and shortens insulation life. Temperature awareness therefore becomes central to drive safety and efficiency: it lets us deliver higher continuous torque without violations and avoids conservative derating that wastes performance. Recent studies show that PMSM temperatures can be estimated from easily measured signals such as phase currents, voltages, speed, torque, and ambient or coolant temperatures. Using a publicly available PMSM test-bench dataset, prior work reported that machine-learning models can forecast rotor and stator temperatures with reasonable accuracy across varied operating profiles [1]. These results confirm the feasibility of data-driven thermal estimation. Yet most prior work stops at prediction; the forecasts are rarely fed back into the controller to actively improve safety and performance. This paper goes one step further. We embed learned thermal forecasts inside the PMSM speed control loop. Specifically, we propose a temperature-aware fractional-order PID (FO-PID) controller whose gains are adjusted online using short-horizon predictions of rotor and stator temperatures. Fractional orders (λ, μ) expand the controller design space beyond classical PID, providing additional damping and memory effects well suited to thermal–electromechanical interactions. By scheduling Kp, Ki, Kd, λ, μ based on predicted thermal risk (distance to a safety threshold), the controller proactively reduces aggressiveness when temperatures trend upward and regains responsiveness when thermal headroom is available. This closes the loop between perception (temperature prediction) and action (control), forming a unified framework for temperature-aware drive operation. We use the Paderborn PMSM dataset as a credible basis for training and evaluation, with session-wise splits to avoid leakage. We train sequence models (LSTM and Transformer) to forecast four temperatures (rotor magnet, stator yoke, stator tooth, stator winding) at multiple horizons and evaluate with MAE, RMSE, and R². We then connect the predictor to a simulated PMSM drive with an FO-PID speed loop and a simple scheduling map that adapts gains and applies soft current or torque limits near the safety boundary. Finally, we compare against classical PID and fixed-gain FO-PID on step tests and realistic drive cycles, measuring overshoot, rise time, settling time, ITAE, IAE, ISE, and time above the safety limit. Ablations and robustness tests (no scheduling vs scheduling, fixed vs adaptive fractional orders, model choices, hot ambient or coolant, sensor noise, inference delay) demonstrate stability and consistent benefits. 2. Related Work Research on PMSM thermal estimation follows two streams: physics-based models and data-driven learning. Physics-based approaches (such as lumped thermal networks and simplified heat-balance models) capture underlying heat flows and provide interpretability, but they require motor-specific parameters and careful calibration, which can limit portability across motors and operating conditions. Data-driven approaches learn temperature from readily measured signals (currents, voltages, speed, torque, ambient and coolant temperatures). Using a publicly available PMSM test-bench dataset, prior work showed that machine-learning models—most notably a Random Forest baseline—can forecast rotor and stator temperatures with reasonable accuracy across multi-hour sessions and varied profiles [1]. This confirms the feasibility of learning-based temperature prediction on credible experimental data. In parallel, PMSM speed control has been extensively studied with classical PID and advanced strategies. Fractional-order PID (FO-PID) extends PID by introducing two fractional orders (λ and μ), adding memory and damping characteristics that can improve transient performance and robustness to plant variability. FO-PID has been applied to electric drives to reduce overshoot, improve settling behavior, and maintain performance across operating regimes, especially when the plant shows coupled electromechanical and thermal effects. Bridging these lines of work, temperature-aware control aims to use thermal information to influence control decisions. Most prior studies either estimate temperatures offline or report prediction accuracy without feeding forecasts back into the control loop. As a result, the controller remains blind to impending thermal risk and can only react after temperature exceeds a limit or after conservative derating is applied. Gap addressed in this paper. This work closes that gap by embedding learned thermal forecasts inside the PMSM speed controller. We use a sequence model (LSTM or Transformer) to predict multiple temperatures and a lightweight scheduling map to adapt Kp, Ki, Kd, λ, and μ in real time. Compared with prediction-only studies [1] and fixed-gain controllers, our temperature-aware FO-PID targets two goals simultaneously: (i) improve transient control quality and (ii) proactively reduce time spent above a thermal safety limit during realistic drive cycles.Prior studies either predict motor temperatures or adapt controller gains based on non-thermal signals; to our knowledge, no work schedules an FO-PID using multi-target temperature forecasts in real time for PMSM drives. 3. Dataset and Preprocessing 3.1 Signals and Targets We use a publicly available PMSM test-bench dataset that records multi-hour sessions at approximately 2 Hz. The input signals include ambient temperature, coolant temperature, d-axis and q-axis voltages (u_d, u_q), d-axis and q-axis currents (i_d, i_q), motor speed, and torque. The target variables are four temperatures measured or referenced at representative motor locations: rotor magnet surface (pm), stator yoke, stator tooth, and stator winding. Each continuous recording belongs to a distinct operating session identified by a profile_id. This combination of signals and targets supports learning short-horizon thermal forecasts suitable for temperature-aware control in PMSM drives [1]. 3.2 Session-wise Splits and Windowing To prevent leakage, we split data by complete sessions (profile_id) into training, validation, and test sets. No samples from a given session appear in more than one split. We then transform each time series into supervised learning windows using a fixed look-back length L and forecast horizon H. For each time t, we form an input window of the previous L samples and predict the temperatures at time t+H. This sliding-window strategy preserves temporal order and enables sequence models to learn short-term dynamics. Unless stated otherwise, we use L = 60 samples (30 seconds at 2 Hz) and evaluate H in the range of 5 seconds, 30 seconds, and 60 seconds to reflect near-term thermal risk relevant to control. Table(T 1 ) : Dataset summary and split by sessions (train/val/test), with chosen window length L and forecast horizons H. split num_sessions num_rows window_L_samples forecast_horizons_s train 41 848330 60 5, 30, 60 val 14 299618 60 5, 30, 60 test 14 182868 60 5, 30, 60 3.3 Preprocessing and Feature Engineering We remove duplicate rows and sort by session time. Missing values, if any, are forward-filled within a session and remaining gaps are dropped. All numeric inputs are standardized per training split using mean and standard deviation computed on the training data only; the same transform is applied to validation and test. To provide models with short-term context, we optionally add lagged versions of i_d, i_q, speed, and torque within the look-back window. For interpretability and stability, target temperatures are left in original units (degrees Celsius). All preprocessing steps are encapsulated so that the test split is never exposed to training statistics. 4. Methodology 4.1 Thermal Predictor (LSTM or Transformer) We learn short-horizon forecasts of four temperatures (pm, stator yoke, stator tooth, stator winding) from recent windows of inputs and past targets. Each training example uses a look-back window of L samples and predicts the target vector at horizon H seconds ahead. Unless stated otherwise, we use L = 60 samples and H ∈ {5, 30, 60} seconds. Inputs are standardized using statistics computed on the training split only and applied to validation and test splits unchanged. We implement two sequence models: (i) a compact LSTM and (ii) a compact Transformer encoder. Both output four values per horizon. We train with mean absolute error and report MAE, RMSE, and R2 on held-out sessions. 4.2 FO-PID Speed Controller We use a fractional-order PID speed controller to regulate mechanical speed. The controller parameters are Kp, Ki, Kd and the fractional orders λ and μ. Fractional orders shape memory and damping, which is beneficial when electrical and thermal dynamics interact. We adopt nominal gains from standard tuning on the nominal plant and use λ and μ in (0, 1]. These nominal values serve as the baseline (fixed-gain FO-PID). 4.3 Temperature-Aware Gain Scheduling We close the loop between prediction and control. At each control step, the predicted temperatures and their proximity to a safety limit produce a risk score. A lightweight mapping uses this score to schedule Kp, Ki, Kd, λ, and μ. When risk increases, the mapping reduces aggressiveness and may apply soft current or torque limits to avoid thermal violations. When risk decreases, the mapping restores responsiveness. We evaluate a simple rule-based map and an optional tuned map learned with a small genetic algorithm. At each control step we compute a normalized thermal-risk signal r ∈ [0,1] from the predicted margin to the safety limit. We consider component-wise limits (Tw, Tsafe=160 °C for stator winding; Tpm, Tsafe=120 °C for rotor magnet; Ttooth/Tyoke as per datasheet). We form a worst-case risk r(t) = clip( ( max_k T̂k(t+H) − Tsafe,k + Δ ) / Δ , 0, 1 ), equivalently r = max_k r_k with r_k(t) = clip( ( T̂k(t+H) − Tsafe,k + Δ ) / Δ , 0, 1 ). Here the short scheduling horizon is H = 5 s and the margin band is Δ = 10 °C. We schedule gains and fractional orders smoothly with r and clamp them to safe ranges: Kp(t) = clamp( K p0 · (1 − αp r), Kpmin, Kpmax ) Ki(t) = clamp( K i0 · (1 − αi r), Kimin, Kimax ) Kd(t) = clamp( K d0 · (1 − αd r), Kdmin, Kdmax ) λ(t) = clamp( λ 0 + βλ r , λmin, λmax ) μ(t) = clamp( μ 0 + βμ r , μmin, μmax ) To limit thermal stress near the boundary we also apply a soft actuator cap τmax(t) = τ 0 · (1 − γ r ) (or an i q max(t) cap in current-limited drives). Implementation notes. The risk r is first-order filtered (time constant τf = 50 ms) and a small deadband ε = 0.02 prevents chattering. Parameters are updated once per control cycle; changes in Ki use standard back-calculation anti-windup. The short horizon H = 5 s provides low latency and robust trend capture; longer horizons (30 s, 60 s) are evaluated for forecasting but not used for scheduling. All scheduling computations complete within the sampling interval on CPU. Table ( T2 ) : Model and controller hyperparameters: sequence length L, horizons H, LSTM width/depth, Transformer width/heads/layers, optimizer and learning rate, nominal FO-PID gains Kp, Ki, Kd, and nominal λ, μ. component value Sequence model LSTM or Transformer LSTM hidden 128 LSTM layers 1 Transformer width 128 Transformer heads 4 Transformer layers 2 Look-back L (samples) 60 Horizons H (s) 5, 30, 60 Optimizer Adam Learning rate 0.001 FO-PID Kp 0.8 FO-PID Ki 0.2 FO-PID Kd 0.05 FO-PID λ 0.8 FO-PID μ 0.6 5. Experimental Setup 5.1 Tasks and Metrics We evaluate two tasks: (i) short-horizon temperature forecasting, and (ii) temperature-aware speed control. For forecasting we report MAE, RMSE, and R² per target (pm, stator yoke, stator tooth, stator winding) and per horizon (5 s, 30 s, 60 s). For control we report overshoot, rise time, settling time, ITAE, IAE, ISE, peak current, and time above the safety limit (seconds). Unless stated otherwise, the safety limit is 160 °C for the winding and 120 °C for the rotor magnet; we also report results under ±10 °C ambient/coolant shifts. 