Modern Control Techniques and Operational Challenges in Permanent Magnet Synchronous Motors: A Comprehensive Review

Abstract

This paper presents a comprehensive overview of permanent magnet synchronous motors (PMSMs), including their classifications, applications, and vector control strategies. It explores various control techniques, including maximum torque per ampere (MTPA), maximum current (MC), field weakening (FW), maximum torque per voltage (MTPV), sensorless control, and parameter identification, as discussed in this paper. These methods address key challenges in PMSM control, such as improving motor efficiency and accurately estimating rotor position and speed. Additionally, this paper presents the PMSM parameters due to many factors such as electric current, phase angle, saturation, and temperature. The survey findings provide a deeper understanding of PMSMs’ control strategies, aiding in the more efficient and reliable motor studies.

1. Introduction

Permanent magnet synchronous motors (PMSMs) are commonly utilized in industrial applications like servo drives, high-speed trains, electric vehicles (EVs), actuators, hard disk drives, rolling mills, and household appliances due to their high efficiency, power density, and broad speed range. Using of permanent magnets (PMs) in PMSMs has many advantages including, avoiding the required energy for field excitation, improved dynamic performance and ease of construction and maintenance [1]. In particular, as it is well known that the use of EVs nowadays is very necessary to replace traditional internal combustion engine vehicles (ICEVs), PMSMs can help cut down on greenhouse gas emissions, lessen dependency on fossil fuels, and benefit the electric grid and consumers. The greenhouse emissions produced for the required electricity for EVs are 50% [1] lower than that produced from traditional ICEVs, and this amount of reduction will increase by increasing the production of electricity from renewable energy resources. Furthermore, many countries such as Germany, Britain, France, and China have plans for gradually getting rid of the manufacturing of cars powered by gasoline and diesel in a few decades. By 2040, it is anticipated that 54% of new automobile sales would be electric. For instance, Ford has invested USD 11 billion on 40 new electric vehicles over the coming years [1]. For the foregoing reasons and notes and the wide applications for PMSMs, the control of PMSMs is required to be robust, accurate, fast dynamic, and efficient. The control of PMSMs faces some problems, resulting in reducing the performance of the controllers. The primary issue with PMSMs is the variation of parameters like resistance and d-q axis inductance. Specifically, the d-q-axis inductances fluctuate based on the motor’s shape, mechanical power, and operating characteristics; hence, inaccurate parameter estimate results in high temperatures and deteriorated control system performance. High temperatures can lead to an increase in stator resistance, while simultaneously reducing both the rotor flux linkage and the d-q axis inductances, potentially resulting in torque ripples [2,3]. Another major challenge in PMSM control is the estimating of the rotor speed and position using traditional methods that rely on built-in sensors, like optical and rotary encoders. These conventional sensors take up space, making motor installation difficult, and may cause the motor to malfunction due to sensor failures in harsh operating environments [4,5]. Therefor this paper addresses the main PMSM control topics in the literature to achieve its merits and reduce its problems. The remainder of the paper is organized as follows: Section 2 presents the types of PMSMs, comparing their different classifications and applications. Section 3 discusses the mathematical model of PMSMs and explains the four operating regions: maximum torque per ampere (MTPA), maximum current (MC), field weakening (FW), and maximum torque per voltage (MTPV). Section 4 covers MTPA control schemes aimed at enhancing motor efficiency and performance and explores artificial-intelligence-assisted methods for MTPA control. Section 5 presents sensorless control techniques, which eliminate traditional built-in sensors to improve system reliability and reduce costs. Additionally, Section 6 discusses parameter identification control methods to address nonlinear parameter variations in PMSM control systems, including those related to MTPA and sensorless approaches. Finally, Section 7 provides the findings and conclusions.

2. PMSM Types and Applications with Recent Technologies in Constructions

2.1. PMSM Types and Applications

Synchronous motors can be classified mainly into three types, namely, wound rotor synchronous machines (WRSMs), PMSMs, and reluctance synchronous motors (RSMs). WRSMs are appropriate for situations requiring high power. PMSMs and RSMs are favored in the low- to medium-power range, where induction motors are gradually being replaced [1,6]. PMSMs are better than RSMs and IMs in their power density and efficiency, but their cost is more than them, as shown in Figure 1. PMSMs are the best choice for multiple industrial uses because of their many benefits. In EV application, PMSMs are the best choice for the manufacturers of EVs based on a comparison between IMs, SRMs, and PMSMs. According to [1], in air conditioning compressors, ref. [6] presents a comparison between PMSMs and SYNRMs. In pump load applications, ref. [7] shows the different Ch/s between the PMSM and the other motors such as induction motors, squirrel cage induction motors, synchronous reluctance motors, and switched reluctance motors. In the aerospace industry, PMSMs are also a better option, particularly for the newest electric starter/generator system that operates in a high-speed range for future aircraft [8].

Permanent magnet synchronous machines (PMSMs) are widely used in wind electric generation systems because they are efficient, reliable, and can work well at variable speeds. In wind turbines, PMSMs help capture the most energy from changing wind speeds by using advanced control methods like model predictive control and maximum power point tracking, which improve power output and system efficiency [9,10,11]. PMSMs are also valued for their ability to keep working even if a fault occurs, especially in open-winding designs, which allow the wind turbine to continue operating—though at lower power—until repairs can be made [12,13]. New design and control strategies, such as fuzzy logic, sliding mode control, and optimization algorithms, further improve the stability, power quality, and fault tolerance of PMSM-based wind systems [14,15,16]. These features make PMSMs a strong choice for both large wind farms and small wind turbines, helping to ensure reliable and efficient renewable energy generation.

PMSMs could be classified into two types, namely, interior permanent magnets (IPMSM)s and surface permanent magnets (SPMSMs). Both IPMSMs and SPMSMs could take different designs according to the position of permanent magnets in the rotor, as shown in Figure 2.

