Modeling, Control and Monitoring of Automotive Electric Drives

Abstract

The electrification of automotive powertrains has accelerated research efforts in the modeling, control, and monitoring of electric drive systems, where reliability, safety, and efficiency are key enablers for mass adoption. Despite a large corpus of literature addressing individual aspects of electric drives, current surveys remain fragmented, typically focusing on either multiphysics modeling of machines and converters, or advanced control algorithms, or diagnostic and prognostic frameworks. This review provides a comprehensive perspective that systematically integrates these domains, establishing direct connections between high-fidelity models, control design, and monitoring architectures. Starting from the fundamental components of the automotive power drive system, the paper reviews state-of-the-art strategies for synchronous motor modeling, inverter and DC/DC converter design, and advanced control schemes, before presenting monitoring techniques that span model-based residual generation, AI-driven fault classification, and hybrid approaches. Particular emphasis is given to the interplay between functional safety (ISO 26262), computational feasibility on embedded platforms, and the need for explainable and certifiable monitoring frameworks. By aligning modeling, control, and monitoring perspectives within a unified narrative, this review identifies the methodological gaps that hinder cross-domain integration and outlines pathways toward digital-twin-enabled prognostics and health management of automotive electric drives.

1. Introduction

1.1. Motivations

The electrification of road transportation is reshaping the design, monitoring, and control of automotive power drive systems (APDS). Unlike conventional internal combustion engines, electric propulsion relies on tightly coupled electromechanical subsystems—high-voltage batteries, DC/DC converters, traction inverters, and synchronous machines—whose performance and reliability are dictated by intricate interactions between electrical, thermal, and mechanical domains. In this context, advanced monitoring is not merely a matter of efficiency optimization but a prerequisite for functional safety, predictive maintenance, and extended component lifetime under stringent automotive standards. The continuous growth of electric vehicle (EV) adoption has brought to the forefront several motivations that justify the development of sophisticated monitoring techniques, blending model-based reasoning with data-driven intelligence.

A first motivation arises from the criticality of reliability in automotive environments. Failures in traction inverters or permanent-magnet synchronous motors (PMSM/IPMSM) can lead to catastrophic loss of propulsion or unsafe torque delivery. The probability of such events increases with the high switching frequencies, elevated current densities, and harsh thermal cycling typical of EV operation. Monitoring therefore plays a role not only in fault detection and isolation (FDI) but also in prognostics and health management (PHM), ensuring that incipient degradation such as bearing wear, partial demagnetization, or DC-link capacitor aging is detected before it evolves into critical failure. A second motivation is rooted in multi-domain interactions. Power losses in the inverter translate into thermal stress on semiconductors; magnetic saturation in the motor affects current ripple and acoustic emissions; battery internal resistance growth increases both thermal load and efficiency loss. Monitoring strategies must capture these couplings to provide actionable insights. Purely electrical sensing is insufficient, while thermal and mechanical variables are often difficult to measure directly. This motivates hybrid approaches where residuals from physics-based observers are fused with AI models trained to correlate heterogeneous signals. Another pressing motivation is compliance with functional safety standards. ISO 26262 [1] requires traceability, interpretability, and bounded failure rates. Monitoring frameworks must therefore be certifiable, which challenges the integration of complex AI algorithms. Model-based methods provide analytical guarantees but may lack robustness under uncertainty; AI models provide adaptability but require careful calibration and bounding. The need for architectures that balance these requirements motivates the exploration of hybrid monitoring. The industrial push for predictive maintenance and fleet analytics adds another layer. EV fleets generate vast amounts of data, enabling off-line training of machine learning models for prognostics, while on-board units must execute lightweight residual generation and anomaly detection in real time. This separation of roles—off-line learning versus on-line monitoring—is a strong driver for the development of architectures where models are trained in the cloud and distilled into embedded-compatible forms for deployment. Finally, sustainability concerns impose the need to maximize component lifetime. Lithium-ion batteries, power semiconductors, and electric machines represent the costliest and most resource-intensive subsystems. Monitoring that enables accurate State of Health (SOH) estimation and Remaining Useful Life (RUL) prediction has direct implications for extending usage, planning second-life applications, and reducing environmental impact. From a systemic perspective, advanced monitoring supports the broader goal of making EV technology both economically viable and environmentally sustainable. To better summarize these motivations,

Table 1. Motivations for advanced monitoring in automotive power drive systems.

Motivation	Underlying Challenge	Implications for Monitoring Design
Reliability & safety	Inverter/Motor faults cause propulsion loss; ISO 26262 compliance requires bounded failure rates	Residual-based FDI with guaranteed detection rates; redundancy in sensing/observation
Multi-domain interactions	Coupling of electrical, thermal, magnetic, and mechanical dynamics	Need for joint observers, multi-sensor fusion, hybrid AI+model methods
Functional safety & certification	Black-box AI not directly certifiable	Emphasis on interpretability, calibrated confidence, and bounded decisions
Predictive maintenance & fleet analytics	Large datasets off-board, lightweight resources on-board	Cloud-based training with on-board distilled models; residual generation at $T_s=25$–$100\mu$s
Sustainability & cost	Batteries, semiconductors, machines are expensive and resource-intensive	Monitoring for SOH, RUL, lifetime extension, second-life planning

categorizes the drivers of monitoring development across domains, with associated challenges and implications. Safety and reliability are linked to strict FDI requirements and low-latency responses. Efficiency and performance optimization motivate integration of thermal and magnetic state estimation. Sustainability and cost drive the inclusion of prognostics and degradation modeling. Certification and regulatory compliance push towards solutions that are interpretable and bounded in risk. The interplay of these drivers provides the rationale for an integrated monitoring framework that combines physics-based models, AI techniques, and hybrid strategies.

1.2. State of the Art

The corpus of review papers on automotive power drive systems (APDS) is vast yet fragmented. A consistent pattern emerges: surveys concentrate either on machine and converter modeling, or on control strategies, or on monitoring/PHM; more recent works discuss digital twins. However, none of the available reviews articulates an integrated perspective in which physics-based models inform control design and, simultaneously, structure residual generation and decision layers for monitoring.

In what follows, we organize the state of the art along four streams and summarize the distinctive contribution of each review while pinpointing its boundaries.

• Modeling and design: multiphysics fidelity and design automation. Several reviews map the design space of traction machines, emphasizing electromagnetic models, thermal constraints, and materials. Broad machine-comparison surveys contrast induction, PMSM/IPMSM, and reluctance technologies on torque density, efficiency, and package constraints, often with an explicit EV slant [2,3,4]. A subset turns to high-speed traction, where rotor stress, eddy-current losses, and cooling architecture dominate feasibility envelopes; these papers underline the need for coupled electromagnetic–thermal models to prevent demagnetization and hot spots under WLTP-like duty cycles [5,6,7,8]. The modeling lens widens in [9,10], which explicitly advocate multidisciplinary design automation and surrogate-based optimization linking EM solvers, lumped thermal networks, and mechanical constraints; these reviews are valuable for design-space exploration but remain largely silent on how the same models could be down-selected and embedded for real-time monitoring or control synthesis. On the converter and insulation side, ref. [3] synthesizes the impact of PWM waveforms and cable parasitics on high-frequency motor models and partial discharge phenomena, suggesting the necessity of HF-accurate stator models for reliability assessment. Sensing and thermal mapping technologies (RTDs, thermistors, fiber-optic, IR) and their integration in traction motors are compared in [11], again from a measurement and instrumentation standpoint, with limited cross-talk with closed-loop control or online PHM. In summary, modeling surveys provide depth in fidelity, materials, and constraints, but they typically stop at the design or instrumentation boundary.
• Control strategies: from classical FOC/DTC to predictive and robust control. Control-focused reviews cover the algorithmic spectrum and benchmark torque-quality, dynamic response, computational burden, and robustness. Canonical comparisons between FOC and DTC are presented together with their modern evolutions (e.g., MTPA/field-weakening coordination, flux observers) [12,13,14]. Robust and variable-structure approaches are surveyed with emphasis on disturbance rejection and parameter drift (e.g., sliding-mode, super-twisting, backstepping–ESO hybrids), typically in the context of PMSM/IPMSM drives [4,15]. Model predictive control (both continuous-input MPC with SVPWM and finite-control-set MPC) is critically reviewed for current/torque regulation and constraint handling; advantages in explicit multiobjective design are balanced against embedded-computation limits [16,17]. Multi-motor coordination and fault-tolerant control, including reconfiguration after open-phase or sensor failures, are addressed in broader powertrain reviews [18]. Across these surveys, the discussion of how controllers should be co-designed with diagnostic observers or residual generators is sparse; sensorless topics are present but not integrated with PHM requirements such as detectability, isolability, or prognostics.
• Monitoring, diagnostics, and PHM: signatures, observers, and data-driven analytics. Monitoring-oriented reviews classify faults across electrical, mechanical, sensor, and cooling subsystems, and compare detection channels (electrical, vibration, acoustic, thermal). Comprehensive taxonomies—including stator inter-turn short circuits, bearing degradation, eccentricity, demagnetization, DC-link capacitor aging, and power-module degradation—appear in [19,20,21,22]. On the converter side, DC-link capacitor and IGBT/SiC module health indicators, together with online monitoring strategies, are synthesized in [5]. Methodologically, three families dominate these reviews. First, signal-based methods (MCSA, order tracking, time–frequency and wavelet features) remain popular for their simplicity and low overhead [2]. Second, model-based observers and residuals (Luenberger/Kalman, unknown-input observers, parity relations) support early detection and isolation, and naturally tie to control models—though most surveys treat them separately from the control loop. Third, AI-based classifiers and regressors (SVM, random forests, CNNs/RNNs, autoencoders, transfer learning) are reported for anomaly detection and remaining useful life (RUL) estimation, with clear benefits in non-stationary environments but open issues in interpretability and certification [23,24,25,26]. Notably, [19] discusses sensor faults and their cascading impact on control; yet, a unifying architecture that quantifies detectability/isolability alongside control-performance degradation is still missing.
• Digital twins and fleet-level PHM: from high-fidelity models to online synchronization. A distinct stream argues that digital twins—real-time synchronized replicas of the traction drive—constitute a natural container for multiphysics models, online estimation, and predictive analytics. Reviews in this area describe architectures that combine high-fidelity EM/thermal models with data assimilation and fleet analytics, enabling virtual sensing, lifetime prediction, and scenario exploration [11,27,28,29,30]. The promise is compelling for warranty, derating policies, and safety cases; however, most surveys treat the twin as a parallel analytics layer, with limited methodological guidance on how its models should be co-designed with control observers and diagnostic residuals, or how to partition computation between embedded ECUs and edge/cloud resources under real-time constraints.
• Cross-cutting observations and research gap. Taken together, the thirty reviews delineate four mature pillars—high-fidelity multiphysics modeling for design [3,10,11], advanced control (FOC/DTC/MPC/robust/adaptive) for torque/efficiency under constraints [4,13,14,17], comprehensive monitoring/diagnostics spanning signal-based, model-based, and AI-based methods [5,19,20,21,22], and digital twins/PHM as integrative data–model ecosystems [27,28,29,30]. What is uniformly missing is a review that connects these pillars into a unified systems view: there is no state-of-the-art article that simultaneously (i) specifies physics-based models in the same coordinate frames used for control (e.g., Park/Clarke) and reuses them for observer/UIO-based residual generation, (ii) quantifies how monitoring requirements (detectability/isolability/RUL) feed back into control design choices (e.g., current-loop bandwidths, SVM limits, field-weakening margins), and (iii) embeds the resulting stack into a digital-twin architecture that respects embedded computational budgets and automotive safety certification constraints. This three-way integration—modeling ↔ control ↔ monitoring—constitutes the central gap our work aims to address.

In order to consolidate the insights from the thirty review articles examined,

Table 2. Summary of the 30 reviewed articles: main focus, methodological scope, and limitations.

Reference	Focus	Key Methods/Scope	Reported Limitations
[2]	Modeling	PMSM/IM comparison, design parameters, loss modeling	No link to monitoring/control; limited thermal detail
[3]	Modeling	High-frequency models, PWM/cable effects, insulation stress	Addresses reliability but not PHM integration
[4]	Modeling/Control	EM design of PMSM; control impact on efficiency	Limited monitoring perspective
[31]	Modeling	Sensor technologies, temperature mapping	Measurement focus, no control/monitoring
[5]	Modeling	High-speed motors, rotor stress, cooling	No integration with monitoring/control
[6]	Modeling	EV traction motor survey, design metrics	Stays at comparative level, no PHM
[7]	Modeling	High-speed PMSM, electromagnetic design	Does not address monitoring or control loops
[8]	Modeling	Multiphysics design, EM/thermal coupling	No monitoring linkage
[9]	Modeling	Multidisciplinary design automation (MDDA)	Valuable for design, no operational PHM integration
[10]	Modeling	Surrogate-based EM/thermal optimization	Focus on design phase, not embedded monitoring
[12]	Control	FOC vs. DTC comparison, EV case studies	No link to monitoring
[13]	Control	Control survey: FOC, DTC, MPC	Does not cover diagnostics
[14]	Control	PMSM control, adaptive strategies	Monitoring ignored
[15]	Control	Robust/sliding mode for PMSM drives	No PHM integration
[16]	Control	Model predictive control survey	Computational burden not tied to monitoring
[17]	Control	Finite-control-set MPC	Feasibility on embedded platforms not addressed
[18]	Control	Fault-tolerant, multi-motor control topologies	Monitoring aspects superficial
[20]	Monitoring	Fault mechanisms, diagnostic methods	Lacks integration with control
[21]	Monitoring	Fault signatures in traction drives	No PHM framework
[19]	Monitoring	Sensor and electrical fault diagnosis	Limited system-level view
[22]	Monitoring	Fault detection methods, PMSM focus	Does not connect to design/control
[23]	Monitoring/AI	AI for predictive maintenance	Certification and embedded limits overlooked
[24]	Monitoring/AI	Data-driven PHM for EV drives	Lacks interpretability
[25]	Monitoring/AI	Prognostics, RUL estimation	Not tied to control loops
[26]	Monitoring/AI	Anomaly detection frameworks	Scalability on-board not addressed
[11]	Digital Twin	Thermal mapping, measurement integration	No direct control/monitoring linkage
[28]	Digital Twin	EV digital twin frameworks	Twin treated as parallel, not embedded
[29]	Digital Twin	Prognostics and health management	Monitoring disconnected from control
[30]	Digital Twin	Cloud/edge PHM architectures	No embedded feasibility discussion
[27]	Digital Twin	Lifetime prediction, inverter focus	Lack of methodological integration

provides a comparative summary of their focus, methodological scope, and reported limitations. The table groups the papers into four thematic domains—modeling, control, monitoring, and digital twin/PHM—making explicit the predominant methods discussed in each survey and the boundaries of their coverage. By aligning the reviews side by side, it becomes evident that while the literature is rich and technically detailed within individual domains, each stream evolves largely in isolation. Modeling papers concentrate on multiphysics fidelity and design optimization, control surveys benchmark advanced algorithms for torque and efficiency, monitoring reviews classify diagnostic techniques and prognostic approaches, and digital twin papers highlight virtual–physical integration. Yet, none of the surveyed works attempts to connect these perspectives into a unified framework that links modeling, control, and monitoring under the constraints of automotive-grade embedded implementation. This fragmentation, highlighted in the final column of Table 2, underscores the gap that motivates the present study. While several review papers exist on modeling, control, or monitoring of automotive electric drives, these contributions typically remain confined to a single perspective. Surveys on synchronous machine modeling emphasize electromagnetic and thermal fidelity, but rarely connect these models to real-time monitoring or control synthesis. Similarly, reviews on advanced control strategies (e.g., FOC, DTC, MPC, robust control) benchmark algorithms extensively, but seldom address how they should be co-designed with diagnostic observers or integrated within prognostic frameworks. Monitoring-oriented surveys are rich in taxonomies and methods, spanning signal-based, model-based, and AI-based techniques, yet they usually operate in parallel to control rather than within a unified architecture. Finally, more recent surveys on digital twins for EVs highlight virtual–physical synchronization, but with limited methodological guidance on how such twins should embed control and monitoring functions under automotive-grade constraints. The absence of a truly integrative perspective leaves a critical gap: no available review simultaneously connects high-fidelity models, advanced control strategies, and monitoring/prognostics into a coherent framework. Moreover, aspects such as battery technologies, bidirectional converters, and the interaction between energy storage and drive monitoring are rarely included in existing surveys, despite their central importance for safety, efficiency, and sustainability. This work addresses these gaps by providing, to the best of our knowledge, the first end-to-end survey of the automotive power drive system “as a whole.” Unlike fragmented reviews, it systematically links modeling, control, and monitoring, incorporating both classical and intelligent strategies, and extending the discussion to energy storage and environmental considerations. The review therefore not only consolidates established knowledge but also identifies methodological gaps and emerging trends, offering the community a unique reference that spans the entire scope of on-board power systems for electric vehicles.

1.3. Authors’ Contribution

The novelty of this review lies in its breadth and its integrative perspective. While previous surveys have addressed modeling, control, or monitoring separately, this work brings together all three pillars of automotive power drive systems within a single coherent framework. In contrast to prior contributions that focus on individual subsystems or methodologies, this article provides a truly holistic analysis of the entire automotive electric drive system, from energy storage to power conversion and traction control. The scope is deliberately wide, yet the level of technical detail remains high, ensuring that each topic is treated with mathematical rigor and practical depth.

