Robust Fault-Tolerant Control of a Five-Phase Permanent Magnet Synchronous Motor under an Open-Circuit Fault

Abstract

This paper introduces a robust fault-tolerant control (FTC) for a five-phase permanent magnet synchronous motor (FPPMSM) affected by the third harmonic under an open-circuit fault (OCF). Using field-oriented control, the proposed method demonstrates how to achieve optimal current references for torque decoupling under healthy and faulty conditions. The proposed speed and current loop controllers are based on sliding mode control (SMC), with a nonlinear extended state observer (NESO) that utilizes a hyperbolic tangent function (HTF) to provide feed-forward compensation to the controllers. The results analysis confirmed that the proposed control could enhance the tracking accuracy and robustness to disturbances under various conditions, substantially reducing torque ripples and speed fluctuations under a fault.

1. Introduction

In the contemporary transportation and aerospace industries, the demand for multi-phase permanent magnet synchronous motors has surged, due to their demonstrated superiority in performance and durability [1]. Vehicles, ships, and aerospace applications mainly rely on these motors due to their higher attributes, such as efficiency, power density, and fault tolerance [2,3]. One of the most favorable aspects of these motors for such applications is their ability to maintain essential performance characteristics during faults, which is directly related to the number of phases in the motor. The more phases the motor has, the more degrees of freedom it can exploit, allowing it to maintain its performance despite faults [4].

Although five-phase permanent magnet synchronous motors (FPPMSMs) are robust, open-circuit faults (OCFs) remain a significant concern. These faults usually occur due to disrupted or disconnected wires in the motor windings [5]. It is crucial to detect these faults quickly and accurately, to maintain motor performance and apply an effective fault-tolerant control (FTC) strategy. A widely used approach for identifying OCFs involves monitoring the motor currents with the help of current sensors. In [6], a fault detection method based on current spectrum analysis was developed with the help of fault signatures. The experimental results showed that OCFs could be quickly detected and localized using current sensors without requiring additional hardware.

In the realm of FPPMSMs, a plethora of techniques have been proposed for FTC to counteract the unbalanced currents induced by OCFs [7,8,9]. These strategies revolve around maintaining the rotational movement of the fundamental magnetic motive force and back electromotive force (EMF) at levels consistent with healthy operation. Although traditional linear regulators, such as a proportional-integral (PI) controller, dominate field-oriented control, the inherent complexity and nonlinearity of FPPMSMs during OCFs pose a significant challenge. This complexity can lead to structural asymmetry in FPPMSMs, resulting in undesirable torque ripples. Extensive research efforts have been devoted to proposing solutions aimed at suppressing these torque ripples in the presence of OCFs, as exemplified in numerous studies, including [7,8,9,10] and many others. Remarkably, research fields have not given adequate focus to nonlinear control techniques and compensations under OCFs of multiphase machinery. Highlighting this gap emphasizes the importance of continued exploration and advancement in this field.

Adopting a nonlinear control method like sliding mode control (SMC) has garnered substantial interest [11]. Its ability to accommodate system uncertainties and disturbances with a high degree of tolerance for model imperfections makes it particularly well-suited for application to permanent magnet synchronous motors during fault conditions [12,13]. For instance, in [12], the authors advocated for employing SMC in the speed loop of an FPPMSM under an open circuit fault (OCF), effectively mitigating the effects of unknown dynamics. Meanwhile, [13] proposed the application of SMC in the current control loops, maintaining the traditional linear PI regulator for speed control. Another study proposed a nonlinear control approach using a backstepping controller for FTC to enhance the five-phase induction motor drive dynamics under an OCF [14]. While backstepping controllers can be effective in certain scenarios, they have limitations. They require full information of states, which causes a high computational burden and potentially produces larger control signals [15]. In contrast, SMC features a simple and easily tunable controller structure and aids in eliminating steady-state errors [16]. Although SMC may suffer from the chattering effect, various methods have been proposed to mitigate this issue [17]. One such approach is the reaching law method, which encompasses techniques like constant rate reaching, constant plus proportional rate reaching, and power rate reaching [18]. Among these, incorporating fast reaching and low chattering strategies within the reaching law method has shown promise for reducing chattering effects in SMC.