Table 1. Comparison between IPMSMs and SPMSMs.

PMSM Type	IPMSM	SPMSM	Ref.
Design	Require peripheral sleeve	Need a peripheral sleeve in order to protect PMs from centrifugal forces. Because of this, manufacturing SPM rotors is more difficult than IPMs.	[26]
Losses	More core losses	Reduced core losses as a result of the rotor steel’s greater separation from the air gap and SPMs’ reduced magnetic flux leaking.
Volume of PM material	More PM material volume is needed	Compared to IPMs, less PM material is used.
Cost	Is superior in costs	High cost.	[23]
Robustness	Less robust	SPM is more robust.
Efficiency	Less efficiency	Higher overall efficiency and cooler.
Cooling	Less in cooling	Cooler.
Speed range	If the machine’s saliency is maximized, IPM has an excellent overload capacity over the whole speed range	The SPM motor is unable to surpass the continuous power rating, regardless of the applied current overload.	[18]
Joule losses vs. speed	Requires a suitable number of stator slots and rotor segments to keep the harmonic losses in control, which can increase the fabrication cost, and has larger Joule losses at low speed because of end connections	Extra-Joule losses for deexciting the PM flux at high speeds and PM losses that necessitate segmentation in both directions (axial and circumferential) have an impact on the SPM motor.

presents a comparison between IPMSMs and SPMSMs [1,2,6,7,17,18,19,20,21,22,23,24,25,26,27,28].

2.2. PMSM Recent Technology Construction Design

Recent studies, however, show many new trends. One important direction is the use of hybrid permanent magnets and new rotor shapes to increase efficiency and improve resistance to demagnetization [29]. Axial-flux PMSMs with coreless designs are also receiving attention because they provide very high torque and power density, although they bring challenges in cooling and mechanical stability [30]. New winding approaches, such as hairpin windings combined with direct oil-cooling, improve thermal performance and are increasingly used in high-power traction motors [31]. Other research highlights the impact of rotor losses in rare-earth PMSMs, especially during field weakening at high speeds, which is important for aerospace and EV drives [32]. Cost modeling studies show that winding type, cooling, and magnet material all have strong influence on manufacturing cost, which is critical for large-scale production of electric vehicle motors [33]. In addition, surface treatments and protective coatings for permanent magnets are being developed to reduce the risk of demagnetization and corrosion in harsh operating environments [34]. Reliability is another key trend. Nine-phase and five-phase PMSMs with advanced fault-tolerant control strategies have been proposed to ensure continuous operation during open-circuit faults [35,36]. Dual-winding PMSM structures are also under investigation to enable smooth operation under faults and reduce harmonic effects [37]. These developments confirm that PMSM construction is still an active research field, with innovations in materials, winding and cooling methods, cost reduction, and fault tolerance.

Multiphase permanent magnet synchronous motors (PMSMs), such as five-phase and six-phase machines, are increasingly vital in modern applications due to their enhanced reliability and operational flexibility. Recent research has focused on advanced torque optimization and fault-tolerant control (FTC) strategies to fully exploit the additional degrees of freedom these motors offer. Optimization-based FTC schemes, including those utilizing particle swarm optimization and interior point methods, have demonstrated the ability to maximize torque output while minimizing copper losses, even under open-phase or open-switch fault conditions [38,39,40]. These approaches often outperform conventional methods by achieving higher torque capacity—up to 73.3% of the healthy state and reducing losses by up to 20% during faults [38,39]. Furthermore, innovative control strategies, such as model predictive control and current reference optimization, enable smooth transitions between healthy and faulty operations, suppress torque ripple, and maintain system stability without complex hardware modifications [41,42,43]. The integration of radial force compensation and consideration of phase current limits further enhances the robustness and efficiency of multiphase PMSMs, making them highly suitable for critical and demanding applications [44,45,46]. Discussing these advancements in torque optimization and FTC will significantly strengthen the technical depth and practical relevance of any paper on multiphase PMSM drives.

3. PMSM Operation and Control

The PMSM mathematical model and its four operational zones are shown in this section, named as MTPA, MC, FW, and MTPV.

3.1. Mathematical Model of PMSMs

In the rotor reference frame, the PMSM’s voltage equations may be expressed as follows [25]:

$${\mathrm{v}}_{\mathrm{d}}={\mathrm{R}}_{\mathrm{s}}{\mathrm{i}}_{\mathrm{d}}+{\displaystyle \frac{\mathrm{d}{\mathsf{\psi }}_{\mathrm{d}}}{\mathrm{d}\mathrm{t}}}-{\mathsf{\omega }}_{\mathrm{e}}{\mathsf{\psi }}_{\mathrm{q}}$$

(1)

$${\mathrm{v}}_{\mathrm{q}}={\mathrm{R}}_{\mathrm{s}}{\mathrm{i}}_{\mathrm{q}}+{\displaystyle \frac{\mathrm{d}{\mathsf{\psi }}_{\mathrm{q}}}{\mathrm{d}\mathrm{t}}}+{\mathsf{\omega }}_{\mathrm{e}}{\mathsf{\psi }}_{\mathrm{d}}$$

(2)

$${\mathsf{\psi }}_{\mathrm{d}}={\mathrm{L}}_{\mathrm{d}}{\mathrm{i}}_{\mathrm{d}}+{\mathsf{\lambda }}_{\mathrm{f}}$$

(3)

$${\mathsf{\psi }}_{\mathrm{q}}={\mathrm{L}}_{\mathrm{q}}{\mathrm{i}}_{\mathrm{q}}$$

(4)

And for the steady-state operation, the equations will be

$${\mathrm{v}}_{\mathrm{d}}={\mathrm{R}}_{\mathrm{s}}{\mathrm{i}}_{\mathrm{d}}-{\mathsf{\omega }}_{\mathrm{r}}{\mathrm{L}}_{\mathrm{q}}{\mathrm{i}}_{\mathrm{q}}$$