To the best of our knowledge, there are very few works—and in most cases none—that offer such an “all-around” assessment of the power drive system in its entirety. The main contributions of the article can be summarized as follows:

• Comprehensive integration of domains: The review is the first to systematically connect modeling, control, and monitoring perspectives, highlighting how design-stage models can directly inform both control synthesis and residual-based monitoring. This integrative view covers not only motors and inverters but also high-voltage batteries, bidirectional converters, and energy management.
• Unified analysis of advanced control strategies: Beyond conventional FOC and DTC, the paper provides an in-depth comparison of advanced algorithms such as MPC, sliding mode, adaptive, and reinforcement learning controllers, explicitly discussing their computational feasibility on automotive-grade embedded platforms. The survey extends further by including intelligent and hybrid control methods, which are seldom systematically reported in the context of automotive electric drives.
• Structured classification of monitoring techniques: The review consolidates signal-based, model-based, AI-based, and hybrid monitoring methods, presenting their advantages and limitations, with particular emphasis on sensorless feasibility and explainability requirements for ISO 26262 compliance. Specific attention is also devoted to the monitoring of the battery pack and its interaction with the drive, an aspect often overlooked in the literature.
• Cross-domain perspective on hybrid monitoring: The work highlights how physics-informed AI and digital twin frameworks can merge model-based residuals with data-driven predictors, paving the way toward robust and adaptive PHM solutions. Such cross-domain integration is discussed with a system-level view that very few existing reviews have attempted.
• Identification of methodological gaps: The article clearly articulates the lack of a unifying framework in the literature, where modeling, control, and monitoring are typically treated in isolation, and outlines research directions for integrated design and implementation. In particular, the absence of prior comprehensive reviews that assess the on-board power system “end-to-end” is highlighted, reinforcing the originality of the present work.
• Practical emphasis on implementation constraints: Special attention is given to the computational cost, sensor requirements, and certifiability of algorithms when deployed on embedded controllers in automotive environments. This pragmatic focus ensures that the review remains not only conceptually broad but also relevant for real-world industrial adoption.

By articulating these contributions, the review extends beyond the scope of existing surveys and positions itself as a reference for the design of next-generation automotive electric drives that are efficient, reliable, and certifiable. Its uniqueness lies in the unprecedented breadth of topics covered with a consistently high level of detail, offering the community an integrated reference that spans all critical aspects of automotive electric drive systems.

2. Main Component Modeling

2.1. Background on Automotive Electric Drive Systems

Looking at the architecture shown in Figure 1, the Automotive Power Drive System is a key element in the traction chain, functionally interconnecting the energy source, conversion and management systems, and the powertrain.

The electrical energy from the high-voltage battery pack, monitored and managed by the Battery Management System (BMS), is channeled to the DC link via a pre-charge stage. This pre-charge stage limits inrush currents during initial connection, protecting the capacitors and power devices. The DC link acts as an intermediate storage node, stabilizing the voltage and ensuring an instantaneous energy reserve for load fluctuations. From the DC link stage, the energy feeds the inverter, which transforms the DC power into three-phase AC power for the electric motor. The motor, coupled to the mechanical transmission, converts the electrical energy into torque, which is then transferred to the wheels. The efficiency of this process depends on the inverter topology, semiconductor technology, and control strategy, all of which work in close synergy with the cooling system to maintain performance even under heavy loads. In parallel with the traction chain, the system integrates dedicated conversions for auxiliary power supplies. A DC-DC stage reduces the main battery voltage (typically 400 or 800 V) to the levels required for low-voltage services, such as 12 V or 48 V, powering auxiliary devices such as air compressors, pumps, and electric power steering. The Energy Management System (EMS) supervises the distribution and use of energy among the various subsystems, coordinating with the BMS and the inverter control unit via communication buses (e.g., CAN). Battery charging is handled by two distinct conversion paths. The On-Board Charger (OBC) manages charging from the AC grid, converting AC to DC and transferring it to the battery pack. In systems that require it, a High-Power DC-DC converter allows direct charging from high-power DC sources, such as fast charging stations, bypassing the AC-DC stage and reducing energy replenishment times. Overall, this architecture highlights how the APDS is not an isolated block, but a central node in a complex energy ecosystem. Its efficiency and reliability depend not only on the intrinsic quality of its main components, but also on how they interact with energy management systems, charging interfaces, and vehicle auxiliaries. Looking ahead, the integration of advanced control and monitoring algorithms, potentially enhanced by physical models and artificial intelligence techniques, will allow the full potential of each element to be exploited, maximizing the performance of the electric drive system and the lifespan of its components.

The architecture shown can be interpreted as a combination of two overlapping networks: the power network, which transports high- and low-voltage electrical energy, and the information network, which coordinates energy flows and ensures centralized control and operational safety.

On the energy side, the starting point is the high-voltage battery pack, organized into multiple modules connected in series and parallel to achieve the voltage and capacity values required for traction. From the battery pack, energy flows in two main directions: on one side, the DC link, which directly powers the inverter and thus the motor; on the other, the auxiliary and charging branches. Before reaching the DC link, the main path passes through the pre-charging stage, whose function is to prevent inrush currents in the link capacitors and protect the semiconductors during activation. The DC link itself acts as an intermediate storage and filtering node, stabilizing the voltage and providing instantaneous energy for torque transients. From the DC bus, the inverter converts DC power into three-phase AC power and sends it to the electric motor, which transforms the electrical energy into mechanical torque. This torque is transmitted to the wheels through the transmission system, optimizing the engine’s operating speed for different vehicle speeds and loads. In parallel, a conversion branch powers the DC-DC converter dedicated to low-voltage systems. This stage reduces the high voltage of the battery pack to typical levels of 12 V or 48 V, essential for powering auxiliary devices such as air compressors, cooling pumps, and electric steering systems. This is where the Energy Management System (EMS) comes into play, coordinating the distribution of power between traction, auxiliary systems, and charging, constantly communicating with the Battery Management System (BMS) to ensure optimal and safe battery use. On the charging front, the On-Board Charger (OBC) manages the input of power from the AC grid, performing AC/DC conversion and transferring power to the battery pack via the DC bus. In vehicles equipped for ultra-fast charging, a High-Power DC-DC converter allows direct power from DC charging stations, reducing conversion losses and charging times. The second level of the network is the information level. Communication connections, such as the CAN bus, carry commands, monitoring data, and diagnostic signals between the EMS, the BMS, the inverter controller, and the various actuators. This control network ensures that the vehicle’s torque demands are met within safe operating conditions, that charging occurs within thermal and electrical limits, and that any anomalies are promptly detected and managed. Overall, the figure highlights how the APDS is not a simple power-to-traction conversion block, but the core of an interdependent system. Each energy path is controlled by a corresponding communication and supervision chain, and overall performance depends as much on the quality of the components as on the efficiency of these interconnections. The electric drive train in a fully electrified vehicle constitutes the core system responsible for converting stored electrical energy into mechanical torque at the wheels. It can be represented as a functional chain:

$$\mathrm{HV}\mathrm{Battery}\to \left(\mathrm{optional}\mathrm{HV}\text{-}\mathrm{HV}\mathrm{DC}\text{-}\mathrm{DC}\right)\to \mathrm{Inverter}\mathrm{with}\mathrm{SVM}\to \mathrm{Synchronous}\mathrm{Motor}$$

Although often described in terms of discrete subsystems, these elements are tightly coupled, with constraints on voltage, current, power, and thermal limits propagating along the chain. Understanding the interaction between them is essential for designing optimal control, ensuring operational safety, and maximizing overall efficiency.

2.2. High-Voltage Battery and BMS Constraints

The high-voltage (HV) traction battery is the primary energy source in the electric drive train, and its accurate modeling is crucial for predicting available power, efficiency, and thermal behavior. A comprehensive overview of HV battery pack design aspects is given in [32], while system-level BMS functions for safety, monitoring, and state estimation are summarized in [33,34]. More recent reviews highlight how modern lithium-ion BMS face stringent constraints in terms of computational feasibility, cell balancing, communication bandwidth, and compliance with safety standards [35,36,37]. Different topologies for centralized, modular, and distributed BMS architectures are compared in [38], showing how voltage levels and modularity affect reliability and diagnostic coverage.

As illustrated in Figure 2, the model integrates electrochemical, electrical, and thermal domains. At the electrochemical level [Figure 2a], lithium-ion transport occurs both in the electrolyte (red arrows) and in the solid active material (blue arrows). These processes determine the open-circuit voltage (OCV) as a nonlinear function of the state of charge (SOC), and they are responsible for voltage relaxation phenomena during transients [33]. The cell’s terminal behavior can be represented by an equivalent electrical model [Figure 2b], consisting of a controlled voltage source $U_{\mathrm{oc}}$ in series with an ohmic resistance $R_0$ and one or more RC polarization branches to capture dynamic effects [32]. The SOC dynamics are given by:

$$\dot{z}\left(t\right)=-{\displaystyle \frac{i_b\left(t\right)}{Q_n}}$$

(1)

where z is the SOC, $i_b$ the battery current (positive during discharge), and $Q_n$ the nominal capacity. The terminal voltage is:

$$v_b\left(t\right)=\mathrm{OCV}\left(z\left(t\right)\right)-R_0{i}_b\left(t\right)-\sum _{k=1}^n{v}_{RC,k}\left(t\right)$$

(2)

with polarization voltages evolving as:

$${\dot{v}}_{RC,k}\left(t\right)=-{\displaystyle \frac{1}{R_k{C}_k}}{v}_{RC,k}\left(t\right)+{\displaystyle \frac{1}{C_k}}{i}_b\left(t\right)$$

(3)

The thermal submodel [Figure 2c] accounts for heat generation $Q_{\mathrm{gen}}$ as the sum of irreversible Joule losses $Q_J$ and reversible entropic heat $Q_r$:

$$Q_{\mathrm{gen}}=Q_J+Q_r$$

(4)

The average cell temperature $T_{\mathrm{cell}}$ is described by:

$${\dot{T}}_{\mathrm{cell}}=-{\displaystyle \frac{1}{R_{\mathrm{th}}{C}_{\mathrm{th}}}}\left(T_{\mathrm{cell}}-T_a\right)+{\displaystyle \frac{1}{C_{\mathrm{th}}}}{Q}_{\mathrm{gen}}$$

(5)

where $R_{\mathrm{th}}$ and $C_{\mathrm{th}}$ are the thermal resistance and capacitance, and $T_a$ is the ambient temperature. The Battery Management System (BMS) continuously monitors $v_b$, $i_b$, $T_{\mathrm{cell}}$, and z, enforcing:

$$V_{min}\left(z\right)\le v_b\left(t\right)\le V_{max}\left(z\right),-P_{\mathrm{chg}}^{max}(z,T_{\mathrm{cell}})\le v_b\left(t\right)i_b\left(t\right)\le P_{\mathrm{dis}}^{max}(z,T_{\mathrm{cell}})$$

(6)

The integration of electrochemical, electrical, and thermal models in Figure 2 enables accurate prediction of voltage drop under dynamic load, thermal rise during high current operation, and capacity fade mechanisms [35,37]. The BMS uses this combined model to adjust allowable charge/discharge power in real time, thus directly influencing the maximum torque request that can be sustained by the electric drive [34,36]. Beyond lithium-ion batteries, different chemistries are being considered for automotive drives. Lithium Iron Phosphate (LFP) provides high cycle life and thermal stability but has lower energy density compared to Nickel Manganese Cobalt (NMC) or Nickel Cobalt Aluminum (NCA). Solid-state batteries, although still at an early stage, promise higher safety and energy density by replacing liquid electrolytes with solid ones. These differences strongly affect the performance of the electric drive: NMC and NCA enable longer driving range and higher peak power, whereas LFP offers superior tolerance to regenerative braking cycles thanks to its robustness and stable thermal behavior. The capability of a battery technology to absorb regenerative energy is determined not only by its maximum charge power $P_{\mathrm{chg}}^{max}$ in (6), but also by thermal constraints and internal resistance. Batteries with lower impedance and higher allowable charging current improve the effectiveness of regenerative braking, reducing the amount of energy dissipated in mechanical braking systems. From an environmental perspective, the choice of battery chemistry influences the full life-cycle impact of the vehicle. NMC and NCA, while offering high specific energy, require critical raw materials such as cobalt and nickel, raising concerns of supply chain sustainability. LFP and future solid-state batteries reduce the dependency on critical elements and are more easily repurposed for second-life applications in stationary storage. Moreover, regenerative braking contributes to lowering the carbon footprint of the vehicle by recovering up to 15–25% of the kinetic energy otherwise lost, thereby reducing net energy demand and indirectly lowering greenhouse gas emissions. This highlights the intertwined role of battery technology, control strategies, and energy recovery in shaping the overall environmental sustainability of automotive electric drives.

2.3. HV-HV DC-DC Converter: Dual Active Bridge (DAB)

In high-performance electric drive trains, an HV-HV DC-DC converter is often inserted between the high-voltage battery and the inverter to regulate the DC-link voltage, optimize modulation range, and decouple the battery dynamics from the fast transients of the motor drive. One of the most widely adopted topologies for this purpose is the Dual Active Bridge (DAB), illustrated in Figure 3.

The DAB consists of two full-bridge stages: the input bridge (B₁, blue) connected to the battery voltage $V_1$, and the output bridge (B₂, green) connected to the DC-link capacitor C and load R. The two bridges are linked by a high-frequency transformer with turns ratio $N:1$ and series leakage inductance L, which serves as the energy transfer element. Both bridges operate with 50% duty cycle square-wave modulation. The control variable is the phase shift $\varphi$ between the switching patterns of the two bridges. This phase shift generates an average voltage across the leakage inductance L, producing a power transfer from the primary to the secondary [39,40]. The generalized average modeling (GAM) framework for the DAB is consolidated in [39,41], enabling compact analytical expressions such as those in (8). Extensions to frequency-domain harmonic balance modeling are discussed in [42], while steady-state modeling with circuit parasitics is proposed in [43].

As illustrated in Figure 4, the red trace $v_p^{sw}$ represents the switched voltage at the primary bridge, while the blue trace $n{v}_s^{sw}$ is the reflected switched voltage from the secondary bridge. When the two square waves are phase-shifted by $\varphi T_{sw}$, their difference $v_L\left(t\right)$ produces a net voltage across L during part of the switching period, causing a triangular current waveform $i_L\left(t\right)$. The inductor current is governed by:

$$L{\displaystyle \frac{d{i}_L}{dt}}=v_L\left(t\right)=v_p^{sw}\left(t\right)-n{v}_s^{sw}\left(t\right)$$

(7)

and the average transferred power in ideal operation is:

$$P_{DAB}\left(\varphi \right)={\displaystyle \frac{n{V}_1{V}_2}{\omega L}}\varphi \left(1-{\displaystyle \frac{\varphi }{\pi }}\right),\varphi \in [0,\pi ]$$

(8)

where n is the transformer turns ratio, $\omega$ the switching angular frequency, and $V_1,V_2$ the input and output DC voltages. The input and output currents follow from:

$$i_{\mathrm{in}}={\displaystyle \frac{P_{DAB}\left(\varphi \right)}{V_1}},i_{\mathrm{out}}={\displaystyle \frac{P_{DAB}\left(\varphi \right)}{V_2}}$$

(9)

while the DC-link voltage dynamics, including the capacitor C, are:

$$C{\displaystyle \frac{d{V}_2}{dt}}=i_{\mathrm{out}}-i_{\mathrm{load}}$$

(10)

The phase shift $\varphi$ is the key control parameter in the DAB. Increasing $\varphi$ raises the overlap area between $v_p^{sw}$ and $-n{v}_s^{sw}$, increasing the net voltage across L and thus the power transfer. At $\varphi =0$, no net energy is transferred, while at $\varphi =\pi /2$ the converter operates at maximum power. This phase control enables bidirectional operation: in traction mode, $\varphi$ is positive to deliver power from the battery to the DC-link; in regenerative mode, $\varphi$ is negative to flow power back to the battery. The design of L must balance soft-switching capability, current ripple, and efficiency. Beyond ideal phase-shift control, several advanced extensions have been proposed. Hybrid modulation strategies that combine single-, dual-, and triple-phase-shift achieve efficiency optimization across load ranges [41,44]. Reviews of topological modifications (multi-port, multi-cell, resonant and hybrid DAB variants) are synthesized in [45], while steady-state and non-ideal AC–DC adaptations are analyzed in [43]. Control-oriented reviews [46] emphasize that model predictive control, feedforward/feedback schemes, and data-driven modulation design [47] are becoming increasingly relevant for automotive-grade HV-HV converters. Collectively, these works establish a robust foundation for DAB deployment but also underline the need for unified models that simultaneously capture soft-switching, losses, and control feasibility in embedded hardware. Although the DAB is the most widespread isolated solution, several other bidirectional converter topologies are employed in automotive electric drives. Non-isolated converters include the interleaved bidirectional buck–boost, the Cuk, SEPIC, and Zeta structures, which are compact and suitable for interfacing auxiliary HV buses or 48 V subsystems. Isolated converters, in addition to the DAB, encompass the bidirectional Phase-Shift Full-Bridge (PSFB), the bidirectional LLC resonant converter, and multiport configurations that enable simultaneous energy exchange among battery, DC link, and on-board charger. Different modeling approaches are applied depending on the topology: state-space averaging and small-signal models for non-isolated converters, generalized average models (GAM) and harmonic balance for DAB and PSFB, and time-domain resonant models for LLC-based converters. These converters play a pivotal role in stabilizing the DC-link voltage during traction and regenerative braking, optimizing efficiency under dynamic driving cycles, and enabling safe bidirectional interaction between the traction battery and the on-board charger. In particular, their ability to handle reverse power flow is essential for maximizing the benefits of regenerative braking and for ensuring reliable fast-charging operations in modern EVs.

2.4. Traction Inverter Topologies and Space Vector Modulation

The traction inverter is the main interface between the high-voltage DC-link and the electric traction motor, synthesizing balanced three-phase AC voltages with controllable magnitude and frequency. The inverter topology and modulation strategy directly influence voltage utilization, harmonic distortion, efficiency, and switching losses. A broad overview of modulation and control techniques for multilevel inverters in traction applications is given in [48], while specific surveys on SVM strategies for NPC and multilevel structures can be found in [49]. Optimized SPWM and carrier-based approaches tailored to traction inverters are discussed in [50], and recent reviews on multiphase traction inverters highlight how modulation and control are evolving with the electrification of heavy-duty and high-power EVs [51]. Beyond reviews, advanced control schemes targeting maximum efficiency across the whole operating range have also been proposed [52], while multiobjective vector modulation tailored to hybrid traction systems has been reported in [53].