Nonetheless, permanent magnet synchronous motors face an array of challenges from external disturbances, encompassing load variations, thermal effects, rotor harmonic flux, inverter non-linearity, parameter uncertainties, and other unknown factors [19]. These disturbances can be collectively termed mismatched disturbances. The traditional technique of SMC may struggle to fully mitigate the impact of these disturbances, due to its sensitivity and susceptibility to the chattering phenomenon [20]. Furthermore, combining controllers with observers proved effective in several works utilizing disturbance observer control strategies to boost controls specifically for addressing mismatched disturbances [17,21,22,23,24]. However, it is worth noting that, while linear strategies have their merits, they may struggle to fully account for the inherent nonlinearities that arise during faults. In comparison, nonlinear controllers and observers tend to offer more robust solutions in such scenarios. Therefore, this paper introduces a comprehensive control strategy that integrates three key elements: optimal current references for the FPPMSM model in both healthy and faulty conditions; a simple SMC design that combines proportional and power rate reaching properties to achieve robust and efficient motor control; and a nonlinear extended state observer (NESO) employing a hyperbolic tangent function (HTF) for feed-forward compensation to the SMC, specifically designed to address mismatched disturbances.

After this introduction, this paper will explore the following subjects: Section 2 details the modeling of FPPMSM under both normal and fault conditions. Section 3 covers the determination of optimal control references for these models. Section 4 outlines the combination of NESO with SMC design. Section 5 presents an analysis of the outcomes of the proposed method. Lastly, Section 6 offers concluding remarks.

2. Dynamic Model of FPPMSM

The FPPMSM model focuses solely on fundamental and third-harmonic components. Additionally, the model simplifies by ignoring cogging, magnetic saturation, and temperature effects on the flux density of the permanent magnets.

2.1. Healthy Model

In the original $abcde$ frame, the equations for the stator voltage can be expressed as follows [12]:

$$v_s=R_s{i}_s+\frac{d}{dt}(L_s{i}_s+{\varphi }_{fm})$$

(1)

where $v_s$ is a voltage vector of the stator ${[v_{sa},v_{sb},v_{sc},v_{sd},v_{se}]}^{\top }$, $i_s$ is a current vector of the stator ${[i_{sa},i_{sb},i_{sc},i_{sd},i_{se}]}^{\top }$, $R_s$ is a diagonal stator resistance $5\times 5$ matrix, $L_s$ is a stator inductance $5\times 5$ matrix that has a diagonal, in which each element corresponds to the self-inductance of each phase, and the off-diagonal corresponds to the mutual inductance between each phase. ${\varphi }_{fm}$ represents the flux-linkage vector of a permanent magnet, including ${\varphi }_{f1}$ and ${\varphi }_{f3}$ (first- and third-order harmonic components of back-EMF) in the $abcde$ frame.

$${\varphi }_{fm}={\varphi }_{f1}\left[\begin{array}{c}cos\theta \\ cos(\theta -\frac{2\pi }{5})\\ cos(\theta -\frac{4\pi }{5})\\ cos(\theta -\frac{6\pi }{5})\\ cos(\theta -\frac{8\pi }{5})\end{array}\right]+{\varphi }_{f3}\left[\begin{array}{c}cos3\theta \\ cos(3\theta -\frac{6\pi }{5})\\ cos(3\theta -\frac{12\pi }{5})\\ cos(3\theta -\frac{18\pi }{5})\\ cos(3\theta -\frac{24\pi }{5})\end{array}\right]$$

(2)

When applying the Clarke and Park transformation (A1), the $d{q}_p$ and $d{q}_s$ frame of the stator voltage equation is given by [13,25]:

$$\left\{\begin{array}{c}\begin{array}{cc}\hfill v_{dp}& =R_s{i}_{dp}+L_{dp}\frac{d{i}_{dp}}{dt}-n_p\mathsf{\Omega }{L}_{qp}{i}_{qp}\hfill \\ \hfill v_{qp}& =R_s{i}_{qp}+L_{qp}\frac{d{i}_{qp}}{dt}+n_p\mathsf{\Omega }(L_{dp}{i}_{dp}+{\varphi }_{f1})\hfill \\ \hfill v_{ds}& =R_s{i}_{ds}+L_{ds}\frac{d{i}_{ds}}{dt}-3n_p\mathsf{\Omega }{L}_{qs}{i}_{qs}\hfill \\ \hfill v_{qs}& =R_s{i}_{qs}+L_{qs}\frac{d{i}_{qs}}{dt}+3n_p\mathsf{\Omega }(L_{ds}{i}_{ds}+{\varphi }_{f3})\hfill \end{array}\hfill \end{array}\right.$$