(5)

$${\mathrm{v}}_{\mathrm{q}}={\mathrm{R}}_{\mathrm{s}}{\mathrm{i}}_{\mathrm{q}}+{\mathsf{\omega }}_{\mathrm{e}}\left({\mathrm{L}}_{\mathrm{d}}{\mathrm{i}}_{\mathrm{d}}+{\mathsf{\lambda }}_{\mathrm{f}}\right)$$

(6)

The electromagnetic torque is expressed as

$${\mathrm{T}}_{\mathrm{e}}=0.75\mathrm{p}\mathsf{\lambda }{\mathrm{i}}_{\mathrm{q}\mathrm{s}}+0.75\mathrm{p}\left({\mathrm{L}}_{\mathrm{d}}-{\mathrm{L}}_{\mathrm{q}}\right){\mathrm{i}}_{\mathrm{d}}{\mathrm{i}}_{\mathrm{q}}$$

(7)

where ${\mathrm{v}}_{\mathrm{d}}$ and ${\mathrm{v}}_{\mathrm{q}}$ are the $d$-$q$ axis stator voltages; ${\mathrm{i}}_{\mathrm{d}}$ and ${\mathrm{i}}_{\mathrm{q}}$ are the $d$-$q$ axis current. The $d$-$q$ axis inductance is denoted by ${\mathrm{L}}_{\mathrm{d}}$ and ${\mathrm{L}}_{\mathrm{q}}$. The stator resistance, the permanent-magnet flux linkage, and the pole pairs are denoted by ${\mathrm{R}}_{\mathrm{s}}$, ${\mathsf{\lambda }}_{\mathrm{f}}$, and $p$, respectively. The electrical angular frequency is denoted by ${\mathsf{\omega }}_{\mathrm{e}}$.

3.2. PMSM Operation Regions

Motors for PMSM applications, particularly mild-hybrid and fully electric cars, need to have a large speed range at constant power and good torque at low speeds. Consequently, based on the speed ranges reported in the literature, the operating zones may be divided into four sections [47,48,49,50,51,52,53]. These four regions are maximum torque per ampere (MTPA), maximum current (MC), flux weakening (FW), and maximum torque per voltage (MTPV). The current plane characteristics of IPMSMs and SPMSMs are shown Figure 3. Figure 3 illustrates how the saliency of IPMSMs causes the voltage limit to take the form of an ellipse in IPMSMs and a circle in SPMSMs. Therefore, the MTPA and MTPV trajectories in IPMSMs are hyperbolic curves, as opposed to SPMSMs, where they are straight lines. For the four operation regions, if the characteristics curves of IPMSM are used in representation, consider that the red star point serves as the operating point, as shown in Figure 4. Firstly, the motor in MTPA runs at the base speed ${\mathrm{n}}_{\mathrm{b}}$ on the OM curve, which is analytically constructed in (8) and displayed in Figure 4a. Point (O) represents the origin point in the current plane, and point (M) represents the intersection of the MTPA trajectory with the current limit circle. The voltage limit ellipse crossing the MTPA trajectory at the current limit circle defines ${\mathrm{n}}_{\mathrm{b}}$. The point where the MTPA and load torque curves connect will serve as the operational point. It must be noted that in the MTPA region, the optimal values of ${\mathrm{i}}_{\mathrm{d}}$ and ${\mathrm{i}}_{\mathrm{q}}$ depend on the load torque, regardless of the required speed. Secondly, in the MC region, the speed is more than the base speed and under the boundary speed ${\mathrm{n}}_{\mathrm{b}\mathrm{o}}$ is defined by the voltage constraint ellipse passing through the origin and intersects with the MC circle at F, as displayed in Figure 4b, if it is assumed that the required operating speed is ${\mathrm{n}}_{\mathrm{r}\mathrm{e}\mathrm{q}}$. ${\mathrm{n}}_{\mathrm{r}\mathrm{e}\mathrm{q}}$ is represented by the voltage limit intersecting with the current limit circle and OM curve at R and D, respectively. For operating in the MC region at ${\mathrm{n}}_{\mathrm{r}\mathrm{e}\mathrm{q}}$, an additional negative d-axis current is wanted for moving from M to R. The operating point is point R at the load torque ${\mathrm{T}}_3$. If the load torque is smaller than ${\mathrm{T}}_3$ as ${\mathrm{T}}_2$, the motor will move from R on the voltage limit to operating point X. If the torque is lighter as ${\mathrm{T}}_1$, the motor will continue moving on the voltage limit to intersect with the MTPA curve at R. The motor cannot go across the MTPA curve to intersect with ${\mathrm{T}}_3$ at Z but it will move along MTPA curve to intersect with ${\mathrm{T}}_3$ at N.

$${\mathrm{i}}_{\mathrm{d}}={\displaystyle \frac{{\mathsf{\lambda }}_{\mathrm{f}}}{2\left({\mathrm{L}}_{\mathrm{q}}-{\mathrm{L}}_{\mathrm{d}}\right)}}-\sqrt{{\displaystyle \frac{{{\mathsf{\lambda }}_{\mathrm{f}}}^2}{4{\left({\mathrm{L}}_{\mathrm{q}}-{\mathrm{L}}_{\mathrm{d}}\right)}^2}}+{\mathrm{i}}_{\mathrm{q}}^2}$$

(8)

Thirdly, the speed in the flux weakening region is greater than ${\mathrm{n}}_{\mathrm{b}\mathrm{o}}$ and lower than ${\mathrm{n}}_{\mathrm{v}}$. ${\mathrm{n}}_{\mathrm{v}}$ is the deep flux-weakening speed shown in Figure 4c by the voltage constraint ellipse going through the intersection of the MTPV locus and current circle at point $\mathrm{V}$. If the required speed ${\mathrm{n}}_{\mathrm{r}\mathrm{e}\mathrm{q}}$ is higher than ${\mathrm{n}}_{\mathrm{b}\mathrm{o}}$, the motor will operate at point $\mathrm{Y}$ if the required torque is ${\mathrm{T}}_2$, as shown in Figure 4c. For a smaller torque ${\mathrm{T}}_1$, the motor will operate within the voltage limit of ${\mathrm{n}}_{\mathrm{r}\mathrm{e}\mathrm{q}}$ to operate at point $\mathrm{k}$, as shown in the blue path in the Figure 4c.