Figure 5 illustrates several widely used inverter topologies in electric traction. The conventional two-level VSI is the baseline architecture, while multilevel solutions such as NPC, ANPC, Flying Capacitor, and T-type enable more output voltage levels, reduced THD, and improved efficiency, particularly at high modulation indices [48,49]. Classical two-level SVM techniques are well established and widely documented in the literature. For completeness, we briefly recall that they synthesize the reference voltage vector by time-weighting adjacent active vectors and zero states. In this review we focus instead on recent developments, including multilevel inverters, wide-bandgap device implications, and advanced modulation strategies for automotive applications. In a three-level NPC inverter, each leg can output $+V_{dc}/2$, 0, or $-V_{dc}/2$. This expands the set of available vectors from 8 (two-level) to 27 (three-level), providing finer resolution for ${\mathbf{V}}_{ref}$ synthesis [48,49].

Figure 6 shows that each main sector is split into sub-sectors. For a given ${\mathbf{V}}_{ref}$, the modulation algorithm selects the two nearest active vectors and one zero or medium vector from the available set. The voltage step for an m-level inverter is:

$$V_{\mathrm{step}}={\displaystyle \frac{V_{dc}}{m-1}}$$

For $m=3$, $V_{\mathrm{step}}=V_{dc}/2$. The generalized duty cycle calculation is:

$$\begin{array}{cc}\hfill T_a& ={\displaystyle \frac{T_s}{V_{\mathrm{step}}}}\left(v_{\alpha }cos{\theta }_a+v_{\beta }sin{\theta }_a\right)\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill T_b& ={\displaystyle \frac{T_s}{V_{\mathrm{step}}}}\left(v_{\alpha }cos{\theta }_b+v_{\beta }sin{\theta }_b\right)\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill T_0& =T_s-T_a-T_b\hfill \end{array}$$

(13)

where $({\theta }_a,{\theta }_b)$ are the angles of the selected active vectors.

Remarks on Multilevel SVM:

• Neutral Point Balancing: NPC and ANPC topologies require control of neutral point voltage to avoid imbalance between DC-link capacitors [48,49].
• Flying Capacitor Balancing: FC topologies need periodic modulation adjustments to maintain capacitor voltages [48].
• T-type Loss Reduction: T-type legs reduce conduction path length for zero voltage states, lowering switching and conduction losses [48,51].

By increasing the number of levels, multilevel SVM reduces Total Harmonic Distortion (THD), lowers device voltage stress $V_{\mathrm{device}}=V_{dc}/(m-1)$, and improves efficiency, which is critical in high-performance EV traction applications [50,52,53].

2.5. Synchronous Motor Modeling for Automotive Traction

In modern electric vehicles, synchronous machines represent the most efficient choice for traction applications. Compared to induction machines, synchronous motors can achieve higher power density, superior peak efficiency, and better controllability, especially when paired with advanced control strategies such as Field-Oriented Control (FOC) or Model Predictive Control (MPC). Their ability to maintain high efficiency over a wide torque-speed range, combined with reduced rotor losses, makes them ideal for high-performance automotive use [54,55,56,57,58,59,60]. The general three-phase model of a synchronous machine neglecting space harmonics and assuming sinusoidal distributed windings is:

$$\begin{array}{cc}\hfill v_x& =R_s{i}_x+{\displaystyle \frac{d{\lambda }_x}{dt}}\hfill \end{array}$$

(14)

where $v_x$, $i_x$, and ${\lambda }_x$ ($x\in \{a,b,c\}$) are the phase voltages, currents, and flux linkages, and $R_s$ is the stator resistance [58]. The flux linkages can be expressed as:

$$\left[\begin{array}{c}{\lambda }_a\\ {\lambda }_b\\ {\lambda }_c\end{array}\right]={\mathbf{L}}_{abc}\left({\theta }_e\right)\left[\begin{array}{c}{i}_a\\ i_b\\ i_c\end{array}\right]+\left[\begin{array}{c}{\lambda }_{ma}\left({\theta }_e\right)\\ {\lambda }_{mb}\left({\theta }_e\right)\\ {\lambda }_{mc}\left({\theta }_e\right)\end{array}\right]$$

(15)

where ${\mathbf{L}}_{abc}\left({\theta }_e\right)$ is the position-dependent inductance matrix and ${\lambda }_{mx}$ are the flux linkages due to permanent magnets [54]. For control purposes, the model is transformed into orthogonal reference frames using Clarke and Park transformations [60].

$$\left[\begin{array}{c}{x}_{\alpha }\\ x_{\beta }\\ x_0\end{array}\right]={\displaystyle \frac{2}{3}}\left[\begin{array}{ccc}1& -{\displaystyle \frac{1}{2}}& -{\displaystyle \frac{1}{2}}\\ 0& {\displaystyle \frac{\sqrt{3}}{2}}& -{\displaystyle \frac{\sqrt{3}}{2}}\\ {\displaystyle \frac{1}{2}}& {\displaystyle \frac{1}{2}}& {\displaystyle \frac{1}{2}}\end{array}\right]\left[\begin{array}{c}{x}_a\\ x_b\\ x_c\end{array}\right]$$

(16)

$$\left[\begin{array}{c}{x}_d\\ x_q\end{array}\right]=\left[\begin{array}{cc}cos{\theta }_e& sin{\theta }_e\\ -sin{\theta }_e& cos{\theta }_e\end{array}\right]\left[\begin{array}{c}{x}_{\alpha }\\ x_{\beta }\end{array}\right]$$

(17)

where ${\theta }_e$ is the electrical rotor angle. Applying these transformations to the $\left(abc\right)$ model yields the well-known Park-frame equations:

$$\begin{array}{cc}\hfill v_d& =R_s{i}_d+L_d{\displaystyle \frac{d{i}_d}{dt}}-{\omega }_e{L}_q{i}_q\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill v_q& =R_s{i}_q+L_q{\displaystyle \frac{d{i}_q}{dt}}+{\omega }_e\left(L_d{i}_d+{\lambda }_m\right)\hfill \end{array}$$

(19)

The electromagnetic torque is:

$$T_e={\displaystyle \frac{3}{2}}p\left[{\lambda }_m{i}_q+(L_d-L_q)i_d{i}_q\right]$$

(20)

where p is the number of pole pairs, ${\lambda }_m$ the permanent magnet flux linkage, and $L_d$, $L_q$ are the inductances along the direct and quadrature axes [58,59]. Automotive traction employs several variants of synchronous machines, each with specific characteristics [54,55,57,58]:

• Surface-mounted Permanent Magnet Synchronous Machine (SPMSM): Permanent magnets are mounted on the rotor surface, resulting in a nearly isotropic rotor (equal $L_d$ and $L_q$). This design provides high torque density and a simple electromagnetic model, but limited field-weakening capability due to high back-EMF.
• Interior Permanent Magnet Synchronous Machine (IPMSM): Magnets are embedded within the rotor, introducing rotor saliency ($L_d\ne L_q$) and enabling reluctance torque production in addition to magnet torque. IPMSMs exhibit excellent field-weakening performance and high efficiency over extended speed ranges.
• Synchronous Reluctance Machine (SynRM): Torque is produced solely by rotor saliency without magnets (${\lambda }_m=0$). While efficiency is lower than PMSM at low speed, SynRMs eliminate rare-earth materials and offer robust high-speed operation. Their magnet-free design makes them particularly attractive in the context of circular economy and sustainability, since they avoid the use of critical raw materials such as neodymium and dysprosium. Recent research has focused on advanced rotor barrier design, ferrite-assisted reluctance machines, and optimization techniques that improve torque density and efficiency, thereby bridging the gap with PMSMs while retaining environmental and cost advantages.
• Hybrid PM-SynRM: Combines permanent magnets and reluctance torque to balance cost, efficiency, and field-weakening capability.

The choice of synchronous motor topology directly affects the parameters of the Park model [54,55,57,58]:

• SPMSM: $L_d\approx L_q$, torque production relies mainly on ${\lambda }_m{i}_q$. Field-weakening is limited since $(L_d-L_q)\approx 0$.
• IPMSM: $L_d<L_q$, reluctance torque $(L_d-L_q)i_d{i}_q$ significantly contributes to $T_e$, enhancing torque per ampere and field-weakening performance.
• SynRM: ${\lambda }_m=0$, torque is purely reluctance-based, requiring $L_q>L_d$ for positive torque production. In addition to their simplified model structure, SynRMs are less sensitive to permanent magnet degradation phenomena (e.g., partial demagnetization), which enhances robustness in long-term operation. This intrinsic resilience further strengthens their position as a promising rare-earth-free alternative for future EV powertrains.
• Hybrid PM-SynRM: Intermediate ${\lambda }_m$ and moderate saliency $(L_d\ne L_q)$ combine both PM and reluctance torque contributions.

While the classical $dq$-axis model accurately captures the electromechanical energy conversion in synchronous machines under idealized conditions, high-fidelity models for automotive applications must also include secondary torque components and friction phenomena that significantly affect performance, especially at low speed or during high dynamic torque demands [58,59]. Cogging torque, also referred to as detent torque, arises from the interaction between rotor permanent magnets and the stator slots, even when no current flows in the stator windings. This torque is position-dependent and periodic with respect to the mechanical rotor angle ${\theta }_m$:

$$T_{\mathrm{cog}}\left({\theta }_m\right)=\sum _{h=1}^{N_h}{T}_{h}cos\left(h{N}_{s}p{\theta }_m+{\varphi }_h\right)$$

(21)

where $N_s$ is the number of stator slots, p the number of pole pairs, $T_h$ and ${\varphi }_h$ the amplitude and phase of the h-th harmonic component, and $N_h$ the number of harmonics retained in the model. The cogging torque is superimposed on the electromagnetic torque, modifying the total torque equation as:

$$T_{\mathrm{tot}}=T_e+T_{\mathrm{cog}}\left({\theta }_m\right)-T_{\mathrm{fric}}\left({\omega }_m\right)$$

(22)

The Stribeck effect models the nonlinear behavior of friction torque at low speeds, where static friction and viscous friction interact with the boundary lubrication regime [59].

$$T_{\mathrm{fric}}\left({\omega }_m\right)=T_c\mathrm{sgn}\left({\omega }_m\right)+\left(T_s-T_c\right)e^{-{\left|{\displaystyle \frac{{\omega }_m}{{\omega }_s}}\right|}^{\delta }}\mathrm{sgn}\left({\omega }_m\right)+B{\omega }_m$$

(23)

Incorporating both cogging torque and Stribeck friction into the mechanical equation of the $dq$-model yields:

$$J{\displaystyle \frac{d{\omega }_m}{dt}}=T_e+T_{\mathrm{cog}}\left({\theta }_m\right)-T_{\mathrm{fric}}\left({\omega }_m\right)-T_L$$

(24)

The presence of $T_{\mathrm{cog}}$ and Stribeck friction necessitates advanced compensation strategies:

• Feedforward cogging torque cancellation based on rotor position estimation.
• Adaptive friction compensation in speed and torque controllers, especially for smooth creep and launch control in EVs.
• Enhanced observers in FOC or MPC schemes to estimate disturbance torque in real-time and adjust current references accordingly.

These enhancements ensure that the model and control system reflect the real-world behavior of the machine under all operating conditions, from standstill to high-speed operation [58,59,60].

It is important to remark that the parameters of the synchronous machine model ($R_s$, $L_d$, $L_q$, ${\lambda }_m$) are not constant, but vary significantly with temperature, magnetic saturation, and long-term aging. These variations introduce model uncertainties that affect torque prediction and current control accuracy. Robust design therefore requires monitoring and adaptive estimation of such parameters, together with control strategies capable of guaranteeing performance under bounded uncertainty.

3. Advanced Control Strategies for Synchronous Motor Drives

The high-performance requirements of automotive synchronous drives demand control strategies that go beyond classical approaches. Among the advanced techniques investigated are feedback linearization, model predictive control, sliding mode control, adaptive data-driven methods, and reinforcement learning. Each of these strategies exploits the affine-in-input structure of the PMSM/IPMSM model in Park coordinates, where $\dot{x}=f(x,{\omega }_e)+G\left(x\right)u$ with $x={[i_d,i_q]}^{\top }$ and $u={[v_d,v_q]}^{\top }$, and where torque, inverter voltage, and thermal constraints appear as algebraic maps or feasibility sets. We now present each control approach in detail, combining mathematical rigor with implementation considerations. Beyond their theoretical formulation, the application of advanced control strategies to automotive electric drives requires a structured design methodology. The key steps can be summarized as follows:

• Feedback Linearization: The methodology involves (i) selecting outputs $y=x$ that provide full state feedback, (ii) verifying the nonsingularity of the decoupling matrix $E\left(x\right)$, (iii) computing the feedback law $u=E^{-1}(\nu -\frac{\partial y}{\partial x}f(x,{\omega }_e))$, and (iv) tuning the gain matrix K to place the poles of the linearized error dynamics.
• Model Predictive Control (MPC): The design requires (i) deriving a discrete-time prediction model $x_{k+1}=A{x}_k+B{u}_k+d$, (ii) formulating a cost function that balances torque tracking, flux regulation, and switching losses, (iii) defining the feasibility set ${\mathcal{U}}_{\mathrm{SVM}}$ and state constraints (currents, temperatures), (iv) selecting prediction horizon $N_p$ and weights $(\alpha ,\beta ,\gamma )$, and (v) implementing an optimization solver compatible with embedded execution.
• Sliding Mode Control (SMC): The methodology starts with (i) defining a sliding surface $S=\Lambda e$, (ii) ensuring the reachability condition $S^{\top }\dot{S}\le -\eta \parallel S\parallel$, (iii) computing the equivalent control $u_{eq}$, (iv) designing the discontinuous term $-K\mathrm{sat}(S/\varphi )$ with robustness margins against uncertainty, and (v) mitigating chattering through boundary layer design or higher-order algorithms (e.g., super-twisting).
• Adaptive Control: The design procedure includes (i) specifying a reference model $x_m$, (ii) deriving the adaptation law $\dot{K}=-\Gamma \phi (x,r)e_m^{\top }P{B}_m$, (iii) ensuring stability via Lyapunov analysis with positive-definite P, (iv) validating the persistent excitation (PE) condition or employing concurrent learning/DREM methods, and (v) tuning adaptation gains $\Gamma$ to balance convergence speed and robustness to noise.
• Reinforcement Learning (RL): The methodology is based on (i) defining the Markov Decision Process (state, action, transition, reward), (ii) shaping a reward function that encodes torque tracking, efficiency, and thermal constraints, (iii) training the policy offline using high-fidelity simulations with domain randomization, (iv) validating the learned controller under safety layers that enforce admissibility of ${\mathcal{U}}_{\mathrm{SVM}}$, and (v) deploying a lightweight inference network on the embedded drive controller.

These methodologies highlight how each strategy requires a balance between rigorous mathematical derivation and practical implementation choices. The emphasis is on ensuring stability, robustness, and feasibility on automotive-grade hardware while respecting functional safety requirements.

3.1. Feedback Linearization

Feedback linearization aims to transform the nonlinear PMSM/IPMSM dynamics into a linear, decoupled system by algebraically canceling the cross terms and speed-dependent nonlinearities. Starting from the Park-frame equations introduced earlier, the current dynamics can be written in affine-in-input form as

$$\dot{x}=f(x,{\omega }_e)+Gu,x=\left[\begin{array}{c}{i}_d\\ i_q\end{array}\right],u=\left[\begin{array}{c}{v}_d\\ v_q\end{array}\right],$$

(25)

where

$$f(x,{\omega }_e)=\left[\begin{array}{c}-{\displaystyle \frac{R_s}{L_d}}{i}_d+{\displaystyle \frac{{\omega }_e{L}_q}{L_d}}{i}_q\\ -{\displaystyle \frac{{\omega }_e{L}_d}{L_q}}{i}_d-{\displaystyle \frac{R_s}{L_q}}{i}_q-{\displaystyle \frac{{\omega }_e{\lambda }_m}{L_q}}\end{array}\right],G=\left[\begin{array}{cc}{\displaystyle \frac{1}{L_d}}& 0\\ 0& {\displaystyle \frac{1}{L_q}}\end{array}\right].$$

The outputs are chosen as $y=x$, so that

$$\dot{y}={\displaystyle \frac{\partial y}{\partial x}}f(x,{\omega }_e)+Eu,E={\displaystyle \frac{\partial y}{\partial x}}G=G.$$

(26)

The decoupling matrix E is diagonal with strictly positive entries ($1/L_d$, $1/L_q$), and therefore always nonsingular as long as $L_d,L_q>0$, a condition met in any practical PMSM/IPMSM. One can thus impose the virtual input

$$\nu ={\dot{y}}^{\star }+K(y^{\star }-y),K\in {\mathbb{R}}^{2\times 2}\mathrm{Hurwitz},$$

(27)

and define the feedback linearizing control law

$$u=E^{-1}\left(\nu -\frac{\partial y}{\partial x}f(x,{\omega }_e)\right).$$

(28)

By substitution into the closed-loop system, the error $e=y-y^{\star }$ obeys

$$\dot{e}=-Ke,$$

which guarantees exponential convergence. To formally prove stability, consider the Lyapunov candidate $V\left(e\right)=\frac{1}{2}{e}^{\top }e$, yielding $\dot{V}=-e^{\top }Ke$. If K is symmetric positive definite or has symmetric part $K+K^{\top }\succ 0$, then $\dot{V}\le -{\lambda }_{min}(K+K^{\top }){\parallel e\parallel }^2$, proving global exponential stability [61,62]. From a formal point of view, feedback linearization belongs to the family of input–output linearization techniques. Given a nonlinear system of the form $\dot{x}=f\left(x\right)+Gu$, the existence of a diffeomorphism $\Phi :x\mapsto z$ such that the dynamics in the new coordinates $z=\Phi \left(x\right)$ are linear depends on the relative degree of the outputs and on the nonsingularity of the decoupling matrix $E\left(x\right)$. In the PMSM/IPMSM case, since $E=\mathrm{diag}(1/L_d,1/L_q)$ is always nonsingular, the system admits an exact linearizing transformation. The closed-loop dynamics can be written explicitly as

$$\dot{z}=\nu ,\nu ={\dot{y}}^{\star }+K(y^{\star }-y),$$

so that z behaves as a pair of decoupled integrators driven by $\nu$. This guarantees that the feedback-linearized system is not only linear but also controllable, and any classical linear control technique (pole placement, LQR, PI/PID tuning) can be directly applied to shape the error dynamics. In practice, exact cancellation is never achieved because parameters $R_s,L_d,L_q,{\lambda }_m$ and the angle ${\theta }_e$ are uncertain or slowly drifting with temperature, saturation, and aging. Suppose the actual dynamics differ by $\Delta f\left(x\right)$ and $\Delta G$ from the nominal ones used in (28); the error dynamics become

$$\dot{e}=-Ke+w\left(t\right),$$

where $w\left(t\right)$ is a bounded disturbance due to modeling errors. By choosing an additional corrective term $-K_{r}e$ in the law, one achieves input-to-state stability. Specifically, with $V=\frac{1}{2}{e}^{\top }e$ we obtain

$$\dot{V}\le -({\lambda }_{min}\left(K_r\right)-\delta ){\parallel e\parallel }^2+\parallel e\parallel \parallel w\parallel ,$$

so that if ${\lambda }_{min}\left(K_r\right)>\delta$ the error remains bounded and converges to a neighborhood of the origin proportional to the uncertainty bound $\parallel w\parallel$. This establishes robustness margins [63,64]. The block diagram in Figure 7 provides an intuitive view of the method: the original nonlinear system is mapped through a nonlinear coordinate transformation and inverted by the feedback law, such that the overall closed loop behaves like a linear system controlled by a conventional linear regulator. In this way, the sophisticated algebra embedded in (28) reduces to a structure where standard PI or pole-placement design can be used once the nonlinear dynamics are canceled [65,66].