(3)

where $L_{dp}$, $L_{qp}$, $i_{dp}$, $i_{qp}$, $v_{dp}$, and $v_{qp}$ are the inductance, current, and voltage in the $d_p\text{-}{q}_p$ frame affected by the first harmonic, respectively; $L_{ds}$, $L_{qs}$, $i_{ds}$, $i_{qs}$, $v_{ds}$, and $v_{qs}$ are the inductance, current, and voltage in the $d_s\text{-}{q}_s$ frame affected by the third harmonic, respectively; and $\mathsf{\Omega }$ and $n_p$ are the mechanical speed and the number of pole pairs, respectively.

The torque produced by the FPPMSM is given as follows [25]:

$$T_e=\frac{5}{2}{n}_p[{\varphi }_{f1}{i}_{qp}+(L_{dp}-L_{qp})i_{dp}{i}_{qp}+3{\varphi }_{f3}{i}_{qs}+3(L_{ds}-L_{qs})i_{ds}{i}_{qs}]$$

(4)

In order to assist the design of control laws for the FPPMSM and improve the comprehension of the model, the voltages (3) can be expressed as follows:

$$\left\{\begin{array}{c}{\dot{i}}_{dp}=c_1{i}_{dp}+c_2\mathsf{\Omega }{i}_{qp}+b_1{v}_{dp}\hfill \\ {\dot{i}}_{qp}=c_3{i}_{qp}-c_4\mathsf{\Omega }{i}_{dp}-c_5\mathsf{\Omega }+b_2{v}_{qp}\hfill \\ {\dot{i}}_{ds}=c_6{i}_{ds}+c_7\mathsf{\Omega }{i}_{qs}+b_3{v}_{ds}\hfill \\ {\dot{i}}_{qs}=c_8{i}_{qs}-c_9\mathsf{\Omega }{i}_{ds}-c_{10}\mathsf{\Omega }+b_4{v}_{qs}\hfill \\ \dot{\mathsf{\Omega }}=a_1{T}_e-a_1{T}_L-a_2\mathsf{\Omega }\hfill \end{array}\right.$$

(5)

where the machine parameters describe the following FPPMSM model components as follows: $c_1=-\frac{R_s}{L_{dp}}$, $c_2=\frac{L_{qp}}{L_{dp}}{n}_p$, $c_3=-\frac{R_s}{L_{qp}}$, $c_4=\frac{L_{dp}}{L_{qp}}{n}_p$, $c_5=\frac{{\varphi }_{f1}}{L_{qp}}{n}_p$, $c_6=-\frac{R_s}{L_{ds}}$, $c_7=\frac{3L_{qs}}{L_{ds}}{n}_p$, $c_8=-\frac{R_s}{L_{qs}}$, $c_9=\frac{3L_{ds}}{L_{qs}}{n}_p$, $c_{10}=\frac{3{\varphi }_{f3}}{L_{qs}}{n}_p$, $b_1=\frac{1}{L_{dp}}$, $b_2=\frac{1}{L_{qp}}$, $b_3=\frac{1}{L_{ds}}$, $b_4=\frac{1}{L_{qs}}$, $a_1=\frac{1}{J}$, $a_2=\frac{F}{J}$.

2.2. Faulty Model under Open Circuit of Phase “a”

For the sake of simplicity, the fault model of FPPMSM in this paper only covers the OCF in phase “a”. When the OCF occurs, the new Clarke transformation and the new Park transformation (A2) can be formulated based on preserving the back-EMF and the fundamental magnetic motive force, to maintain them the same as in a healthy model [12].