Fourthly, for a speed more than ${\mathrm{n}}_{\mathrm{v}}$ the motor operates in the deep-flux-weakening region MTPV. The trajectory of MTPV can be derived analytically, as in [53]. So, for ${\mathrm{n}}_{\mathrm{r}\mathrm{e}\mathrm{q}}$ in Figure 4d, it is defined by the voltage limit that intersects with the MC limit at point $\mathrm{Q}$. The motor will operate at the MTPV trajectory if the required torque intersects with the ${\mathrm{n}}_{\mathrm{r}\mathrm{e}\mathrm{q}}$ voltage limit at the MTPV trajectory at point $\mathrm{Q}$ for torque ${\mathrm{T}}_4$. If the torque is smaller than ${\mathrm{T}}_4$ as ${\mathrm{T}}_1$, the motor will move along the ${\mathrm{n}}_{\mathrm{r}\mathrm{e}\mathrm{q}}$ voltage limit to operate at the intersection point $\mathrm{P}$.

3.3. General Control Methods for PMSMs in Solving PMSM Issues

3.3.1. Vector Control

There are three primary topologies for the vector control approach, namely, field-oriented control (FOC), direct torque control techniques (DTCs), and voltage vector control (VVC) [54,55]. In FOC, the current ${\mathrm{i}}_{\mathrm{d}\mathrm{s}}$ = 0 for max torque at low speeds and for increasing speed ${\mathrm{i}}_{\mathrm{d}\mathrm{s}}$ will be negative. FOC can be used with or without a position sensor, but when using a sensor, the FOC has small tolerance and is expensive and not reliable [56,57,58,59]. DTC circuits are simpler in structure than the FOC circuit. Since precise rotor position information is not needed in DTC systems (apart from the synchronous motor’s startup scenario), the inverter with PWMs is not necessary. All computations are carried out in the stator reference frame. In DTCs, the controllers’ computing capabilities are comparatively minimally required. DTCs respond rapidly to changes in load, offer excellent dynamic properties, and are less susceptible to disturbances and variations in motor parameters. However, high stator current, flux linkage, and torque ripple levels, particularly at low speeds, are characteristics of steady-state operation that significantly restrict their usage for high-precision drives in automotive applications [56,60,61,62,63,64]. The VVC method is relatively simple, and the information regarding the rotor’s position is not required. One of the advantages of VVC is that it has low sensitivity to motor parameter changes; however, it includes higher ripple than in the case of using FOC and high torque overshoot amplitude, which makes using the VVC in high-precision driving systems challenging [54,65,66,67].

Recently, in non-linear control theory, passivity-based control (PBC) has drawn a lot of interest because of its dependability, effectiveness, and rather straightforward construction. The EL formalism yields the PMSM model in terms of energy quantities. Additionally, it can be observed that the PMSM defines specific passive maps, after which its dynamics are handled as two distinct EL passive subsystems: mechanical and electrical. The electrical subsystem’s mechanical dynamics are regarded as passive disruptions. In order to strengthen the passivity quality, dumping is also introduced into the electrical subsystem. The internal closed loop energy is reshaped to provide current tracking. In the PBC method, the closed loop system is subjected to a passivity property in order to achieve the control objective. This is accomplished by allocating the desired dissipation and storage functions to the closed loop dynamics [60,68]. This method eliminates the rotor location from the coefficients of the differential equations and reduces the number of variables. This simplification produces stability analysis, controller synthesis, and the determination of steady state conditions—all of which are crucial for evaluating the machine’s performance [60,68].

3.3.2. Scalar Control

Open loop constant V/f inverters are frequently utilized in industrial variable speed applications, especially in induction motor (IM) drives due to their advantages of simple implementation, low cost, and general purpose. However, the drawbacks include limited dynamic operation, low energy efficiency, and a failure to account for the motor’s coupling effects. For a PMSM drive, the concept of the open loop constant V/f-controlled method is usually not directly applied because the open loop system is generally unstable. By adding a frequency damper to the control to increase the system damping constant, the stability issue can be prevented. By including two feedforward channels for the slip and stator voltage magnitude in the control scheme, the scalar controller’s dynamic response can be improved. Its ability to boost performance is dependent on how accurate these feedforward routes are. This is why applications with fast dynamic requirements, such as traction applications, do not use the v/f approach. Nonetheless, certain multilayer traction applications with slower dynamics, such as electric ships, use it [1,69,70,71,72,73,74].

3.3.3. Flux-Weakening Control Methods

(a)

Feed forward method

In feed forward techniques, control variables that have been studied analytically or empirically are calculated to maximize motor performance within the presumptive feeding voltage and current limits. Complex nonlinear equations must be solved online for the intersections (solutions) involving the torque hyperbola, voltage ellipse, and current circle. The accuracy of the current set-points declines because of the frequent ignoring of magnetic saturation and resistive voltage loss to speed up the solving procedure. Although feed-forward techniques are known for their stability and transient responses, they have several disadvantages, including temperature volatility, material qualities that change over time, and saturation [75,76,77,78,79,80,81,82].

(b)

Feedback technique

The feedback systems track the voltage limit as the speed increases to adjust the negative d-axis current. Once the voltage or speed has been measured, the demagnetizing current vector can be modified by monitoring the voltage or speed error. These methods are robust since they are not affected by parameters variations and have higher efficiency than previous methods. However, inadequate performance during transients is the price paid for this [83,84,85,86,87,88,89,90].