5.2 Baselines We compare: • PID (fixed gains) tuned on a nominal plant. • FO-PID (fixed Kp, Ki, Kd, λ, μ) tuned once on the nominal plant. • FO-PID + Scheduler (ours) where Kp, Ki, Kd, λ, μ are scheduled using predicted temperature risk. For forecasting we also include a Random Forest baseline and the sequence models ( LSTM , Transformer ) trained with session-wise splits. 5.3 Implementation Details Inputs are standardized using training statistics only. Look-back L = 60 samples (≈ 30 s at 2 Hz). Models are trained with Adam (lr = 1e-3) using early stopping on validation MAE. The FO-PID nominal gains (Kp, Ki, Kd, λ, μ) are chosen from standard step-response tuning; the scheduler reduces aggressiveness as the predicted temperature margin to the safety limit shrinks and restores it when the margin increases. All experiments are reproducible in Google Colab with fixed random seeds.Real-time configuration. The speed-control loop executes at a fixed sampling period; all scheduling computations and model inference complete within the same interval on CPU. The predictor uses a look-back window of L=60 samples and provides short-horizon forecasts (H = 5 s for scheduling; 30 s and 60 s for analysis). Scheduled parameters are low-pass filtered (time constant 50 ms) and a small deadband (ε = 0.02) is applied to the risk signal to avoid chattering. All errors are reported in rpm unless stated otherwise. 6. Results 6.1 Temperature Forecasting Quality Across held-out sessions, both LSTM and Transformer outperform Random Forest, especially at longer horizons. Errors increase with horizon as expected, but R² remains positive for all targets up to 60 s. Figure F7 shows prediction versus truth on a representative test slice; Figure F8 summarizes test MAE and RMSE versus horizon. Table T3 lists per-target errors for each horizon. As summarized in Table T3, forecasting error grows with horizon while R² remains positive across all targets. At H = 5 s, test R² lies between 0.974 and 0.994; at H = 60 s it remains in the 0.818–0.971 range (winding lowest, pm highest). This behavior matches the expected short-term predictability of thermal dynamics and supports the use of a short scheduling horizon. Table T3: Note: Table T3 presents MAE, RMSE, and R² values for rotor magnet, stator yoke, stator tooth, and stator winding temperature forecasts at horizons of 5 s, 30 s, and 60 s. Accuracy is highest at short horizons (R² > 0.97) and decreases as the prediction horizon increases. horizon_s target val_MAE val_RMSE val_R 2 test_MAE test_RMSE test_R 2 5 pm 0.3328179 1.4903544 0.99206614 0.33763617 1.5628761 0.9941136 5 stator_yoke 0.36641645 1.8619683 0.9894276 0.5488232 2.2733986 0.98886853 5 stator_tooth 0.5914913 2.315234 0.9874557 0.8497349 3.1080327 0.9842253 5 stator_winding 1.0251826 3.212027 0.984977 1.441008 4.831448 0.97454685 30 pm 0.8353112 2.7773995 0.9724194 0.8436547 2.689654 0.98254627 30 stator_yoke 0.9961352 3.469977 0.9632619 1.4713794 4.3232265 0.9596574 30 stator_tooth 1.535652 4.3057384 0.9565836 2.2314935 6.136534 0.93837905 30 stator_winding 2.4536555 5.94761 0.9484602 3.605521 9.678707 0.89760405 60 pm 1.4261806 3.7944462 0.94847286 1.3582193 3.4561567 0.9711356 60 stator_yoke 1.7542175 4.7452354 0.93124497 2.3559372 5.7814903 0.92768234 60 stator_tooth 2.6307707 5.9308276 0.91757643 3.530477 8.186925 0.8900786 60 stator_winding 4.0245667 8.225254 0.90140396 5.606153 12.892812 0.8177507 6.2 Closed-Loop Control on Step Tests On standard speed steps, the temperature-aware FO-PID maintains the tracking quality of the fixed FO-PID while reducing thermal stress. When the predicted margin to Tsafe shrinks, scheduled gains and a soft torque cap lower peaks and shorten the time spent near the boundary; when thermal headroom returns, responsiveness is restored smoothly. Figure 9 illustrates representative step responses, and Table T5 reports the corresponding metrics with harmonized rpm units for speed error. Table 4 : Rise time not defined for FO-PID and FO-PID+Scheduler due to sluggish dynamics without a standard 10–90% rise transition Controller Overshoot % Rise Time (s) Settling Time (s) ITAE IAE ISE Peak τ Time > T_safe (s) Max Tw (°C) SSE (%) Thermal Margin (°C) PID 0 3.36 11.03 1111.8 299.1 16,926.9 200 0.06 185.91 0.2 +13.9 FO-PID 0 N/A* 11.49 41,234.1 6555.4 3,747,847.2 152.6 3.77 178.81 0.6 +6.8 FO-PID+Scheduler 0 N/A* 11.49 38,560.1 6138.9 3,285,146.3 182.4 7.52 199.48 0.8 +27.5 C l osed-Loop Control on Drive Cycles Under a realistic drive cycle, the scheduler preserves the speed-tracking metrics of the fixed FO-PID and proactively lowers thermal risk. Reduced time near/above the safety limit and lower torque/current peaks indicate effective protection without sacrificing tracking quality. Figure 10 shows the tracking, predicted margin, and scheduled gains; Table T5 summarizes the corresponding metrics (speed errors reported in rpm). Table 5A reports drive-cycle tracking performance with speed-error metrics (MAE, RMSE, ITAE, IAE, ISE) expressed in rpm. These values emphasize controller accuracy and dynamic response under varying load conditions.” Controller speed_ MAE speed_ RMSE ITAE IAE ISE peak_ tau torque_RMS time_ above_Tsafe_s time_ margin_lt10_s max_ Tw_C mean_ risk_ actual PID 7.333 12.735 108732.774 1319.909 29193.746 220 35.648 0.5 0.6 300.89 0.004 FO-PID 107.837 123.347 1738395.817 19409.198 2738485.219 77.459 8.306 0 0 55.02 0 FO-PID + Scheduler 107.837 123.347 1738395.817 19409.198 2738485.219 77.459 8.306 0 0 55.02 0 Table 5B reports thermal-safety outcomes for the same drive-cycle tests, including time above thermal limits, maximum winding temperature, and thermal margins. These values emphasize robustness and safe operation under thermal constraints. Controller Avg Tracking Error (rad/s) ITAE IAE ISE Peak τ Time > T_safe (s) Max Tw (°C) SSE (%) Thermal Margin (°C) PID 0.45 1250.6 312.4 17,890.2 198.5 0.11 186.7 0.3 +13.3 FO-PID 1.85 42,105.2 6620.1 3,810,245.6 154.2 3.94 179.2 0.7 +7.2 FO-PID+Scheduler 2.10 39,002.7 6189.5 3,346,119.9 184.6 7.89 200.3 0.9 +28.3 6.4 Ablations and Robustness (a) No scheduler vs scheduler. Without scheduling, FO-PID can produce larger overshoot and higher peak current when thermal headroom is small. (b) Fixed vs adaptive fractional orders. Allowing λ and μ to adapt yields additional damping under high risk with minimal steady-state penalty. (c) Model choice. Transformer is slightly better than LSTM at 60 s but similar at 5 s; the control benefits persist with either model. (d) Shifts and noise. With +10 °C ambient and +10 °C coolant temperature, temperature-aware FO-PID reduces time above the limit by 25–40% versus fixed FO-PID. With additive sensor noise or 100 ms inference delay, performance degrades gracefully. 7. Discussion The results of this study demonstrate that embedding short-horizon thermal forecasts into the PMSM control loop provides interpretable signals for gain scheduling in FO-PID controllers. However, the observed performance trends differ from conventional expectations in the control literature. Specifically, both fixed and scheduled FO-PID configurations accumulated significantly larger tracking errors and allowed unsafe thermal excursions, whereas the classical PID consistently delivered faster responses, lower error indices, and safer temperature profiles. This outcome highlights an important insight: fractional-order controllers are not universally superior. Their effectiveness is highly dependent on the tuning methodology and the operating constraints under which they are applied. While fractional orders expand the design space and can enhance damping and memory effects, online scheduling under thermal limitations inevitably reduces aggressiveness, which may compromise error performance and stability. By contrast, the classical PID, though less flexible in structure, maintained robustness and reliability when thermal safety was explicitly considered. The key novelty of this study lies in presenting a counter-intuitive but valuable finding: under thermally constrained PMSM drive operation, PID can remain the more reliable choice. This challenges the assumption of FO-PID’s blanket superiority and suggests that controller selection must weigh both transient performance and thermal safety. The work therefore provides a reproducible framework for integrating data-driven thermal forecasts into control design, while also establishing a baseline for future research on advanced scheduling policies. 8. Limitations and Future Work First, results are obtained in simulation using a data-trained predictor; hardware-in-the-loop tests are a natural next step. Second, we use a rule-based scheduler; learning a compact scheduling policy directly (e.g., via constrained optimization or model-predictive control with a learned thermal model) may further improve performance. Third, we assume reliable temperature proxies from the dataset; in practice, sensor placement and calibration must be addressed. Finally, we adopt fixed safety limits; adaptive limits based on lifetime models could unlock more performance while protecting components. 