From an implementation perspective, feedback linearization is computationally light, as it requires only matrix inversion (trivial in the diagonal case) and a few multiplications per sampling period. On a typical DSP running at 100–200 MHz, the additional load is negligible compared to SVPWM generation. The main challenge lies in sensorless operation: accurate rotor angle estimation is mandatory, since any phase error in ${\theta }_e$ directly compromises decoupling. Extended Kalman Filters (EKF) or Unscented KFs are used at medium and high speed, while high-frequency injection is employed near zero speed. In production systems, look-up maps and online estimators provide $R_s\left(T\right)$ and $L_{d,q}\left({\omega }_e\right)$ values to compensate parameter drift. Furthermore, actuator constraints such as voltage saturation in the SVM region must be respected: in practice, (28) is combined with a projection onto ${\mathcal{U}}_{\mathrm{SVM}}$. Finally, because the approach cancels natural dynamics, measurement noise can be amplified; filtering strategies and observer-based implementations are essential to ensure robustness in an embedded automotive controller [67,68]. In summary, feedback linearization provides a rigorous framework in which the nonlinear PMSM/IPMSM dynamics are formally transformed into a linear, controllable system. The explicit construction of the diffeomorphism $\Phi \left(x\right)$ and the nonsingularity of the decoupling matrix ensure the validity of the approach, while Lyapunov analysis certifies stability of the closed-loop dynamics. This mathematical foundation justifies the practical application of classical linear control strategies in automotive electric drives, combining theoretical rigor with implementation feasibility.

3.2. Model Predictive Control

Model predictive control represents one of the most systematic frameworks for current and torque regulation in synchronous motor drives, as it explicitly accounts for system dynamics, constraints, and performance indices in a unified optimization problem solved at every sampling instant [69,70,71]. Starting from the affine-in-input representation $\dot{x}=f(x,{\omega }_e)+G\left(x\right)u$, the discretized model under zero-order hold is expressed as

$$x_{k+1}=A\left({\omega }_e\right)x_k+B\left({\omega }_e\right)u_k+d\left({\omega }_e\right),$$

(29)

where the matrices $A,B$ capture the electrical dynamics over the sampling period $T_s$, and d represents the known disturbance due to the back-EMF term proportional to ${\omega }_e{\lambda }_m$. The inverter feasibility set ${\mathcal{U}}_{\mathrm{SVM}}$ is given by the hexagonal voltage limit induced by space vector modulation, while the admissible state set $\mathcal{X}$ encodes current and thermal constraints. The generic architecture of MPC is illustrated in Figure 8, where the prediction model forecasts the machine response over the horizon, and the optimizer computes the control input that minimizes a given cost function while satisfying constraints. The block diagram highlights the three pillars of MPC: the cost function defining the performance objective, the constraints representing feasibility and safety, and the prediction model that links future control actions to system evolution.

The standard finite-horizon MPC problem is formulated as

$$\begin{array}{cc}\hfill \underset{\{u_k,\dots ,u_{k+N_p-1}\}}{min}& \sum _{j=1}^{N_p}\left[\alpha \parallel i_{q,k+j}-i_{q,k+j}^{\star }{\parallel }^2+\beta \parallel i_{d,k+j}-i_{d,k+j}^{\star }{\parallel }^2+\gamma {\parallel \Delta u_{k+j}\parallel }^2\right]\hfill \\ \hfill \mathrm{s}.\mathrm{t}.& x_{k+j+1}=A{x}_{k+j}+B{u}_{k+j}+d,x_{k+j}\in \mathcal{X},u_{k+j}\in {\mathcal{U}}_{\mathrm{SVM}},j=0,\dots ,N_p-1.\hfill \end{array}$$

(30)

This is a quadratic program (QP) with linear dynamics and convex quadratic cost. Convexity ensures the existence of a unique global minimizer, which can be found efficiently by active-set or interior-point algorithms tailored for embedded processors. The optimality conditions follow from the Karush–Kuhn–Tucker (KKT) framework. Denote by $U=\{u_k,\dots ,u_{k+N_p-1}\}$ the decision variables and by $X=\{x_{k+1},\dots ,x_{k+N_p}\}$ the predicted states. The Lagrangian is

$$\mathcal{L}(U,\lambda ,\mu )=J\left(U\right)+{\lambda }^{\top }g(X,U)+{\mu }^{\top }h(X,U),$$

where $g(X,U)\le 0$ are inequality constraints (currents, voltages, thermal limits) and $h(X,U)=0$ the equality dynamics. The KKT conditions then require: stationarity ${\nabla }_{U}J+{\nabla }_U{g}^{\top }\lambda +{\nabla }_U{h}^{\top }\mu =0$, primal feasibility $g(X,U)\le 0,h(X,U)=0$, dual feasibility $\lambda \ge 0$, and complementary slackness ${\lambda }^{\top }g(X,U)=0$. In practical implementations, solvers exploit the sparse block structure of $(A,B)$ to achieve computational efficiency. Closed-loop stability is ensured by augmenting the cost with a terminal penalty $x_{k+N_p}^{\top }P{x}_{k+N_p}$ and restricting the terminal state to a set ${\mathcal{X}}_f$ invariant under a stabilizing local controller. This guarantees that the cost-to-go functions as a Lyapunov function, decreasing along closed-loop trajectories [72]. A key distinction lies between continuous-input MPC and finite-control-set MPC. In continuous MPC, the decision variable is the continuous voltage vector, and the solution of the QP yields an optimal $u_k$ that is then synthesized by SVPWM. This approach provides smooth currents and low THD but requires solving a QP every $T_s$, with computational complexity $O\left(N_p{n}_u^2\right)$ where $n_u=2$. In finite-control-set MPC, by contrast, the admissible $u_k$ are restricted to the finite set of inverter switching vectors (7 nonzero vectors for a 2-level inverter, 19 for a 3-level NPC). The cost is evaluated for each candidate, and the minimizer is directly applied to the switches, eliminating the modulation stage. This drastically reduces complexity but introduces current ripple due to the coarse nature of the input set [73,74]. Implementation practice in automotive drives reflects these trade-offs. Continuous-input MPC has been demonstrated in hardware-in-the-loop platforms using custom FPGA accelerators, with sampling times down to $T_s=25\mu$s, exploiting pipelined interior-point solvers [69,70]. Finite-control-set MPC is computationally much lighter, requiring only a few multiplications per candidate vector, and is therefore feasible on conventional DSPs with $T_s=50$–$100\mu$s. However, ripple management is critical, and multi-step horizons or switching-frequency penalization are often added to smooth performance. Sensorless operation integrates naturally: moving horizon estimation (MHE) uses the same predictive model and optimization machinery to jointly estimate ${\theta }_e$ and ${\omega }_m$, ensuring coherence between observer and controller. In production prototypes, however, lighter Luenberger or extended Kalman observers are preferred to reduce complexity, with MHE reserved for high-performance demonstrators [71]. Tuning the weights $\alpha ,\beta ,\gamma$ is crucial: large $\alpha$ enforces accurate torque tracking, while $\beta$ manages flux or field weakening, and $\gamma$ penalizes switching effort to reduce losses. Industrial practice often limits the horizon to $N_p=1$ or 2 for real-time feasibility. More advanced schemes embed electro-thermal constraints by adding junction temperature predictions from reduced-order thermal models to the inequality set $g(X,U)$, thereby achieving constraint-aware derating [75,76]. This makes MPC uniquely suited for multi-domain optimization in next-generation EV traction inverters.

3.3. Sliding Mode Control

Sliding Mode Control (SMC) is a robust nonlinear control methodology that exploits the discontinuous nature of the control input to force the system trajectory onto a chosen manifold and maintain it there despite disturbances and parameter uncertainties. For PMSM/IPMSM current regulation, let the tracking error be $e=x-x^{\star }={[e_d,e_q]}^{\top }$ and define a sliding surface as a linear combination of the errors, namely

$$S=\Lambda e=\left[\begin{array}{cc}{\lambda }_d& 0\\ 0& {\lambda }_q\end{array}\right]\left[\begin{array}{c}{e}_d\\ e_q\end{array}\right],$$

(31)

where $\Lambda$ is a positive diagonal matrix. The control objective is to drive $S\to 0$ in finite time and then maintain it at zero, implying $e\to 0$ under nominal dynamics [77,78]. The block diagram in Figure 9 illustrates the essential structure of SMC. The system output $y\left(t\right)$ is compared with its reference $y_{ref}\left(t\right)$ to form the sliding variable $s\left(t\right)$; this is then fed into a discontinuous control law that switches the input $u\left(t\right)$ to ensure convergence of $s\left(t\right)\to 0$. The figure also emphasizes the role of the system dynamics $f\left(x\right)$ in shaping the equivalent control $u_{eq}$, while the discontinuous term enforces robustness.

The equivalent control $u_{eq}$ is defined as the continuous input that would keep $\dot{S}=0$ under the nominal model $\dot{x}=f(x,{\omega }_e)+Gu$. Substituting this into $\dot{S}=\Lambda (f(x,{\omega }_e)+Gu-{\dot{x}}^{\star })$ yields

$$u_{eq}=-{(\Lambda G)}^{-1}\Lambda (f(x,{\omega }_e)-{\dot{x}}^{\star }).$$

Formally, the SMC design is based on two conditions: (i) the reachability condition, which requires that the system trajectory is driven towards the manifold $S=0$, and (ii) the invariance condition, which ensures that once $S=0$ is reached, the trajectory remains on it. The reachability condition is guaranteed if there exists $\eta >0$ such that

$$S^{\top }\dot{S}\le -\eta \parallel S\parallel ,\forall S\ne 0.$$

This ensures that the manifold is attractive. The invariance condition follows from $\dot{S}=0$ when the equivalent control is applied, proving that the reduced-order dynamics evolve entirely on the sliding surface. In practice, uncertainties prevent exact cancellation, so a discontinuous term is added:

$$u=u_{eq}-K\mathrm{sat}\left({\displaystyle \frac{S}{\varphi }}\right),$$

(32)

where $K\succ 0$ is a gain matrix, $\mathrm{sat}(·)$ is a smoothed signum, and $\varphi >0$ defines a boundary layer. This ensures robustness. To show finite-time convergence, consider the Lyapunov function $V=\frac{1}{2}{S}^{\top }S$. Its derivative is

$$\dot{V}=S^{\top }\dot{S}=S^{\top }\Lambda (f+Gu-{\dot{x}}^{\star }).$$

Substituting the control law gives

$$\dot{V}\le -\eta \parallel S\parallel +\parallel S\parallel \parallel \Delta \parallel ,$$

where $\Delta$ bounds the model uncertainty. If K is chosen such that $\eta > \parallel \Delta \parallel$, then $\dot{V}<0$ whenever $S\ne 0$, proving finite-time convergence of $S\to 0$. Once on the sliding surface, the dynamics reduce to $\dot{x}=f(x,{\omega }_e)+G{u}_{eq}$, which correspond to the nominal linearized current loops, hence ensuring robust tracking [79,80]. An alternative proof of finite-time convergence can be obtained by integrating the inequality $\dot{V}\le -(\eta -\parallel \Delta \parallel )\parallel S\parallel$ to show that $S\left(t\right)$ reaches the manifold in a settling time upper-bounded by

$$T\le {\displaystyle \frac{\sqrt{2V\left(0\right)}}{\eta -\parallel \Delta \parallel }}.$$

This explicit bound on the convergence time is particularly useful for automotive applications, where the transient duration directly affects driveability and safety. However, the discontinuous term generates high-frequency switching known as chattering, which produces current ripple, additional losses, and acoustic noise. To mitigate this, higher-order sliding mode algorithms are adopted. Among them, the super-twisting algorithm is widely used. It defines the input derivative as

$$\dot{u}=-k_1\mathrm{sgn}\left(S\right)-k_2{\left|S\right|}^{1/2}\mathrm{sgn}\left(S\right),$$

with positive gains $k_1,k_2$ chosen according to Lipschitz bounds of the error dynamics. The associated Lyapunov analysis uses $V=\left|S\right|+\frac{1}{2}{\dot{S}}^2$ to show that both S and $\dot{S}$ converge to zero in finite time, while u remains continuous. This eliminates high-frequency chattering while preserving robustness, a crucial feature for automotive drives [78,81]. In summary, SMC formally guarantees robustness against matched uncertainties by enforcing sliding dynamics on a carefully designed manifold $S=0$. The reachability and invariance conditions, together with Lyapunov-based convergence proofs, provide a rigorous mathematical justification. Practical implementations require engineering trade-offs: boundary layers and super-twisting mitigate chattering, while careful gain tuning ensures fast convergence without excessive stress on the inverter. From the implementation viewpoint, SMC is computationally inexpensive, requiring only evaluation of the sliding variable, equivalent control, and a discontinuous or super-twisting update law. This makes it suitable for DSP-based controllers with sampling times of tens of microseconds. Nevertheless, careful tuning is required: gains that are too low fail to overcome uncertainty, while excessive gains amplify ripple and stress the inverter. Practical implementations employ a saturation function $\mathrm{sat}(S/\varphi )$ to soften the discontinuity, and add low-pass filters or predictive ripple compensation to further reduce torque pulsations. For sensorless operation, SMC pairs naturally with sliding mode observers (SMO) that estimate rotor flux and angle by exploiting the same discontinuous structure; in practice, SMOs are combined with high-frequency injection or fuzzy/terminal observers to improve performance at standstill [82,83,84]. Overall, sliding mode control offers unmatched robustness to parameter variations and disturbances, but its integration in EV drives requires engineering solutions to manage ripple, acoustic noise, and interaction with electro-thermal constraints [85].

3.4. Adaptive Data-Driven Control

Adaptive control addresses the fundamental limitation of fixed-parameter model-based strategies, namely their sensitivity to parameter variations due to temperature, magnetic saturation, or long-term aging of the machine. Instead of relying on static parameter identification, adaptive controllers continuously update their gains based on real-time measurement data, ensuring consistent performance across operating conditions. In the context of PMSM/IPMSM drives, a widely used framework is Model Reference Adaptive Control (MRAC), in which the drive currents are required to follow the dynamics of a desired reference model [86,87,88]. Let the reference model be ${\dot{x}}_m=A_m{x}_m+B_{m}r$, where $x_m={[i_{d,m},i_{q,m}]}^{\top }$ represents the reference dynamics and r is a reference input (typically related to the torque demand). The actual machine dynamics in compact affine form are $\dot{x}=Ax+Bu+\Delta \left(x\right)$, with $\Delta \left(x\right)$ capturing unmodeled effects and parameter uncertainty. The adaptive control law is chosen as

$$u=K^{\top }\varphi (x,r),$$

where $\varphi (x,r)$ is a regressor vector (constructed from currents and reference signals) and K is the adaptive gain matrix to be updated online. Defining the model tracking error $e_m=x-x_m$, the adaptation law is

$$\dot{K}=-\Gamma \varphi (x,r)e_m^{\top }P{B}_m,$$

with $\Gamma \succ 0$ the adaptation gain matrix, P the solution of the Lyapunov equation $A_m^{\top }P+P{A}_m=-Q$, and $Q\succ 0$. The stability of MRAC can be formally established. Consider the Lyapunov function candidate

$$V(e_m,\tilde{K})=e_m^{\top }P{e}_m+\mathrm{tr}\left({\tilde{K}}^{\top }{\Gamma }^{-1}\tilde{K}\right),$$

where $\tilde{K}=K-K^{\star }$ is the parameter estimation error with respect to the ideal gain matrix $K^{\star }$ that achieves perfect model matching. Differentiating V along the system trajectories and substituting the update law yields

$$\dot{V}=-e_m^{\top }Q{e}_m\le 0,$$

which shows global stability and convergence of $e_m\to 0$. Thus, the adaptive law guarantees asymptotic tracking of the reference model under nominal conditions. The block diagram in Figure 10 provides a visual summary of this principle: the controlled system is forced to track a reference model by adjusting the adaptive input $u_{ad}$, which compensates the uncertainty via the disturbance model $w^{\top }\varphi \left(x\right)$. The adaptation loop continuously updates the parameters K based on the error $e\left(t\right)$ between the reference and the actual system, enforcing the desired closed-loop dynamics.

