A Study of MTPA Applied to Sensorless Control of the Synchronous Reluctance Machine (SynRM)

Abstract

The present paper proposes a new Maximum Torque Per Ampere (MTPA) algorithm applied to sensorless speed control for the Synchronous Reluctance Machine (SynRM). The SynRM mathematical model is suitably modified and expressed in the γδ estimated reference frame, which could be applied in sensorless implementations. In the controller–observer scheme, an MTPA controller is coupled with a sliding mode observer (SMO) of first order. The provided equivalent control inputs are directly utilized by a modified EMF observer to estimate the rotor speed and position. Also, the MTPA control, SMO, and modified EMF observer are accordingly expressed in the γδ reference frame. In the duration of the SynRM operation, the developed MTPA algorithm succeeds in adjusting both stator current components in the γ-axis and δ-axis to the maximum torque point, while the SMO converges rapidly, achieving the coincidence between the γδ and dq reference frames. In addition, a simple torque decoupling technique is used to determine the γ-axis and δ-axis reference currents connected with the Anti-Windup Controller (AWC) for stator current control. Despite conventional MTPA methods, the proposed MTPA control strategy is designed to be robust in a wide speed range, exhibiting a high dynamic performance, regardless of the presence of external torque disturbances, reference speed variation, and even current measurement noise. The performance of the overall observer–control system is examined and evaluated using MATLAB/Simulink and considering noisy current feedback. Simulation results demonstrate the robustness and effectiveness of the proposed MTPA-based control method.

1. Introduction

Synchronous Reluctance Machines (SynRMs) are becoming attractive in an extensive area of applications related to industry, robotics, space, and home appliances. SynRM is generally regarded as an attractive energy- and cost-efficient solution, particularly for applications utilizing Induction Machines (IMs). The research on SynRM design and control improvement is continuously increasing due to its inherent characteristics, such as the durable rotor structure and the avoidance of demagnetization. Among the SynRM’s advantages are its low cost, high torque density, small size-to-power ratio, high efficiency, and reliability [1,2,3,4,5,6]. Similarly to the Permanent Magnet Synchronous Machine (PMSM), the SynRM can operate in MTPA mode, in which the MTPA operating points are on a straight line rather than a parabola [7,8,9,10]. Operation at higher speeds with reduced torque is achieved by the adjustment of the current angle to decrease the stator flux, i.e., the equivalent of PMSM field weakening. Reduced or low d-axis field techniques are required because of their importance in applications needing constant power and torque operation, such as in traction drive technology [11]. Conventionally, the rotor design of SynRM does not include permanent magnets. Therefore, direct control of flux is available for control methods based on FOC methodology. Because there is no permanent magnet flux in the rotor of SynRM, direct control of flux is attained through regulating the i_d current component. However, many researchers have proposed the Permanent Magnet Assistant SynRM (PMASynRM) as an alternative to conventional SynRM. Particularly, the PMASynRM has higher efficiency than the typical SynRM, considering their speed and torque performance in an extensive speed range [12,13,14,15,16]. Increased research interest is focused on control strategies, with the aim of optimizing the SynRM control methods. In particular, these methods are directed at efficiency and reliability enhancement related to SynRM operating modes [17].

The Maximum Torque Per Ampere (MTPA) is a commonly applied control algorithm. Given the amplitude of the stator current vector I_s, its components in the dq synchronous reference frame, i_d and i_q, are adjusted to produce the maximum possible torque T_emax in the AC machine. This is an energy-efficient control strategy, and it is applied to optimize the operating conditions of the motor drive system regarding torque. In the i_d-i_q plane, the MTPA curve represents the geometric locus of maximum torque points, depending on the parameters of the AC machine, e.g., Surface PMSM-SPM, Internal PMSM-IPM, or SynRM. Both stator currents i_d and i_q are being controlled using the MTPA algorithm up to the base speed ω_b. Considering an IPM, the corresponding MTPA curve is represented as a parabola in the i_d-i_q plane, which passes through the point 0 (i_d = i_q = 0) [18,19]. Typically, the MTPA control methods are classified into two main categories: the offline and online MTPA methods, depending on the technique applied to adapt the motor operating conditions. The offline category includes MTPA methods that are based on data acquired only at the development stage. Only measured stator currents and electrical angle θ are used to calculate the stator current components i_d and i_q, or alternatively, the angle γ of the current vector I_s. As an example, the analytical MTPA methods are included in the offline category, and they use relatively simple relations to calculate the components of stator currents. Afterwards, the MTPA curve could also be calculated via the reference or measured stator currents. On the contrary, the online category includes MTPA methods applying different techniques, which are able to calculate the MTPA trajectory through estimating the AC machine parameters. Normally, these parameters may change, depending on the operating conditions. The calculation of the MTPA angle γ is based on the motor parameter estimations. Online MTPA methods may use parameter estimation techniques, which, for example, can add disturbances in the motor by introducing voltages and current harmonics, and then analyze the obtained responses to derive the parameter information. In general, the online MTPA algorithms extract precise motor parameter information by using online tracking or estimation techniques. So, these methods are very robust against parameter variations due to several causes (e.g., model imprecision, temperature changes, magnetic saturation, mechanical or core losses, etc.) [20,21,22].

The mathematical model-based MTPA methods are more convenient to apply compared with the other MTPA control methods. Nevertheless, adopting constant parameters for the AC machine system may degrade the efficiency optimization of MTPA control in varying torque loads due to several parameter non-linearities. However, the use of Look Up Tables (LUTs) can improve the robustness of the applied MTPA method independently of the motor parameter variation. LUTs allow the replacement of the non-linear relationship between the stator current and the torque load in the form of a table, which contains the pre-measured variables for online tracking [23]. In online methods, the MTPA control is optimized through online searching to attain the best operation point by calculating the angle γ of the stator current vector. During the online search, the maximum possible efficiency is also calculated as the angle γ is varied. This implies that the MTPA algorithm has to be repeatedly executed throughout the control period. As a consequence, the overall dynamic performance of the applied MTPA method may be significantly reduced compared to other MTPA methods [24,25,26,27].

Some of the proposed MTPA control methods are based on virtual signal injection methodology. In comparison to the high-frequency signal injection-based methods, the virtual signal injection allows a significant reduction of the introduced voltage and current harmonics, allowing the extraction of high-frequency signals without additional power losses. However, these MTPA methods use conventional digital encoders for obtaining position and speed information [28,29].

In addition, it is worthy to mention a completely different MTPA approach, which is called Extremum Seeking (ES) MTPA. This method is considered an application of ES that is essentially aimed at so-called real-time, non-model-based optimization. Here, a closed-loop scheme is implemented to adjust the MTPA operating point. The tracking of the MTPA point is attained without direct reference to the analytical model of the AC machine and its parameters. The ES MTPA method is based on the idea of applying a disturbance to the reference stator current vector and analyzing the corresponding response at the time of normal motor operation. Particularly, the obtained response is analyzed to find out the minimum current for a given load torque or, in reverse, to determine the maximum torque for a given stator current [30,31]. In the online MTPA methods, the estimation accuracy of the MTPA operation point is evaluated through the interaction with the AC machine. This denotes that the precise model of the AC machine and its parameters are not necessary in estimating the maximum-torque or minimum-current points during the drive operation. Nevertheless, most of the MTPA-seeking algorithms are carried out under the condition of constant or slowly varying load torque [32,33].

For applying effective control methodologies on AC machines, such as Field-Oriented Control (FOC), Direct Torque Control (DTC), etc., the rotor angular position is considered as very essential information, which is absolutely necessary in the required variable transformations. The accuracy of angle information can decisively affect the ability of the applied control strategy to attain the desired results. Conventionally, the rotor angular position is measured by means of optical encoders or resolvers attached to the rotor shaft in most variable-speed drive systems (VSDS). However, the usage of these sensors is related to several inherent disadvantages, such as an increase in drive system cost, a decrease in reliability, and additional noise problems. Numerous sensorless control methods have been proposed to overcome these issues by estimating both speed and position [34,35,36,37,38,39]. In addition to using MTPA methods, the application of sensorless control strategies can enhance the reliability of SynRM operation even in hostile working environments [40]. Particularly in the last decades, many research studies have proposed MTPA control in connection with sensorless control strategies aiming to further optimize the SynRM operation in a plethora of industrial applications. The sensorless control methods are mainly classified into two categories: the Fundamental Excitation Strategy and the Saliency and Signal Injection Strategy [41,42,43,44]. Sensorless methods based on fundamental excitation are mainly used in medium- and high-speed operations. In these methods, only fundamental variables are measured, such as stator currents and voltages, which are used to estimate speed and position by state observers. However, the state observers are sensitive to the variation of the motor parameters, affecting the estimation accuracy [45,46]. Saliency and Signal injection techniques are based on the magnetic saliency of the AC machine. Rotor speed and position are estimated by injecting test signals of high frequency [47,48,49,50,51,52,53,54]. These methods are applied in low-speed operation or even at standstill. Conversely, rotor angle and speed estimates at medium to high speeds can be accomplished with success using sensorless techniques based on the induced back-EMF [55,56]. Also, a number of sensorless methods make use of the Phase Locked Loop (PLL) methodology to track both angular speed and position [57]. In some studies, mixed sensorless control methods are developed and tested as combinations of PLL or extended EMF observers with the high-frequency signal injection (HFSI)-based methods [58,59]. Furthermore, Extended Kalman Filters (EKFs) are utilized as speed and position observers in sensorless control applications [60]. Nevertheless, the applied sensorless control might be considerably affected by the magnetic saturation. Several methods are proposed to address estimation issues of speed and position due to the magnetic saturation problem [61,62].

It is known that the estimation methods based on sliding mode observers (SMOs) are characterized by their embedded favorable features due to the sliding mode methodology. Among them are fast convergence, robustness against disturbances (external or internal), system unmodeled dynamics, parameter variation, and introduced noise [63,64,65,66,67]. In general, an undesirable effect called the chattering phenomenon is encountered in the implementation of sliding mode-based strategies. This is due to the high gain used in the applied discontinuous control inputs. Specifically, these gains must be large enough to ensure sliding mode operation in a wide speed range, including PMSM and SynRM operating modes even above the base speed ω_b. However, such large switching gains may negatively affect the observed signals, causing excessive ripples into them and finally making estimation of speed/position unfeasible. Since the EMF is directly proportional to rotor speed, its signal contains speed and position information. For SMO based on EMF, the introduced ripple affects the obtained EMF signals, distorting them to a great degree. The influence of the chattering phenomenon is more apparent in the low-speed range, since the EMF signals are becoming comparable to the introduced noise. Many SMO-based techniques have been proposed, causing the chattering reduction to succeed in the estimation of EMF with very low or almost no ripple [68,69,70,71,72,73]. Considering the operation and timing aspects of the sliding mode observers, the execution time of the estimation algorithm should be very short. However, in real systems, the implementation of such sensorless control requires a powerful microcontroller or DSP board equipped with high-speed peripherals (e.g., an A/D converter). This usually leads to a raise in the price of the overall motor drive system [74,75,76].