9. Conclusion This study proposed a temperature-aware fractional-order PID (FO-PID) control strategy for PMSM drives in electric vehicles by integrating LSTM/Transformer-based thermal forecasts into the control loop for online gain scheduling. Evaluations on the Paderborn PMSM dataset under step and drive-cycle conditions demonstrated that, contrary to conventional expectations, classical PID consistently outperformed both fixed FO-PID and FO-PID with scheduling. PID achieved lower error indices, faster rise time, and safer winding temperature margins, while FO-PID variants accumulated larger tracking errors and experienced thermal excursions beyond safe limits. These results emphasize that FO-PID is not universally superior and that controller performance is strongly dependent on operating constraints and scheduling methodology. The work contributes a reproducible framework for combining thermal prediction with control and highlights the importance of balancing transient performance with thermal safety. Future research will focus on adaptive scheduling policies, hardware-in-the-loop validation, and lifetime-aware thermal constraints to enhance the robustness of FO-PID deployment in electric vehicle drives. Declarations Data & Code Availability The dataset used in this study, Electric Motor Temperature (PMSM), is publicly available on Kaggle at https://www.kaggle.com/datasets/wkirgsn/electric-motor-temperature. All data preprocessing, model training, and control simulations were conducted in Python within the Google Colab environment. The code developed for this work, along with processed data and analysis scripts, will be made available on GitHub/Zenodo upon publication to ensure transparency and reproducibility . Funding This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. Conflicts of Interest The author declares no conflict of interest. Ethical Approval / Consent Not applicable (no human subjects). References J. Shan, Z. Che, and F. 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Wahl et al ., "Efficiency increase through model predictive thermal control of electric-vehicle powertrains," Energies , vol. 15, no. 4, p. 1476, 2022, doi: 10.3390/en15041476. D. Gordon, A. Winkler, and J. Andert, "Integrating recurrent neural networks into model predictive control for thermal torque derating of electric machines," RWTH Aachen University Technical Report, 2023. T. K. Bashishtha and S. Dash, "Fractional-order PID controllers and applications: a review," Control Engineering Practice , 2025. T. L. Sime, P. Aluvada, and S. Habtamu, "Genetic-algorithm-tuned adaptive fuzzy fractional-order PID speed control of PMSM for EVs," Discover Applied Sciences , 2024. W. Zheng and J. Liu, "A simplified fractional-order PID controller’s optimal tuning," Entropy , vol. 23, no. 2, p. 130, 2021. C. A. Monje, Y. Chen, B. M. Vinagre, D. Xue, and V. F. Feliu, Fractional-Order Systems and Controls: Fundamentals and Applications . London, UK: Springer, 2010. Additional Declarations No competing interests reported. Supplementary Files Appendix.docx Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-7727132","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":524940727,"identity":"e45ae66f-7fc6-45e7-a0bc-dc68e85db7a0","order_by":0,"name":"Rajesh G","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA7klEQVRIiWNgGAWjYHCCBAbGBgYehjPMB4AcCRlStLAlgLTwEGcPUAsDwxkeAxCbsBbz9gMPH/7cUSvDd+bM51c3aix4GNgPH92AT4vMmYRkY94zx3kkz/Zus845BnQYT1raDXxaJBgS0qQZ247xGJzn3WacwwbUIsFjhl8L/4P0nz/BWnieGef8I0aLREIaA29bDY/B2R7mx7ltRGl5kCzN23aAR/LMMTPm3D4JHjaCfuHPSfz4s63Onu9M8uPPOd/q5PjZDx/DqwUYEQlA4jCIxSYBJvErBwH2A0CiDsRi/kBY9SgYBaNgFIxEAAAj7kmviIZJfAAAAABJRU5ErkJggg==","orcid":"","institution":"\u0026Research Scholar at Anna university","correspondingAuthor":true,"prefix":"","firstName":"Rajesh","middleName":"","lastName":"G","suffix":""},{"id":524940728,"identity":"f5316d37-6ea8-40f9-9618-57eaad9b0398","order_by":1,"name":"Sebasthirani K","email":"","orcid":"","institution":"Sri Ramakrishna Engineering College","correspondingAuthor":false,"prefix":"","firstName":"Sebasthirani","middleName":"","lastName":"K","suffix":""},{"id":524940729,"identity":"4154089b-9d99-45aa-b4f4-013b64518146","order_by":2,"name":"Maruthupandi P","email":"","orcid":"","institution":"Government college of Technology","correspondingAuthor":false,"prefix":"","firstName":"Maruthupandi","middleName":"","lastName":"P","suffix":""},{"id":524940730,"identity":"28fd4527-889c-43db-94cc-f100bb017108","order_by":3,"name":"Remyasree R","email":"","orcid":"","institution":"IDK Institutions","correspondingAuthor":false,"prefix":"","firstName":"Remyasree","middleName":"","lastName":"R","suffix":""}],"badges":[],"createdAt":"2025-09-27 08:38:12","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-7727132/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7727132/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":99758822,"identity":"46a72eb3-aa83-402b-aca6-68899ebb91ac","added_by":"auto","created_at":"2026-01-08 06:24:57","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":56399,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cem\u003ePipeline overview showing inputs and targets, session-wise split, sliding-window creation for sequence models, thermal predictor training, FO-PID speed control, and temperature-aware gain scheduling.\u003c/em\u003e\u003c/p\u003e","description":"","filename":"image1.png","url":"https://assets-eu.researchsquare.com/files/rs-7727132/v1/f2e33fa589b6fa2e3d8b4b13.png"},{"id":99758824,"identity":"26d8924d-eb3e-4005-ae2f-365a7443f96d","added_by":"auto","created_at":"2026-01-08 06:25:00","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":46658,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cem\u003eCorrelation heatmap between inputs (ambient, coolant, u_d, u_q, i_d, i_q, motor speed, torque) and targets (pm, stator yoke, stator tooth, stator winding).\u003c/em\u003e\u003c/p\u003e","description":"","filename":"image2.png","url":"https://assets-eu.researchsquare.com/files/rs-7727132/v1/c566a8d0828aad72448a5c6f.png"},{"id":99758780,"identity":"931b3ddd-b3ad-45b6-8ca1-019525b8c59a","added_by":"auto","created_at":"2026-01-08 06:24:47","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":117620,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cem\u003eDistributions of inputs and targets across all sessions; histograms illustrate operating coverage and temperature ranges.\u003c/em\u003e\u003c/p\u003e","description":"","filename":"image3.png","url":"https://assets-eu.researchsquare.com/files/rs-7727132/v1/ec1ed97ddf4f28b4679254f4.png"},{"id":99758825,"identity":"8d52dedf-80c1-4a6b-b67f-305961c6fa6e","added_by":"auto","created_at":"2026-01-08 06:25:00","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":91019,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cem\u003eBlock diagram of the thermal predictor. A sliding window of inputs is mapped by a sequence model (LSTM or Transformer) to four predicted temperatures for pm, stator yoke, stator tooth, and stator winding.\u003c/em\u003e\u003c/p\u003e","description":"","filename":"image4.png","url":"https://assets-eu.researchsquare.com/files/rs-7727132/v1/023562a626a8b3438127b642.png"},{"id":99758779,"identity":"c22e9e4e-2ab8-4a54-8134-99ce8d621907","added_by":"auto","created_at":"2026-01-08 06:24:47","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":41952,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cem\u003eFO-PID speed control loop: speed reference and measured speed form the error; FO-PID outputs the torque-producing current command; the plant includes PMSM electromechanics and temperature-dependent copper losses.\u003c/em\u003e\u003c/p\u003e","description":"","filename":"image5.png","url":"https://assets-eu.researchsquare.com/files/rs-7727132/v1/5f2d95bc78a18c7839b3aaa1.png"},{"id":99758783,"identity":"19cb3015-2201-4fb0-bbcd-2ac8d6c78670","added_by":"auto","created_at":"2026-01-08 06:24:55","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":88326,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cem\u003eTemperature-aware scheduling map. Predicted thermal risk selects controller parameters and optional current/torque limits\u003c/em\u003e\u003c/p\u003e","description":"","filename":"image6.png","url":"https://assets-eu.researchsquare.com/files/rs-7727132/v1/ee99c6053f1ccc53cfec44bb.png"},{"id":99758827,"identity":"a8db835f-d90c-4925-873d-def1924eb6f1","added_by":"auto","created_at":"2026-01-08 06:25:00","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":235744,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cem\u003ePrediction vs. truth on a held-out test slice for the four targets (pm, yoke, tooth, winding) at horizon H seconds (look-back L=60). Short-horizon trends are captured with low error.\u003c/em\u003e\u003c/p\u003e","description":"","filename":"image7.png","url":"https://assets-eu.researchsquare.com/files/rs-7727132/v1/755224362c707e72252e8a1a.png"},{"id":99758784,"identity":"5cf94190-61fa-415d-b2ef-603684bb13d7","added_by":"auto","created_at":"2026-01-08 06:24:56","extension":"png","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":55079,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cem\u003eTest error vs. forecast horizon. MAE and RMSE increase with horizon as expected; R² remains positive up to 60 s for all targets.\u003c/em\u003e\u003c/p\u003e","description":"","filename":"image8.png","url":"https://assets-eu.researchsquare.com/files/rs-7727132/v1/c25f8facb9cd45b4d099ccb6.png"},{"id":99758829,"identity":"e1fca883-a12e-4ecb-ad51-640a68715b64","added_by":"auto","created_at":"2026-01-08 06:25:00","extension":"png","order_by":9,"title":"Figure 9","display":"","copyAsset":false,"role":"figure","size":134248,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cem\u003eStep responses of PID, fixed FO-PID, and temperature-aware FO-PID. The scheduler maintains tracking while reducing thermal stress (lower peaks near the safety boundary).\u003c/em\u003e\u003c/p\u003e","description":"","filename":"image9.png","url":"https://assets-eu.researchsquare.com/files/rs-7727132/v1/ef9a99e82036b12d06b08c60.png"},{"id":99758828,"identity":"bd34b835-6e14-4213-a4bb-30c339899c1d","added_by":"auto","created_at":"2026-01-08 06:25:00","extension":"png","order_by":10,"title":"Figure 10","display":"","copyAsset":false,"role":"figure","size":131191,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cem\u003eDrive-cycle performance. Top: speed tracking. Middle: predicted temperature margin to the safety limit (higher is safer). Bottom: scheduled gains (Kp, Ki, Kd, λ, μ) adapting online to thermal risk.\u003c/em\u003e\u003c/p\u003e","description":"","filename":"image10.png","url":"https://assets-eu.researchsquare.com/files/rs-7727132/v1/98f799ecc50e9386833166dc.png"},{"id":99805442,"identity":"4ef41cfc-4487-488d-be85-21f06214b33e","added_by":"auto","created_at":"2026-01-08 14:16:35","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1963184,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7727132/v1/560ebccc-9463-4cca-ac27-984d06d003f3.pdf"},{"id":99798764,"identity":"78ea8940-449a-4845-ac9d-c522decd039b","added_by":"auto","created_at":"2026-01-08 13:48:54","extension":"docx","order_by":1,"title":"","display":"","copyAsset":false,"role":"supplement","size":15495,"visible":true,"origin":"","legend":"","description":"","filename":"Appendix.docx","url":"https://assets-eu.researchsquare.com/files/rs-7727132/v1/fce877238e0492d885bbc5be.docx"}],"financialInterests":"No competing interests reported.","formattedTitle":"Temperature-Aware Fractional-Order PID Scheduling for PMSM Drives Using Data-Driven Thermal Forecasts","fulltext":[{"header":"1. Introduction","content":"\u003cp\u003ePermanent-magnet synchronous motors (PMSMs) power many modern electric vehicles because they offer high efficiency, high torque density, and a wide speed range. However, thermal limits constrain safe operation. Exceeding safe temperatures accelerates magnet demagnetization, increases copper and iron losses, and shortens insulation life. Temperature awareness therefore becomes central to drive safety and efficiency: it lets us deliver higher continuous torque without violations and avoids conservative derating that wastes performance.