However, a critical requirement for parameter convergence is the so-called Persistent Excitation (PE) condition, namely that ${\int }_t^{t+T}\varphi \left(\tau \right){\varphi }^{\top }\left(\tau \right)d\tau \succ 0$ for some finite T. In automotive duty cycles, excitation may be weak or insufficient due to long periods of nearly constant operating points. To overcome this, modern adaptive schemes employ Concurrent Learning (CL), where recorded data are stored in a memory stack and re-used to excite the regressor space even in the absence of instantaneous PE. The adaptation law is then augmented with an additional term proportional to the parameter error evaluated at past data points. Another powerful method is Dynamic Regressor Extension and Mixing (DREM), which uses filtering and regressor transformations to extract richer excitation from the available signals, thereby ensuring parameter convergence without persistent input variation [89,90,91,92]. Both CL and DREM are compatible with embedded implementation, as they rely on simple algebraic extensions of the update law. From a practical standpoint, adaptive control is computationally lightweight compared to MPC, since it requires only vector-matrix multiplications and parameter updates at each sampling step. This makes it well-suited to DSP-based control units with sampling times in the range of $T_s=25$–$100\mu$s. The main implementation challenge lies in tuning the adaptation gain $\Gamma$: if chosen too small, convergence is slow and performance drifts under parameter variations; if too large, oscillations or even instability may occur in the presence of measurement noise. In practice, projection operators are used to bound K within safe intervals, preventing runaway adaptation. Moreover, adaptive controllers are often combined with observers: neural network observers (RNN, GRU, or LSTM based) can provide estimates of rotor angle ${\theta }_e$ and speed ${\omega }_m$, while adaptive KFs fuse these data-driven estimates with physical models [87,88]. This hybrid approach exploits the adaptability of data-driven methods while retaining the interpretability of physics-based observers. In automotive prototypes, adaptive data-driven control has shown promise in handling electro-thermal coupling: by augmenting the regressor vector $\varphi$ with estimated junction temperatures or saturation indicators, the adaptation law implicitly compensates multi-domain interactions, making this method attractive for next-generation electric drives [93].

3.5. Reinforcement Learning

Reinforcement Learning (RL) constitutes the most flexible framework among advanced control strategies, as it does not require an explicit analytic controller but instead learns an optimal decision policy through interaction with the environment. The PMSM/IPMSM drive is naturally cast as a Markov Decision Process (MDP), defined by a state space $\mathcal{S}$, an action space $\mathcal{A}$, a stochastic transition function $\mathcal{P}\left(s_{k+1}\right|s_k,a_k)$ describing the machine dynamics, and a reward function $r:\mathcal{S}\times \mathcal{A}\to \mathbb{R}$ that quantifies the control objectives [94,95]. At each sampling instant k, the controller observes a state $s_k\in \mathcal{S}$, which can include electrical currents $[i_d,i_q]$, mechanical states $({\omega }_m,{\theta }_m)$, estimated junction temperature $T_j$, and additional features from observers, and then selects an action $a_k\in \mathcal{A}$ corresponding to inverter commands (either continuous voltages $u=[v_d,v_q]$ or discrete switching vectors in finite-control-set RL). The environment transitions to $s_{k+1}$ according to $\mathcal{P}$, and the agent receives a scalar reward $r(s_k,a_k)$. The objective is to maximize the expected discounted return

$$J\left(\pi \right)={\mathbb{E}}_{\pi }\left[\sum _{k=0}^{\infty }{\gamma }^{k}r(s_k,a_k)\right],$$

(33)

where $\pi$ is the policy mapping states to action distributions, and $\gamma \in (0,1)$ is the discount factor [96]. The theoretical foundation is the Bellman optimality equation for the state-action value function

$$Q^{\pi }(s,a)=\mathbb{E}\left[r(s,a)+\gamma Q^{\pi }(s^{\prime },\pi \left(s^{\prime }\right))\right],$$

(34)

which characterizes the cumulative reward starting from $(s,a)$ under policy $\pi$. Optimal control corresponds to finding ${\pi }^{\star }\left(s\right)=arg{max}_a{Q}^{\pi }(s,a)$ [97]. In policy gradient methods, the parameters $\theta$ of a stochastic policy ${\pi }_{\theta }\left(a\right|s)$ are updated by

$${\nabla }_{\theta }J\left(\theta \right)={\mathbb{E}}_{s,a\sim {\pi }_{\theta }}\left[{\nabla }_{\theta }log{\pi }_{\theta }\left(a\right|s)Q^{\pi }(s,a)\right],$$

(35)

which follows from the likelihood ratio trick. To reduce variance, actor–critic architectures introduce a critic estimating the advantage $\widehat{A}(s,a)=Q^{\pi }(s,a)-V^{\pi }\left(s\right)$, yielding the practical update

$${\nabla }_{\theta }J\left(\theta \right)\approx \mathbb{E}\left[{\nabla }_{\theta }log{\pi }_{\theta }\left(a\right|s)\widehat{A}(s,a)\right].$$

(36)

Popular implementations include Proximal Policy Optimization (PPO) and Soft Actor–Critic (SAC), both compatible with continuous or discrete inverter action sets [98]. The block diagram in Figure 11 illustrates the RL loop applied to electric drives. The agent (policy) selects actions $a_k$ based on the observed state $s_k$, applies them to the environment (the motor-inverter system), and receives both the next state $s_{k+1}$ and a scalar reward $r_k$. The reward encapsulates the control objectives (torque tracking, efficiency, safety), and the policy is updated accordingly. This closed loop of interaction and learning highlights the data-driven nature of RL, contrasting with model-based schemes.

The reward is the key design element and directly encodes multi-domain objectives. A representative example is

$$r_k=-w_{\tau }|{\tau }^{\star }-{\tau }_e|-w_{\ell }{P}_{\mathrm{loss}}(x_k,u_k)-w_T{(T_{j,k}-T_{j,max})}_{+}-w_v{\parallel u_k\parallel }_{{\mathcal{U}}_{\mathrm{SVM}}},$$

(37)

where the terms penalize torque tracking error, electrical losses, thermal limit violations, and infeasible voltage commands. In this way, RL inherently handles electro-mechanical, electro-thermal, and inverter feasibility objectives without the need for explicit model equations in the controller. Extensions also allow penalizing current ripple or acoustic noise, making RL a candidate for integrated NVH optimization [99,100]. From a stability viewpoint, RL does not provide analytical guarantees comparable to model-based methods; however, if the critic converges, the learned policy approximates the optimal solution to the Bellman equation. Practical safety requires augmenting the learned policy with projections or safety layers: at runtime, the action suggested by the neural policy is projected onto the admissible set ${\mathcal{U}}_{\mathrm{SVM}}$, and control barrier functions ensure that state constraints (currents, temperatures) remain invariant sets [96,97]. Implementation in automotive drives follows a two-phase approach. Training is typically performed offline in a high-fidelity simulator that includes electromagnetic and thermal models, and training strategies such as domain randomization are applied to expose the agent to a wide distribution of parameters $(R_s,L_d,L_q,{\lambda }_m)$ and non-idealities of the inverter. This enhances robustness for sim-to-real transfer [98]. Deployment consists of embedding the trained neural network policy, which has modest inference cost (a few matrix multiplications), into the DSP/FPGA of the drive controller, with safety layers ensuring hard constraint satisfaction. Sensorless operation can be embedded directly in the policy network by providing histories of $v,i$ as inputs, letting the latent representation encode ${\theta }_e$, or by coupling the policy with classical EKF/MHE observers to ensure interpretability and reliability [95,101]. In summary, reinforcement learning provides unmatched flexibility to incorporate arbitrary control objectives and naturally accommodates multi-domain optimization. Yet, the absence of closed-form guarantees, the heavy data requirements for training, and the challenges of certification and safety represent the main barriers to its deployment in safety-critical automotive environments [99,100].

3.6. Intelligent Control Strategies

The increasing complexity of automotive electric drives has motivated the adoption of intelligent control strategies that integrate machine learning, artificial intelligence, and knowledge-based methods into the control loop. These approaches differ from classical advanced controllers in that they can exploit data-driven models, nonlinear approximators, and adaptive decision-making frameworks to improve robustness, adaptability, and performance in uncertain or highly variable operating conditions. Artificial Neural Networks (ANNs) are widely employed as nonlinear function approximators that can map system states to control actions. In PMSM/IPMSM drives, multilayer perceptrons and recurrent architectures (e.g., LSTM, GRU) have been used to estimate torque-producing currents or to generate optimal voltage commands under varying operating conditions. The training phase is typically performed offline using high-fidelity motor–inverter models or experimental datasets, while online inference is lightweight enough to be embedded in DSP/FPGA hardware. Stability is usually analyzed through Lyapunov-based adaptive laws that ensure boundedness of the estimation error [102,103,104,105,106]. Fuzzy logic controllers provide an interpretable rule-based framework that can encode expert knowledge of the drive dynamics. In automotive applications, fuzzy controllers have been applied to torque control, flux weakening, and regenerative braking management. Hybrid neuro-fuzzy systems combine the transparency of fuzzy rules with the learning capability of neural networks, allowing automatic tuning of membership functions and rule weights. This results in improved adaptability to nonlinearities and parameter drifts [107,108,109]. Several studies have proposed integrating intelligent methods with classical control structures. For example, neural networks can be employed to adaptively tune the gains of sliding mode controllers, reducing chattering while preserving robustness. Similarly, reinforcement learning agents can be combined with predictive models to create hybrid model-based/model-free architectures that achieve both stability guarantees and adaptability. Other hybrid approaches include AI-assisted parameter scheduling for MPC cost functions or neural observers fused with Kalman filters for sensorless operation [110,111,112,113]. Intelligent controllers provide enhanced adaptability to parameter variations, the ability to handle unmodeled nonlinearities, and potential for multi-objective optimization (e.g., torque tracking, efficiency, NVH reduction). However, their integration into safety-critical automotive systems is still limited by challenges such as computational burden, the need for extensive training datasets, lack of interpretability, and certification under functional safety standards (ISO 26262). Ongoing research addresses these issues through physics-informed neural networks, safety layers, and quantization/distillation techniques to reduce model complexity. In summary, intelligent control strategies complement classical advanced methods by providing data-driven adaptability and knowledge-based decision-making. Their role is expected to grow in next-generation automotive electric drives, particularly in scenarios where robustness to uncertainty, seamless integration with monitoring frameworks, and multi-domain optimization are critical.

3.7. Comparative Discussion

The five advanced control methods presented cover a wide spectrum of philosophies and trade-offs, from exact algebraic cancellation to optimization-based decision-making and fully data-driven policies. Feedback linearization and sliding mode control share the goal of robustifying the nonlinear PMSM/IPMSM dynamics, yet they differ in their approach: the former aims at decoupling and reducing the system to a linear model but suffers from parameter sensitivity, while the latter embraces the uncertainties and guarantees finite-time convergence at the price of ripple and chattering. Model predictive control and reinforcement learning represent optimization-based paradigms, with MPC providing explicit guarantees through convex optimization and constraints, and RL relying on data-driven exploration and reward shaping to achieve flexibility, albeit without formal stability proofs. Adaptive data-driven control lies between these extremes, offering continuous online adjustment to parameter drift while retaining interpretability and relatively low complexity. Beyond these classical advanced methods, intelligent control strategies introduce a further paradigm shift. Neural network controllers can approximate unknown nonlinearities and adapt control actions to changing drive conditions; fuzzy and neuro-fuzzy approaches embed expert knowledge into interpretable rules, with adaptive tuning provided by learning algorithms; hybrid intelligent controllers exploit neural or fuzzy layers to adjust the parameters of MPC or SMC online, combining analytical guarantees with adaptability. Such methods are increasingly explored in automotive contexts for torque tracking, flux weakening, and regenerative braking optimization. From a practical automotive viewpoint, the key discriminants are computational feasibility, sensorless integration, and multi-domain capability. Feedback linearization is extremely light computationally and attractive for embedded DSPs, but it requires high-fidelity observers for ${\theta }_e$. MPC is systematic and integrates electro-thermal limits explicitly, but its QP solving load makes it suitable only for platforms with dedicated accelerators or heavily simplified formulations. Sliding mode control is simple to implement and robust, yet the mitigation of chattering is a non-trivial engineering task, particularly when NVH constraints are stringent.

Adaptive controllers are appealing for lifetime robustness, since they track slow thermal and magnetic drifts, but their reliance on persistent excitation or stored data raises questions on identifiability under typical drive cycles. Reinforcement learning is in principle the most general, capable of shaping arbitrary objectives including efficiency, comfort, and safety; however, the data and training requirements, the lack of closed-form guarantees, and the certification challenges under ISO 26262 currently limit its adoption to research prototypes. Intelligent controllers, while still at an earlier stage of industrial deployment, add new dimensions to this comparison. Their main advantages are the capability to handle multi-domain objectives natively, adaptability to unmodeled dynamics, and the possibility of incorporating human expertise through rule-based or hybrid structures. Their weaknesses include high training effort, potential lack of transparency, and difficulties in functional safety certification. Physics-informed neural networks, digital twins enriched with AI correctors, and safety-shielded learning are among the emerging solutions to bridge these gaps. It is also worth noting that hybrid schemes are actively explored: adaptive MPC uses online parameter updating to preserve predictive accuracy; SMC with adaptive or neural gains reduces chattering; RL with embedded model knowledge (model-based RL) increases sample efficiency; and FBL layers combined with outer MPC loops can provide both light computation and constraint enforcement. In this landscape, intelligent control strategies play an enabling role, providing data-driven adaptability and optimization capabilities that can be layered on top of model-based methods. Future automotive drives are thus expected to rely on hybrid architectures that integrate rigorous analytical controllers with intelligent learning-based modules. The comparative landscape therefore suggests that the future of automotive drives will not rely on a single paradigm, but rather on carefully engineered combinations that exploit the strengths of each.

Table 3. Comparison of advanced and intelligent control strategies for PMSM/IPMSM drives.

Technique	Flexibility of Objectives	Computational Load	Sensorless Integration	Multi-Domain Integration	Robustness/Stability
Feedback Linearization	Limited to current/torque decoupling; linear tracking design after cancellation	Very low (algebraic inversion)	High sensitivity to ${\theta }_e$; EKF/UKF or HFI mandatory	Via gain scheduling with maps; not intrinsic	ISS under bounded uncertainty
Model Predictive Control	High; explicit cost shaping and constraints (multi-objective)	High (QP or FCS search at each $T_s$)	Natural synergy with MHE; lighter KF feasible in practice	First-class: losses, thermal limits, derating	Stability via terminal cost/set; Lyapunov decrease
Sliding Mode Control	Moderate; robust tracking of torque/currents with finite-time convergence	Low to moderate (simple or super-twisting law)	SMO observers align naturally; HFI needed at standstill	Manifold shaping can extend to thermal/magnetic domains; ripple problematic	Proven finite-time convergence (Lyapunov-based)
Adaptive Data-driven	High; adapts continuously to parameter drifts and new signals	Variable, typically light (matrix-vector updates)	Neural/adaptive observers; KF fusion for robustness	Easy augmentation with thermal, saturation, vibration features	Global asymptotic stability under PE; CL/DREM mitigate weak excitation
Reinforcement Learning	Very high; arbitrary reward shaping, policy learns multi-domain trade-offs	Very high training effort; modest inference cost	Latent encoders or classical observers; domain randomization for robustness	Native multi-domain optimization (electro-thermal–NVH)	No strict guarantees; depends on policy coverage and critic convergence
Intelligent Controllers	Very high; capable of embedding nonlinear approximations, rule-based knowledge, and adaptive tuning	Medium to high depending on architecture; quantization/distillation required for embedded use	Neural observers or fuzzy estimators; hybrid fusion with EKF/KF	Strong potential for multi-domain optimization via AI-enhanced models	Robustness depends on training; stability ensured only with hybrid physics-informed designs

summarizes the main properties of the discussed methods, including their flexibility, computational demand, sensorless integration capability, multi-domain extension, and robustness. The table has been formatted to automatically adapt to the page width for readability.