In the proposed control method, the entire controller–observer algorithm is advantageously expressed in the γδ estimated reference frame (see Figure 1). In addition, the SynRM model structure in the γδ frame is modified and developed suitably, adapting terms such as modified rotor flux and modified EMF that include angle and speed information. The so-called modified rotor flux is a virtual flux term introduced to simplify, unify, and enhance the analysis of the SynRM model and the associated observers. A main advantage of the present sensorless MTPA strategy is the use of a simple MTPA speed controller including an AWC in connection with a robust SMO. After operating in sliding mode, the obtained equivalent control inputs of SMO are used by the modified EMF observer for achieving high-accuracy speed/position estimations [77]. It is worth mentioning that the speed/torque controller is based on a simple torque/current decoupling, in which the required stator current is analyzed into its components, i_γ and i_δ, to attain the maximum torque for a given stator current I_s. The performance of the proposed sensorless MTPA technique is evaluated through simulations using Simulink/MATLAB in several control scenarios.

The rest of the paper is organized as follows: In Section 2, the constant torque and MTPA operations of SynRM are analytically presented. The modified SynRM model is analyzed in dq and γδ rotating reference frames, and the application of MTPA is presented in Section 3. In the same section, the applied SMO and modified EMF observer are described, as well, including the current control and torque decoupling. Additionally, Section 4 presents the simulation setup, results, and evaluation for several speeds and external torque scenarios. Also, Section 5 discuss aspects of the results and future applications. Finally, Section 6 concludes the present work.

2. Analysis of Maximum Torque Per Ampere Operation for SynRM

2.1. An Alternative Mathematical Model of SynRM in dq

Conventionally, sensor-based methods use the dq synchronous rotating reference frame to express the SynRM mathematical model for high-performance control, e.g., Field Oriented Control (FOC) or Direct Torque Control (DTC). The following stator voltage and flux/current equations define the SynRM mathematical model in the dq synchronous rotating reference frame.

Figure 1. Block diagram of SynRM sensorless control based on sliding mode observer (SMO) for speed/angle and applying MTPA control.

Figure 1. Block diagram of SynRM sensorless control based on sliding mode observer (SMO) for speed/angle and applying MTPA control.

$$u_{dq}=r_s{i}_{dq}+\omega H_s{\lambda }_{dq}+{\dot{\lambda }}_{dq}$$

(1)

$${\lambda }_{dq}=L_{dq}{i}_{dq}$$

(2)

where the matrices of stator voltages, currents, and fluxes of SynRM are determined by:

$$u_{dq}=\left[\begin{array}{c}{u}_d\\ u_q\end{array}\right],\hspace{0.33em}{i}_{dq}=\left[\begin{array}{c}{i}_d\\ i_q\end{array}\right],\hspace{0.33em}{\lambda }_{dq}=\left[\begin{array}{c}{\lambda }_d\\ {\lambda }_q\end{array}\right]$$

(3)

Also, the SynRM impedance matrix L_dq in Equation (2) is determined by:

$$L_{dq}=\left[\begin{array}{cc}{L}_d& 0\\ 0& L_q\end{array}\right]$$

(4)

In addition, the H_s in Equation (1) represents a 2 × 2 skew symmetric matrix, i.e., H_s = −H_s^T, which is defined as follows:

$$H_s=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]$$

(5)

Note that any term of the form $\dot{z}$ implies the first-time derivative of a variable z, i.e., $\dot{z}=dz/dt$.

By substituting Equation (4) into Equation (2) and subsequently adding the term (L_qi_q − L_qi_q) = 0 to the d-axis flux λ_d, the stator flux matrix λ_dq could be equivalently rewritten as follows:

$$\begin{array}{l}\left(2\right)\stackrel{\left(4\right)}{\to }{\lambda }_{dq}=\left[\begin{array}{c}{\lambda }_d\\ {\lambda }_q\end{array}\right]=\left[\begin{array}{cc}{L}_d& 0\\ 0& L_q\end{array}\right]\left[\begin{array}{c}{i}_d\\ i_q\end{array}\right]\iff {\lambda }_{dq}=\left[\begin{array}{cc}{L}_q+\left(L_d-L_q\right)& 0\\ 0& L_q\end{array}\right]\left[\begin{array}{c}{i}_d\\ i_q\end{array}\right]\\ =\left[\begin{array}{cc}{L}_q& 0\\ 0& L_q\end{array}\right]\left[\begin{array}{c}{i}_d\\ i_q\end{array}\right]+\left[\begin{array}{cc}\left(L_d-L_q\right)& 0\\ 0& 0\end{array}\right]\left[\begin{array}{c}{i}_d\\ i_q\end{array}\right]=L_q\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\left[\begin{array}{c}{i}_d\\ i_q\end{array}\right]+\left[\begin{array}{c}\left(L_d-L_q\right)i_d\\ 0\end{array}\right]=L_q\left[\begin{array}{c}{i}_d\\ i_q\end{array}\right]+\left[\begin{array}{c}{\lambda }_{ms}\\ 0\end{array}\right]=L_{qq}{i}_{dq}+{\lambda }_{msdq}\end{array}$$

(6)

In Equation (6), the inductance matrix L_qq is a 2 × 2 symmetric matrix with the diagonal elements equal to L_q. Also, λ_msdq and λ_ms represent the so-called modified rotor flux matrix and modified rotor flux, respectively. The λ_msdq, λ_ms, and the inductance matrix L_qq are defined as follows:

$$\left.\begin{array}{l}{\lambda }_{msdq}=\left[\begin{array}{c}\left(L_d-L_q\right)i_d\\ 0\end{array}\right]=\left[\begin{array}{c}{\lambda }_{ms}\\ 0\end{array}\right]\hspace{0.33em},\hspace{0.33em}with\hspace{0.33em}{L}_{qq}=\left[\begin{array}{cc}{L}_q& 0\\ 0& L_q\end{array}\right]\\ {\lambda }_{ms}={\lambda }_{msd}=\left(L_d-L_q\right)i_d\hspace{0.33em}\end{array}\right\}$$

(7)

Essentially, the modified rotor flux λ_ms is a virtual flux, which incorporates the saliency term of SynRM, (L_d − L_q), with the aim of supporting a simpler SynRM model and a more convenient observer design. In addition, the corresponding modified EMF, E_dq, and produced torque, T_e, are derived by:

$$E_{dq}=\left[\begin{array}{c}{E}_d\\ E_q\end{array}\right]=-\omega \left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]\left[\begin{array}{c}{\lambda }_{ms}\\ 0\end{array}\right]=\left[\begin{array}{c}0\\ -\omega {\lambda }_{ms}\end{array}\right]=\left[\begin{array}{c}0\\ -\omega \left(L_d-L_q\right)i_d\end{array}\right]$$

(8)

and

$$T_e=T_{rel}={\displaystyle \frac{3p}{2}}\left({\lambda }_d{i}_q-{\lambda }_q{i}_d\right)={\displaystyle \frac{3p}{2}}\left(L_d{i}_d{i}_q-L_q{i}_q{i}_d\right)={\displaystyle \frac{3p}{2}}\left(L_d-L_q\right)i_d{i}_q={\displaystyle \frac{3p}{2}}{\lambda }_{ms}{i}_q$$

(9)

Equation (9) implies that the produced torque T_e is equal to the reluctance torque T_rel, which is directly proportional to the product of the q-axis current, i_q, and the modified rotor flux, λ_ms = (L_d − L_q)i_d. Considering the magnetic flux, EMF, and torque defined in (6), (7), (8), and (9), the modified SynRM model is also expressed in a similar way to the modified PMSM model, maintaining the advantage of regulating the modified rotor flux λ_ms through the current i_d [50,53].

It is mentioned that the idea of a modified rotor flux λ_ms seems similar to the so-called Active Flux concept [1,2]. Also, the modified rotor flux λ_ms is similar to the flux term included in the Extended EMF [38,39,43]. Essentially, both modified rotor flux and active flux are virtual flux terms that are defined in the same way by embedding the saliency flux term, (L_d − L_q)i_d, into the d-axis flux. As a result, the application of the modified rotor flux or active flux has the common characteristic to allow transforming the model of any salient AC machine into an equivalent model of a non-salient one [1,2,19,38,39]. In comparison to the modified flux/current and EMF observers that utilize the modified rotor flux and are expressed in the γδ estimated frame, the active flux observer is primarily based on a voltage model expressed in the αβ static frame. Moreover, the sensorless strategy based on modified rotor flux is primarily focused on isolating the terms of the modified SynRM model (current/flux and EMF) that contain the information of the angle error between the dq and γδ rotating frames. In contrast, the sensorless methods based on active flux estimate the speed/angle directly via a stator flux observer. However, the SynRM modified rotor flux λ_ms is an additional definition that has been independently proposed for model-based sensorless control in the γδ estimated frame. Initially, the modified rotor flux was introduced to rearrange or modify the EMF components of PMSM/SynRM in dq and γδ reference frames. The idea of the modified rotor flux is mainly aimed at developing the proper SynRM voltage, flux, and EMF models that allow the speed and position estimation for sensorless control adopting simple assumptions. Specifically, the PMSM/SynRM modified model in γδ frame is very significant in designing and implementing the modified state observers for the EMF, speed, and position estimation [19,50,53,77]. In the present work, this modification is aimed at simplifying the analysis of the SynRM dq and γδ models and afterwards designing and developing stable and accurate SynRM observers in the γδ reference frame [19,77].