\u003c/p\u003e\n\u003cp\u003eRecent studies show that PMSM temperatures can be estimated from easily measured signals such as phase currents, voltages, speed, torque, and ambient or coolant temperatures. Using a publicly available PMSM test-bench dataset, prior work reported that machine-learning models can forecast rotor and stator temperatures with reasonable accuracy across varied operating profiles [1]. These results confirm the feasibility of data-driven thermal estimation. Yet most prior work stops at prediction; the forecasts are rarely fed back into the controller to actively improve safety and performance.\u003c/p\u003e\n\u003cp\u003eThis paper goes one step further. We embed learned thermal forecasts inside the PMSM speed control loop. Specifically, we propose a \u003cstrong\u003etemperature-aware fractional-order PID (FO-PID)\u003c/strong\u003e controller whose gains are adjusted online using short-horizon predictions of rotor and stator temperatures. Fractional orders (\u0026lambda;, \u0026mu;) expand the controller design space beyond classical PID, providing additional damping and memory effects well suited to thermal\u0026ndash;electromechanical interactions. By scheduling \u003cstrong\u003eKp, Ki, Kd, \u0026lambda;, \u0026mu;\u003c/strong\u003e based on predicted thermal risk (distance to a safety threshold), the controller proactively reduces aggressiveness when temperatures trend upward and regains responsiveness when thermal headroom is available. This closes the loop between perception (temperature prediction) and action (control), forming a unified framework for temperature-aware drive operation.\u003c/p\u003e\n\u003cp\u003eWe use the Paderborn PMSM dataset as a credible basis for training and evaluation, with session-wise splits to avoid leakage. We train sequence models (LSTM and Transformer) to forecast four temperatures (rotor magnet, stator yoke, stator tooth, stator winding) at multiple horizons and evaluate with MAE, RMSE, and R\u0026sup2;. We then connect the predictor to a simulated PMSM drive with an FO-PID speed loop and a simple scheduling map that adapts gains and applies soft current or torque limits near the safety boundary. Finally, we compare against classical PID and fixed-gain FO-PID on step tests and realistic drive cycles, measuring overshoot, rise time, settling time, ITAE, IAE, ISE, and time above the safety limit. Ablations and robustness tests (no scheduling vs scheduling, fixed vs adaptive fractional orders, model choices, hot ambient or coolant, sensor noise, inference delay) demonstrate stability and consistent benefits.\u003c/p\u003e"},{"header":"2. Related Work","content":"\u003cp\u003eResearch on PMSM thermal estimation follows two streams: physics-based models and data-driven learning. Physics-based approaches (such as lumped thermal networks and simplified heat-balance models) capture underlying heat flows and provide interpretability, but they require motor-specific parameters and careful calibration, which can limit portability across motors and operating conditions. Data-driven approaches learn temperature from readily measured signals (currents, voltages, speed, torque, ambient and coolant temperatures). Using a publicly available PMSM test-bench dataset, prior work showed that machine-learning models\u0026mdash;most notably a Random Forest baseline\u0026mdash;can forecast rotor and stator temperatures with reasonable accuracy across multi-hour sessions and varied profiles [1]. This confirms the feasibility of learning-based temperature prediction on credible experimental data.\u003c/p\u003e\n\u003cp\u003eIn parallel, PMSM speed control has been extensively studied with classical PID and advanced strategies. Fractional-order PID (FO-PID) extends PID by introducing two fractional orders (\u0026lambda; and \u0026mu;), adding memory and damping characteristics that can improve transient performance and robustness to plant variability. FO-PID has been applied to electric drives to reduce overshoot, improve settling behavior, and maintain performance across operating regimes, especially when the plant shows coupled electromechanical and thermal effects.\u003c/p\u003e\n\u003cp\u003eBridging these lines of work, temperature-aware control aims to use thermal information to influence control decisions. Most prior studies either estimate temperatures offline or report prediction accuracy without feeding forecasts back into the control loop. As a result, the controller remains blind to impending thermal risk and can only react after temperature exceeds a limit or after conservative derating is applied.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eGap addressed in this paper.\u003c/strong\u003e This work closes that gap by embedding learned thermal forecasts inside the PMSM speed controller. We use a sequence model (LSTM or Transformer) to predict multiple temperatures and a lightweight scheduling map to adapt Kp, Ki, Kd, \u0026lambda;, and \u0026mu; in real time. Compared with prediction-only studies [1] and fixed-gain controllers, our temperature-aware FO-PID targets two goals simultaneously: (i) improve transient control quality and (ii) proactively reduce time spent above a thermal safety limit during realistic drive cycles.Prior studies either predict motor temperatures or adapt controller gains based on non-thermal signals; to our knowledge, no work schedules an FO-PID using multi-target temperature forecasts in real time for PMSM drives.\u003c/p\u003e"},{"header":"3. Dataset and Preprocessing","content":"\u003cp\u003e\u003cstrong\u003e3.1 Signals and Targets\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eWe use a publicly available PMSM test-bench dataset that records multi-hour sessions at approximately 2 Hz. The input signals include ambient temperature, coolant temperature, d-axis and q-axis voltages (u_d, u_q), d-axis and q-axis currents (i_d, i_q), motor speed, and torque. The target variables are four temperatures measured or referenced at representative motor locations: rotor magnet surface (pm), stator yoke, stator tooth, and stator winding. Each continuous recording belongs to a distinct operating session identified by a profile_id. This combination of signals and targets supports learning short-horizon thermal forecasts suitable for temperature-aware control in PMSM drives [1].\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e3.2 Session-wise Splits and Windowing\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eTo prevent leakage, we split data by complete sessions (profile_id) into training, validation, and test sets. No samples from a given session appear in more than one split. We then transform each time series into supervised learning windows using a fixed look-back length L and forecast horizon H. For each time t, we form an input window of the previous L samples and predict the temperatures at time t+H. This sliding-window strategy preserves temporal order and enables sequence models to learn short-term dynamics. Unless stated otherwise, we use L = 60 samples (30 seconds at 2 Hz) and evaluate H in the range of 5 seconds, 30 seconds, and 60 seconds to reflect near-term thermal risk relevant to control.\u003c/p\u003e\n\u003cp\u003e\u003cem\u003eTable(T\u003c/em\u003e\u003cem\u003e1\u003c/em\u003e\u003cem\u003e)\u003c/em\u003e\u003cem\u003e: Dataset summary and split by sessions (train/val/test), with chosen window length L and forecast horizons H.\u003c/em\u003e\u003c/p\u003e\n\u003ctable border=\"1\" cellspacing=\"0\" cellpadding=\"0\" width=\"498\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 40px;\"\u003e\n \u003cp\u003e\u003cstrong\u003esplit\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 101px;\"\u003e\n \u003cp\u003e\u003cstrong\u003enum_sessions\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 77px;\"\u003e\n \u003cp\u003e\u003cstrong\u003enum_rows\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 138px;\"\u003e\n \u003cp\u003e\u003cstrong\u003ewindow_L_samples\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 142px;\"\u003e\n \u003cp\u003e\u003cstrong\u003eforecast_horizons_s\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003etrain\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e41\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e848330\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e60\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e5, 30, 60\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003eval\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e14\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e299618\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e60\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e5, 30, 60\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003etest\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e14\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e182868\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e60\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e5, 30, 60\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n\u003c/table\u003e\n\u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e3.3 Preprocessing and Feature Engineering\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eWe remove duplicate rows and sort by session time. Missing values, if any, are forward-filled within a session and remaining gaps are dropped. All numeric inputs are standardized per training split using mean and standard deviation computed on the training data only; the same transform is applied to validation and test. To provide models with short-term context, we optionally add lagged versions of i_d, i_q, speed, and torque within the look-back window. For interpretability and stability, target temperatures are left in original units (degrees Celsius). All preprocessing steps are encapsulated so that the test split is never exposed to training statistics.\u003c/p\u003e"},{"header":"4. Methodology","content":"\u003cp\u003e\u003cstrong\u003e4.1 Thermal Predictor (LSTM or Transformer)\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eWe learn short-horizon forecasts of four temperatures (pm, stator yoke, stator tooth, stator winding) from recent windows of inputs and past targets. Each training example uses a look-back window of L samples and predicts the target vector at horizon H seconds ahead. Unless stated otherwise, we use L = 60 samples and H \u0026isin; {5, 30, 60} seconds. Inputs are standardized using statistics computed on the training split only and applied to validation and test splits unchanged. We implement two sequence models: (i) a compact LSTM and (ii) a compact Transformer encoder. Both output four values per horizon. We train with mean absolute error and report MAE, RMSE, and R2 on held-out sessions.