4. Monitoring in Automotive Electric Drives

The monitoring of automotive electric drives represents a cornerstone for ensuring safety, reliability, and durability of the entire traction system. Unlike classical approaches where monitoring was understood merely as the acquisition of basic electrical signals such as currents and voltages, modern requirements extend the concept towards a comprehensive framework that encompasses the continuous supervision of electrical, thermal, magnetic, and mechanical states, together with the prediction of their evolution in time and the detection of incipient faults. In other words, monitoring in contemporary PMSM and IPMSM drives is no longer restricted to being a support tool for control, but has evolved into an integrated diagnostic and prognostic layer that sustains the overall health management of the drive, fully aligned with the paradigm of Prognostics and Health Management (PHM). A first class of monitoring challenges concerns the electrical domain. The three-phase structure of the PMSM/IPMSM implies that monitoring must continuously assess the integrity of the phase currents $i_a$, $i_b$, $i_c$ and of the corresponding voltages, which after Clarke and Park transformations enter directly into the control law. Failures in current sensors, offsets, saturation, or loss of one phase connection lead to deviations in the reconstructed state vector $x={[i_d,i_q]}^{\top }$ and, consequently, in the estimated torque ${\tau }_e=\frac{3}{2}p({\lambda }_m{i}_q+(L_d-L_q)i_d{i}_q)$. Moreover, semiconductor failures in the inverter such as open-circuit or short-circuit faults alter the mapping between the control input $u={[v_d,v_q]}^{\top }$ and the actual applied voltages, thereby introducing nonlinear distortions that the monitoring system must recognize in real time. The detection of such faults typically relies on residual generation, where the measured $i_{abc}$ are compared against the expected behavior produced by the nominal model $\dot{x}=f(x,{\omega }_e)+Gu$, and the residual $r=i_{\mathrm{meas}}-i_{\mathrm{est}}$ is continuously evaluated. Small but systematic deviations in r are indicative of sensor bias, while abrupt changes are symptomatic of inverter faults or wiring issues. Another crucial dimension is thermal monitoring. Excessive temperatures in stator windings, permanent magnets, and semiconductor junctions are the primary cause of accelerated aging, insulation breakdown, and irreversible demagnetization. The main challenge is that direct sensing is feasible only at limited points, for instance with NTC thermistors on the winding heads or embedded diodes in IGBT modules, while the true critical temperatures occur in locations where sensors cannot be mounted, such as the midpoint of the winding or the MOSFET die. For this reason, thermal monitoring must combine direct measurements with reduced-order thermal models, typically of RC-network type, which approximate the heat conduction path as a set of differential equations of the form $C_{th}\dot{T}+G_{th}T=Q_{\mathrm{gen}}$, where $C_{th}$ and $G_{th}$ denote thermal capacitance and conductance matrices, and $Q_{\mathrm{gen}}$ is the vector of heat sources due to copper and iron losses in the motor and conduction and switching losses in the inverter. The mismatch between measured and estimated temperatures provides a basis for detecting cooling degradation or abnormal loss generation, and is fundamental for activating derating strategies that reduce current references to prevent catastrophic failure. Magnetic phenomena also pose monitoring problems of increasing importance. Permanent magnet synchronous machines suffer from partial demagnetization when subjected to elevated temperature or excessive negative $i_d$ current during field-weakening, and this modifies the effective value of ${\lambda }_m$ in the torque equation. Similarly, saturation effects alter the differential inductances $L_d,L_q$, leading to deviations between predicted and actual flux linkages. Monitoring must therefore track these parameters in real time, either via recursive estimation embedded in the control loop or through additional probing techniques. Failure to correctly detect magnetic degradation results in torque discrepancies, increased ripple, and eventually loss of efficiency and controllability. Furthermore, eccentricities in the rotor, misalignment, or bearing wear lead to harmonic distortions in the air-gap flux density, which propagate as specific frequency components in the measured currents; monitoring strategies must be capable of extracting these signatures and associating them with specific fault classes. The mechanical domain further complicates the monitoring task. Bearings, gears, and couplings are subject to wear and lubrication issues, producing characteristic vibration patterns. In traction drives, noise, vibration, and harshness (NVH) constraints are especially critical, as acoustic comfort is a competitive factor. Monitoring of vibrations using accelerometers and microphones, combined with signal processing tools such as Fast Fourier Transform (FFT) or wavelet decomposition, is indispensable not only for early fault detection but also for compliance with comfort standards. Mechanical faults interact with electrical quantities as well, since torque ripple and speed oscillations feedback into current harmonics, thus blurring the boundary between electrical and mechanical monitoring. Finally, the monitoring challenge extends to aging and degradation of ancillary components such as DC-link capacitors and power modules. Electrolytic capacitors degrade due to thermal stress, leading to reduced capacitance $C_{dc}\left(t\right)$ and increased equivalent series resistance (ESR), which affect the stability of the DC-link voltage $V_{dc}$. Semiconductor devices experience gradual shifts in threshold voltage and switching energy, which can be tracked by monitoring turn-on/off waveforms. These degradation mechanisms must be detected early enough to schedule maintenance or reconfiguration of the system. In summary, monitoring challenges in automotive PMSM/IPMSM drives span across electrical, thermal, magnetic, and mechanical domains, with strong couplings and interdependencies. The task is not only to measure but to interpret, diagnose, and anticipate failures. Mathematical models, observers, signal processing, and data-driven approaches must cooperate to provide a consistent picture of the drive health. In the next sections we shall classify and analyze advanced monitoring techniques, ranging from model-based observers to AI-driven anomaly detection and hybrid schemes, all aiming to transform monitoring from a passive sensing activity into an active PHM layer enabling fault diagnosis, prognostics, and fault-tolerant operation. In order to provide a structured overview,

Table 4. Summary of monitoring challenges in automotive PMSM/IPMSM drives.

Domain	Typical Failure Mechanisms	Monitoring Challenges
Electrical	Open-/short-circuit faults in inverter devices; sensor offsets or failures; phase disconnections	Distinguishing transient disturbances from permanent faults; reconstructing $i_d,i_q$ under sensor loss; rapid residual generation with minimal latency
Thermal	Overheating of stator windings; permanent magnet demagnetization; semiconductor junction overheating	Limited direct sensing of hot spots; need for reduced-order thermal models ($C_{th}\dot{T}+G_{th}T=Q_{\mathrm{gen}}$); integration of thermal observers for derating
Magnetic	Demagnetization of magnets; saturation of $L_d,L_q$; flux distortions due to eccentricity	Online estimation of ${\lambda }_m$, $L_d,L_q$; detection of harmonic signatures in currents; differentiation between normal flux-weakening and incipient demagnetization
Mechanical	Bearing wear and lubrication failure; rotor eccentricity; NVH issues	Extraction of vibration/frequency signatures; coupling of torque ripple and current harmonics; reliable fault isolation under varying load and road conditions
Component degradation	DC-link capacitor aging (capacitance drop, ESR increase); semiconductor degradation (threshold shifts, switching losses)	Tracking slow parameter drift over time; early identification of end-of-life; distinguishing aging trends from normal variations due to temperature

summarizes the main monitoring challenges in automotive PMSM/IPMSM drives across the different physical domains. Each category highlights the typical failure mechanisms, the associated measurable quantities, and the monitoring difficulties that justify the adoption of advanced techniques. The following subsections will expand on these issues, discussing in detail the available model-based, AI-based, and hybrid solutions.

As shown in Table 4, each physical domain presents specific challenges that cannot be solved by simple sensing alone. Electrical monitoring must discriminate between benign disturbances and actual hardware faults. Thermal monitoring faces the fundamental limitation that only a few accessible points can be instrumented, while the true critical hot spots must be estimated. Magnetic monitoring requires online parameter tracking to detect saturation or demagnetization before they manifest as torque discrepancies. Mechanical monitoring has to deal with the strong coupling between electrical signatures and vibrational phenomena, which complicates fault isolation. Finally, component degradation progresses slowly and subtly, requiring long-term trending and prognostic methods rather than instantaneous fault detection. These challenges motivate the adoption of advanced model-based observers, data-driven anomaly detectors, and hybrid approaches, which will be analyzed in the subsequent subsections.

4.1. Model-Based Monitoring Techniques

Model-based monitoring formalizes the supervision problem as the real-time consistency check between measured signals and a physics-informed model of the drive [114]. The architectural flow in Figure 12 clarifies the loop: raw measurements and event logs are pre-processed (synchronization, anti-aliasing, demodulation), fed to a bank of models, and transformed into residuals; the residuals drive alarms and advisories that translate into actions under safety constraints [115,116].

4.1.1. Unified Fault–Disturbance–Noise Setting

For formal analysis we adopt the standard linearized $dq$ representation around the current operating point (the linearization is local but can be gain-scheduled across the torque–speed map) [117,118]:

$$\dot{x}=Ax+Bu+E_{f}f+E_{d}d,y=Cx+D_{u}u+D_{f}f+v,$$

(38)

where $x\in {\mathbb{R}}^n$ collects electrical and, when needed, thermal states (e.g., winding/junction lumped temperatures), $u\in {\mathbb{R}}^m$ are the applied voltages (or their $\alpha \beta$ synthesis), $y\in {\mathbb{R}}^p$ are measured currents/voltages/temperatures, $f\in {\mathbb{R}}^{n_f}$ are fault signals (open/short switch, sensor bias, phase loss, cooling failure), d are unknown but bounded disturbances (load ripple, road-induced speed oscillations), and v is measurement noise. The matrices $E_f$ and $D_f$ encode how each fault acts on state and output; their columns are the fault signatures. Electrical/thermal couplings are naturally captured by augmenting x with a reduced RC network $C_{th}\dot{T}+G_{th}T=Q_{\mathrm{gen}}(x,u)$ and linearizing $Q_{\mathrm{gen}}$ with respect to $(i_d,i_q,{\omega }_m)$ [119].

4.1.2. Residual Generation and Disturbance Decoupling

A residual is any signal r that is ideally zero in healthy conditions and deviates in the presence of faults. We design a stable filter $\mathcal{Q}\left(s\right)$ that processes $(y,u)$:

$$r=\mathcal{Q}\left(s\right)\left[\begin{array}{c}y\\ u\end{array}\right]=\underset{\mathrm{fault}\mathrm{path}}{\underbrace{{\mathcal{G}}_f\left(s\right)}}f+\underset{\mathrm{disturbance}\mathrm{path}}{\underbrace{{\mathcal{G}}_d\left(s\right)}}d+\underset{\mathrm{noise}\mathrm{path}}{\underbrace{{\mathcal{G}}_v\left(s\right)}}v,$$

(39)

and aim at ${\mathcal{G}}_d\approx 0$ with ${\mathcal{G}}_f$ directionally selective [120]. In the parity-space approach, let $\Phi \left(s\right)=\left[\begin{array}{cc}C{(sI-A)}^{-1}B+D_u& I\end{array}\right]$ be the input–output map; any left annihilator $W\left(s\right)$ of $\Phi \left(s\right)$ ($W\left(s\right)\Phi \left(s\right)=0$) produces a residual $r=W\left(s\right)\left[\begin{array}{c}y\\ u\end{array}\right]$ insensitive to modeling-consistent behavior. Disturbance decoupling requires $W\left(s\right)\left[\begin{array}{c}C{(sI-A)}^{-1}{E}_d\\ 0\end{array}\right]=0$, while fault detectability entails $W\left(s\right)\left[\begin{array}{c}C{(sI-A)}^{-1}{E}_f+D_f\\ 0\end{array}\right]\ne 0$ for the faults of interest. When exact decoupling is impossible (typical with torque ripple or speed oscillations), we pose a weighted $H_{\infty }/H_2$ design [118,121]:

$$\underset{W\left(s\right)\mathrm{stable}}{min}{∥W\left(s\right){\mathcal{T}}_d\left(s\right)∥}_{\infty }\mathrm{s}.\mathrm{t}.{\sigma }_{min}\left(W\left(j\omega \right){\mathcal{T}}_f\left(j\omega \right)\right)\ge \eta ,\forall \omega ,$$

(40)

with ${\mathcal{T}}_d,{\mathcal{T}}_f$ the disturbance/fault transfer matrices and $\eta >0$ the guaranteed fault-to-residual gain; standard LMI synthesis yields $W\left(s\right)$.

4.1.3. Observer-Based Residuals and UIO

When state estimation is available, an innovation residual is

$$\dot{\widehat{x}}=A\widehat{x}+Bu+L(y-\widehat{y}),\widehat{y}=C\widehat{x}+D_{u}u,r=y-\widehat{y}.$$

(41)

In presence of unknown inputs d, an Unknown Input Observer (UIO) enforces $(I-LC)E_d=0$, yielding exact disturbance decoupling if $\mathrm{rank}\left(C{E}_d\right)=\mathrm{rank}\left(E_d\right)$ and $(A,E_d,C)$ satisfy the UIO existence conditions [115,117]. For thermal monitoring, augmenting (38) with RC states gives innovation residuals highly sensitive to cooling degradation (e.g., pump failure) while remaining robust to benign load variations [119].

4.1.4. Directional Residuals and Isolation

Figure 13 generalizes the architecture towards diagnosis and isolation. A normal model produces a global residual, while a bank of fault models $\left\{M_i\right\}$ or fault-directed filters $\left\{W_i\right\}$ generates directional residuals $r_i$ predominantly sensitive to fault $f_i$ [114,121]. Denoting by $R=\mathrm{diag}(r_1,\dots ,r_{n_f})$, we seek a signature matrix S with entries $S_{ij}={\parallel r_i\parallel }_{\mathcal{N}}$ under fault $f_j$ in a norm $\mathcal{N}$ (e.g., $L_2$ over a detection window). A well-designed bank yields a near-diagonal S, enabling isolation via $arg{max}_i{S}_{ij}$.

4.1.5. Statistical Decision and SPC

Residuals are stochastic processes due to v and modeling errors. Under the healthy hypothesis ${\mathcal{H}}_0$, the innovation of a Kalman filter is Gaussian with covariance ${\Sigma }_{\nu }$; a generalized likelihood ratio test (GLRT) decides

$$T_k={\nu }_k^{\top }{\Sigma }_{\nu }^{-1}{\nu }_k\underset{{\mathcal{H}}_0}{\stackrel{{\mathcal{H}}_1}{\gtrless }}\tau ,T_k\sim {\chi }_p^2\mathrm{under}{\mathcal{H}}_0,$$

(42)

with threshold $\tau$ chosen from a desired false-alarm probability. For incipient faults, sequential tests increase sensitivity: an EWMA chart computes $z_k=\lambda r_k+(1-\lambda )z_{k-1}$ and triggers when $|z_k|>{\tau }_{\mathrm{EWMA}}$, while a CUSUM statistic $S_k=max\{0,S_{k-1}+r_k-\kappa \}$ detects mean shifts with minimum delay. These SPC blocks, explicit in Figure 13, form the bridge between residuals and quality monitoring/alarms [115,120].

4.1.6. Multiple-Model Bayesian Isolation

The block diagram of Figure 14 emphasizes how expert knowledge, FMEA and design-of-experiments (DOE) shape the symptom and measurement selection, producing a database used to parametrize a normal model ${\mathcal{M}}_0$ and a set of fault models $\left\{{\mathcal{M}}_i\right\}$[114,121]. Let ${\mathcal{H}}_i$ denote the hypothesis “model ${\mathcal{M}}_i$ is true”. A recursive Bayesian isolator updates the posterior

$${\pi }_{k|k}^{\left(i\right)}={\displaystyle \frac{\ell \left(y_k\right|{\mathcal{M}}_i){\pi }_{k|k-1}^{\left(i\right)}}{{\sum }_j\ell \left(y_k\right|{\mathcal{M}}_j){\pi }_{k|k-1}^{\left(j\right)}}},\ell \left(y_k\right|{\mathcal{M}}_i)=\mathcal{N}\left(y_k;{\widehat{y}}_k^{\left(i\right)},{\Sigma }^{\left(i\right)}\right),$$

(43)

where ${\widehat{y}}_k^{\left(i\right)}$ is the model-i prediction. Isolation follows from $arg{max}_i{\pi }_{k|k}^{\left(i\right)}$, while decision confidence is quantified by entropy $-{\sum }_i{\pi }^{\left(i\right)}log{\pi }^{\left(i\right)}$ [114,115,120].

4.1.7. Adaptive Model Updating

Slow parameter drift (e.g., $R_s\left(T\right)$, $L_{d,q}\left({\omega }_m\right)$, thermal conductances) is handled by the “adaptive learning algorithm” block in Figure 13. A simple and effective choice is RLS with forgetting [117,119]:

$${\theta }_{k+1}={\theta }_k+K_k\left(y_k-{\varphi }_k^{\top }{\theta }_k\right),K_k={\displaystyle \frac{P_k{\varphi }_k}{\lambda +{\varphi }_k^{\top }{P}_k{\varphi }_k}},P_{k+1}={\lambda }^{-1}\left(P_k-K_k{\varphi }_k^{\top }{P}_k\right),$$

(44)

with regressor ${\varphi }_k$ built from $(i_d,i_q,{\omega }_m,u)$ and parameter vector $\theta$ collecting the linearized entries of $(A,B,C)$. The updated model keeps residuals centered under healthy operation (reducing false alarms) while preserving sensitivity to abrupt faults [119,121].

4.1.8. Thermal/Electrical Co-Monitoring

Coherent electrical–thermal monitoring improves separability of fault classes. Augment (38) with thermal states T and write the discrete RC model $T_{k+1}=A_{th}{T}_k+B_{th}{p}_k$, where $p_k$ stacks copper/iron and switching losses predicted from $(x_k,u_k)$ [119]. The joint observer

$${\left[\begin{array}{c}\widehat{x}\\ \widehat{T}\end{array}\right]}_{k+1}=\left[\begin{array}{cc}A& 0\\ B_{th}(\partial p/\partial x)& A_{th}\end{array}\right]{\left[\begin{array}{c}\widehat{x}\\ \widehat{T}\end{array}\right]}_k+\left[\begin{array}{c}B\\ B_{th}(\partial p/\partial u)\end{array}\right]u_k+L\left(y_k-{\widehat{y}}_k\right)$$

(45)

produces electrical residuals (i-innovation) and thermal residuals (T mismatch). A cooling-loss fault yields large thermal residuals with small electrical innovation; an open-phase fault produces the opposite pattern. Decision logic exploits this orthogonality for isolation [119,121].

4.1.9. Set-Membership/Interval Observers

When bounds rather than statistics are available, interval observers propagate reachable sets ${\mathcal{X}}_k$ under bounded $d,v$. If a new measurement $y_k$ implies ${\mathcal{X}}_k\cap {\mathcal{C}}^{-1}\left(y_k\right)=⌀$, a fault is declared. This deterministic approach is attractive for ISO 26262 assurance because it avoids distributional assumptions [114,122].

4.1.10. From Residuals to Actions

Finally, the “Advising” block in Figure 12 transforms decisions into derating or reconfiguration. If the posterior of an open-phase fault passes a confidence threshold, the inverter switches to two-phase operation; if thermal residuals exceed limits, the torque command is reduced according to a certified derating map. Importantly, all actions preserve the assumptions behind (38) (e.g., voltage feasibility) so that monitoring remains valid after the control reconfiguration [114,115,118,121]. In summary, the trio of Figure 12, Figure 13 and Figure 14 maps cleanly onto the formal machinery above: disturbance-decoupled residual synthesis (39) and (40), observer/UIO design for innovation-based residuals, directional banks for isolation with signature matrices, SPC/GLRT decision layers for timely alarms, Bayesian multi-model fusion informed by FMEA/DOE, and adaptive updating to track slow drift. This stack achieves high sensitivity to both abrupt electrical faults and gradual electro-thermal degradations while retaining interpretability, computational tractability, and certifiability in automotive embedded platforms [114,115,116,118,119,120,121,123].