By including an open-circuit permanent magnet flux linkage vector in the magnetic co-energy derivative relative to the electrical angle and applying the new transformations (A2), the torque can be obtained as follows [10]:

$$T_{ef}=T_{ef1}+T_{ef3}$$

(6)

$$T_{ef1}=\frac{5}{2}{n}_p{\varphi }_{f1}{i}_{qp}+\frac{5}{2}{n}_p{i}_{dp}{i}_{qp}\left(L_{dp}-L_{qp}\right)$$

(7)

$$\begin{array}{cc}\hfill T_{ef3}& =\frac{15}{4}{n}_p{\varphi }_{f3}\left[-i_{qp}(cos\left(2\theta \right)-cos\left(4\theta \right))\right.\hfill \\ \hfill & \left.+i_{dp}(sin\left(2\theta \right)+sin\left(4\theta \right))+2i_{\beta s}cos\left(3\theta \right)\right]\hfill \end{array}$$

(8)

where $T_{ef}$ is the total torque in the presence of an OCF. $T_{ef1}$ represents the torque influenced by the fundamental harmonic, corresponding to healthy conditions. $T_{ef3}$ represents the torque influenced by the third-order harmonic, which produces torque ripples. Furthermore, the FPPMSM dynamic model under the open circuit to phase “a” can be derived as follows [10]:

$$\left\{\begin{array}{c}{\dot{i}}_{dp}=c_1{i}_{dp}+c_2\mathsf{\Omega }{i}_{qp}+b_1{v}_{dp}\hfill \\ {\dot{i}}_{qp}=c_3{i}_{qp}-c_4\mathsf{\Omega }{i}_{dp}-c_5\mathsf{\Omega }+b_2{v}_{qp}\hfill \\ {\dot{i}}_{\beta s}=c_{11}{i}_{\beta s}-c_{12}\mathsf{\Omega }cos\left(3\theta \right)+b_{\beta s}{v}_{\beta s}\hfill \\ \dot{\mathsf{\Omega }}=a_1{T}_{ef}-a_1{T}_L-a_2\mathsf{\Omega }\hfill \end{array}\right.$$

(9)

where $c_{11}=-\frac{R_s}{L_{ls}},c_{12}=\frac{3{\varphi }_{f3}}{L_{ls}}{n}_p,b_{\beta s}=\frac{1}{L_{ls}}$. $L_{ls}$ and $\theta$ are the leakage inductance and the electrical angle, respectively. $i_{\beta s}$ and $v_{\beta s}$ are the current and voltage in the ${\beta }_s$-axis, respectively. It should be noted that when OCFs occur, the FPPMSM loses a degree of freedom of the ${\alpha }_s$-axis, which was removed in (A2).

3. Optimal References for the Currents

3.1. References in Healthy Conditions

To achieve decoupling during healthy operation of the torque (4), the field-oriented control strategy controls the currents $i_{dp}$ and $i_{ds}$ to zero. Hence, the torque equation is expressed as follows:

$$T_e=\frac{5}{2}{n}_p({\varphi }_{f1}{i}_{qp}+3{\varphi }_{f3}{i}_{qs})$$

(10)

As the current references are related to the properties of the back-EMF, the ratio between the $i_{qp}$ and $i_{qs}$ currents is given by [26]:

$$\epsilon =\frac{i_{qs}}{i_{qp}}=\frac{3{\varphi }_{f3}}{{\varphi }_{f1}}$$

(11)

where $\epsilon$ is the ratio of harmonic components between ${\varphi }_{f1}$ and ${\varphi }_{f3}$, which will ensure that copper losses are kept to a minimum.

From (10) and (11), we obtain the electromagnetic torque reference as follows:

$$T_e^{*}=k_T{i}_{qp}^{*}$$

(12)

where $k_T=\frac{5}{2}{n}_p{\varphi }_{f1}(1+{\epsilon }^2)$. Based on (10) and (12), the current references for pre-fault operation are given by $i_{dp}^{*}=0$, $i_{qp}^{*}=\frac{T_e^{*}}{k_T}$, $i_{ds}^{*}=0$, and $i_{qs}^{*}=\epsilon i_{qp}^{*}$.

3.2. References in Faulty Conditions

The reluctance torque arises due to the inequality between the d- and q-axis inductances, leading to undesirable torque ripples. To minimize these ripples, the $i_{dp}$ current should be set to zero to eliminate the term representing the difference in inductances in Equation (6). Similarly, the presence of $i_{\beta s}$ also contributes to torque ripples, particularly in third-order pulsations. Calculating the optimal $i_{\beta s}^{*}$ current reference depends on the chosen control criteria, which involves a trade-off between maximizing torque output (MTO) and minimizing copper loss (MCL).

3.2.1. Criteria for the MCL Strategy

To minimize copper loss, an optimal control strategy is to drive as many currents to zero as possible, while ensuring that the required output torque can be produced with the least number of currents without affecting performance. Figure 1 illustrates the structure of the MCL control criteria.