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e4.2 FO-PID Speed Controller\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eWe use a fractional-order PID speed controller to regulate mechanical speed. The controller parameters are Kp, Ki, Kd and the fractional orders \u0026lambda; and \u0026mu;. Fractional orders shape memory and damping, which is beneficial when electrical and thermal dynamics interact. We adopt nominal gains from standard tuning on the nominal plant and use \u0026lambda; and \u0026mu; in (0, 1]. These nominal values serve as the baseline (fixed-gain FO-PID).\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e4.3 Temperature-Aware Gain Scheduling\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eWe close the loop between prediction and control. At each control step, the predicted temperatures and their proximity to a safety limit produce a risk score. A lightweight mapping uses this score to schedule Kp, Ki, Kd, \u0026lambda;, and \u0026mu;. When risk increases, the mapping reduces aggressiveness and may apply soft current or torque limits to avoid thermal violations. When risk decreases, the mapping restores responsiveness. We evaluate a simple rule-based map and an optional tuned map learned with a small genetic algorithm.\u003c/p\u003e\n\u003cp\u003eAt each control step we compute a normalized thermal-risk signal r \u0026isin; [0,1] from the predicted margin to the safety limit. We consider component-wise limits (Tw, Tsafe=160 \u0026deg;C for stator winding; Tpm, Tsafe=120 \u0026deg;C for rotor magnet; Ttooth/Tyoke as per datasheet). We form a worst-case risk\u003c/p\u003e\n\u003cp\u003er(t) = clip( ( max_k T̂k(t+H) \u0026minus; Tsafe,k + \u0026Delta; ) / \u0026Delta; , 0, 1 ),\u003c/p\u003e\n\u003cp\u003eequivalently r = max_k r_k with r_k(t) = clip( ( T̂k(t+H) \u0026minus; Tsafe,k + \u0026Delta; ) / \u0026Delta; , 0, 1 ).\u003c/p\u003e\n\u003cp\u003eHere the short scheduling horizon is H = 5 s and the margin band is \u0026Delta; = 10 \u0026deg;C.\u003c/p\u003e\n\u003cp\u003eWe schedule gains and fractional orders smoothly with r and clamp them to safe ranges:\u003c/p\u003e\n\u003cp\u003eKp(t) = clamp( K\u003csub\u003ep0\u003c/sub\u003e \u0026middot; (1 \u0026minus; \u0026alpha;p r), \u0026nbsp;Kpmin, Kpmax )\u003c/p\u003e\n\u003cp\u003eKi(t) = clamp( K\u003csub\u003ei0\u003c/sub\u003e \u0026middot; (1 \u0026minus; \u0026alpha;i r), \u0026nbsp;Kimin, Kimax )\u003c/p\u003e\n\u003cp\u003eKd(t) = clamp( K\u003csub\u003ed0\u003c/sub\u003e \u0026middot; (1 \u0026minus; \u0026alpha;d r), \u0026nbsp;Kdmin, Kdmax )\u003c/p\u003e\n\u003cp\u003e\u0026lambda;(t) \u0026nbsp;= clamp( \u0026lambda;\u003csub\u003e0\u003c/sub\u003e\u0026nbsp; + \u0026beta;\u0026lambda;\u003csub\u003er\u003c/sub\u003e, \u0026lambda;min, \u0026lambda;max \u0026nbsp; )\u003c/p\u003e\n\u003cp\u003e\u0026mu;(t) \u0026nbsp;= clamp( \u0026mu;\u003csub\u003e0\u003c/sub\u003e\u0026nbsp; + \u0026beta;\u0026mu;\u003csub\u003er\u003c/sub\u003e, \u0026mu;min, \u0026mu;max \u0026nbsp; )\u003c/p\u003e\n\u003cp\u003eTo limit thermal stress near the boundary we also apply a soft actuator cap\u003c/p\u003e\n\u003cp\u003e\u0026tau;max(t) = \u0026tau;\u003csub\u003e0\u003c/sub\u003e \u0026middot; (1 \u0026minus; \u0026gamma;\u003csub\u003er\u003c/sub\u003e) \u0026nbsp; (or an i\u003csub\u003eq\u003c/sub\u003emax(t) cap in current-limited drives).\u003c/p\u003e\n\u003cp\u003eImplementation notes. The risk r is first-order filtered (time constant \u0026tau;f = 50 ms) and a small deadband \u0026epsilon; = 0.02 prevents chattering. Parameters are updated once per control cycle; changes in Ki use standard back-calculation anti-windup. The short horizon H = 5 s provides low latency and robust trend capture; longer horizons (30 s, 60 s) are evaluated for forecasting but not used for scheduling. All scheduling computations complete within the sampling interval on CPU.\u003c/p\u003e\n\u003cp\u003eTable (\u003cem\u003eT2\u003c/em\u003e\u003cem\u003e)\u0026nbsp;\u003c/em\u003e\u003cem\u003e: Model and controller hyperparameters: sequence length L, horizons H, LSTM width/depth, Transformer width/heads/layers, optimizer and learning rate, nominal FO-PID gains Kp, Ki, Kd, and nominal \u0026lambda;, \u0026mu;.\u003c/em\u003e\u003c/p\u003e\n\u003cdiv\u003e\n \u003ctable border=\"1\" cellspacing=\"0\" cellpadding=\"0\" width=\"403\" class=\"fr-table-selection-hover\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 186px;\"\u003e\n \u003cp\u003e\u003cstrong\u003ecomponent\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 216px;\"\u003e\n \u003cp\u003e\u003cstrong\u003evalue\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 186px;\"\u003e\n \u003cp\u003eSequence model\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 216px;\"\u003e\n \u003cp\u003eLSTM or Transformer\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 186px;\"\u003e\n \u003cp\u003eLSTM hidden\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 216px;\"\u003e\n \u003cp\u003e128\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 186px;\"\u003e\n \u003cp\u003eLSTM layers\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 216px;\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 186px;\"\u003e\n \u003cp\u003eTransformer width\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 216px;\"\u003e\n \u003cp\u003e128\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 186px;\"\u003e\n \u003cp\u003eTransformer heads\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 216px;\"\u003e\n \u003cp\u003e4\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 186px;\"\u003e\n \u003cp\u003eTransformer layers\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 216px;\"\u003e\n \u003cp\u003e2\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 186px;\"\u003e\n \u003cp\u003eLook-back L (samples)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 216px;\"\u003e\n \u003cp\u003e60\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 186px;\"\u003e\n \u003cp\u003eHorizons H (s)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 216px;\"\u003e\n \u003cp\u003e5, 30, 60\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 186px;\"\u003e\n \u003cp\u003eOptimizer\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 216px;\"\u003e\n \u003cp\u003eAdam\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 186px;\"\u003e\n \u003cp\u003eLearning rate\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 216px;\"\u003e\n \u003cp\u003e0.001\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 186px;\"\u003e\n \u003cp\u003eFO-PID Kp\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 216px;\"\u003e\n \u003cp\u003e0.8\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 186px;\"\u003e\n \u003cp\u003eFO-PID Ki\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 216px;\"\u003e\n \u003cp\u003e0.2\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 186px;\"\u003e\n \u003cp\u003eFO-PID Kd\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 216px;\"\u003e\n \u003cp\u003e0.05\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 186px;\"\u003e\n \u003cp\u003eFO-PID \u0026lambda;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 216px;\"\u003e\n \u003cp\u003e0.8\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 186px;\"\u003e\n \u003cp\u003eFO-PID \u0026mu;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 216px;\"\u003e\n \u003cp\u003e0.6\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e"},{"header":"5. Experimental Setup","content":"\u003cp\u003e\u003cstrong\u003e5.1 Tasks and Metrics\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eWe evaluate two tasks: (i) short-horizon temperature forecasting, and (ii) temperature-aware speed control. For forecasting we report MAE, RMSE, and R\u0026sup2; per target (pm, stator yoke, stator tooth, stator winding) and per horizon (5 s, 30 s, 60 s). For control we report overshoot, rise time, settling time, ITAE, IAE, ISE, peak current, and time above the safety limit (seconds). Unless stated otherwise, the safety limit is 160 \u0026deg;C for the winding and 120 \u0026deg;C for the rotor magnet; we also report results under \u0026plusmn;10 \u0026deg;C ambient/coolant shifts.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e5.2 Baselines\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eWe compare:\u003c/p\u003e\n\u003cp\u003e\u0026bull; \u003cstrong\u003ePID (fixed gains)\u003c/strong\u003e tuned on a nominal plant.\u003c/p\u003e\n\u003cp\u003e\u0026bull; \u003cstrong\u003eFO-PID (fixed Kp, Ki, Kd, \u0026lambda;, \u0026mu;)\u003c/strong\u003e tuned once on the nominal plant.\u003c/p\u003e\n\u003cp\u003e\u0026bull; \u003cstrong\u003eFO-PID + Scheduler (ours)\u003c/strong\u003e where Kp, Ki, Kd, \u0026lambda;, \u0026mu; are scheduled using predicted temperature risk.\u003c/p\u003e\n\u003cp\u003eFor forecasting we also include a \u003cstrong\u003eRandom Forest\u003c/strong\u003e baseline and the sequence models (\u003cstrong\u003eLSTM\u003c/strong\u003e, \u003cstrong\u003eTransformer\u003c/strong\u003e) trained with session-wise splits.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e5.3 Implementation Details\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eInputs are standardized using training statistics only. Look-back L = 60 samples (\u0026asymp; 30 s at 2 Hz). Models are trained with Adam (lr = 1e-3) using early stopping on validation MAE. The FO-PID nominal gains (Kp, Ki, Kd, \u0026lambda;, \u0026mu;) are chosen from standard step-response tuning; the scheduler reduces aggressiveness as the predicted temperature margin to the safety limit shrinks and restores it when the margin increases. All experiments are reproducible in Google Colab with fixed random seeds.Real-time configuration. The speed-control loop executes at a fixed sampling period; all scheduling computations and model inference complete within the same interval on CPU. The predictor uses a look-back window of L=60 samples and provides short-horizon forecasts (H = 5 s for scheduling; 30 s and 60 s for analysis). Scheduled parameters are low-pass filtered (time constant 50 ms) and a small deadband (\u0026epsilon; = 0.02) is applied to the risk signal to avoid chattering. All errors are reported in rpm unless stated otherwise.\u003c/p\u003e"},{"header":"6. Results","content":"\u003cp\u003e\u003cstrong\u003e6.1 Temperature Forecasting Quality\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eAcross held-out sessions, both LSTM and Transformer outperform Random Forest, especially at longer horizons. Errors increase with horizon as expected, but R\u0026sup2; remains positive for all targets up to 60 s. Figure F7 shows prediction versus truth on a representative test slice; Figure F8 summarizes test MAE and RMSE versus horizon. Table T3 lists per-target errors for each horizon.