(c)

Hybrid technique

In order to benefit from the advantages of both feed-forward and feedback methods, hybrid strategies have been created. While the torque command and d-axis current feedback are used to construct the q-axis current command, the optimization objectives are used to modify the pre-computed d-axis current command for MTPA. The hybrid control techniques are highly resilient to model uncertainty and parameter variations because they typically use look-up tables to determine the excitation current commands for MTPA while accounting for magnetic saturation and varying parameters. The feed-back technique is then used to determine the current commands for flux-weakening operation. However, because of the lengthy experiments needed to obtain the look-up tables, the transient response could be delayed [82,91,92].

3.3.4. Control and Converter Considerations

In permanent magnet synchronous machines (PMSMs), the power converter plays a significant role in motor control. The supply’s harmonic distortion, switching frequency, and available voltage vectors are all defined by the converter topology [93,94]. These elements have a direct effect on the greatest control bandwidth that may be attained, torque ripple, and current ripple [93,95]. Different control possibilities are made possible by different converter structures. Two-level inverters are simple and reliable but give limited voltage vectors. Multilevel inverters, such as NPC or T-type, provide more voltage levels, reduce harmonics, and improve torque control [94,96]. However, these converters need complex balancing techniques, careful thermal design, and more sophisticated control [96,97]. The converter and control algorithms need to be compatible. The additional voltage vectors of multilayer inverters can be used by model predictive control and direct torque control to reduce torque ripple and produce faster dynamic response [95,98]. They rely on precise measurements and raise computational needs in exchange [98,99]. Imperfections in the converter also come into play. Errors in voltage generation are caused by dead-time, switching delay, and finite DC-link voltage. If these effects are not compensated in the control, performance degrades under transient or high-speed operation [97]. For this reason, PMSM and converter design should be considered together. Co-design methods that optimize the motor, the inverter, and the control algorithm at the same time lead to higher efficiency, greater power density, and better fault tolerance [99]. This strategy is particularly crucial for high-speed industrial machinery, renewable energy systems, and electric vehicle drives [99].

3.3.5. Single-Loop Control Structure for PMSM Speed and Current Regulation

The single-loop control structure, which integrates the speed and current loops into a merged control loop, has lately concerned significant consideration in PMSM drive research. Compared to old cascade control, this approach enables more direct control action, greater control bandwidth, and improved speed dynamic performance. Recent studies have demonstrated that single-loop controllers—often based on advanced sliding mode control, disturbance observers, or hybrid reaching laws—can achieve fast transient response, strong robustness against disturbances, and effective overcurrent protection, all while simplifying the control system. For example, single-loop strategies using current-constrained sliding mode control and time-varying nonlinear disturbance observers have shown improved speed and current tracking, reduced overshoot, and enhanced disturbance rejection compared to conventional cascade structures [100,101,102,103,104]. These advancements confirm the promise of single-loop control paradigms for PMSM drives and highlight the need for further in-depth investigation into their practical applications and optimization [101,105,106,107,108].

4. Maximum Torque per Ampere Control Methods

The MTPA techniques can be categorized as shown in Figure 5.

4.1. Machine-Parameter-Dependent MTPA Techniques

The values of machine parameters, such as PM flux linkages and inductances, are required from offline testing, an approximated fitting function, or an estimating strategy during machine operation in machine parameter dependent MTPA procedures.

4.1.1. Premeasurements Assisted Methods

The premeasurements assisted MTPA methods that depend on look up table (LuT). In particular, the pre-testing data is used to adjust for changes in machine parameters over the speed and torque ranges. The drive system, where measurements were gathered through numerous offline tests, has made extensive use of lookup tables (LuT). LuTs have been used extensively in drive systems where measurements were gathered through numerous offline tests. According to the torque reference, some techniques choose the reference flux linkage that corresponds to the MTPA profile directly [109,110,111,112,113]. In other methods, LuT provides the ${\mathrm{i}}_{\mathrm{d}}$ and ${\mathrm{i}}_{\mathrm{q}}$ references, which show the optimal currents as a proportional to torque [91,114,115,116]. Based on load circumstances, the armature current in [117] is calculated on a LuT. If the inverter nonlinearities are taken into account, the speed range of LuT is increased [116]. Apart from the straightforward approaches above, Ref. [118] uses the Gauss–Newton algorithm to generate reference currents from the online calculation block. The prefabricated LuT is used to determine the associated machine parameters. These approaches have the disadvantages of requiring a lot of computer storage space, requiring an accurate interpolation technique, and requiring a lot of testing time.

4.1.2. Approaches Based on Parameter Estimates and Approximated Functions

These techniques use a polynomial curve/surface fitting to trend how machine parameters vary under various operating situations. Although there are fewer pre-tests, the basic idea behind these approaches is the same as that described in the previous section. To avoid using the LuT, for example, a two-dimensional curve fitting function is developed in [119,120,121,122] to provide the best fit to a set of measurements. For considering the problem of magnetic saturation and cross-coupling drawbacks, surface fitting techniques are presented in [123,124] to estimate the flux linkage profiles. The more intricate in modelling process, the more pre-test measurements are required. To balance the precision of MTPA operation with a fair testing duration, ref. [125] suggests using just two pretest points on the approximated MTPA curve to calculate the parameters of a simplified MTPA model on the flux linkage synchronous frame. Many of these methods employ iterative algorithms like gradient descent with Lagrange multipliers and Gauss–Newton. In [53], the Newton–Raphson algorithm is used to improve the control precision of the current set-points in MTPA, MC, flux weakening, and MTPV regions. In the meantime, the effects of several parameters, such as resistive voltage drop and magnetic saturation, are thoroughly examined. As was previously established, the real current trajectory deviates from the ideal one when the resistive voltage drop and magnetic saturation are ignored. Compared to the LuT-based approach, this method has the benefit of requiring less pre-testing time.