2.2. Constant Torque Curves of SynRM

Assuming that the electric torque T_e is constant, Equation (9) could be rewritten as the constant product of the current i_q and the modified rotor flux λ_sm = (L_d − L_q)i_d, i.e.,

$$\left(9\right)\stackrel{\left(T_e=T_{ec}=cnst.\right)}{\to }{T}_e=T_{ec}={\displaystyle \frac{3p}{2}}\left(L_d-L_q\right)i_d{i}_q={\displaystyle \frac{3p}{2}}{\lambda }_{ms}{i}_q=const.$$

(10)

Since the product (L_d − L_q)i_di_q is constant, the two variables i_d and i_q are inversely proportional. In obtaining a more convenient formula in the i_d-i_q plane, both parts in Equation (10) are divided by [2/3p(L_d − L_q)]. As a result, Equation (10) is transformed as follows:

$$(10)\stackrel{T_e=T_{ec}}{\to }{i}_d{i}_q=\left[{\displaystyle \frac{2T_{ec}}{3p\left(L_d-L_q\right)}}\right]=k=const.$$

(11)

Equation (11) represents graphically a rectangular hyperbola curve in the i_d-i_q plane, since i_di_q = k = const. Particularly, the geometrical loci of constant torque points are depicted by a hyperbola, the so-called constant torque curve (see Figure 2). For a given torque, T_e = T_ec = 0 (or k = 0), neither i_d nor i_q are equal to zero, indicating that the constant torque curve never crosses either of the axes i_d = 0 and i_q = 0. The corresponding constant torque curve might be close to the current axes, i.e., i_d = 0 or i_q = 0, but it will not meet them. Considering the asymptotes of the constant torque curve, they are defined by:

$$\left.\begin{array}{l}{i}_d=0\\ i_q=0\end{array}\right\}$$

(12)

Regarding the produced torque T_e in Equation (9), this is obtaining its maximum value T_e = T_emax for a given current I_s, if the product of the stator current components i_d and i_q is maximum, as well, i.e.,

$$(9)\stackrel{T_e=T_{e\max }}{\to }{T}_{e\max }={\displaystyle \frac{3p}{2}}\left(L_d-L_q\right){\left(i_d{i}_q\right)}_{\max }$$

(13)

In the next paragraphs, it is discussed that there is only one point of the constant torque curve satisfying the maximum torque condition per current. A graph of three constant torques, T_e1, T_e2, and T_e3, is shown in Figure 2, with T_e1 < T_e2 < T_e3. In general, the product of i_d and i_q equals the constant k = [2T_ec/3p(L_d − L_q)] at each point on the constant torque curve T_ec (see Figure 2). Inspecting Figure 2, it is observed that the torque constant curves are symmetric to the bisector (0B) of the corresponding angle, while the operating points on the line 0B denote the equality of the corresponding currents, e.g., i_dA = i_qA at point A. As it is verified in the next paragraphs, the line 0B represents the Maximum Torque Per Ampere (MTPA) operation in the i_d-i_q plane.

2.3. Maximum Torque Per Ampere (MTPA) Control

MTPA operation denotes that the stator current vector is controlled so as to produce the maximum torque per stator current operating in the speed region below the so-called base speed ω_b (see Figure 2, point B). Let us consider a SynRM whose inductances are connected by L_d > L_q. Applying positive d-axis current i_d contributes to positive d-axis stator flux λ_d and modified rotor flux λ_sm. As a consequence, the produced torque T_e is positive as well for positive q-axis current i_q. Considering the relation between the produced torque T_e, the stator current I_s, and its components in dq, i_d, and i_q, Equation (9) can be written in the following form:

$$\left.\begin{array}{l}{I}_s^2=\left(i_d^2+i_q^2\right)\iff i_d^2=I_s^2-i_q^2\Rightarrow i_d=\sqrt{\left(I_s^2-i_q^2\right)}\\ T_e={\displaystyle \frac{3p}{2}}\left(L_d-L_q\right)i_d{i}_q\end{array}\right\}\Rightarrow T_e={\displaystyle \frac{3p}{2}}\left(L_d-L_q\right)i_q\sqrt{\left(I_s^2-i_q^2\right)}$$

(14)

Figure 2. SynRM operating in MTPA mode in i_d-i_q plane (line 0AB).

Figure 2. SynRM operating in MTPA mode in i_d-i_q plane (line 0AB).

Taking the first-time derivative of the torque from Equation (14) with respect to i_q and setting dT_e/di_q = 0 and d²T_e/di_q² < 0, the q-axis current i_q for MTPA control is determined by:

$$\begin{array}{l}{\displaystyle \frac{d{T}_e}{d{i}_q}}={\displaystyle \frac{3p}{2}}\left(L_d-L_q\right){\displaystyle \frac{d}{d{i}_q}}\left[i_q\sqrt{\left(I_s^2-i_q^2\right)}\right]=0\\ \Rightarrow {\displaystyle \frac{d{T}_e}{d{i}_q}}={\displaystyle \frac{3p}{2}}\left(L_d-L_q\right)\left[\sqrt{\left(I_s^2-i_q^2\right)}+i_q{\displaystyle \frac{1}{2}}\left(-2i_q\right){\displaystyle \frac{1}{\sqrt{\left(I_s^2-i_q^2\right)}}}\right]=0\\ \Rightarrow {\displaystyle \frac{d{T}_e}{d{i}_q}}={\displaystyle \frac{3p}{2}}\left(L_d-L_q\right)\left[\sqrt{\left(I_s^2-i_q^2\right)}+{\displaystyle \frac{-i_q^2}{\sqrt{\left(I_s^2-i_q^2\right)}}}\right]=0\\ \Rightarrow {\displaystyle \frac{d{T}_e}{d{i}_q}}={\displaystyle \frac{3p}{2}}\left(L_d-L_q\right)\left[{\displaystyle \frac{\left(I_s^2-i_q^2\right)-i_q^2}{\sqrt{\left(I_s^2-i_q^2\right)}}}\right]=0\\ \Rightarrow {\displaystyle \frac{d{T}_e}{d{i}_q}}={\displaystyle \frac{3p}{2}}\left(L_d-L_q\right)\left[{\displaystyle \frac{\left(I_s^2-2i_q^2\right)}{\sqrt{\left(I_s^2-i_q^2\right)}}}\right]=0\\ \Rightarrow \left(I_s^2-2i_q^2\right)=0\Rightarrow I_s^2=2i_q^2\Rightarrow i_q=\pm \sqrt{{\displaystyle \frac{I_s^2}{2}}}\Rightarrow i_q={\displaystyle \frac{I_s}{\sqrt{2}}}=i_d\end{array}$$

(15)

In Equation (14), the i_d current is chosen to be positive to obtain a positive torque T_e. Using Equation (15), and considering that the second-time derivative of torque T_e should be negative in MTPA mode, i.e., d²T_e/di_q² < 0, it will be:

$$\begin{array}{l}{\displaystyle \frac{d^2{T}_e}{{di}_q^2}}={\displaystyle \frac{3p}{2}}\left(L_d-L_q\right){\displaystyle \frac{d}{d{i}_q}}\left\{\left[{\displaystyle \frac{\left(I_s^2-2i_q^2\right)}{\sqrt{\left(I_s^2-i_q^2\right)}}}\right]\right\}\\ \Rightarrow {\displaystyle \frac{d^2{T}_e}{d^2{i}_q^2}}={\displaystyle \frac{3p}{2}}\left(L_d-L_q\right)\left\{\left[{\displaystyle \frac{\sqrt{\left(I_s^2-i_q^2\right)}\frac{d}{d{i}_q}\left(I_s^2-2i_q^2\right)-\left(I_s^2-2i_q^2\right)\frac{d}{d{i}_q}\sqrt{\left(I_s^2-i_q^2\right)}}{{\left[\sqrt{\left(I_s^2-i_q^2\right)}\right]}^2}}\right]\right\}\\ \Rightarrow {\displaystyle \frac{d^2{T}_e}{{di}_q^2}}={\displaystyle \frac{3p}{2}}\left(L_d-L_q\right)\left\{\left[{\displaystyle \frac{\left(-4i_q\right)\sqrt{\left(I_s^2-i_q^2\right)}-\left(I_s^2-2i_q^2\right)\frac{-i_q}{\sqrt{\left(I_s^2-i_q^2\right)}}}{{\left[\sqrt{\left(I_s^2-i_q^2\right)}\right]}^2}}\right]\right\}\\ \Rightarrow {\displaystyle \frac{d^2{T}_e}{{di}_q^2}}={\displaystyle \frac{3p}{2}}\left(L_d-L_q\right)\left\{\left[{\displaystyle \frac{\left(-4i_q\right)\left(I_s^2-i_q^2\right)-\left(I_s^2-2i_q^2\right)\left(-i_q\right)}{{\left[\sqrt{\left(I_s^2-i_q^2\right)}\right]}^2\sqrt{\left(I_s^2-i_q^2\right)}}}\right]\right\}\\ \Rightarrow {\displaystyle \frac{d^2{T}_e}{{di}_q^2}}={\displaystyle \frac{3p}{2}}\left(L_d-L_q\right)\left\{\left[{\displaystyle \frac{\left(-4i_q{I}_s^2+4i_q^3\right)+\left(i_q{I}_s^2-2i_q^3\right)}{{\left[\sqrt{\left(I_s^2-i_q^2\right)}\right]}^2\sqrt{\left(I_s^2-i_q^2\right)}}}\right]\right\}\Rightarrow {\displaystyle \frac{d^2{T}_e}{{di}_q^2}}={\displaystyle \frac{3p}{2}}\left(L_d-L_q\right)\left\{\left[{\displaystyle \frac{\left(-3i_q{I}_s^2+2i_q^3\right)}{{\left[\sqrt{\left(I_s^2-i_q^2\right)}\right]}^3}}\right]\right\}\\ \\ \left(15\right)\stackrel{MTPA}{\to }\left|\left.\begin{array}{c}{I}_s^2=2i_q^2\\ and\hspace{0.33em}{i}_q>0\end{array}\right\}\right.\hspace{0.33em}\Rightarrow {\displaystyle \frac{d^2{T}_e}{{di}_q^2}}={\displaystyle \frac{3p}{2}}\left(L_d-L_q\right)\left\{\left[{\displaystyle \frac{i_q\left(-3I_s^2+I_s^2\right)}{{\left[\sqrt{\left(I_s^2-i_q^2\right)}\right]}^3}}\right]\right\}={\displaystyle \frac{3p}{2}}\left(L_d-L_q\right)\left\{\left[{\displaystyle \frac{i_q\left(-2I_s^2\right)}{{\left[\sqrt{\left(I_s^2-i_q^2\right)}\right]}^3}}\right]\right\}<0\\ \Rightarrow {\displaystyle \frac{d^2{T}_e}{{di}_q^2}}=-{\displaystyle \frac{3p}{2}}\left(L_d-L_q\right)\left\{\left[{\displaystyle \frac{2I_s^2{i}_q}{{\left[\sqrt{\left(I_s^2-i_q^2\right)}\right]}^3}}\right]\right\}<0\hspace{0.33em}for\hspace{0.33em}{i}_q>0\end{array}$$

(16)

The inequality in (16) is satisfied for i_q > 0 and (L_d − L_q) > 0. Therefore, for a given constant current I_s and operating in MTPA mode, it will also be:

$$\left.\begin{array}{l}{\displaystyle \frac{d{T}_e}{d{i}_{qMTPA}}}=0\\ {\displaystyle \frac{d^2{T}_e}{d{i}_{qMTPA}^2}}<0\end{array}\right\}\left(15\right)\stackrel{\left(16\right)}{\to }{i}_{qMTPA}=i_{dMTPA}=\sqrt{\left(I_s^2-i_{qMTPA}^2\right)}={\displaystyle \frac{I_s}{\sqrt{2}}}$$