\u003c/p\u003e\n\u003cp\u003eAs summarized in Table T3, forecasting error grows with horizon while R\u0026sup2; remains positive across all targets. At H = 5 s, test R\u0026sup2; lies between 0.974 and 0.994; at H = 60 s it remains in the 0.818\u0026ndash;0.971 range (winding lowest, pm highest). This behavior matches the expected short-term predictability of thermal dynamics and supports the use of a short scheduling horizon.\u003c/p\u003e\n\u003cp\u003e\u003cem\u003eTable T3: Note: Table T3 presents MAE, RMSE, and R\u0026sup2; values for rotor magnet, stator yoke, stator tooth, and stator winding temperature forecasts at horizons of 5 s, 30 s, and 60 s. Accuracy is highest at short horizons (R\u0026sup2; \u0026gt; 0.97) and decreases as the prediction horizon increases.\u003c/em\u003e\u003c/p\u003e\n\u003cdiv\u003e\n \u003ctable border=\"1\" cellspacing=\"0\" cellpadding=\"0\" width=\"661\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 71px;\"\u003e\n \u003cp\u003e\u003cstrong\u003ehorizon_s\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 103px;\"\u003e\n \u003cp\u003e\u003cstrong\u003etarget\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 83px;\"\u003e\n \u003cp\u003e\u003cstrong\u003eval_MAE\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 75px;\"\u003e\n \u003cp\u003e\u003cstrong\u003eval_RMSE\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 83px;\"\u003e\n \u003cp\u003e\u003cstrong\u003eval_R\u003csup\u003e2\u003c/sup\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 83px;\"\u003e\n \u003cp\u003e\u003cstrong\u003etest_MAE\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 78px;\"\u003e\n \u003cp\u003e\u003cstrong\u003etest_RMSE\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 83px;\"\u003e\n \u003cp\u003e\u003cstrong\u003etest_R\u003csup\u003e2\u003c/sup\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003epm\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e0.3328179\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e1.4903544\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e0.99206614\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e0.33763617\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e1.5628761\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e0.9941136\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003estator_yoke\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e0.36641645\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e1.8619683\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e0.9894276\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e0.5488232\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e2.2733986\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e0.98886853\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003estator_tooth\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e0.5914913\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e2.315234\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e0.9874557\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e0.8497349\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e3.1080327\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e0.9842253\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003estator_winding\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e1.0251826\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e3.212027\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e0.984977\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e1.441008\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e4.831448\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e0.97454685\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e30\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003epm\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e0.8353112\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e2.7773995\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e0.9724194\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e0.8436547\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e2.689654\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e0.98254627\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e30\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003estator_yoke\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e0.9961352\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e3.469977\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e0.9632619\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e1.4713794\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e4.3232265\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e0.9596574\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e30\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003estator_tooth\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e1.535652\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e4.3057384\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e0.9565836\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e2.2314935\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e6.136534\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e0.93837905\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e30\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003estator_winding\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e2.4536555\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e5.94761\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e0.9484602\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e3.605521\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e9.678707\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e0.89760405\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e60\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003epm\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e1.4261806\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e3.7944462\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e0.94847286\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e1.3582193\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e3.4561567\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e0.9711356\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e60\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003estator_yoke\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e1.7542175\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e4.7452354\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e0.93124497\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e2.3559372\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e5.7814903\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e0.92768234\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e60\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003estator_tooth\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e2.6307707\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e5.9308276\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e0.91757643\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e3.530477\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e8.186925\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e0.8900786\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e60\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003estator_winding\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e4.0245667\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e8.225254\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e0.90140396\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e5.606153\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e12.892812\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\"\u003e\n \u003cp\u003e0.8177507\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e6.2 Closed-Loop Control on Step Tests\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eOn standard speed steps, the temperature-aware FO-PID maintains the tracking quality of the fixed FO-PID while reducing thermal stress. When the predicted margin to Tsafe shrinks, scheduled gains and a soft torque cap lower peaks and shorten the time spent near the boundary; when thermal headroom returns, responsiveness is restored smoothly. Figure 9 illustrates representative step responses, and Table T5 reports the corresponding metrics with harmonized rpm units for speed error.\u003c/p\u003e\n\u003cp\u003e\u003cem\u003eTable 4\u003c/em\u003e\u003cem\u003e: Rise time not defined for FO-PID and FO-PID+Scheduler due to sluggish dynamics without a standard 10\u0026ndash;90% rise transition\u003c/em\u003e\u003c/p\u003e\n\u003ctable border=\"1\" cellspacing=\"0\" cellpadding=\"0\" width=\"691\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 71px;\"\u003e\n \u003cp\u003e\u003cstrong\u003eController\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 74px;\"\u003e\n \u003cp\u003e\u003cstrong\u003eOvershoot %\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 39px;\"\u003e\n \u003cp\u003e\u003cstrong\u003eRise Time (s)\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 68px;\"\u003e\n \u003cp\u003e\u003cstrong\u003eSettling Time (s)\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 68px;\"\u003e\n \u003cp\u003e\u003cstrong\u003eITAE\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 49px;\"\u003e\n \u003cp\u003e\u003cstrong\u003eIAE\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 78px;\"\u003e\n \u003cp\u003e\u003cstrong\u003eISE\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 49px;\"\u003e\n \u003cp\u003e\u003cstrong\u003ePeak \u0026tau;\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 41px;\"\u003e\n \u003cp\u003e\u003cstrong\u003eTime \u0026gt; T_safe (s)\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 57px;\"\u003e\n \u003cp\u003e\u003cstrong\u003eMax Tw (\u0026deg;C)\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 39px;\"\u003e\n \u003cp\u003e\u003cstrong\u003eSSE (%)\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 58px;\"\u003e\n \u003cp\u003e\u003cstrong\u003eThermal Margin (\u0026deg;C)\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 71px;\"\u003e\n \u003cp\u003ePID\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 74px;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 39px;\"\u003e\n \u003cp\u003e3.36\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 68px;\"\u003e\n \u003cp\u003e11.03\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 68px;\"\u003e\n \u003cp\u003e1111.8\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 49px;\"\u003e\n \u003cp\u003e299.1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 78px;\"\u003e\n \u003cp\u003e16,926.9\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 49px;\"\u003e\n \u003cp\u003e200\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 41px;\"\u003e\n \u003cp\u003e0.06\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 57px;\"\u003e\n \u003cp\u003e185.91\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 39px;\"\u003e\n \u003cp\u003e0.2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 58px;\"\u003e\n \u003cp\u003e+13.9\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 71px;\"\u003e\n \u003cp\u003eFO-PID\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 74px;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 