When both $i_{dp}^{*}$ and $i_{\beta s}^{*}$ are zero, the reference torque under an OCF according to the MCL criteria is simplified as follows:

$$T_{ef}^{*}=k_f{i}_{qp}^{*}(1-0.5\epsilon cos\left(2\theta \right)+0.5\epsilon cos\left(4\theta \right))$$

(13)

where $k_f=\frac{5}{2}{n}_p{\varphi }_{f1}$.

3.2.2. Criteria for the MTO Strategy

Figure 2 depicts the structure of the MTO control criteria. To achieve the maximum output torque from the inverter, the phase currents should have equal amplitudes. Assuming that the stator magnetomotive force is constant, the authors in [7] suggested choosing the currents of the other phases after phase “a” is open-circuited as $i_b=-i_d$ and $i_c=-i_e$. Furthermore, from (A2), we obtain the following:

$$\begin{array}{cc}\hfill i_{\beta p}& =\frac{2}{5}\left[(sin\frac{2\pi }{5}-sin\frac{6\pi }{5})i_b+(sin\frac{4\pi }{5}-sin\frac{8\pi }{5})i_c\right]\hfill \\ \hfill & =\frac{2}{5}(sin\frac{2\pi }{5}-sin\frac{6\pi }{5})(i_b+i_c)\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill i_{\beta s}& =\frac{2}{5}\left[(sin\frac{6\pi }{5}-sin\frac{18\pi }{5})i_b+(sin\frac{12\pi }{5}-sin\frac{24\pi }{5})i_c\right]\hfill \\ \hfill & =\frac{2}{5}(sin\frac{6\pi }{5}-sin\frac{18\pi }{5})(i_b+i_c)\hfill \end{array}$$

(15)

Dividing (15) by (14) yields

$$\begin{array}{cc}\hfill i_{\beta s}& =\frac{sin\frac{6\pi }{5}-sin\frac{18\pi }{5}}{sin\frac{6\pi }{5}-sin\frac{18\pi }{5}}{i}_{\beta p}=0.236i_{\beta p}\hfill \\ \hfill & =0.236(-i_{dp}sin\theta +i_{qp}cos\theta )\hfill \end{array}$$

(16)

By setting $i_{dp}^{*}$ to zero, the reference for the ${\beta }_s$-axis current is given by the following:

$$i_{\beta s}^{*}=0.236i_{qp}^{*}cos\theta$$

(17)

Hence, substituting (17) into the torque equation in the presence of an OCF (6) yields

$$T_{ef}^{*}=k_f{i}_{qp}^{*}(1-0.382\epsilon cos\left(2\theta \right)+0.618\epsilon cos\left(4\theta \right))$$

(18)

Following the MTO strategy for the reference torque (18), all of the amplitudes of the phases are expected to be the same and increased by $38.2\%$.

It is evident that the ratio $\epsilon$ appears with nonlinear terms in the torque reference and cannot be set to zero, as the third-order harmonic component is considered to enhance the torque performance. Moreover, nonlinear controllers are used in this paper to address the unmodeled and nonlinear terms in the motor dynamics.

4. Design Combination of SMC and NESO

Designing SMC with observers is not a new concept. Recently, several novel observer-based SMC methods have emerged, aiming to reduce the complexity in uncertain systems and faults [27,28]. Moreover, addressing mismatched disturbances in FPPMSM under OCF makes these techniques particularly valuable.

4.1. NESO Design

The FPPMSM model (5) can be expressed as follows, taking into account the perturbation parameters, load, and unknown terms as disturbances:

$$\left\{\begin{array}{c}{\dot{i}}_{dp}=f_{dp}+b_1{v}_{dp}+d_1\left(t\right)\hfill \\ {\dot{i}}_{qp}=f_{qp}+b_2{v}_{qp}+d_2\left(t\right)\hfill \\ {\dot{i}}_{ds}=f_{ds}+b_3{v}_{ds}+d_3\left(t\right)\hfill \\ {\dot{i}}_{qs}=f_{qs}+b_4{v}_{qs}+d_4\left(t\right)\hfill \\ \dot{\mathsf{\Omega }}=a_1{T}_e+d_5\left(t\right)\hfill \end{array}\right.$$

(19)