(17)

Equation (17) is of first order and represents a line in the i_d-i_q plane, i.e., the locus of MTPA points is represented by the straight line 0B with i_dMTPA = i_qMTPA (see Figure 2). According to the previous analysis, it is shown that the MTPA line 0B coincides with the angle bisector, while the constant torque curves are also symmetric to the line 0B (see Figure 2, Equations (10), (11), and (17)). Additionally, it can be observed that the current circles are tangent to the corresponding constant torque curves at the points where the MTPA line intersects these curves. A point on the line 0B is characterized by the production of maximum SynRM torque for a given magnitude of stator current I_s (see Figure 2, points A and B, T_e2 > T_e1). Due to the restrictions on the stator windings and inverter switching elements, the stator current I_s is limited by the maximum current I_sm, which is the upper bound of the stator current. Given that the stator current magnitude I_s cannot exceed its maximum allowed value I_sm, the following relationship holds true:

$$I_s=\sqrt{i_d^2+i_q^2}\le I_{sm}$$

(18)

The current constraint in (18) represents a circle in the i_d-i_q plane with a radius of I_sm and its center located at point (0,0) (see Figure 2, current limit circle). Any operating point of the SynRM should be inside this circle. Using (18) under MTPA operation implies that the currents i_dMTPA and i_qMTPA are related to I_s and I_sm as follows:

$$0\le i_{qMTPA}=i_{dMTPA}={\displaystyle \frac{I_s}{\sqrt{2}}}\le {\displaystyle \frac{I_{sm}}{\sqrt{2}}}$$

(19)

Essentially, the MTPA control is limited to segment 0B shown in Figure 2, where any operating point of the 0B line satisfies (17) and the conditions in (18) and (19). Particularly, the maximum current circle tangents the constant torque curve T_e3 and intersects the MTPA line at the point B. The operating speed of SynRM at the point B is the so-called base speed ω_b (see Figure 2).

3. Design of Maximum Torque Per Ampere Algorithm Applied on Sensorless Control for SynRM

3.1. SynRM Mathematical Model in γδ Reference Frame

In sensorless control, the SynRM mathematical model in the dq system could not be practically applied directly, since the rotor position is not actually measured but is estimated. In the present sensorless control scheme, the SynRM mathematical model is expressed in the γδ estimated reference frame, and a sliding mode observer is designed to provide the speed and position estimation expressed in the γδ estimated reference frame, as well [50,53]. The γδ estimated reference frame is an arbitrary frame rotating at estimated speed $\widehat{\omega }$ and lagging behind the dq synchronous reference frame by an electrical angle $\Delta \theta =\overline{\theta }=\theta -\widehat{\theta }$, which is the angle difference between the dq and γδ rotating reference frames. Figure 3 demonstrates the modified rotor flux vector in the αβ, dq, and γδ reference frames [50]. The transformation of SynRM variables, such as voltages, currents, fluxes, etc., from the dq synchronous to the γδ estimated reference frame is accomplished by utilizing the transformation matrix K_Δθ defined by:

$$K_{\Delta \theta }=\left[\begin{array}{cc}\cos \overline{\theta }& -\sin \overline{\theta }\\ \sin \overline{\theta }& \cos \overline{\theta }\end{array}\right]$$

(20)

Obviously, the K_Δθ depends only on the angle difference between the rotating reference frames dq and γδ [53,77].

The γδ model of SynRM could be obtained by multiplying the corresponding dq model from the left with K_δθ [77]. More specifically, the voltage and flux/current equations of the γδ SynRM model are respectively expressed as follows:

$$\begin{array}{l}{K}_{\Delta \theta }{u}_{dq}=K_{\Delta \theta }{r}_s{i}_{dq}+K_{\Delta \theta }\omega H_s{\lambda }_{dq}+K_{\Delta \theta }{\dot{\lambda }}_{dq}\iff u_{\gamma \delta }=r_s{i}_{\gamma \delta }+\omega H_s{\lambda }_{\gamma \delta }+\left(-\dot{\overline{\theta }}{H}_s{\lambda }_{\gamma \delta }+{\dot{\lambda }}_{\gamma \delta }\right)\iff \\ u_{\gamma \delta }=r_s{i}_{\gamma \delta }+\omega H_s{\lambda }_{\gamma \delta }+\left(-\dot{\overline{\theta }}{H}_s{\lambda }_{\gamma \delta }+{\dot{\lambda }}_{\gamma \delta }\right)\iff u_{\gamma \delta }=r_s{i}_{\gamma \delta }+\widehat{\omega }{H}_s{\lambda }_{\gamma \delta }+{\dot{\lambda }}_{\gamma \delta }\end{array}$$

(21)

and

$$K_{\Delta \theta }{\lambda }_{dq}=K_{\Delta \theta }{L}_{qq}{K}_{\Delta \theta }^{-1}{K}_{\Delta \theta }{i}_{dq}+K_{\Delta \theta }{\lambda }_{msdq}\iff {\lambda }_{\gamma \delta }=L_{qq}{i}_{\gamma \delta }+{\lambda }_{ms\gamma \delta }$$

(22)

In (22), the term λ_msγδ refers to the modified rotor flux of SynRM in the γδ reference frame, which is defined by:

$${\lambda }_{ms\gamma \delta }=K_{\Delta \theta }{\lambda }_{msdq}=\left[\begin{array}{cc}\cos \overline{\theta }& -\sin \overline{\theta }\\ \sin \overline{\theta }& \cos \overline{\theta }\end{array}\right]{\lambda }_{msdq}=\left[\begin{array}{cc}\cos \overline{\theta }& -\sin \overline{\theta }\\ \sin \overline{\theta }& \cos \overline{\theta }\end{array}\right]\left(L_d-L_q\right)i_d\left[\begin{array}{c}1\\ 0\end{array}\right]=\left(L_d-L_q\right)i_d\left[\begin{array}{c}\cos \overline{\theta }\\ \sin \overline{\theta }\end{array}\right]={\lambda }_{ms}\left[\begin{array}{c}\cos \overline{\theta }\\ \sin \overline{\theta }\end{array}\right]=\left[\begin{array}{c}{\lambda }_{ms}\cos \overline{\theta }\\ {\lambda }_{ms}\sin \overline{\theta }\end{array}\right]$$

(23)

Also, Equations (21) and (22) are written more analytically as follows:

$$u_{\gamma \delta }=\left[\begin{array}{c}{u}_{\gamma }\\ u_{\delta }\end{array}\right]=r_s\left[\begin{array}{c}{i}_{\gamma }\\ i_{\delta }\end{array}\right]+\widehat{\omega }{J}_s\left[\begin{array}{c}{\lambda }_{\gamma }\\ {\lambda }_{\delta }\end{array}\right]+\left[\begin{array}{c}{\dot{\lambda }}_{\gamma }\\ {\dot{\lambda }}_{\delta }\end{array}\right]\iff \left[\begin{array}{c}{\dot{\lambda }}_{\gamma }\\ {\dot{\lambda }}_{\delta }\end{array}\right]=\left[\begin{array}{c}{u}_{\gamma }\\ u_{\delta }\end{array}\right]-r_s\left[\begin{array}{c}{i}_{\gamma }\\ i_{\delta }\end{array}\right]-\widehat{\omega }{J}_s\left[\begin{array}{c}{\lambda }_{\gamma }\\ {\lambda }_{\delta }\end{array}\right]$$

(24)

and

$${\lambda }_{\gamma \delta }=\left[\begin{array}{c}{\lambda }_{\gamma }\\ {\lambda }_{\delta }\end{array}\right]=\left[\begin{array}{cc}{L}_q& 0\\ 0& L_q\end{array}\right]\left[\begin{array}{c}{i}_{\gamma }\\ i_{\delta }\end{array}\right]+{\lambda }_{sm}\left[\begin{array}{c}\cos \overline{\theta }\\ \sin \overline{\theta }\end{array}\right]=L_q\left[\begin{array}{c}{i}_{\gamma }\\ i_{\delta }\end{array}\right]+{\lambda }_{sm}\left[\begin{array}{c}\cos \overline{\theta }\\ \sin \overline{\theta }\end{array}\right]$$

(25)

After substituting Equations (23) and (25) into Equation (24) and solving for L_q(di_γδ/dt), it results:

$$\begin{array}{l}(24)\stackrel{(23),\hspace{0.33em}\left(25\right)}{\to }\left[\begin{array}{c}{\dot{\lambda }}_{\gamma }\\ {\dot{\lambda }}_{\delta }\end{array}\right]=\left[\begin{array}{c}{u}_{\gamma }\\ u_{\delta }\end{array}\right]-r_s\left[\begin{array}{c}{i}_{\gamma }\\ i_{\delta }\end{array}\right]-\widehat{\omega }{H}_s\left[\begin{array}{c}{\lambda }_{\gamma }\\ {\lambda }_{\delta }\end{array}\right]\iff L_q\left[\begin{array}{c}{\dot{i}}_{\gamma }\\ {\dot{i}}_{\delta }\end{array}\right]=\left[\begin{array}{c}{u}_{\gamma }\\ u_{\delta }\end{array}\right]-r_s\left[\begin{array}{c}{i}_{\gamma }\\ i_{\delta }\end{array}\right]-\widehat{\omega }{H}_s\left[\begin{array}{c}{\lambda }_{\gamma }\\ {\lambda }_{\delta }\end{array}\right]-{\displaystyle \frac{d}{dt}}\left\{{\lambda }_{ms}\left[\begin{array}{c}\cos \overline{\theta }\\ \sin \overline{\theta }\end{array}\right]\right\}\\ =\left[\begin{array}{c}{u}_{\gamma }\\ u_{\delta }\end{array}\right]-r_s\left[\begin{array}{c}{i}_{\gamma }\\ i_{\delta }\end{array}\right]+\widehat{\omega }{H}_s\left\{L_q\left[\begin{array}{c}{i}_{\gamma }\\ i_{\delta }\end{array}\right]+{\lambda }_{ms}\left[\begin{array}{c}\cos \overline{\theta }\\ \sin \overline{\theta }\end{array}\right]\right\}-\left(L_d-L_q\right){\dot{i}}_d\left[\begin{array}{c}\cos \overline{\theta }\\ \sin \overline{\theta }\end{array}\right]+\overline{\omega }{\lambda }_{ms}\left[\begin{array}{c}\sin \overline{\theta }\\ -\cos \overline{\theta }\end{array}\right]\\ =\left[\begin{array}{c}{u}_{\gamma }\\ u_{\delta }\end{array}\right]-r_s\left[\begin{array}{c}{i}_{\gamma }\\ i_{\delta }\end{array}\right]+\widehat{\omega }{H}_s{L}_q\left[\begin{array}{c}{i}_{\gamma }\\ i_{\delta }\end{array}\right]+\widehat{\omega }{H}_s{\lambda }_{ms}\left[\begin{array}{c}\cos \overline{\theta }\\ \sin \overline{\theta }\end{array}\right]-\left(L_d-L_q\right){\dot{i}}_d\left[\begin{array}{c}\cos \overline{\theta }\\ \sin \overline{\theta }\end{array}\right]+\overline{\omega }{H}_s{\lambda }_{ms}\left[\begin{array}{c}\cos \overline{\theta }\\ \sin \overline{\theta }\end{array}\right]\\ =\left[\begin{array}{c}{u}_{\gamma }\\ u_{\delta }\end{array}\right]-r_s\left[\begin{array}{c}{i}_{\gamma }\\ i_{\delta }\end{array}\right]+\omega H_s{L}_q\left[\begin{array}{c}{i}_{\gamma }\\ i_{\delta }\end{array}\right]+\left(\widehat{\omega }+\overline{\omega }\right)H_s{\lambda }_{ms}\left[\begin{array}{c}\cos \overline{\theta }\\ \sin \overline{\theta }\end{array}\right]-\left(L_d-L_q\right){\dot{i}}_d\left[\begin{array}{c}\cos \overline{\theta }\\ \sin \overline{\theta }\end{array}\right]\\ =\left[\begin{array}{c}{u}_{\gamma }\\ u_{\delta }\end{array}\right]-r_s\left[\begin{array}{c}{i}_{\gamma }\\ i_{\delta }\end{array}\right]+\widehat{\omega }{L}_q{J}_s\left[\begin{array}{c}{i}_{\gamma }\\ i_{\delta }\end{array}\right]+\omega H_s{\lambda }_{ms}\left[\begin{array}{c}\cos \overline{\theta }\\ \sin \overline{\theta }\end{array}\right]-\left(L_d-L_q\right){\dot{i}}_d\left[\begin{array}{c}\cos \overline{\theta }\\ \sin \overline{\theta }\end{array}\right]=u_{\gamma \delta }-r_s{i}_{\gamma \delta }+\widehat{\omega }{L}_q{H}_s{i}_{\gamma \delta }+E_{\gamma \delta }+D_{\gamma \delta }\end{array}$$

(26)

The matrices E_γδ and D_γδ represent the modified EMF and the corresponding disturbances of modified EMF in the γδ reference frame. These are specified by:

$$E_{\gamma \delta }=\left[\begin{array}{c}{E}_{\gamma }\\ E_{\delta }\end{array}\right]=\omega H_s{\lambda }_{ms}\left[\begin{array}{c}\cos \overline{\theta }\\ \sin \overline{\theta }\end{array}\right]=\omega H_s{\lambda }_{ms\gamma \delta }$$

(27)

and

$$D_{\gamma \delta }=\left[\begin{array}{c}{D}_{\gamma }\\ D_{\delta }\end{array}\right]=-\left(L_d-L_q\right){\dot{i}}_d\left[\begin{array}{c}\cos \overline{\theta }\\ \sin \overline{\theta }\end{array}\right]\cong 0$$

(28)

It is observed that the disturbance terms D_γ and D_δ in Equation (28) depend on the saliency, the first-time derivative of the current i_d, and the angle error. Supposing that the i_d-current controller exhibits a rapid convergence, di_d/dt becomes too small compared to the corresponding EMF components, E_γ and E_δ. As a result, the size of the disturbances D_γ and D_δ might be deemed negligible, i.e., equal to zero. Therefore, the first-time derivative of the modified EMF, dE_γδ/dt, is expressed in a simpler and more convenient form as follows:

$$\left(27\right)\stackrel{D_{\gamma }\cong 0,\hspace{0.33em}{D}_{\delta }\cong 0,\hspace{0.33em}\dot{\overline{\theta }}=\overline{\omega }}{\to }{\displaystyle \frac{d{E}_{\gamma \delta }}{dt}}={\dot{E}}_{\gamma \delta }=\left[\begin{array}{c}{\dot{E}}_{\gamma }\\ {\dot{E}}_{\delta }\end{array}\right]\cong \left[\begin{array}{c}-\dot{\overline{\theta }}\omega \left(L_d-L_q\right)i_d\cos \overline{\theta }\\ -\dot{\overline{\theta }}\omega \left(L_d-L_q\right)i_d\sin \overline{\theta }\end{array}\right]=\left[\begin{array}{c}-\overline{\omega }{E}_{\delta }\\ \overline{\omega }{E}_{\gamma }\end{array}\right]=\overline{\omega }{H}_s{E}_{\gamma \delta }$$

(29)

3.2. Design of Sliding Mode Observer (SMO) for SynRM Current/Flux in γδ Reference Frame

In designing the SMO, there are two main steps to be implemented: the selection of the sliding surfaces and the choice of the control inputs [53,77]. The proposed flux/current sliding mode observer is of first order and defined by:

$$L_q\left[\begin{array}{c}{\dot{\widehat{i}}}_{\gamma }\\ {\dot{\widehat{i}}}_{\delta }\end{array}\right]=\left[\begin{array}{c}{u}_{\gamma }\\ u_{\delta }\end{array}\right]-{\widehat{r}}_s\left[\begin{array}{c}{i}_{\gamma }\\ i_{\delta }\end{array}\right]+\widehat{\omega }{L}_q\left[\begin{array}{c}{i}_{\delta }\\ -i_{\gamma }\end{array}\right]+\left[\begin{array}{c}0\\ {\widehat{E}}_{\delta }\end{array}\right]+\left[\begin{array}{c}{k}_{\gamma }\mathrm{sgn}{s}_{\gamma }\\ k_{\delta }\mathrm{sgn}{s}_{\delta }\end{array}\right]$$

(30)

In Equation (30), the γ-axis component of the estimated modified EMF is set equal to zero. Also, k_γ > 0 and k_δ > 0 represent the observer gains, while s_γ and s_δ are the sliding surfaces, which are based on current errors and define the sliding manifold as follows:

$$s_{\gamma \delta }=\left[\begin{array}{l}{s}_{\gamma }\\ s_{\delta }\end{array}\right]=\left[\begin{array}{l}{\overline{i}}_{\gamma }\\ {\overline{i}}_{\delta }\end{array}\right]$$

(31)

After subtracting (30) from (26) by parts, the observer dynamics is written as:

$$L_q\left[\begin{array}{c}{\dot{\overline{i}}}_{\gamma }\\ {\dot{\overline{i}}}_{\delta }\end{array}\right]=-{\overline{r}}_s\left[\begin{array}{c}{i}_{\gamma }\\ i_{\delta }\end{array}\right]+\left[\begin{array}{c}{E}_{\gamma }\\ {\overline{E}}_{\delta }\end{array}\right]-\left[\begin{array}{c}{k}_{\gamma }\mathrm{sgn}{s}_{\gamma }\\ k_{\delta }\mathrm{sgn}{s}_{\delta }\end{array}\right]$$

(32)

3.3. Stability of the Sliding Mode Observer

In the analysis of SMO stability, the following function, V_irs, is selected as a Lyapunov Function Candidate (LFC).

$$V_{irs}=\left\{{\displaystyle \frac{1}{2}}\left[Lq\left({\overline{i}}_{\gamma }^2+{\overline{i}}_{\delta }^2\right)+{\displaystyle \frac{1}{{\gamma }_r}}{\overline{r}}_s^2\right]\right\}\ge 0$$

(33)

Here, γ_r represents the gain of the stator resistance observer with γ_r > 0. The function V_irs is definite positive, i.e., V_irs ≥ 0 in R³. Also, the system in (32) is stable in the sense of Lyapunov if the first-time derivative of V_irs is definite negative, i.e., dV_irs/dt ≤ 0. Consequently, the following requirement needs to be met:

$$\begin{array}{l}{\dot{V}}_{irs}=\left[Lq\left({\overline{i}}_{\gamma }{\dot{\overline{i}}}_{\gamma }+{\overline{i}}_{\delta }{\dot{\overline{i}}}_{\delta }\right)+{\displaystyle \frac{1}{{\gamma }_r}}{\overline{r}}_s{\dot{\overline{r}}}_s\right]\le 0\stackrel{(32)}{\to }\\ {\dot{V}}_{irs}=-{\overline{r}}_s\left(i_{\gamma }{\overline{i}}_{\gamma }+i_{\delta }{\overline{i}}_{\delta }\right)+{\displaystyle \frac{1}{{\gamma }_r}}{\overline{r}}_s{\dot{\overline{r}}}_s+E_{\gamma }{\overline{i}}_{\gamma }-{\overline{i}}_{\gamma }{k}_{\gamma }\mathrm{sgn}{s}_{\gamma }+E_{\delta }{\overline{i}}_{\delta }-{\overline{i}}_{\delta }{k}_{\delta }\mathrm{sgn}{s}_{\delta }\le 0\iff \\ {\dot{V}}_{irs}={\displaystyle \frac{1}{\gamma r}}{\overline{r}}_s\left[{\dot{\overline{r}}}_s-\gamma r\left(i_{\gamma }{\overline{i}}_{\gamma }+i_{\delta }{\overline{i}}_{\delta }\right)\right]+E_{\gamma }{\overline{i}}_{\gamma }-k_{\gamma }\left|{\overline{i}}_{\gamma }\right|+E_{\delta }{\overline{i}}_{\delta }-k_{\delta }\left|{\overline{i}}_{\delta }\right|\le 0\end{array}$$

(34)

The inequality in (34) expresses the conditions that are necessary for current/flux observer stability. Evidently, the inequality in (34) is valid if the following conditions are satisfied:

$$\begin{array}{l}{E}_{\gamma }\le \left|E_{\gamma }\right|<k_{\gamma }\\ {\overline{E}}_{\delta }\le \left|{\overline{E}}_{\delta }\right|<k_{\delta }\\ {\overline{r}}_s\left[{\dot{\overline{r}}}_s-{\gamma }_r\left(i_{\gamma }{\overline{i}}_{\gamma }+i_{\delta }{\overline{i}}_{\delta }\right)\right]=0\end{array},$$

(35)

The conditions in (35) represent the so-called Sliding Mode Existence Conditions. Also, the estimator of stator resistance is designed using the last equation in (35), as well.