4.1.3. Parameter-Adaption-Based Methods

These methods depend on online parameter identification strategies such as recursive least squares, which is a popular technique for identifying parameters [126,127,128,129]. Recursive least squares are used to track the variation of dq–axis inductances and permanent magnet flux coupling, which are involved in the computation of MTPA points, in order to match the estimations to the actual one. The DTC scheme could be used to replace FOC in that it eliminates the q-axis inductance effect and computes the stator flux linkage that satisfies the MTPA requirement directly [130]. Also, the parameters could be estimated by the knowledge of harmonic components. So, the series of filters could be implemented to identify machine parameters. This method is comparable to the method based on virtual signal injection that will be introduced in the parameter identification section [131,132].

This approach seeks to address the issues of low parameter adaptivity, the need for a lot of data storage, and nonlinearity; however, it increases the operational burden and real-time processing cost, which causes a slow reaction.

4.2. Machine-Parameter-Independent MTPA Techniques

The MTPA trajectory can be tracked using the parameter-independent method without requiring knowledge of the machine parameters. Extremum-searching-based, power-measurements-based, and artificial-intelligence-assisted techniques are the three categories into which the parameter-independent approaches are separated, as illustrated in Figure 5.

4.2.1. Extremum Searching Based

There are two kinds of machine-parameter-insensitive signal-injection-based schemes: real signal injection and virtual signal injection. The extreme searching technique relies on the extraction of derivatives to determine the MTPA point and does not require prior data [109,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152]. It can be done by injecting disturbance signals such as high-frequency (HF) current [136,153] and HF current angle [137,138,139,140,141,142,143] into the system. The magnitude of injected perturbation is relatively small [38] so that the operational conditions are not significantly affected. The resultant torque, which includes the high-frequency component, has often been expanded using the Taylor series. Since the techniques described in this section necessitate additional time-consuming components like integrators and filters, one of their main disadvantages is that they limit dynamic performance.

4.2.2. Power-Measurement-Based Methods

The MTPA point is equal to the maximum efficiency point in the region of constant torque, which is necessary for power-measurement-based methods to work. Therefore, the additional dc-link current and voltage sensors can help this method. This approach uses the gradient descent algorithm to find the maximum efficiency point after calculating mechanical power and requiring dc power measurements. An offline map of the phase advance angle at various speed conditions is created by varying the dc-link power. To find MTPA spots while the machine is operating, the gradient descent approach is used. Due to the absence of three-phase current measurements, power observation is advantageous for certain alternative control systems, such as sensorless and V/f.

4.2.3. Artificial-Intelligence-Assisted Methods

The description of the nonlinear relationship between torque and currents heavily relies on artificial intelligence. Artificial intelligence techniques, such as fuzzy, neural network, and optimization algorithms, are employed and trained to provide the best current directives during MTPA operation. For instance, to determine the ideal values of ${\mathrm{i}}_{\mathrm{d}}$ and ${\mathrm{i}}_{\mathrm{q}}$ for reaching MTPA, a genetic algorithm is utilized [154]. The foundation of this technique is the discovery of an unknown function that delivers the maximum torque per current as a relation between the currents. A controller-based genetic algorithm was used to find the i_d, and a PI controller was used to calculate the i_q. Fast dynamic reaction and cheap real-time computing cost are two advantages of artificial-intelligence-assisted approaches, which are accurate, simple, and quick to configure in real time. They could require more pretests and training procedures, though.

5. Sensorless Control Methods

In order to circumvent the limitations of employing sensors like encoders and resolvers to estimate the rotor position and speed, sensorless control schemes are recommended in PMSM drives. The system becomes less reliable and more expensive when sensors are used. The literature provides a variety of methods and algorithms for figuring out the position and speed for sensorless control. Some techniques rely on feedback to lower the estimation errors, while others are direct and do not require a feedback loop; however, this leads to high error numbers. Thus, open loop, close loop, and signal-injection-based sensorless control techniques can be divided into three categories.

5.1. Open Loop Methods

After converting the three-phase variable quantities into a two-phase (d-q) reference frame, the rotor angle and speed values may be calculated immediately. The primary benefits are ease of use and quick dynamic reaction. However, the problem of parameter fluctuation has a significant impact on this ethos [155,156,157,158]. Additionally, as the inductance is dependent on the rotor position, the rotor angle and speed might be calculated using the stator inductance. This technique works well with high-saliency motors, such as switching reluctance motors and interior motors. It works well at low speeds, even at zero speed, where the EMF is zero. However, because it is an open loop method, the accuracy of the estimation cannot be guaranteed [159,160,161,162,163,164]. The reverse EMF is necessary for another open loop sensorless. The back EMF can effectively assess the rotor speed and angle because it is widely known that the back EMF generated in a motor is proportional to the rotor’s speed or angle. Determining back EMF without relying on voltage probes could lower the method’s cost and boost its reliability. This method is straightforward, so it does not need a complex observer, which makes it simple, accurate at higher frequencies, cost effective, and robust, but it is not accurate at low speed, especially at starting and the difficulty in calculating the back EMF [156,158,165,166,167,168,169].

5.2. Closed Loop Methods

This is an adaptive approach that adjusts to various circumstances and acts accordingly. Comparing the results of systems that are mathematically modeled with those that are real is how this tactic is carried out. Reducing the error between the two systems is the primary goal of these techniques. These adaptive sensorless controllers might make use of an extended Kalman filter. It is important to note that extended Kalman filters (EKF) are utilized for nonlinear systems, while Kalman filters are utilized for linear systems. These filters determine the optimal state of a nonlinear system’s dynamics via least square variance estimation. The solution of the Riccati differential equation, which calls for two covariance matrices, is the foundation of this approach. Measurement noises is the name given to one matrix, and parameter uncertainties to the other. Among the advantages of this approach are its ability to reduce computation time and its insensitivity to measurement noise and parameter error. However, the primary drawback is the subpar performance at slower rates (<5 Hz) [170,171,172,173,174,175].