In the faulty conditions model (9), the currents of $i_{dp}$ and $i_{qp}$ remain the same as in the model under healthy conditions, except for the ${\beta }_s$-axis, which is given by

$${\dot{i}}_{\beta s}=f_{\beta s}+b_{\beta s}{v}_{\beta s}+d_6\left(t\right)$$

(20)

With

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & f_{dp}=c_1{i}_{dp}+c_2\mathsf{\Omega }{i}_{qp},f_{qp}=c_3{i}_{qp}-c_4\mathsf{\Omega }{i}_{dp}-c_5\mathsf{\Omega }\hfill \\ \hfill & f_{ds}=c_6{i}_{ds}+c_7\mathsf{\Omega }{i}_{qs},f_{qs}=c_8{i}_{qs}-c_9\mathsf{\Omega }{i}_{ds}-c_{10}\mathsf{\Omega }\hfill \\ \hfill & f_{\beta s}=c_{11}{i}_{\beta s}-c_{12}\mathsf{\Omega }cos\left(3\theta \right)\hfill \end{array}\end{array}$$

where $d_i\left(t\right)$ ($i=1,\dots ,6$) represent the mismatched disturbances corresponding to models (19) and (20). It is assumed that the mismatched disturbances and their derivatives are bounded as follows:

$${\rho }_i>max\left(\right|{\dot{d}}_i\left(t\right)\left|\right),{\delta }_i>max\left(\right|d_i\left(t\right)\left|\right)$$

(21)

Assuming that the equation describing the state of a nonlinear system is given as

$$\dot{x}=f(x,t)+bu+d\left(t\right)$$

(22)

where $f(x,t)$ is a known nonlinear function of x, u is the control input, b is a constant input gain, and $d\left(t\right)$ is the disturbance influencing the state.

The estimation of disturbance $d\left(t\right)$ in system (22) is conducted using a NESO designed as follows:

$$\left\{\begin{array}{c}{\dot{z}}_1=z_2-h(z_1-x)+f(x,t)+bu\hfill \\ {\dot{z}}_2=-h^{2}tanh(z_1-x)\hfill \end{array}\right.$$

(23)

Figure 3 shows the NESO diagram, where $z_1$ and $z_2$ are the estimates of x and $d\left(t\right)$, respectively; h is a positive design parameter.

The estimation errors, denoted by $e_1=z_1-x$ and $e_2=z_2-d\left(t\right)$, are transformed into ${\zeta }_1=e_1$ and ${\zeta }_2=e_2-h{e}_1$ considering the stable error dynamics at equilibrium. The dynamics of this transformation, derived from system (22) and the extended state observer (23), are as follows:

$$\left\{\begin{array}{c}{\dot{\zeta }}_1={\zeta }_2\hfill \\ {\dot{\zeta }}_2={\dot{e}}_2-h{\zeta }_2=-h^{2}tanh\left({\zeta }_1\right)-\eta -h{\zeta }_2\hfill \end{array}\right.$$

(24)

where $\eta =\dot{d}\left(t\right)$.

Consider the Lyapunov function shown below:

$$V_{\zeta }=h^2{\int }_0^{{\zeta }_1}tanh\left({\zeta }_1\right)d{\zeta }_1+\frac{1}{2}{\zeta }_2^2$$

(25)

Equation (24) provides the Lyapunov function derivative, as follows:

$$\begin{array}{cc}\hfill {\dot{V}}_{\zeta }& =\frac{\partial V_{\zeta }}{\partial {\zeta }_1}{\dot{\zeta }}_1+\frac{\partial V_{\zeta }}{\partial {\zeta }_2}{\dot{\zeta }}_2\hfill \\ \hfill & =h^{2}tanh\left({\zeta }_1\right){\zeta }_2+{\zeta }_2(-h^{2}tanh\left({\zeta }_1\right)-\eta -h{\zeta }_2)\hfill \\ \hfill & =-{\zeta }_2(\eta +h{\zeta }_2)\hfill \end{array}$$

(26)

If $\eta =0$, then ${\dot{V}}_{\zeta }=-h{\zeta }_2^2\le 0$. Despite the derivative of the zero solution, the Lyapunov function remains stable due to its construction from the system (24). If $\eta \ne 0$ and assuming that $\left|\eta \right|\le \rho$, the derivative of the Lyapunov function (26) can be given as follows:

$${\dot{V}}_{\zeta }\le -h{\left({\zeta }_2+\frac{\eta }{2h}\right)}^2+\frac{{\rho }^2}{4h}$$