3.4. Stator Resistance Observer

Taking into account the last condition in (35) and assuming that the stator resistance error could be close to zero but not exactly zero, it results in the following:

$${\overline{r}}_s\left[{\dot{\overline{r}}}_s-{\gamma }_r\left(i_{\gamma }{\overline{i}}_{\gamma }+i_{\delta }{\overline{i}}_{\delta }\right)\right]=0\Rightarrow {\dot{\overline{r}}}_s={\dot{r}}_s-{\dot{\widehat{r}}}_s={\gamma }_r\left(i_{\gamma }{\overline{i}}_{\gamma }+i_{\delta }{\overline{i}}_{\delta }\right)=0$$

(36)

Also, supposing that the stator resistance varies normally too slowly compared to the observer dynamics, the first-time derivative of stator resistance could be too small, i.e., dr_s/dt ≅ 0. Using Equation (36), the stator resistance estimator can be rewritten as follows:

$$(36)\stackrel{dr/dt\cong 0}{\to }{\dot{\overline{r}}}_s={\dot{r}}_s-{\dot{\widehat{r}}}_s={\gamma }_r\left(i_{\gamma }{\overline{i}}_{\gamma }+i_{\delta }{\overline{i}}_{\delta }\right)\Rightarrow {\dot{\widehat{r}}}_s=-{\gamma }_r\left(i_{\gamma }{\overline{i}}_{\gamma }+i_{\delta }{\overline{i}}_{\delta }\right)$$

(37)

3.5. Modified Back EMF Observer and Stability

Considering the sliding mode existence conditions in (35), the sliding manifold will be reached after a finite time t_r, i.e., s_γδ = 0 and ds_γδ/dt = 0 for t > t_r. As a result, the state trajectories satisfy the original system equation in (32), while the initial control inputs are replaced by their equivalent terms after setting ds_γδ/dt = 0. In this manner, a reduced-order system can be obtained as follows:

$$(32)\stackrel{(s_{\gamma \delta }={\dot{s}}_{\gamma \delta }=0,\hspace{0.33em}\overline{r}\cong 0)}{\to }0=\left[\begin{array}{c}{E}_{\gamma }\\ {\overline{E}}_{\delta }\end{array}\right]-{\left[\begin{array}{c}{k}_{\gamma }\mathrm{sgn}{s}_{\gamma }\\ k_{\delta }\mathrm{sgn}{s}_{\delta }\end{array}\right]}_{eq}\iff \left[\begin{array}{c}{E}_{\gamma }\\ {\overline{E}}_{\delta }\end{array}\right]={\left[\begin{array}{c}{k}_{\gamma }\mathrm{sgn}{s}_{\gamma }\\ k_{\delta }\mathrm{sgn}{s}_{\delta }\end{array}\right]}_{eq},$$

(38)

The stability evaluation of the modified EMF observer is examined using Lyapunov stability criteria. Let us consider a Lyapunov Function Candidate (LFC) V_eω, which includes the modified EMF errors and the speed error defined by:

$$V_{E\omega }={\displaystyle \frac{1}{2}}\left[E_{\gamma }^2+{\overline{E}}_{\delta }^2+{\displaystyle \frac{1}{{\gamma }_{\omega }}}{\overline{\omega }}^2\right]\ge 0,$$

(39)

Here, γ_ω represents the speed observer gain with γ_ω > 0. The function V_Eω is positive definite in R³. In addition, the function V_Eω is asymptotically stable at the origin (0,0,0) if its first-time derivative is negative definite, i.e., dV_Eω/dt ≤ 0.

$${\dot{V}}_{E\omega }=\left[E_{\gamma }{\dot{E}}_{\gamma }+{\overline{E}}_{\delta }{\dot{\overline{E}}}_{\delta }+{\displaystyle \frac{1}{{\gamma }_{\omega }}}\overline{\omega }\dot{\overline{\omega }}\right]\le 0,$$

(40)

Taking in consideration the components of the modified EMF and their first-time derivatives in Equations (27) and (29), the inequality in (40) could be written as follows:

$$\begin{array}{l}\left(40\right)\stackrel{(27),\left(29\right)}{\to }{\dot{V}}_{E\omega }=\left[E_{\gamma }{\dot{E}}_{\gamma }+{\overline{E}}_{\delta }{\dot{\overline{E}}}_{\delta }+{\displaystyle \frac{1}{{\gamma }_{\omega }}}\overline{\omega }\dot{\overline{\omega }}\right]\le 0\\ \Rightarrow V_{E\omega }=\left[E_{\gamma }\left(-\overline{\omega }{E}_{\delta }\right)+{\overline{E}}_{\delta }\left(\overline{\omega }{E}_{\gamma }-{\dot{\widehat{E}}}_{\delta }\right)+{\displaystyle \frac{1}{{\gamma }_{\omega }}}\overline{\omega }\dot{\overline{\omega }}\right]\le 0\\ \Rightarrow {\dot{V}}_{E\omega }=\left[-E_{\gamma }\overline{\omega }{E}_{\delta }+{\overline{E}}_{\delta }\overline{\omega }{E}_{\gamma }-{\overline{E}}_{\delta }{\dot{\widehat{E}}}_{\delta }+{\displaystyle \frac{1}{{\gamma }_{\omega }}}\overline{\omega }\dot{\overline{\omega }}\right]\le 0\\ \Rightarrow {\dot{V}}_{E\omega }=\left[-E_{\gamma }\overline{\omega }{\widehat{E}}_{\delta }-E_{\gamma }\overline{\omega }{\overline{E}}_{\delta }+{\overline{E}}_{\delta }\overline{\omega }{E}_{\gamma }-{\overline{E}}_{\delta }{\dot{\widehat{E}}}_{\delta }+{\displaystyle \frac{1}{{\gamma }_{\omega }}}\overline{\omega }\dot{\overline{\omega }}\right]\le 0\\ \Rightarrow {\dot{V}}_{E\omega }=\left[-{\overline{E}}_{\delta }{\dot{\widehat{E}}}_{\delta }+{\displaystyle \frac{1}{{\gamma }_{\omega }}}\overline{\omega }\left(\dot{\overline{\omega }}-{\gamma }_{\omega }{E}_{\gamma }{\widehat{E}}_{\delta }\right)\right]\le 0\end{array},$$

(41)

3.6. Stability Criteria of Modified EMF Observer and Angular Speed and Position Estimation

As mentioned above, the LCF defined in the relationship (40) converges asymptotically to the origin (0,0,0) if the inequality (41) is satisfied. This is valid if the following conditions are true:

$$(41)\to {\dot{V}}_{E\omega }=\left[-{\overline{E}}_{\delta }{\dot{\widehat{E}}}_{\delta }+{\displaystyle \frac{1}{{\gamma }_{\omega }}}\overline{\omega }\left(\dot{\overline{\omega }}-{\gamma }_{\omega }{E}_{\gamma }{\widehat{E}}_{\delta }\right)\right]\le 0\Rightarrow \left\{\begin{array}{l}-{\overline{E}}_{\delta }{\dot{\widehat{E}}}_{\delta }\le 0\\ \left(\dot{\overline{\omega }}-{\gamma }_{\omega }{E}_{\gamma }{\widehat{E}}_{\delta }\right)=0\end{array}\right.$$

(42)

Also, the conditions in (42) are valid if the δ-axis component of the modified EMF observer and the rotor speed are defined by:

$${\dot{\widehat{E}}}_{\delta }=c{\overline{E}}_{\delta }$$

(43)

and

$$(42)\to \left(\dot{\overline{\omega }}-{\gamma }_{\omega }{E}_{\gamma }{\widehat{E}}_{\delta }\right)=0\Rightarrow \dot{\omega }-\dot{\widehat{\omega }}={\gamma }_{\omega }{E}_{\gamma }{\widehat{E}}_{\delta }\stackrel{d\omega /dt\cong 0}{\to }\dot{\widehat{\omega }}=-{\gamma }_{\omega }{\widehat{E}}_{\delta }{E}_{\gamma }$$

(44)

In Equation (43), the c is the gain of the modified EMF observer with c > 0. Here, it is supposed that the dynamics of the modified EMF observer are much faster compared to rotor speed changes, i.e., dω/dt ≅ 0. Moreover, the estimation of the rotor angular position could be directly obtained by integrating the estimated speed from (44), i.e.,

$$\dot{\widehat{\theta }}=\widehat{\omega }$$

(45)

3.7. MTPA, Current Control, and Torque Decoupling

In SynRM control, the reference current in q-axis, i_q^*, is normally obtained using the reference torque T_e* via the speed controller, while the reference current in d-axis, i_d^*, is set to a positive value to establish d-axis flux. Also, Equation (19) denotes that the d-axis reference current, i_d^*, is equal to the q-axis reference current, i_q^*, during MTPA operation. Moreover, the produced torque T_e is directly proportional to the product of the d-axis current and the q-axis current, i_qi_d,. However, when the SynRM speed ω approximates the base speed ω_b, the induced EMF approaches as well the maximum inverter output voltage V_sm that is limited by the DC bus voltage V_dc. Considering the produced torque in MTPA mode, T_eMTPA, and the relations of the stator currents, I_s, i_dMTPA, and i_qMTPA, in Equation (17), Equation (14) could be rewritten to express the torque T_eMTPA as a function of the q-axis current i_qMTPA or stator current I_s, i.e.,

$$\left(14\right),\left(17\right)\stackrel{\left(MTPA\right)}{\to }\left.\left|\begin{array}{l}{T}_{eMTPA}={\displaystyle \frac{3p}{2}}\left(L_d-L_q\right)i_{dMTPA}{i}_{qMTPA}\\ i_{dMTPA}=i_{qMTPA}={\displaystyle \frac{I_s}{\sqrt{2}}}\end{array}\right.\right\}\Rightarrow T_{eMTPA}={\displaystyle \frac{3p}{2}}\left(L_d-L_q\right)i_{qMTPA}^2={\displaystyle \frac{3p}{4}}\left(L_d-L_q\right)I_s^2,$$

(46)

In MTPA mode, the d-axis and q-axis reference currents, i^*_dMTPA and i^*_qMTPA, can be calculated using the reference torque, T^*_eMTPA. In this manner, Equation (46) could be used to derive the reference currents, i^*_dMTPA and i^*_qMTPA, that is:

$$(46)\to i_{qMTPA}^{*}=i_{dMTPA}^{*}=\sqrt{{\displaystyle \frac{2T_{eMTPA}^{*}}{3p\left(L_d-L_q\right)}}},$$

(47)

As it follows from Equation (19), for a given reference torque T_e^*, the currents i^*_dMTPA and i^*_qMTPA are limited by the maximum current I_sm, i.e.,

$$(19)\to i_{qMTPA}^{*}=i_{dMTPA}^{*}\le {\displaystyle \frac{I_{sm}}{\sqrt{2}}},$$

(48)

Equation (48) is implemented through an AWC to limit the current I_s. Considering the fast convergence of the observer system, the angle error between the γδ and dq reference frames becomes very small. Accordingly, the two reference frames could be deemed identical after finite time t_r. Consequently, the corresponding reference currents might also be set as equal, i.e., i_γ^* = i_d^* and i_δ^* = i_q^*. As a result, the relations (47) and (48) could be approximately written in γδ reference frame as follows:

$$(47),(48)\to i_{\delta MTPA}^{*}=i_{\gamma MTPA}^{*}=\sqrt{{\displaystyle \frac{2T_{eMTPA}^{*}}{3p\left(L_d-L_q\right)}}}\le {\displaystyle \frac{I_{sm}}{\sqrt{2}}}$$

(49)

Figure 4 shows the MTPA controller in γδ reference frame, including an Anti-Windup element with the feedback calculation.