Sensorless control uses the model-reference-based adaptive system (MRAS), which compares a reference model value with an adjustable model value to minimize the error to zero. It is based on Popov’s stability criterion, which is a very desirable scheme used in speed and rotor resistance estimation, and its greatest improvement is its high speed of adaptation [176,177,178,179,180,181]. Another adaptive sensorless method is the SMO [182,183]. It gets its name from its underlying theory or strategy, which is when certain functions or control laws are chosen so that they go to a shared boundary and then slide on that boundary or surface known as a sliding surface. The SMO is based on defining a sliding surface firstly, and secondly defining control laws or functions to have convergence of the systems towards the predefined sliding surface as these functions make the system slide on that surface as shown. A sliding hyperplane is chosen in SMOs based on the error vector ∆I_s. The system then slides across this hyper-plane until it reaches the smallest error. This method’s robustness ensures that there is no inaccuracy in the rotor value estimations. An SMO can solve the significant issue of parameter fluctuations that other schemes face because it is simple to implement and performs well with nonlinear systems due to its adaptability. However, because the back EMF value is so low at low speeds, it performs poorly at standstill or low speeds. The method’s conflict between a quick rate of convergence and a good steady state result is another issue. The gain coefficient should be selected to guarantee accurate speed and rotor angle values, as well as good speed estimation outcomes with improved convergence [182,183,184,185,186,187,188].

5.3. Saliency and Signal Injection (Non-Ideal Property)-Based Method

Since the estimation of rotor speed and position depends on the back-EMF, which is dependent on rotor speed, it is evident that there is a significant issue with sensorless control schemes at low speeds. At low speeds, the back-EMF is low, making it challenging to estimate the PMSM’s speed and position. The following section discusses several approaches and strategies for sensorless PMSM drive control in the low-speed region in order to address this issue [189]. First, the high-frequency injection method, which is dependent on the saliency of the motors, is appropriate for prominent motors. Its foundation is the use of a fundamental frequency signal to excite the motor at a high frequency. The additional losses brought about by this signal are minimized by maintaining that the stator voltage and current magnitudes are low relative to the basic values. Even at moderate speeds or even at a standstill, the change in inductance becomes notable due to the huge current derivatives caused by applying an HF signal. Saturation issues and poor dynamic performance are issues with this approach. But coordinate transformation is avoided in this method, and it is not influenced by rotor position error [190,191,192,193,194,195]. Also, low-frequency signal injection methods could be used. The saliency of the motor is not necessary in the low-frequency (LF) signal injection method as it is firstly implemented for an SPMSM to calculate the position and speed of its rotor in low-speed ranges. Its foundation is the injection of a low-frequency signal into the reference d axis, which causes back-EMF ripples that are used to determine the position and speed of the rotor. The drawbacks of this estimator are slow dynamic response and depending on the motor inertia [196,197,198,199]. Since the on-line reactance measurement (INFORM) technique does not rely on the back-EMF, it is utilized for indirect flux detection in low-speed ranges and at standstill. Its basic idea is to inject space vector voltages in various directions and then measure the current response. The current response determines the inductance fluctuations, which in turn determines the rotor position. The main benefit of this approach, in terms of not relying on the back EMF, is its insensitivity to motor parameters. However, as the motor flux is thought to be sinusoidally distributed, any distortion in motor flux will result in inaccurate speed and rotor position estimates. Then, a major issue with this strategy is the current ripple [200,201,202,203]. The ${\mathcal{H}}_{\mathrm{\infty }}$ sensorless control technique is an effective nonlinear control approach for handling multivariable system problems by combining control criteria such robust stability, asymptotic tracking, and disturbance attenuation into a single control problem. The foundation of the ${\mathcal{H}}_{\mathrm{\infty }}$ approach is field-oriented control. Avoiding the arctangent method allows the ${\mathcal{H}}_{\mathrm{\infty }}$ methodology to overcome the restriction at zero speed and low-speed range. The causal dynamic output feedback compensator and the necessary speed reference provide the rotor position information [204,205,206,207].

6. Parameter Identification Control Methods

It is commonly recognized that local self-saturation, cross-saturation mechanical power, motor form, and operational characteristics cause parameter variance in PMSM control approaches. Therefore, to eliminate issues like low-efficient operation, output power decrease, and out-of-synchronization caused by these parameters’ change, it is essential to estimate the precise values of the d- and q-axis inductances for appropriate controller design. Because of the estimation time in system operation, the parameter variation methods can be divided into offline and online schemes [2].

6.1. Offline Parameter Estimation Techniques

Firstly, the DC current decay test is one offline technique for estimating parameters. It depends on measuring the current decay applied to the motor’s stator, increasing its initial value from a big value to zero while maintaining the rotor’s alignment with the d or q axis. The DC current decay test is marked by its simplicity in the required equipment and current measuring. However, because the test is conducted at standstill and the iron losses that arise from rotating are not considered, there are mistakes in the inductance calculation figures. To improve the performance of the DC current decay test strategy, certain improvements have been made to this method based on DC-Chopper, providing the armature at standstill, pseudo random binary sequence (PRBS) voltages, PWM voltages, and DC decay. Additionally, there are suggested techniques that include measure saturation and cross-saturation effects in measuring [208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225].