(27)

According to (27), it becomes evident that when $|{\zeta }_2| \le \frac{\left|\eta \right|}{h}\le \frac{\rho }{h}$, then ${\dot{V}}_{\zeta }<0$, indicating the stability of system (24), which will reduce at a steady state as follows:

$$\left\{\begin{array}{c}{\dot{\zeta }}_1={\zeta }_2=0\hfill \\ {\dot{\zeta }}_2=-h^{2}tanh\left({\zeta }_1\right)-\eta -h{\zeta }_2=0\hfill \end{array}\right.$$

(28)

As the HTF is monotonically increasing, we can deduce from Equation (28) that, by taking its absolute value, it will be upper bounded as $|{\zeta }_1| \le {\tanh }^{-1}\left(\frac{\rho }{h^2}\right)$. This bound will ensure that the NESO errors of the estimation meet these conditions:

$$\begin{array}{ccc}\hfill & |e_1| \le {tanh}^{-1}\left(\frac{\rho }{h^2}\right),\hfill & \hfill |e_2| \le h{tanh}^{-1}\left(\frac{\rho }{h^2}\right)\end{array}$$

(29)

By selecting an appropriate parameter $h>0$, the estimation errors $e_1$ and $e_2$ will eventually approach the vicinity of zero.

According to (23), the NESO based on the HTF to estimate the mismatched disturbance for each state of the FPPMSM model is given as follows:

• Defining x, u, $z_1$, and $z_2$ as $i_{dp}$, $v_{dp}$, ${\widehat{i}}_{dp}$, and ${\widehat{d}}_1\left(t\right)$, respectively, $d_1\left(t\right)$ can be estimated by

  $$\left\{\begin{array}{c}{\dot{\widehat{i}}}_{dp}={\widehat{d}}_1\left(t\right)-h_1({\widehat{i}}_{dp}-i_{dp})+f_{dp}+b_1{v}_{dp}\hfill \\ {\dot{\widehat{d}}}_1\left(t\right)=-h_1^{2}tanh({\widehat{i}}_{dp}-i_{dp})\hfill \end{array}\right.$$

  (30)
• Defining x, u, $z_1$, and $z_2$ as $i_{qp}$, $v_{qp}$, ${\widehat{i}}_{qp}$, and ${\widehat{d}}_2\left(t\right)$, respectively, $d_2\left(t\right)$ can be estimated by

  $$\left\{\begin{array}{c}{\dot{\widehat{i}}}_{qp}={\widehat{d}}_2\left(t\right)-h_2({\widehat{i}}_{qp}-i_{qp})+f_{qp}+b_2{v}_{qp}\hfill \\ {\dot{\widehat{d}}}_2\left(t\right)=-h_2^{2}tanh({\widehat{i}}_{qp}-i_{qp})\hfill \end{array}\right.$$

  (31)
• Defining x, u, $z_1$, and $z_2$ as $i_{ds}$, $v_{ds}$, ${\widehat{i}}_{ds}$, and ${\widehat{d}}_3\left(t\right)$, respectively, $d_3\left(t\right)$ can be estimated by

  $$\left\{\begin{array}{c}{\dot{\widehat{i}}}_{ds}={\widehat{d}}_3\left(t\right)-h_3({\widehat{i}}_{ds}-i_{ds})+f_{ds}+b_3{v}_{ds}\hfill \\ {\dot{\widehat{d}}}_3\left(t\right)=-h_3^{2}tanh({\widehat{i}}_{ds}-i_{ds})\hfill \end{array}\right.$$

  (32)
• Defining x, u, $z_1$, and $z_2$ as $i_{qs}$, $v_{qs}$, ${\widehat{i}}_{qs}$, and ${\widehat{d}}_4\left(t\right)$, respectively, $d_4\left(t\right)$ can be estimated by

  $$\left\{\begin{array}{c}{\dot{\widehat{i}}}_{qs}={\widehat{d}}_4\left(t\right)-h_4({\widehat{i}}_{qs}-i_{qs})+f_{qs}+b_4{v}_4\hfill \\ {\dot{\widehat{d}}}_4\left(t\right)=-h_4^{2}tanh({\widehat{i}}_{qs}-i_{qs})\hfill \end{array}\right.$$