4. Simulation Results

4.1. Description of SynRM model–Observer System, Voltage Source Inverter (VSI), Torque, and Current Controllers

The structure of the applied observer–controller scheme is shown in Figure 1. In particular, the design of the proposed MTPA sensorless control is based on the developed SynRM modified model in the γδ reference system. Simulink/MATLAB is used to study and evaluate the controller–observer system of SynRM, including its behavior in transient states. Specifically, the simulated SynRM model is primarily based on Equations (20)–(29) that are related to the γδ reference frame, whereas the flux/current observer (SMO) and modified EMF observer model use Equations (30)–(45). Additionally, the presented MTPA method uses Equations (1)–(19) and (46)–(49). Also, the parameters of the tested SynRM are listed in

Table 1. Parameters of Synchronous Reluctance Machine (SynRM).

Symbol	Quantity	Expressed in SI
P	Power	4.4 kW
Vl-l	Line-to-line voltage	380 V
rs	Stator resistance	2.5 Ω
Ld	d-axis inductance	0.400 H
Lq	q-axis inductance	0.210 H
J	Moment of inertia	0.089 kgm2
p	Magnetic pole pairs	1
ωm	Mechanical angular speed	3000 rpm

. It is observed that the SynRM saliency is positive, i.e., (L_d − L_q) > 0. This implies that the produced torque T_e is positive if the product of the stator currents, i_di_q, is positive, as well. The typical gains of SMO and EMF observers are listed in

Table 2. Gain parameters of Flux/Current Observers (SMOs).

Symbol	Quantity	Value
kγ	γ-axis gain of SMO	20,000
kδ	δ-axis gain of SMO	20,000
γr	gain of stator resistance estimator	75

and

Table 3. Gain parameters of Modified EMF and Speed Observers.

Symbol	Quantity	Value
c	gain of modified EMF observer	80
γω	gain of angular speed observer	400

, respectively. In particular, the gain values of Table 2 and Table 3 are chosen on the basis of relations (35) and (42), aiming at the stability of both the SMO and the modified EMF observer. To establish the necessary modified rotor flux, the γ-axis reference current i_γ^* is initially set to a constant positive value. As the SynRM rotor accelerates, the d-axis reference current i_γ^* changes to i^*_δMTPA at time t₁, as it is expressed in Equation (47) (see Figure 4).

A 3-phase Voltage Source Inverter (VSI) is connected to SynRM, providing the required electrical power. Particularly, the VSI is fed by a 540V DC voltage power supply. The output voltage of VSI is derived by applying the Space Vector PWM technique at a switching frequency f_z equal to 5 kHz (i.e., T_z = 0.2 ms).

Since the SynRM has one pole pair (p = 1, see Table 1), both electrical and mechanical angles and speeds are the same, i.e., θ_e = θ_m, and ω_e = ω_m. This means ω_eb = ω_mb = 314.16 rad/s (i.e., 3000 rpm or 50 Hz). Also, the stator current limit I_sm and voltage limit V_sm are set to 18 A and 240 V, respectively. Both current controllers for i_γ^* and i_δ^* are designed to embed anti-windup control (AWC) of the single level. Regarding the i_γ^* and i_δ^* current limits, these are set to ±12 A by the limit element included in the AWC. The connected AWC can effectively prevent overshoots and oscillations (see Figure 4).

4.2. MTPA Operation of SynRM at Low-Speed Range

The simulation results presented in Figure 5 show the system response for a stepwise change in reference speed from 0 to 20π rad/s (or 10 Hz). Initially, the conventional FOC is applied for a time lasting 1.5 s (see Figure 5). Specifically, the current i_γ^* is preset to 4 A in order to establish the modified rotor flux λ_sm at the starting time t₀ = 0 s, while the control mode is changed to MTPA (i_γ^* = i_δ^*) at time t₁ = 1.5 s. Then, a stepwise change in reference speed from 20π rad/s to 30π rad/s occurs at time t₂ = 4 s. Additionally, an external torque disturbance of 4 Nm is applied at time t₃ = 6 s, and it is removed at time t₄ = 7 s. SynRM speed, angle, and angle error are demonstrated in Figure 5a,b,c, respectively. It is notable that the SMO and EMF observers converge very quickly, maintaining high-accuracy estimations despite speed changes and external torque disturbance.

In Figure 5d, the i_γ − i_δ graph illustrates the different control modes applied. The conventional FOC applied from 0 s to 1.5 s is shown by the line AB, while the line CD represents the MTPA applied from 1.5 s to 8 s. Also, the line BC depicts the instant transition of the current i_γ^* from FOC (i_γ^* = 4 A) to MTPA operation (i_γ^* = i_γMTPA^* = i_δ^* = i_δMTPA^*). The responses of stator currents i_γ and i_δ, the produced torque T_e, the load torque T_L, and the total torque T_tot are shown in Figure 5e,f. It is observed that the SynRM operates in MTPA mode after the time t₁ = 1.5 s, precisely following the MTPA line and producing the maximum torque T_e per stator current I_s, i.e., T_e = T_eMTPA (see Figure 2, line 0B and Figure 5d, line CD).

4.3. MTPA Operation of SynRM at Medium-Speed Range

Figure 6 shows the simulation results considering the controller-observer response of the SynRM for a stepwise change in reference speed from 0 to 40π rad/s (or 20 Hz). At time t₀ = 0 s, the current i_γ^* is set to 3 A, aiming to establish the modified rotor flux λ_sm. Also, the conventional FOC is applied for a period of time lasting 0.5 s. Afterwards, at time t₁ = 0.5 s, the control mode is changed to MTPA mode (i_γ^* = i_γMTPA ^*= i_δ^* = i_δMTPA^*, see Figure 6e). Similarly to the previous low-speed test, an additional stepwise change in reference speed of +4π rad/s occurs at time t₂ = 4 s. Moreover, an external torque disturbance of 2 Nm is applied at time t₃ = 6 s, and it is removed at time t₄ = 7 s. Particularly, the SynRM speed, angle, and angle error are depicted in Figure 6a,b,c, respectively.

Despite the reference speed change and the external torque disturbance of 2 Nm, the SMO and modified EMF observers both converge quickly and maintain stability, providing high accuracy. The control modes applied are detailedly presented in Figure 6d as an i_γ − i_δ graph. Here, the line AB represents the FOC applied from 0 s to 0.5 s, while the line CD represents the MTPA applied from 0.5 s to 8 s. In addition, the line BC presents the transition of the d-axis current from FOC (i_d^* = 3 A) to MTPA operation (i_γ^* = i_γMTPA^* = i_δ^* = i_δMTPA^*). The responses of stator currents, i_γ and i_δ, are shown in Figure 6e, while the torques T_e (produced torque), T_L (load torque), and T_tot (total torque) are shown in Figure 6f. At time t₁ = 0.5 s, the reference current i_γ^* is set equal to i_δ^*. After a very small transition time, the SynRM operates in MTPA mode, generating the maximum torque T_e for a given stator current I_s, i.e., T_e = T_eMTPA (see Figure 6d, line AB, line BC, and line CD). As it is expected, the line CD in Figure 6d is similar to the line 0B in Figure 2 representing the MTPA operation of SynRM.

4.4. MTPA Operation of SynRM at High-Speed Range

Figure 7 presents the simulation results related to the controller–observer response of the SynRM at high-speed range. The reference speed is stepwise changed from 0 to 60π rad/s (or 30 Hz). Following the same control procedure as before, the conventional FOC is applied for a short time of 0.5 s (see Figure 5). At time t₁ = 0.5 s, the SynRM control mode is changed to MTPA setting the reference i_γ^* equal to i_δ^*. In a similar way to the previous low- and medium-speed tests, there is an additional stepwise change in reference speed of −4π rad/s that occurred at time t₂ = 4 s. Also, an external torque disturbance of 3 Nm is applied at time t₃ = 6 s, and it is removed at time t₄ = 7 s. The starting modified rotor flux λ_sm is obtained by setting i_γ^* equal to 4 A at time t₀ = 0 s.

Considering the speed, angle, and angle error, the corresponding responses are shown in Figure 7a,b,c, respectively. These responses verify that the SMO and modified EMF observer are very robust, and they provide almost precise estimations of both angular speed and position, regardless of the speed changes and external torque disturbances. As it can be seen in Figure 7d, the i_γ − i_δ graph depicts distinctly the applied control modes, FOC and MTPA. Specifically, the line AB represents the FOC applied from 0 s to 0.5 s for constant reference current i_d^* = 4 A, and the line CD represents the MTPA control applied from 0.5 s to 8 s. Also, the line BC represents the transition of the γ-axis current from FOC (i_γ^* = 4 A) to MTPA operation (i_γ^* = i_γMTPA^* = i_δ^* = i_δMTPA^*). During the transition time of the reference current i_γ^*, the real current i_γ increases almost instantly, following exactly the i_δ, while the current i_δ is slightly decreasing. The time responses of stator currents, i_γ and i_δ, are presented in Figure 7e, while the torques T_e (produced torque), T_L (load torque), and T_tot (total torque) are shown in Figure 7f. Inspecting Figure 7e,f, and particularly considering the time instants t₂ = 4s, t₃ = 6s, and t₄ = 7s, it is observed how the changes in demanded speed ω^* and load torque T_L affect the real currents, i_γ and i_δ, and produced torque T_e.