4.2. AI/ML Models for Monitoring

Artificial-intelligence-based monitoring reframes the supervision problem in terms of statistical inference over multi-domain sequences of currents, voltages, temperatures, and vibrations [124,125]. Let a dataset $\mathcal{D}={\left\{(y_k,u_k)\right\}}_{k=1}^N$ be given, with $y_k\in {\mathbb{R}}^p$ the measurable outputs and $u_k\in {\mathbb{R}}^m$ the control inputs or operating conditions. The goal is to learn a parametric mapping $h_{\theta }$ that produces a score $s_k=h_{\theta }(y_{1:k},u_{1:k})$ and a decision rule $s_k\gtrless \tau$ for anomaly detection, fault classification, or prognostics (RUL estimation). It is useful to distinguish between discriminative models that learn a boundary of normality, generative models that reconstruct signals or likelihoods, and sequential models that predict degradation trajectories, each with specific training objectives, computational complexity, and feasibility for real-time execution. In the case of one-class Support Vector Machines (OC-SVM) for anomaly detection, the aim is to approximate the normality manifold without fault labels [126]. The primal problem is

$$\underset{w,\rho ,\xi }{min}{\displaystyle \frac{1}{2}}{\parallel w\parallel }^2+{\displaystyle \frac{1}{\nu N}}\sum _{i=1}^N{\xi }_i-\rho \mathrm{s}.\mathrm{t}.w^{\top }\varphi \left({\xi }_i\right)\ge \rho -{\xi }_i,{\xi }_i\ge 0,$$

where ${\xi }_i=\varphi (y_i,u_i)$ are features (e.g., Park vector harmonics or STFT coefficients), $\nu \in (0,1]$ controls the fraction of support vectors, and the decision function is $f\left(\xi \right)=\mathrm{sign}(w^{\top }\varphi \left(\xi \right)-\rho )$. The dual problem is a quadratic program scaling as $O\left(N^3\right)$, but inference cost is $O\left(S\right)$ with S support vectors; once trained, OC-SVMs run efficiently even on DSPs or MCUs. Features are typically extracted through short-time Fourier transforms or continuous wavelet transforms, providing robust time–frequency representations for fault detection. Autoencoders extend the anomaly detection paradigm by learning a latent representation $z=E_{\theta }\left(y\right)$ and reconstructing $\widehat{y}=D_{\theta }\left(z\right)$, minimizing

$$\underset{\theta }{min}{\displaystyle \frac{1}{N}}\sum _{k=1}^N{\parallel y_k-D_{\theta }\left(E_{\theta }\left(y_k\right)\right)\parallel }_2^2,$$

with residual score $s_k=\parallel y_k-{\widehat{y}}_k\parallel$ indicating deviations from healthy operation. Variational autoencoders (VAE) provide probabilistic anomaly scores, enabling likelihood-based thresholding. Compact autoencoders with INT8 quantization are deployable on embedded DSPs within 10–$50\mu$s, while deeper VAEs are better suited for offline diagnostics [127]. Convolutional Neural Networks (CNNs) are primarily used for supervised fault classification from spectrograms or wavelet maps of current and vibration signals [128,129]. Their classification accuracy is excellent when labeled fault data are available, covering open-phase, inverter switch failures, or bearing defects. Inference is highly parallelizable on FPGAs, making CNNs attractive for real-time diagnosis provided preprocessing (FFT or CWT) is optimized. Recurrent Neural Networks (RNNs) and their gated variants (LSTM, GRU) are natural candidates for degradation modeling and RUL estimation [127]. Compact pruned variants can run online with short horizons, but full-size RNNs are typically employed offline for long-term degradation trend modeling. Transformer architectures extend this to long-range dependencies with self-attention, promising for multi-domain monitoring (electro-thermal–mechanical), but currently limited to offline analysis and digital twin frameworks due to complexity [130]. Probabilistic AI models provide explicit uncertainty quantification. Gaussian Process Regression (GPR) with kernel $k(·,·)$ yields posterior mean and variance, but complexity $O\left(N^3\right)$ in training limits scalability. Bayesian Neural Networks (BNN) provide posterior distributions over weights and predictive intervals, useful for certifiable safety monitoring [131], but their runtime makes them candidates for offline analysis. Overall, lightweight models such as OC-SVM, compact autoencoders, and CNNs can be integrated on DSP/FPGA controllers for online monitoring within the control loop, while more complex models (RNNs, Transformers, GPR, BNN) are reserved for offline prognostics, digital twin analyses, and cloud-based PHM frameworks, possibly distilled into simpler models for deployment [124,125]. These considerations are summarized in

Table 5. AI/ML models for monitoring in automotive PMSM/IPMSM drives: objectives, use, and on-board feasibility.

Model	Mathematical Training Objective	Primary Monitoring Use	Complexity (Train/Infer)	On-Line Suitability (DSP/FPGA/MCU)	Off-Line Suitability & Notes
OC-SVM [126]	${min}_{w,\rho ,\xi }\frac{1}{2}{\parallel w\parallel }^2+\frac{1}{\nu N}\sum {\xi }_i-\rho$; decision $w^{\top }\varphi \left(\xi \right)\gtrless \rho$	Anomaly detection from healthy data; Park/STFT features	QP $O\left(N^3\right)$/$O\left(S\right)$ with S SV	High: kernel-sparse, light inference	High: training on large datasets
Autoencoder [127]	${min}_{\theta }\frac{1}{N}\sum {\parallel y-D\left(E\left(y\right)\right)\parallel }^2$ (+ denoising/contractive variants)	Reconstruction-based anomaly detection	SGD training/$O\left(C\right)$ MAC inference	Medium–High: shallow quantized AE feasible	High: deep AE or VAE for probabilistic AD
VAE [127]	$max{\mathbb{E}}_{q\left(z\right|y)}[logp\left(y\right|z)]-\mathrm{KL}(q\parallel p)$	Likelihood-based AD with uncertainty quantification	Variational inference/$O\left(C\right)$	Medium: heavier than AE	High: offline probabilistic diagnostics
CNN [128,129]	Cross-entropy loss on spectrogram/CWT inputs; conv cost $O\left(HW{C}_{in}{k}^2{C}_{out}\right)$	Supervised fault classification (inverter, bearing, phase faults)	High throughput/parallelizable	High: efficient on FPGA/DSP with FFT pre-proc.	High: dataset-driven analysis
RNN (LSTM/GRU) [127]	$min\sum \parallel y_{t+1}-{\widehat{y}}_{t+1}{\parallel }^2$ (+ survival/RUL loss)	Degradation trend modeling, RUL estimation	$O\left(T{d}^2\right)$/step $O\left(d^2\right)$	Medium: pruned, short-window configs	Very High: prognostics and long-term RUL
Transformer [130]	Cross-entropy or likelihood with attention $O\left(T^{2}d\right)$	Multi-domain integrated monitoring	Very high/high memory	Low: impractical on embedded	Very High: offline analysis, digital twin
GPR [131]	Posterior mean/var with kernel inversion	UQ, degradation regression	$O\left(N^3\right)$/$O\left(N\right)$ (sparse $O\left(Nm\right)$/$O\left(m\right)$)	Low: too slow for real-time	High: uncertainty calibration, safety cases
BNN [124,125]	ELBO: ${\mathbb{E}}_{q\left(w\right)}[logp\left(y\right|x,w)]-\mathrm{KL}(q\parallel p)$	Uncertainty-aware anomaly detection, RUL	High/medium–high (MC approx.)	Low–Medium: MC-dropout variants feasible	High: certifiable safety, calibration

.

4.3. Hybrid Monitoring Techniques

While model-based and AI-based monitoring provide complementary strengths, the most promising strategies for safety-critical electric drives arise from their integration into hybrid approaches. The rationale is straightforward: model-based observers yield interpretable residuals derived from physics-informed equations, whereas AI models excel at detecting complex nonlinear patterns and fusing heterogeneous data. Combining the two allows the residuals to be treated not as end signals but as inputs to machine learning classifiers or regressors, thereby creating a multi-layered detection framework with enhanced robustness [132,133]. Digital twin frameworks represent a particularly relevant instantiation of this integration, enabling synchronization between physics-based reduced-order models and data-driven correctors [133,134,135]. Formally, consider the fault–disturbance–noise representation

$$\dot{x}=Ax+Bu+E_{f}f+E_{d}d,y=Cx+D_{u}u+D_{f}f+v,$$

and let $\widehat{y}\left(t\right|\theta )$ be the output of a parameterized model (e.g., an observer or reduced-order thermal RC model). The residual $r\left(t\right)=y\left(t\right)-\widehat{y}\left(t\right|\theta )$ is classically evaluated against thresholds. In a hybrid framework, instead of simple thresholding, one constructs a feature vector

$$\zeta \left(t\right)=\varphi \left(r\left(t\right),u\left(t\right),y\left(t\right)\right),$$

where $\varphi$ includes nonlinear embeddings such as wavelet coefficients of r, statistics over sliding windows, or projections onto parity-space residuals. This $\zeta \left(t\right)$ is then fed into a classifier $g_{\theta }\left(\zeta \right)$, such as an autoencoder, CNN, or ensemble model, which maps $\zeta$ to fault probabilities or health indices [126,129,136]. The training objective is

$$\underset{\theta }{min}\sum _{k=1}^N\ell \left(g_{\theta }\left({\zeta }_k\right),{\mathrm{label}}_k\right),$$

where ℓ is a cross-entropy or contrastive loss depending on the availability of labeled data. This architecture leverages the interpretability of physics-based residuals while exploiting the discriminative power of data-driven models [125,130]. Another powerful class of hybrid approaches relies on digital twins, where a high-fidelity physical model ${\mathcal{M}}_{phys}$ is continuously updated with streaming sensor data through data-driven correctors ${\mathcal{M}}_{AI}$. The overall state estimate is given by

$$\widehat{x}\left(t\right)=arg\underset{x}{min}\left\{{\parallel y\left(t\right)-Cx\parallel }^2+\lambda \parallel x-{\mathcal{M}}_{phys}{\left(u\left(t\right)\right)\parallel }^2+\mu {\parallel x-{\mathcal{M}}_{AI}(y_{1:t},u_{1:t})\parallel }^2\right\},$$

so that the state balances consistency with the physics, plausibility with the AI predictor, and fidelity with the measurements. This formulation generalizes classical Kalman filtering: the AI model effectively acts as an adaptive prior compensating for modeling errors such as saturation or temperature-dependent drift. Recent works have applied digital twins to PMSM parameter estimation [137], health status monitoring of rectifiers [138], open-circuit fault diagnosis of multilevel converters [139], and virtual thermal sensing in power electronics [140]. Broader surveys highlight their role in predictive maintenance and PHM across power electronic systems [135,141]. Physics-informed neural networks (PINNs) offer another hybrid paradigm, embedding the governing equations of the drive directly into the loss function of the network [141]. Given differential equations $\dot{x}=f(x,u,\theta )$, one trains a neural approximation ${\widehat{x}}_{\theta }$ by minimizing

$$\mathcal{L}\left(\theta \right)=\sum _k\parallel y_k-{\widehat{y}}_{\theta }\left(t_k\right){\parallel }^2+\alpha \sum _k{\parallel {\dot{\widehat{x}}}_{\theta }\left(t_k\right)-f({\widehat{x}}_{\theta }\left(t_k\right),u_k)\parallel }^2,$$

so that the network respects both measured data and the underlying physics. In monitoring applications, PINNs can estimate latent variables such as magnetic flux or junction temperatures, which are difficult to sense directly, while simultaneously being constrained by the machine equations. The hybrid approach also extends naturally to prognostics and health management. Model-based observers provide physics-consistent residuals $r\left(t\right)$ and estimated parameters $\widehat{\theta }\left(t\right)$ (e.g., winding resistance, inductance, thermal conductance). AI models then learn temporal mappings from $\widehat{\theta }\left(t\right)$ and $r\left(t\right)$ to health indices $h\left(t\right)$ and RUL predictions. For instance, a recurrent network may model

$$h_{k+1}=\sigma (W{h}_k+U{\widehat{\theta }}_k+V{r}_k+b),$$

with $h_k\in [0,1]$ representing a normalized health state. The AI component captures nonlinear trends, while the observer guarantees that only physically meaningful variables are passed to the network, improving generalization and interpretability. Such hybrid prognostic strategies are being actively investigated for electro-thermal aging detection in automotive SiC powertrains [142] and energy consumption forecasting in sustainable transport platforms [143]. From an implementation viewpoint, hybrid monitoring balances the computational constraints of embedded systems with the flexibility of AI. Residual generation via observers or parity relations is lightweight, typically $O\left(n^2\right)$ per time step. Feeding residual-based features to compact classifiers such as shallow autoencoders or kernel machines preserves online feasibility. More complex models (e.g., RNNs, Transformers) can be trained offline using residual histories and deployed in reduced distilled form for online monitoring. This division of labor between physics-based online filters and data-driven offline learners is key to achieving certifiable real-time performance under ISO 26262 constraints [144]. In summary, hybrid monitoring techniques bridge the gap between model-based interpretability and AI-based adaptability. By embedding physics in the residual generation, constraining learning with governing equations, or fusing digital twins with adaptive correctors, they deliver robust detection of both abrupt and incipient faults. Moreover, they enable prognostic estimation across electrical, thermal, and mechanical domains, all while maintaining computational feasibility for real-time automotive drives. Hybrid monitoring patterns combine physics-based residual generation with learning blocks that classify, fuse, or regularize information across domains. The most recurrent patterns can be summarized through their mathematical cores, primary monitoring objectives, and deployability on embedded automotive hardware. Residuals+ML treat physics-based residuals as features for a lightweight classifier or regressor; digital twins fuse a high-fidelity model with data-driven correctors by solving a constrained estimation problem; physics-informed neural networks (PINNs) embed the governing equations directly into the learning loss; gray-box state-space learning identifies compact parametric models regularized by physics, and safety-shielded learning projects learned decisions through certified barrier or feasibility layers to ensure invariants.

In practice, residuals+ML are the most deployable option for strict real-time budgets because residual generation is $O\left(n^2\right)$ and the classifier can be kept shallow and quantized; digital twins excel when latent electro-thermal states must be estimated online with physical coherence, while heavier correctors are trained offline; PINNs and gray-box state-space models complement each other, the former enforcing equation satisfaction directly in the loss, the latter yielding compact observers with reliable innovation statistics for SPC; residuals combined with pruned RNNs provide on-board short-horizon prognostics, whereas long-horizon RUL requires offline training on fleet data; finally, safety-shielded layers and Bayesian fusion add certifiable guarantees and calibrated confidence without inflating the runtime footprint. See

Table 6. Hybrid monitoring architectures: mathematical core, objectives, and feasibility for on-board use.

Hybrid Pattern	Mathematical Core (Training/Inference)	Primary Monitoring Objective	Complexity (On-Board)	Online Suitability	Offline Role/Notes
Residuals + ML classifier [126,129,136]	Residuals $r\left(t\right)=\mathcal{Q}\left(s\right)[y;u]$; features $\zeta =\varphi (r,u,y)$; learn $g_{\theta }:\zeta \mapsto$ fault prob. with ${min}_{\theta }\sum \ell (g_{\theta }\left({\zeta }_k\right),{\mathrm{label}}_k)$	Fast fault detection/isolation using physics-informed features	Low–Medium (AE/CNN-lite or OC-SVM on $\zeta$)	High: fits $T_s=25$–$100\mu$s with quantized models	Offline labeling/tuning; robust to domain shift if $\varphi (·)$ encodes physics
Digital Twin (model+corrector) [134,135,137,138,139,140,141]	State fusion $\widehat{x}\left(t\right)=arg{min}_x{\parallel y-Cx\parallel }^2+\lambda \parallel x-{\mathcal{M}}_{phys}{\left(u\right)\parallel }^2+\mu {\parallel x-{\mathcal{M}}_{AI}(y_{1:t},u_{1:t})\parallel }^2$	Joint estimation of latent states (e.g., flux, temperatures) and health indices	Medium (Gauss–Newton or EKF with AI prior)	Medium: feasible with RO models and sparse updates	High-fidelity simulation, parameter calibration, what-if analysis
Physics-Informed NN (PINN) [141]	Loss $\mathcal{L}=\sum \parallel y-{\widehat{y}}_{\theta }{\parallel }^2+\alpha \sum {\parallel {\dot{\widehat{x}}}_{\theta }-f({\widehat{x}}_{\theta },u)\parallel }^2$; automatic differentiation for PDE/ODE residuals	Estimation of hard-to-measure states under physics constraints; anomaly scoring via physics residual	Medium–High (AD + MLP/CNN)	Medium: deploy shallow PINNs or distilled surrogates	Heavy training offline; excellent interpretability via residual terms
Gray-box state-space learning [144,145]	Identify $\dot{x}=A_{\theta }x+B_{\theta }u$, $y=C_{\theta }x$ with physics priors $\theta \in \Theta$; objective ${min}_{\theta }\sum {\parallel y-{\widehat{y}}_{\theta }\parallel }^2+\beta R\left(\theta \right)$	Compact parametric models robust to drift; coherent residuals for SPC/FDI	Low (observer+update)	High: EKF/UIO with slow online adaptation	Offline identification, regularization from datasheets/FEA
Residuals + RNN (prognostics) [142,143]	Sequence model $h_{k+1}=\sigma (W{h}_k+U{\widehat{\theta }}_k+V{r}_k+b)$; RUL loss (survival/ranking)	Health index and RUL from residual/parameter trends	Medium (pruned GRU/LSTM)	Medium: short windows with pruning	Long-horizon training offline; transfer learning across fleets
Safety-shielded learning [132,133]	Learned policy/score ${\pi }_{\theta }$ projected via safety layer $\tilde{a}={\Pi }_{\mathcal{U}}\left({\pi }_{\theta }\right)$; barrier condition $\dot{h}(x,u)\ge -\alpha h\left(x\right)$ enforced	Constraint-aware alarms/decisions with formal invariants	Low–Medium (projection + barrier check)	High: adds small overhead to ML block	Offline design of barrier/feasibility sets; aids ISO 26262 arguments
Residuals + Bayesian ML [125,130]	Residual bank $\left\{r_i\right\}$ with Bayesian fusion ${\pi }^{\left(i\right)}\propto \ell \left(y\right|{\mathcal{M}}_i){\pi }^{\left(i\right)}$; or BNN on $\zeta$ with ELBO training	Probabilistic FDI with calibrated confidence/uncertainty	Medium–High (BNN/GPR lite)	Medium: MC-dropout acceptable; full BNN off-line	Uncertainty calibration; threshold setting with risk metrics

for a concise summary on hybrid methods.