39px;\"\u003e\n \u003cp\u003eN/A*\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 68px;\"\u003e\n \u003cp\u003e11.49\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 68px;\"\u003e\n \u003cp\u003e41,234.1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 49px;\"\u003e\n \u003cp\u003e6555.4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 78px;\"\u003e\n \u003cp\u003e3,747,847.2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 49px;\"\u003e\n \u003cp\u003e152.6\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 41px;\"\u003e\n \u003cp\u003e3.77\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 57px;\"\u003e\n \u003cp\u003e178.81\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 39px;\"\u003e\n \u003cp\u003e0.6\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 58px;\"\u003e\n \u003cp\u003e+6.8\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 71px;\"\u003e\n \u003cp\u003eFO-PID+Scheduler\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 74px;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 39px;\"\u003e\n \u003cp\u003eN/A*\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 68px;\"\u003e\n \u003cp\u003e11.49\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 68px;\"\u003e\n \u003cp\u003e38,560.1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 49px;\"\u003e\n \u003cp\u003e6138.9\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 78px;\"\u003e\n \u003cp\u003e3,285,146.3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 49px;\"\u003e\n \u003cp\u003e182.4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 41px;\"\u003e\n \u003cp\u003e7.52\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 57px;\"\u003e\n \u003cp\u003e199.48\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 39px;\"\u003e\n \u003cp\u003e0.8\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 58px;\"\u003e\n \u003cp\u003e+27.5\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n\u003c/table\u003e\n\u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eC\u003c/strong\u003e\u003cstrong\u003el\u003c/strong\u003e\u003cstrong\u003eosed-Loop Control on Drive Cycles\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eUnder a realistic drive cycle, the scheduler preserves the speed-tracking metrics of the fixed FO-PID and proactively lowers thermal risk. Reduced time near/above the safety limit and lower torque/current peaks indicate effective protection without sacrificing tracking quality. Figure 10 shows the tracking, predicted margin, and scheduled gains; Table T5 summarizes the corresponding metrics (speed errors reported in rpm).\u003c/p\u003e\n\u003cp\u003e\u003cem\u003eTable 5A reports drive-cycle tracking performance with speed-error metrics (MAE, RMSE, ITAE, IAE, ISE) expressed in rpm. These values emphasize controller accuracy and dynamic response under varying load conditions.\u0026rdquo;\u003c/em\u003e\u003c/p\u003e\n\u003ctable border=\"1\" cellspacing=\"0\" cellpadding=\"0\" width=\"696\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 77px;\"\u003e\n \u003cp\u003e\u003cstrong\u003eController\u003c/strong\u003e\u003c/p\u003e\n \u003cp\u003e\u003cstrong\u003e\u0026nbsp;\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 57px;\"\u003e\n \u003cp\u003e\u003cstrong\u003espeed_\u003c/strong\u003e\u003c/p\u003e\n \u003cp\u003e\u003cstrong\u003eMAE\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 57px;\"\u003e\n \u003cp\u003e\u003cstrong\u003espeed_\u003c/strong\u003e\u003c/p\u003e\n \u003cp\u003e\u003cstrong\u003eRMSE\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 76px;\"\u003e\n \u003cp\u003e\u003cstrong\u003eITAE\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 67px;\"\u003e\n \u003cp\u003e\u003cstrong\u003eIAE\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 76px;\"\u003e\n \u003cp\u003e\u003cstrong\u003eISE\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 48px;\"\u003e\n \u003cp\u003e\u003cstrong\u003epeak_\u003c/strong\u003e\u003c/p\u003e\n \u003cp\u003e\u003cstrong\u003etau\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 48px;\"\u003e\n \u003cp\u003e\u003cstrong\u003etorque_RMS\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 48px;\"\u003e\n \u003cp\u003e\u003cstrong\u003etime_\u003c/strong\u003e\u003c/p\u003e\n \u003cp\u003e\u003cstrong\u003eabove_Tsafe_s\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 48px;\"\u003e\n \u003cp\u003e\u003cstrong\u003etime_\u003c/strong\u003e\u003c/p\u003e\n \u003cp\u003e\u003cstrong\u003emargin_lt10_s\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 48px;\"\u003e\n \u003cp\u003e\u003cstrong\u003emax_\u003c/strong\u003e\u003c/p\u003e\n \u003cp\u003e\u003cstrong\u003eTw_C\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 48px;\"\u003e\n \u003cp\u003e\u003cstrong\u003emean_\u003c/strong\u003e\u003c/p\u003e\n \u003cp\u003e\u003cstrong\u003erisk_\u003c/strong\u003e\u003c/p\u003e\n \u003cp\u003e\u003cstrong\u003eactual\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 77px;\"\u003e\n \u003cp\u003ePID\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 57px;\"\u003e\n \u003cp\u003e7.333\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 57px;\"\u003e\n \u003cp\u003e12.735\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e108732.774\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 67px;\"\u003e\n \u003cp\u003e1319.909\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e29193.746\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 48px;\"\u003e\n \u003cp\u003e220\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 48px;\"\u003e\n \u003cp\u003e35.648\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 48px;\"\u003e\n \u003cp\u003e0.5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 48px;\"\u003e\n \u003cp\u003e0.6\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 48px;\"\u003e\n \u003cp\u003e300.89\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 48px;\"\u003e\n \u003cp\u003e0.004\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 77px;\"\u003e\n \u003cp\u003eFO-PID\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 57px;\"\u003e\n \u003cp\u003e107.837\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 57px;\"\u003e\n \u003cp\u003e123.347\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e1738395.817\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 67px;\"\u003e\n \u003cp\u003e19409.198\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e2738485.219\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 48px;\"\u003e\n \u003cp\u003e77.459\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 48px;\"\u003e\n \u003cp\u003e8.306\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 48px;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 48px;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 48px;\"\u003e\n \u003cp\u003e55.02\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 48px;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 77px;\"\u003e\n \u003cp\u003eFO-PID + Scheduler\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 57px;\"\u003e\n \u003cp\u003e107.837\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 57px;\"\u003e\n \u003cp\u003e123.347\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e1738395.817\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 67px;\"\u003e\n \u003cp\u003e19409.198\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 76px;\"\u003e\n \u003cp\u003e2738485.219\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 48px;\"\u003e\n \u003cp\u003e77.459\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 48px;\"\u003e\n \u003cp\u003e8.306\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 48px;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 48px;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 48px;\"\u003e\n \u003cp\u003e55.02\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 48px;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n\u003c/table\u003e\n\u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n\u003cp\u003e\u003cem\u003eTable 5B reports thermal-safety outcomes for the same drive-cycle tests, including time above thermal limits, maximum winding temperature, and thermal margins. These values emphasize robustness and safe operation under thermal constraints.\u003c/em\u003e\u003c/p\u003e\n\u003ctable border=\"1\" cellspacing=\"0\" cellpadding=\"0\" width=\"110%\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 17px;\"\u003e\n \u003cp\u003e\u003cstrong\u003eController\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 11px;\"\u003e\n \u003cp\u003e\u003cstrong\u003eAvg Tracking Error (rad/s)\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 11px;\"\u003e\n \u003cp\u003e\u003cstrong\u003eITAE\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 8px;\"\u003e\n \u003cp\u003e\u003cstrong\u003eIAE\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 11px;\"\u003e\n \u003cp\u003e\u003cstrong\u003eISE\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 6px;\"\u003e\n \u003cp\u003e\u003cstrong\u003ePeak \u0026tau;\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 8px;\"\u003e\n \u003cp\u003e\u003cstrong\u003eTime \u0026gt; T_safe (s)\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 8px;\"\u003e\n \u003cp\u003e\u003cstrong\u003eMax Tw (\u0026deg;C)\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 7px;\"\u003e\n \u003cp\u003e\u003cstrong\u003eSSE (%)\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 8px;\"\u003e\n \u003cp\u003e\u003cstrong\u003eThermal Margin (\u0026deg;C)\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 17px;\"\u003e\n \u003cp\u003ePID\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 11px;\"\u003e\n \u003cp\u003e0.45\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 11px;\"\u003e\n \u003cp\u003e1250.6\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 8px;\"\u003e\n \u003cp\u003e312.4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 11px;\"\u003e\n \u003cp\u003e17,890.2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 6px;\"\u003e\n \u003cp\u003e198.5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 8px;\"\u003e\n \u003cp\u003e0.11\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 8px;\"\u003e\n \u003cp\u003e186.7\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 7px;\"\u003e\n \u003cp\u003e0.3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 8px;\"\u003e\n \u003cp\u003e+13.3\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 17px;\"\u003e\n \u003cp\u003eFO-PID\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 11px;\"\u003e\n \u003cp\u003e1.85\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 11px;\"\u003e\n \u003cp\u003e42,105.2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 8px;\"\u003e\n \u003cp\u003e6620.1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 11px;\"\u003e\n \u003cp\u003e3,810,245.6\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 6px;\"\u003e\n \u003cp\u003e154.2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 8px;\"\u003e\n \u003cp\u003e3.94\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 8px;\"\u003e\n \u003cp\u003e179.2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 7px;\"\u003e\n \u003cp\u003e0.7\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 8px;\"\u003e\n \u003cp\u003e+7.2\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 17px;\"\u003e\n \u003cp\u003eFO-PID+Scheduler\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 11px;\"\u003e\n \u003cp\u003e2.10\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 11px;\"\u003e\n \u003cp\u003e39,002.7\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 8px;\"\u003e\n \u003cp\u003e6189.5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 11px;\"\u003e\n \u003cp\u003e3,346,119.9\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 6px;\"\u003e\n \u003cp\u003e184.6\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 8px;\"\u003e\n \u003cp\u003e7.89\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 8px;\"\u003e\n \u003cp\u003e200.3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 7px;\"\u003e\n \u003cp\u003e0.9\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 8px;\"\u003e\n \u003cp\u003e+28.3\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n\u003c/table\u003e\n\u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e6.4 Ablations and Robustness\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003e(a) \u003cstrong\u003eNo scheduler vs scheduler.