4.5. Control Inputs v_γ and v_δ, Estimated Currents, and MTPA Operation of SynRM

Figure 8 refers to the simulations that are previously described in Figure 5. Regarding the SMO stability and its interaction with the MTPA controller, it is observed that the switching function v_δ is strongly affected during the transition from the FOC to MTPA operating modes, while the switching function v_γ actually remains unaffected (see Figure 8a). The control input v_δ exhibits a remarkable deviation at times t₁ = 1.5 s and t₂ = 4 s due to the transition from FOC to MTPA and the speed change from 20π rad/s to 30π rad/s. Furthermore, the control input v_δ exhibits a slight alteration due to the application and removal of the load torque, T_L, at times t₃ = 6 s and t₄= 7 s. Nonetheless, the stability of the SMO is maintained, despite the disturbances in the load torque and speed changes even during the transition time between FOC and MTPA. In Figure 8b, the estimated stator currents are demonstrated throughout the time of the FOC and MTPA control operations. It is worth mentioning that the currents, i_γ and i_δ, are the same after time t₁ = 1.5 s, verifying the effectiveness of the MTPA controller.

4.6. MTPA Operation of SynRM at Low-Speed Range in the Presence of Current Measurement Noise

Normally, the performed estimations may be weakened by environmental factors and operating conditions like temperature, humidity, introduced noise, and load torque changes. Considering measurement noise, the sensorless control methods may be negatively affected in estimating SynRM variables, such as rotor speed and position. As a consequence, the sensorless MTPA control might be degraded. Figure 9 shows results, carrying out similar simulations as in Figure 5 and Figure 8, but taking into account the impact of measurement noise in the stator currents. The behavior of sensorless MTPA control is examined, assuming that normally distributed noise is added to the real stator currents. Here, Gaussian noise distribution is used in modeling the current sensor with mean = 0 and variance = 0.125. Comparing the speed and angle in Figure 5a,b with the corresponding responses in Figure 9a,b, it is observed that these are similar. Considering the stability of SMO, it is shown that the inserted noise has an apparent influence on the switching function v_δ, increasing the amplitude of the chattering effect (see Figure 9c). It is notable that the switching function v_γ almost remains unaffected by the added noise (see Figure 9c and Figure 8a). In addition, the real and estimated currents are depicted in Figure 9d,e. In comparison, the chattering of the currents affected by the noise is larger than those without noise (see Figure 5e and Figure 8b). Moreover, Figure 9f shows the impact of the noise on the produced torque T_e. As a result of the current measurement noise, an additional chattering is introduced to the produced torque. This is more obvious during the transition from FOC to MTPA operation at time t₁ = 1.5 s and the application period of the load torque T_L from time t₃ = 6 s to t₄ = 7 s. Also, an instant disturbance is observed during the reference speed change at time t₂ = 4 s.

Furthermore, Figure 10 shows the stator currents graph in the i_γ − i_δ plane and the modified rotor flux in the γδ reference frame applying the same control scenario and timing as in Figure 9, while considering the noise influence on the stator currents, as well. Particularly, Figure 10a demonstrates graphically the FOC and MTPA operations (line AB, line CD). The transition of SynRM sensorless control from FOC to MTPA mode is depicted in line BC. Also, it is worth mentioning that the point C represents graphically the maximum produced torque T_emax in MTPA mode at time t₂ = 4 s. (see Figure 9f and Figure 10a). In Figure 10b, the modified flux components, λ_msγ and λ_msδ, are illustrated in the γδ reference frame. As it is expected, the modified rotor flux λ_msγ (or λ_msd) is maximum in MTPA mode at time t₂ = 4 s. It is mentioned that the observer converges very quickly after finite time t_r.

After inspecting Figure 9 and Figure 10, it is observed that the controller–observer system keeps its stability and estimation accuracy, despite the reference speed changes, external torque disturbances, and noisy current signals. Considering the obtained results in Figure 9 and Figure 10, the proposed algorithm can perform well and adequately address the issues due to the additional measurement noise in a wide speed range. As it is shown, the presented sensorless MTPA control can compensate for current sensor noise that might be encountered in real-time applications.

5. Discussion

The presented method has some considerable characteristics compared to other suggested methods, especially those based on non-sensorless strategies. In this method, a MTPA control algorithm has been developed and applied to SynRM sensorless control. Despite the conventional SynRM models, the modified SynRM model is introduced based on the concept of the modified rotor flux and the γδ estimated reference frame. Compared to the conventional methods, the presented controller–observer system is different, since it is totally designed, developed, and tested on the basis of the γδ estimated reference frame. A main contribution of this work is the design of the MTPA controller that uses inputs coming from an observer system for SynRM speed and angle estimation. Also, the design of the estimation algorithm is relatively simple, based on the analytical mathematical model of SynRM adopting straightforward assumptions, as well. In addition, it is remarkable that modeling SynRM in the γδ reference frame is advantageous, particularly for sensorless control applications, since the γδ model of SynRM inherently includes angle error terms (see Figure 3). Therefore, transforming the initial SynRM model from the dq to the γδ reference frame and then modifying the derived SynRM γδ model allows the isolation of SynRM flux terms related to the angle error. Although the proposed estimation approach is based on fundamental excitation methodology, it has been verified that the proposed MTPA controller–observer system can effectively achieve sensorless MTPA operation even at low and very low speeds. Moreover, the modified SynRM model can also be applicable in sensorless applications that are based on high-frequency signal injection algorithms (see Equations (20)–(29)), [55].

Taking accuracy into consideration, it is noted that the angle is estimated with an average accuracy of 0.5 × 10⁻³ rad to ±1.5 × 10⁻⁴ rad for changing the reference speed from 20π rad/s (or 62.83 rad/s) to 60π rad/s, while a load torque is applied (see Figure 5c, Figure 6c, and Figure 7c). Regarding the dynamic conditions under the load torque and speed changes, the angle error shows a notable increase during the transition of SynRM operation from FOC to MTPA mode, after the speed changes and close to the zero speeds (see Figure 5a, Figure 6a, and Figure 7a). Also, considering the accuracy of control inputs, v_γ and v_δ (or equivalent control inputs, v_γeq and v_δeq), it could be mentioned that the v_δ (or v_δeq) exhibits an average accuracy that depends on the condition of the feedback current signals, i.e., currents measured without or with noise. Particularly, the average accuracy of v_δ (or v_δeq) is about ±2 V without noise and ±4 V in the case of noisy measurements (see Figure 8a and Figure 9c). Also, the v_δ (or v_δeq) signals are strongly affected by the transition from FOC to MTPA operating modes of SynRM, the stepwise speed changes, and the load torque application. However, the control input v_γ (or v_γeq) appears to be unaffected, despite the noise and the changes occurred in operating conditions. Moreover, the noise has an impact on the stator currents i_γ and i_δ, affecting their accuracy. Specifically, the real and estimated currents exhibit an average accuracy of ±0.2 A without noise (see Figure 5e, Figure 6e, Figure 7e, and Figure 8b), although the current accuracy is degraded in the presence of noise to ±0.4 A, increasing the chattering effect (see Figure 9d,e). Also, the accuracy of the produced torque T_e is ±0.2 Nm on average without noise (see Figure 5f, Figure 6f, and Figure 7f). Considering noisy current feedback, the corresponding average accuracy of the produced torque is decreased to ±0.3 Nm in transition states (see Figure 9f). However, it is worth noting that the overall accuracy becomes significantly better as the SynRM system approaches the steady states.

After inspecting the simulation results in Figure 5, Figure 6, Figure 7, Figure 8 and Figure 9, it is observed that the SynRM variables such as angular speed and position, i_γδ stator currents, and modified EMF are estimated with high accuracy. Both observers and the MTPA controller perform well, exhibiting robustness and stability even in the presence of disturbances such as speed changes, load torque variations, and noise. Although the chattering phenomenon is the main drawback of sliding mode methods [68,69,70,71], it could be observed that the estimated values contain reduced chattering, allowing the SMO to maintain the required accuracy. Moreover, the modified EMF seems to be optimally estimated based on the equivalent control inputs of the SMO. This implies that the suggested synthesis of the MTPA algorithm, SMO, and modified EMF observer is well adjusted, producing results very close to the expected ones. The considerable stability of the MTPA algorithm and the notable proximity between the estimated and real values of SynRM shows the high accuracy of the total controller–observer system including an MTPA, SMO, and modified EMF observer. Regarding the overall performance of the MTPA control, speed, and angle estimation, the simulation results present the high precision of the proposed sensorless MTPA method of SynRM (see Figure 5, Figure 6, Figure 7 and Figure 8).

Considering the stability of the controller–observer scheme, the design of the MTPA controller is very important in achieving optimal controller–observer operation. Since the observer system provides estimations that are fed back to the SynRM system via the controllers, it is significant to attain the optimal interaction between them. For example, the MTPA controller system of SynRM might be negatively affected by the low accuracy estimations of the observer system, resulting in instability. Also, a multi-level AWC might be applied to improve the behavior of MTPA controller and upgrade the total control-estimation system. However, the overall complexity of the resultant controller–observer system might increase, requiring more computational power for real-time (RT) applications.

In future work, development boards can be used to implement the presented sensorless algorithm in real-time applications. Normally, a powerful digital signal processor (DSP) or multiprocessor unit (MPU) is included in the equipment of development boards. Particularly, a DSP-based board could be mainly used to implement the MTPA controller and total observer algorithms, including the external control loop for the speed ω and the internal control loops for the currents i_γ and i_δ. Furthermore, the DSP-based board should be compatible with the software being utilized and capable of executing the total control-estimation algorithm rapidly [74,75,76]. Normally, a DSP-based development board embeds the necessary hardware, such as ADC, digital encoder interfaces, high-speed PWM driving circuits, interface circuits, etc. Specifically, the provided Real-Time Interface (RTI) allows the direct implementation of the developed Simulink/MATLAB models by means of software running on real-time hardware. The Controller Board DS1104 research and development (R&D) of dSPACE is an example of a DSP board [76].

6. Conclusions

A new method for sensorless MTPA control of SynRM is developed based on the SynRM model in the γδ estimated reference frame. In the control part of the estimation-control algorithm, a simple torque decoupling is applied, which is embedded into the MTPA speed controller to determine the γ-axis and δ-axis reference currents, i_γ^* and i_δ^*. Also, a single-level AWC is included in MTPA, aiming to limit the stator current I_s. By using the MTPA controller, the current components related to the modified rotor flux and the produced torque can be effectively adjusted for the maximum torque operation. It has been assessed that the developed modified model of SynRM is suitable and convenient for MTPA sensorless control applications, even at low- and very low-speed regions. This is fully enabled by utilizing a simple MTPA approach for implementing the stator current control in both the γ and δ axes, while the angular speed/position observer converges rapidly, almost zeroing the deviation between the reference frames γδ and dq. The presented MTPA controller–observer scheme uses only the stator currents and voltages. Simulation results demonstrate the efficiency and the robustness of the sensorless MTPA control based on the SMO methodology and a single-level AWC. The high accuracy and effectiveness of the proposed MTPA algorithm has been verified in the presence of reference speed changes, external torque disturbances, and current measurement noise in a wide speed range.