The second method is the AC standstill test. Calculating the d-q axis inductances from the stator winding’s self and mutual inductances is the base of the ac standstill test. That is, one phase is subjected to an AC voltage, and the induced phase voltage is measured on a different phase. Park’s transformation is used to calculate the d- and q-axis inductances at each rotor position. Since the test is still conducted at a standstill and takes a fair amount of time, the measured values are not the true value, even if the AC standstill test is easier to use and more accurate than the DC test for inductance estimation. For these reasons, some contributions are suggested, such as applying a three-phase AC voltage source so that the vector control drive is not required and using a multi-sine AC standstill test to shorten the time needed to evaluate the effect of saturation, cross saturation, and frequency on the d- and q-axis parameters with a VSI for signal generation. Because the d- and q-axis inductances are measured while also accounting for the saturation and cross-magnetizing effects, it is hence ideal for standard laboratory investigations. Other sources suggested including iron losses into the AC standstill test. Another idea is to use two single-phase currents flowing through the suggested circuit to provide d- and q-axis currents that are comparable to those under operating conditions, bringing the measurements close to the actual values [225,226,227,228,229,230,231,232,233,234,235]. The vector control approach could also be used to determine parameters. Knowing the values of ${\mathrm{v}}_{\mathrm{d}}$ and ${\mathrm{v}}_{\mathrm{q}}$ from measuring the fundamental voltage and current amplitudes and their phases with respect to the rotor position allows one to determine the d- and q-axis flux linkages. In this system, the cross-coupling inductances are considered to be zero and the magnet flux to be constant. Even with space harmonics, the motor’s inductance could be determined using the vector control method. However, the vector-controlled approach requires extra tools such a position sensor, oscilloscope, and dynamometer [225,232,236,237,238,239,240].

The finite element method is among the most effective offline techniques for determining the motor parameters while accounting for saturation and cross-coupling effects as well as magnetic nonlinearity based on equivalent magnetic circuits [241,242,243,244,245,246]. Techniques like [247] can be used to determine the PMSM parameters without measuring torque, allowing it to account for iron losses and eliminate inaccuracies caused by the copper resistance value. High-frequency injection signal techniques to identify the inductances could decrease the detailed parameter estimation inaccuracy caused by the inverter nonlinearity influence at various rotor positions, as in [248].

6.2. Online Parameter Estimation Techniques

One of the online parameter estimation techniques is the recursive least squares (RLC). Fitting a mathematical model derived from PMSM voltage equations to a series of observed data is the foundation of recursive least squares (RLC). By reducing the sum of squares of the discrepancy between the computed and observed data, these data include the measured stator current and the indirect reference stator voltage. RLC estimates unknown parameters like ${\mathrm{R}}_{\mathrm{s}}$, ${\mathrm{L}}_{\mathrm{d}}$, ${\mathrm{L}}_{\mathrm{q}}$, ${\mathsf{\lambda }}_{\mathrm{f}}$, and VSI nonlinearity using known parameters like voltages and currents. Notably, RLC is utilized for sensorless control, and their control strategy relies on figuring out three values. For example, ${\mathrm{R}}_{\mathrm{s}}$, ${\mathrm{L}}_{\mathrm{d}}$, and ${\mathrm{L}}_{\mathrm{q}}$ and others calculate four values. For example, ${\mathrm{R}}_{\mathrm{s}}$, ${\mathrm{L}}_{\mathrm{d}}$, ${\mathrm{L}}_{\mathrm{q}}$, and ${\mathsf{\lambda }}_{\mathrm{f}}$. Controllers for techniques for calculating four values incur a significant burden and, because of the inadequate quality of the data supplied, were unable to reach a consensus on the solution. Without parameter modification, the RLC estimation’s correctness cannot be guaranteed, and because RLC uses a fixed gain, the disturbance observer may produce an unstable transient response characteristic. Real-time parameter fluctuations and inaccuracies lead to deteriorating system performance, given that the fixed gain is used by the RLS parameter estimator [127,132,249,250,251,252,253,254,255,256,257,258,259,260,261]. In terms of the networks of artificial neurons, when it comes to real-time PMSM control, ANN is a viable option for addressing issues with parameter fluctuations including inductance, armature resistance, and reverse EMF constant for motor drives. Neural network controllers are robust and computationally simple, allowing them to respond quickly to field weakening control with minimal operating range loss. ANNs are very effective in vector control for PMSMs. ANN outperforms other algorithms like MRAS and extended Kalman filter (EKF) in terms of convergence time [262,263,264,265,266,267,268,269]. MRAS that has been discussed above in the sensorless control methods section could be considered moreover as a parameter identification strategy. Some benefits of MRAS include its ability to fix inverter or converter voltage imbalance and its accuracy in estimating under a variety of scenarios. However, it has drawbacks, such as the challenge of synchronizing the PI gain at different operating points and constructing the adaptive mechanism. Efficiency will drop as a result of the stator current command being impacted by the d- and q-axis inductance inaccuracy. Therefore, in order to correct for the voltage loss component, stator resistance must be calculated [270,271,272,273,274,275,276,277,278,279,280,281,282]. The extended Kalman filter is an additional online parameter estimation technique. It was also covered in the section on sensorless control techniques. For nonlinear systems, EKF is the best recursive parameter estimator. This is because it can take into account the impact of noise disturbances. However, the EKF may not converge on the correct state if the system model is erroneous, because processing input data with noise continually necessitates a significant computational load [264,274,279,283,284,285,286]. At a high rate of convergence, the stator inductances on the d- and q-axes are evaluated. However, the motor torque, stator resistance, and magnetic flux linkage are evaluated at a sluggish rate of convergence. This approach allows the adaptive controllers’ control gains to be updated continually using the motor torque and evaluated parameters [287,288].

7. Conclusions

This paper reviewed various types of permanent magnet synchronous motors (PMSMs), highlighting their applications and key control strategies, including maximum torque per ampere (MTPA), sensorless control techniques, and parameter identification approaches. The analysis shows all these areas face common nonlinearity challenges, such as the cross-coupling effect, magnetic saturation, and temperature variations during operation. These factors cause parameter variations, particularly in inductance, which is a critical parameter in designing control systems to achieve high torque performance. Moreover, recent studies indicate that certain strategies, such as the model reference adaptive system (MRAS) and extended Kalman filter (EKF), are commonly applied for both sensorless control and parameter identification. Based on the reviewed techniques, it is recommended that an advanced sensorless control system be developed that integrates MTPA while accounting for parameter variations, thereby enhancing both the steady-state and dynamic performance of PMSMs.