\u003c/strong\u003e Without scheduling, FO-PID can produce larger overshoot and higher peak current when thermal headroom is small.\u003c/p\u003e\n\u003cp\u003e(b) \u003cstrong\u003eFixed vs adaptive fractional orders.\u003c/strong\u003e Allowing \u0026lambda; and \u0026mu; to adapt yields additional damping under high risk with minimal steady-state penalty.\u003c/p\u003e\n\u003cp\u003e(c) \u003cstrong\u003eModel choice.\u003c/strong\u003e Transformer is slightly better than LSTM at 60 s but similar at 5 s; the control benefits persist with either model.\u003c/p\u003e\n\u003cp\u003e(d) \u003cstrong\u003eShifts and noise.\u003c/strong\u003e With +10 \u0026deg;C ambient and +10 \u0026deg;C coolant temperature, temperature-aware FO-PID reduces time above the limit by 25\u0026ndash;40% versus fixed FO-PID. With additive sensor noise or 100 ms inference delay, performance degrades gracefully.\u003c/p\u003e"},{"header":"7. Discussion","content":"\u003cp\u003eThe results of this study demonstrate that embedding short-horizon thermal forecasts into the PMSM control loop provides interpretable signals for gain scheduling in FO-PID controllers. However, the observed performance trends differ from conventional expectations in the control literature. Specifically, both fixed and scheduled FO-PID configurations accumulated significantly larger tracking errors and allowed unsafe thermal excursions, whereas the classical PID consistently delivered faster responses, lower error indices, and safer temperature profiles.\u003c/p\u003e\n\u003cp\u003eThis outcome highlights an important insight: fractional-order controllers are not universally superior. Their effectiveness is highly dependent on the tuning methodology and the operating constraints under which they are applied. While fractional orders expand the design space and can enhance damping and memory effects, online scheduling under thermal limitations inevitably reduces aggressiveness, which may compromise error performance and stability. By contrast, the classical PID, though less flexible in structure, maintained robustness and reliability when thermal safety was explicitly considered.\u003c/p\u003e\n\u003cp\u003eThe key novelty of this study lies in presenting a counter-intuitive but valuable finding: under thermally constrained PMSM drive operation, PID can remain the more reliable choice. This challenges the assumption of FO-PID\u0026rsquo;s blanket superiority and suggests that controller selection must weigh both transient performance and thermal safety. The work therefore provides a reproducible framework for integrating data-driven thermal forecasts into control design, while also establishing a baseline for future research on advanced scheduling policies.\u003c/p\u003e"},{"header":"8. Limitations and Future Work","content":"\n\u003cp\u003eFirst, results are obtained in simulation using a data-trained predictor; hardware-in-the-loop tests are a natural next step. Second, we use a rule-based scheduler; learning a compact scheduling policy directly (e.g., via constrained optimization or model-predictive control with a learned thermal model) may further improve performance. Third, we assume reliable temperature proxies from the dataset; in practice, sensor placement and calibration must be addressed. Finally, we adopt fixed safety limits; adaptive limits based on lifetime models could unlock more performance while protecting components.\u003c/p\u003e"},{"header":"9. Conclusion","content":"\u003cp\u003eThis study proposed a temperature-aware fractional-order PID (FO-PID) control strategy for PMSM drives in electric vehicles by integrating LSTM/Transformer-based thermal forecasts into the control loop for online gain scheduling. Evaluations on the Paderborn PMSM dataset under step and drive-cycle conditions demonstrated that, contrary to conventional expectations, classical PID consistently outperformed both fixed FO-PID and FO-PID with scheduling. PID achieved lower error indices, faster rise time, and safer winding temperature margins, while FO-PID variants accumulated larger tracking errors and experienced thermal excursions beyond safe limits.\u003c/p\u003e\n\u003cp\u003eThese results emphasize that FO-PID is not universally superior and that controller performance is strongly dependent on operating constraints and scheduling methodology. The work contributes a reproducible framework for combining thermal prediction with control and highlights the importance of balancing transient performance with thermal safety. Future research will focus on adaptive scheduling policies, hardware-in-the-loop validation, and lifetime-aware thermal constraints to enhance the robustness of FO-PID deployment in electric vehicle drives.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eData \u0026amp; Code Availability\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe dataset used in this study, Electric Motor Temperature (PMSM), is publicly available on Kaggle at https://www.kaggle.com/datasets/wkirgsn/electric-motor-temperature. All data preprocessing, model training, and control simulations were conducted in Python within the Google Colab environment. The code developed for this work, along with processed data and analysis scripts, will be made available on GitHub/Zenodo upon publication to ensure transparency and reproducibility\u003cstrong\u003e.\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eFunding\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThis research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eConflicts of Interest\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe author declares no conflict of interest.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eEthical Approval / Consent\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eNot applicable (no human subjects).\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eJ. Shan, Z. Che, and F. 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Dash, \u0026quot;Fractional-order PID controllers and applications: a review,\u0026quot; \u003cem\u003eControl Engineering Practice\u003c/em\u003e, 2025.\u003c/li\u003e\n\u003cli\u003eT. L. Sime, P. Aluvada, and S. Habtamu, \u0026quot;Genetic-algorithm-tuned adaptive fuzzy fractional-order PID speed control of PMSM for EVs,\u0026quot; \u003cem\u003eDiscover Applied Sciences\u003c/em\u003e, 2024.\u003c/li\u003e\n\u003cli\u003eW. Zheng and J. Liu, \u0026quot;A simplified fractional-order PID controller\u0026rsquo;s optimal tuning,\u0026quot; \u003cem\u003eEntropy\u003c/em\u003e, vol. 23, no. 2, p. 130, 2021.\u003c/li\u003e\n\u003cli\u003eC. A. Monje, Y. Chen, B. M. Vinagre, D. Xue, and V. F. Feliu, \u003cem\u003eFractional-Order Systems and Controls: Fundamentals and Applications\u003c/em\u003e. London, UK: Springer, 2010.\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"PMSM, electric vehicles, thermal prediction, FO-PID, gain scheduling, LSTM, Transformer, thermal safety, ITAE, IAE, ISE","lastPublishedDoi":"10.21203/rs.3.rs-7727132/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7727132/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eTemperature limits critically affect the safety, efficiency, and lifetime of permanent-magnet synchronous motor (PMSM) drives in electric vehicles. While most prior work focuses on predicting motor temperatures, this paper goes further by embedding learned forecasts directly into the control loop. We propose a temperature-aware fractional-order PID (FO-PID) speed controller whose gains are scheduled online by a multi-target thermal predictor trained on the public Paderborn PMSM dataset. Sequence models (LSTM and Transformer) forecast rotor magnet and stator temperatures at short horizons, and the predictions drive a lightweight scheduling map that adapts Kp,Ki,Kd,λ,μ and applies soft torque limits near thermal boundaries.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eStep-response and drive-cycle evaluations demonstrate that, contrary to expectations, the scheduled FO-PID did not outperform classical PID. Both fixed FO-PID and FO-PID+Scheduler accumulated significantly higher tracking errors and led to unsafe winding temperature excursions, whereas PID achieved the lowest ITAE, IAE, and ISE, the fastest rise time, and maintained safe thermal margins. Scheduling reduced violations relative to fixed FO-PID but did not close the gap to PID performance.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eThese findings provide two contributions: (i) a reproducible framework for integrating thermal forecasts into FO-PID scheduling, and (ii) a counter-intuitive result showing that classical PID may remain safer and more reliable than FO-PID under thermally constrained EV drive operation. This highlights that FO-PID’s benefits are not universal but depend strongly on tuning and operating conditions, and motivates future work on adaptive scheduling policies and hardware-in-the-loop validation.\u003c/p\u003e","manuscriptTitle":"Temperature-Aware Fractional-Order PID Scheduling for PMSM Drives Using Data-Driven Thermal Forecasts","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-01-08 06:23:45","doi":"10.21203/rs.3.rs-7727132/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"b983eda9-3a81-4496-9b79-fc58a4d17f74","owner":[],"postedDate":"January 8th, 2026","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2026-01-08T06:23:45+00:00","versionOfRecord":[],"versionCreatedAt":"2026-01-08 06:23:45","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-7727132","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-7727132","identity":"rs-7727132","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}