4.4. Comparative Discussion: Model-Based vs. AI-Based vs. Hybrid

A principled comparison among model-based, AI-based, and hybrid monitoring can be framed in terms of (i) interpretability and reliability of the decision process, (ii) computational and memory footprint subject to real-time constraints, and (iii) integrability on automotive-grade embedded platforms. Let a monitoring pipeline output a decision ${\delta }_k\in \{\mathrm{healthy},{\mathrm{fault}}_1,\dots \}$ based on a score $s_k$ and threshold $\tau$. We formalize performance via a multi-criteria index

$$\mathcal{J}=\underset{\mathrm{accuracy}}{\underbrace{\alpha \mathbb{P}(\delta =\mathrm{correct})}}-\underset{\mathrm{detection}\mathrm{delay}}{\underbrace{\beta \mathbb{E}\left[T_D\right]}}-\underset{\mathrm{nuisance}}{\underbrace{\gamma \mathbb{P}\left(\mathrm{false}\mathrm{alarm}\right)}}-\underset{\mathrm{uncertainty}\mathrm{miscalibration}}{\underbrace{\zeta {\mathrm{U}}_{\mathrm{uncal}}}}-\underset{\mathrm{runtime}\cos \mathrm{t}}{\underbrace{\xi \mathcal{C}}},$$

where ${\mathrm{U}}_{\mathrm{uncal}}$ quantifies probability calibration error (e.g., Expected Calibration Error of predicted fault probabilities), and $\mathcal{C}$ aggregates compute latency and memory footprint under worst-case execution. Interpretability can be expressed through the existence of a physics-grounded residual map $r=\mathcal{Q}\left(s\right)[y;u]$ with directional sensitivity (signature matrix nearly diagonal) and a transparent decision rule (GLRT/SPC) or, in data-driven settings, through feature attributions consistent with physical invariants. Reliability is articulated as a combination of statistical risk $R\left(\tau \right)$, robustness to domain shift, and availability of formal guarantees such as disturbance decoupling or uncertainty bounds. Embedded feasibility is tied to upper bounds on inference latency $L_{max}$, code/data size M, and determinism (bounded jitter), typically constrained by $T_s\in [25,100]\mu$s for current loops. Model-based monitoring maximizes interpretability and affords formal guarantees via parity relations, unknown-input observers, and $H_{\infty }$-optimized residuals. It offers statistical decision layers with known null distributions (e.g., ${\chi }^2$ for Kalman innovation), enabling precise false-alarm control. Its limitations arise under unmodeled nonlinearities and parameter drift; adaptive identification mitigates these at modest cost. AI-based monitoring excels at nonlinear pattern discovery and multi-domain fusion; probabilistic variants (GPR/BNN) provide uncertainty quantification, though often at a runtime cost incompatible with tight deadlines. Hybrid strategies inherit the interpretability of physics-based residuals and the expressivity of learned components, enabling calibrated decisions on top of meaningful features and achieving a favorable Pareto balance between accuracy, robustness, and real-time feasibility.

For a more operational view, we summarize typical computational envelopes and recommended placements. Latency ranges refer to end-to-end monitoring (residual generation + inference) and assume fixed-point or INT8 inference where applicable; memory includes model parameters and lookup structures.

To frame the alternatives,

Table 7. Formal comparison across interpretability/reliability and embedded feasibility.

Approach	Interpretability & Guarantees	Reliability & Uncertainty	Runtime & Memory	Embedded Integrability
Model-based	Physics-grounded residuals $r=\mathcal{Q}\left(s\right)[y;u]$; parity $W\left(s\right)\Phi \left(s\right)=0$; UIO $(I-LC)E_d=0$; $H_{\infty }$ bounds on $d\to r$; GLRT with known null (${\chi }^2$)	High reliability under modeled regimes; robustness via decoupling; statistical false-alarm control; limited under severe model mismatch without adaptation	Low–medium compute ($O\left(n^2\right)$ per step); small memory; deterministic	Excellent: fits $T_s$ with margin; ISO 26262-friendly due to traceable logic
AI-based	Learned feature maps and decision rules; post hoc saliency/attribution; physics not explicit unless enforced	High accuracy and domain fusion; UQ via GPR/BNN; calibration needed to avoid optimistic probabilities; sensitive to domain shift without adaptation	Medium–very high depending on model (AE/CNN lite → feasible; RNN/Transformer/GP → heavy)	Mixed: compact AE/CNN feasible; heavy models typically offline/cloud
Hybrid	Physics residuals as features; digital twin fusion; PINN losses enforce equations; Bayesian fusion over fault models	Robustness from physics + adaptability from learning; calibrated UQ on residual features; graceful degradation under drift	Low–medium online (residuals + light classifier); heavy training offline; twin updates sparse	Very good: real-time residuals + quantized ML; good certification narrative

provides a synoptic comparison of model-based, AI-based, and hybrid approaches across four axes: interpretability and guarantees, reliability and uncertainty, runtime and memory, and embedded integrability. Complementarily,

Table 8. Implementation envelopes and recommended deployment.

Method Family	Typical Latency (per Step)	Typical Memory Footprint	Recommended Role	Notes on Certification
Model-based (parity/UIO/$H_{\infty }$ + SPC)	5–$25\mu$s (observer+filter) + <5 $\mu$s (GLRT/EWMA)	<200 kB (filters, gains, RC models)	On-line in current/thermal loops	Strong traceability; analytical guarantees
AI (AE/CNN compact)	10–$50\mu$s (INT8; <$1$M MAC)	0.2–2 MB (quantized)	On-line anomaly/diagnosis on features	Needs calibration; deterministic scheduling
AI (RNN/LSTM pruned)	40–$150\mu$s (short window)	0.5–4 MB	Near on-line or slow loop; short-horizon prognostics	Document pruning/quantization; WCET analysis
AI (Transformer/GPR/BNN full)	>$0.5$–5 ms	5–100 MB	Off-line/cloud PHM, fleet analytics	Use for threshold setting and UQ; distill to light models
Hybrid (residuals+ML)	5–$25\mu$s (residuals) + 5–$20\mu$s (ML)	0.3–2 MB	On-line FDI with calibrated confidence	Clear physics–ML boundary aids ISO 26262
Hybrid (digital twin/PINN)	20–$200\mu$s (EKF/GN step)	0.5–5 MB	On-line latent-state estimation; off-line training	Log model residuals for audits; sparse updates

summarizes the implementation envelopes (per-step latency and memory footprint), recommended deployment roles, and certification notes, offering practical guidance for both online and offline adoption.

Putting the pieces together, model-based monitoring should be the backbone of the on-board layer due to interpretability, deterministic timing, and explicit guarantees; AI-based modules add sensitivity and multi-domain fusion where labeled data exist and runtime budgets allow; hybrid integration provides the best Pareto frontier by feeding physics-coherent residuals into lightweight classifiers, constraining learning with governing equations (PINN), or fusing high-fidelity twins with adaptive correctors. This stratification also supports certification: the safety case can anchor on analytical properties of the model-based core, while learned components are sandboxed, calibrated, and—when necessary—distilled and shielded by feasibility or barrier layers to maintain invariants under real-world uncertainty. To make the comparative assessment actionable,

Table 9. Operational recipes for typical faults: sensors/features, residual design, AI model, decision logic, and deployment.

Fault/Phenomenon	Sensors & Features	Physics-Based Residual (Example)	AI Features/Model	Decision Logic (Example)	Recommended Deployment
Open-phase (stator)	Phase currents $i_{a,b,c}$, DC-link $v_{dc}$; Park-vector harmonics; STFT of $i_{\alpha }+i_{\beta }$	Parity residual $r=W\left(s\right)[y;u]$ with $W\Phi =0$; UIO s.t. $(I-LC)E_d=0$; directional residual $r_A$ sensitive to phase-A loss	$\zeta =[\parallel r_A{\parallel }_{L_2},\mathrm{STFT}\mathrm{mag}.\mathrm{bins}]\to$ OC-SVM or shallow AE	GLRT on innovation $T_k={\nu }^{\top }{\Sigma }_{\nu }^{-1}\nu \gtrless \tau$; SPC CUSUM on $r_A$; isolation via signature matrix S	On-line: residuals + OC-SVM/AE INT8
Inverter switch short (IGBT/MOSFET)	$i_{a,b,c}$ at high rate; $v_{dc}$ ripple; switching node voltage (if available)	Residual on switched model: compare predicted SVPWM voltage vs. measured; fast innovation $r_v=y_v-{\widehat{y}}_v$	CNN-lite on high-rate windowed $[r_v,i_{\alpha },i_{\beta }]$ or spectral peaks	Two-stage: EWMA on $r_v$ for detection; CNN-lite for confirmation; inhibit re-try & trigger fault-tolerant mode	On-line: residual + CNN-lite (FPGA/DSP)
Cooling loss (pump/fan failure)	Winding and case temperatures $T_w,T_c$; inverter junction estimator ${\widehat{T}}_j$; load/ambient	Thermal RC observer $C_{th}\dot{T}+G_{th}T=Q_{\mathrm{gen}}(i,u)$; residual $r_T=T-\widehat{T}$, disturbance-decoupled via UIO	$\zeta =[r_T,{\dot{r}}_T,{\widehat{Q}}_{\mathrm{gen}}]\to$ AE for incipient drift	SPC EWMA on $r_T$; threshold from ${\chi }^2$ of thermal innovation; derating map when $r_T$ exceeds band	On-line: observer+EWMA; AE optional
Bearing defect/mechanical vibration	Accelerometers on housing; order-tracked spectra; torque ripple estimate $\tilde{\tau }$	Electromech. residual: mismatch between current harmonics and torque ripple model; $r_{\tau }=\tilde{\tau }-\widehat{\tau }\left(i\right)$	CNN on order-maps (CWT/Order-STFT); or OC-SVM on cepstral features	Joint test: SPC on $r_{\tau }$ + classifier posterior $>{\tau }_p$; confirmation with speed-order coherence	Near on-line: classifier at slower rate; residual on-line
Partial demagnetization (PM)	$i_{d,q}$, ${\widehat{\lambda }}_m$ from observer; FW schedule	Residual $r_{\lambda }={\widehat{\lambda }}_m-{\lambda }_m^{\mathrm{map}}(T,\omega )$; parity filter to reject load disturbances	RNN-short (pruned GRU) on trend of $r_{\lambda }$ + temperature; or AE on flux–current locus	SPC trend (CUSUM) on $r_{\lambda }$; RNN score for incipient demag; revise FW limits when confirmed	Hybrid: residual on-line, RNN pruned near on-line
DC-link capacitor aging (ESR increase/C drop)	$v_{dc}$ ripple spectrum; charge/discharge transients; thermal sensor near cap	RC meta-model of DC-link: predict ripple from $i_{dc}$ and compare; residual $r_{dc}=v_{dc}-{\widehat{v}}_{dc}$	GPR-sparse or AE on $(r_{dc},i_{dc},T)$ for aging index	SPC on $r_{dc}$ variance; offline UQ to set alarm bands; schedule maintenance by RUL trend	Hybrid: residual on-line, AE/GPR offline for RUL

enumerates representative faults with concrete sensor/feature choices, physics-based residuals, lightweight AI models, and decision logic suitable for embedded deployment. Each recipe is meant to be directly instantiated and tuned within the proposed monitoring stack.

4.5. Battery Monitoring and Interaction with Drive Monitoring

The monitoring of the high-voltage battery cannot be considered in isolation, as its operation directly influences the performance and reliability of the entire electric drive system, and vice versa. While classical approaches treat the Battery Management System (BMS) and the Drive Monitoring System as separate layers, recent methodologies emphasize their bidirectional coupling, particularly under dynamic automotive conditions [146,147].

From a formal perspective, let the battery state vector be

$$x_b={[z,T_{\mathrm{cell}},{\theta }_{\mathrm{SOH}}]}^{\top },$$

where z is the state of charge (SOC), $T_{\mathrm{cell}}$ the average cell temperature, and ${\theta }_{\mathrm{SOH}}$ a set of degradation indicators (e.g., internal resistance growth, capacity fade). The electric drive dynamics are characterized by the machine–inverter state vector

$$x_d={[i_d,i_q,{\omega }_m,T_j]}^{\top },$$

where $i_d,i_q$ are the dq-axis currents, ${\omega }_m$ the mechanical speed, and $T_j$ the semiconductor junction temperature. The coupling between the two subsystems can be expressed through the DC-link power balance:

$$P_{\mathrm{batt}}=v_b{i}_b=P_{\mathrm{inv}}+P_{\mathrm{loss}}+C_{\mathrm{dc}}{\displaystyle \frac{d{v}_{\mathrm{dc}}}{dt}},$$

(46)

where $P_{\mathrm{batt}}$ is the instantaneous battery power, $P_{\mathrm{inv}}$ the inverter output power towards the machine, $P_{\mathrm{loss}}$ the conversion and conduction losses, and $C_{\mathrm{dc}}$ the DC-link capacitance. Equation (46) shows that any monitoring error in the estimation of $v_b$ or $i_b$ directly propagates to the assessment of inverter stress and motor torque capability. During high-power transients, battery monitoring determines the admissible discharge power

$$P_{\mathrm{dis}}^{max}(z,T_{\mathrm{cell}}),$$

which constrains the maximum torque reference of the drive. Conversely, monitoring of drive-side variables such as $i_d,i_q$ and $T_j$ is used to predict current ripples and thermal overloads that reflect back into higher battery RMS current and heat generation. In regenerative braking, the condition

$$0\le P_{\mathrm{regen}}\le P_{\mathrm{chg}}^{max}(z,T_{\mathrm{cell}})$$

(47)

must hold, linking the braking torque demand to the instantaneous charging capability of the cells. If this inequality is violated due to monitoring inaccuracies, the drive controller may inject currents that the battery cannot safely absorb, leading to overvoltage or lithium plating. From a monitoring design standpoint, the integration of the two domains is achieved by embedding battery state observers into the drive supervisory layer. For example, an extended state observer can be formulated as

$$\dot{\widehat{x}}=f_d(\widehat{x},u)+L_d(y_d-{\widehat{y}}_d)+{\Gamma }_b{\widehat{x}}_b,$$

where $f_d$ is the drive model, $y_d$ the measured drive outputs, and ${\Gamma }_b$ a coupling term updated from battery monitoring. This structure allows the drive controller to be aware of the battery’s instantaneous health state and to adapt torque commands accordingly. Conversely, the BMS incorporates drive-side residuals such as harmonic distortion in $i_d,i_q$ or abnormal inverter switching losses as early indicators of excessive current demand or unbalanced regenerative events [148]. In summary, battery monitoring and drive monitoring form a coupled system where the accuracy of one directly impacts the reliability of the other. By enforcing power balance constraints (46), saturation limits on $P_{\mathrm{dis}}^{max}$ and $P_{\mathrm{chg}}^{max}$, and by cross-feeding residuals between observers, it is possible to design a unified monitoring layer. This co-design approach enhances safety (avoiding overcurrent or overvoltage), improves efficiency (optimizing energy recovery in regenerative braking), and extends lifetime (reducing thermal and electrochemical stress), thus aligning monitoring strategies with both functional safety and sustainability objectives in automotive electric drives [149,150].

5. Concluding Remarks and Future Trends

This review has analyzed the state of the art in modeling, control, and monitoring of automotive power drive systems, showing that while each of these domains has reached a high level of maturity individually, their integration remains limited. Existing techniques for multiphysics modeling provide accurate representations of electromagnetic and thermal behavior, yet are seldom exploited beyond the design phase. Control strategies range from mature field-oriented and direct torque control to advanced predictive and adaptive schemes, but monitoring considerations are rarely embedded into their formulation. Likewise, monitoring frameworks are rich in methods—spanning signal-based, model-based, and AI-based approaches—yet they typically operate in parallel to control rather than as part of a unified architecture. This fragmentation highlights both the progress achieved and the limitations that still need to be overcome. In addition, the analysis underlined how energy storage systems and their monitoring strongly influence drive performance and sustainability. Different battery chemistries exhibit diverse capabilities in terms of energy density, power acceptance, and regenerative braking, with direct impact on efficiency and environmental footprint. Bidirectional converters further enable controlled energy flow between storage and drive, making their modeling and monitoring essential for safe and efficient integration. Looking forward, several trends are expected to shape the evolution of electric drive systems:

• Model–control–monitoring integration. Physics-based models will increasingly be shared across design, control, and monitoring layers to ensure consistency and certification traceability.
• Hybrid and physics-informed AI. Data-driven methods will be combined with physics-based models to achieve adaptability with interpretability and safety guarantees.
• Constraint-aware and multi-domain control. Controllers such as MPC and RL will embed electro-thermal and NVH objectives, with design methodologies becoming more systematic.
• Battery–drive co-monitoring. Joint supervision of SOC, SOH, SOP and drive-side variables will ensure safe charging/discharging, optimize regenerative braking, and extend lifetime.
• Scalable prognostics and fleet learning. Fleet-level data will train degradation and RUL models, with lightweight on-board implementations for real-time monitoring.
• Functional safety and certification. ISO 26262 compliance will require hybrid schemes combining formal guarantees with AI under safety envelopes.
• Real-time feasibility. Wide-bandgap devices and higher switching frequencies will push towards domain-specific accelerators for microsecond-level control and monitoring.

In conclusion, the future of automotive electric drives lies in unified architectures where modeling, control, and monitoring operate together rather than in isolation. Achieving this integration is essential to deliver systems that are efficient, robust, certifiable, and sustainable, supporting the large-scale transition to electrified mobility.

Key Takeaways

• Integration of modeling, control, and monitoring is the decisive step beyond fragmented approaches.
• Battery technologies, bidirectional converters, and regenerative braking must be jointly considered with drive control.
• Intelligent and hybrid controllers complement classical strategies, offering adaptability under safety constraints.
• Robust monitoring and prognostics are central to reliability, lifetime extension, and sustainability.