  (33)
• Defining x, u, $z_1$, and $z_2$ as $\mathsf{\Omega }$, $T_e$, $\widehat{\mathsf{\Omega }}$, and ${\widehat{d}}_5\left(t\right)$, respectively, $d_5\left(t\right)$ can be estimated by

  $$\left\{\begin{array}{c}\dot{\widehat{\mathsf{\Omega }}}={\widehat{d}}_5\left(t\right)-h_5(\widehat{\mathsf{\Omega }}-\mathsf{\Omega })+a_1{T}_e\hfill \\ {\dot{\widehat{d}}}_5\left(t\right)=-h_5^{2}tanh(\widehat{\mathsf{\Omega }}-\mathsf{\Omega })\hfill \end{array}\right.$$

  (34)
• Defining x, u, $z_1$, and $z_2$ as $i_{\beta s}$, $v_{\beta s}$, ${\widehat{i}}_{\beta s}$, and ${\widehat{d}}_6\left(t\right)$, respectively, $d_6\left(t\right)$ can be estimated by

  $$\left\{\begin{array}{c}{\dot{\widehat{i}}}_{\beta s}={\widehat{d}}_6\left(t\right)-h_6({\widehat{i}}_{\beta s}-i_{\beta s})+f_{\beta s}+b_{\beta s}{v}_{\beta s}\hfill \\ {\dot{\widehat{d}}}_6\left(t\right)=-h_6^{2}tanh({\widehat{i}}_{\beta s}-i_{\beta s})\hfill \end{array}\right.$$

  (35)

where $h_i>0$ ($i=1,\dots ,6$) represent the constant gains of the observer.

4.2. SMC Design

This paper introduces a method that combines proportional rate reaching with power rate reaching to accelerate the attainment of the sliding surface [18]. Moreover, the NESO is designed to improve convergence performance against mismatched disturbances and reduce steady-state error. This allows for a reduction in the switching gain in SMC, focusing primarily on the speed of reaching the sliding surface, thereby decreasing chattering. The proposed SMC approach is expected to be more effective in increasing the speed of reaching and is detailed below:

$$\left\{\begin{array}{c}\dot{s}=-k|s{|}^{\alpha }\mathrm{sign}(s)-ms\hfill \\ 0<\alpha <1,k>0,m>0\hfill \end{array}\right.$$

(36)

where $\alpha$, k, and m are the design parameters that satisfy the conditions in the previous equation, while the sliding surface is denoted by s.

The proposed SMC diagram is shown in Figure 4. The sliding surfaces for each of the six controllers are described below:

$$\left\{\begin{array}{c}{s}_1={\widehat{i}}_{dp}-i_{dp}^{*}\hfill \\ s_2={\widehat{i}}_{qp}-i_{qp}^{*}\hfill \\ s_3={\widehat{i}}_{ds}-i_{ds}^{*}\hfill \\ s_4={\widehat{i}}_{qs}-i_{qs}^{*}\hfill \\ s_5=\widehat{\mathsf{\Omega }}-{\mathsf{\Omega }}^{*}\hfill \\ s_6={\widehat{i}}_{\beta s}-i_{\beta s}^{*}\hfill \end{array}\right.$$

(37)

where “*” denotes the reference state, while “^” denotes the state observer estimated by the NESO.

Based on (36) and (37), the laws of discontinuous control for each controller are given below:

$$\left\{\begin{array}{c}{u}_i=-k_i|s_i{|}^{{\alpha }_i}\mathrm{sign}(s_i)-m_i{s}_i\hfill \\ i=1,\dots ,6\hfill \end{array}\right.$$

(38)

where ${\alpha }_i$, $k_i$, and $m_i$ ($i=1,\dots ,6$) are positive design parameters.

The combination of (38) with the laws of equivalent control calculated from the derivative of the sliding surfaces allows us to obtain the following control laws:

$$\begin{array}{cc}\hfill & \left\{\begin{array}{cc}\hfill v_{dp}^{*}& =\frac{1}{b_1}(-{\widehat{f}}_{dp}-{\widehat{d}}_1\left(t\right)+{\dot{i}}_{dp}^{*})+u_1\hfill \\ \hfill v_{qp}^{*}& =\frac{1}{b_2}(-{\widehat{f}}_{qp}-{\widehat{d}}_2\left(t\right)+{\dot{i}}_{qp}^{*})+u_2\hfill \\ \hfill v_{ds}^{*}& =\frac{1}{b_3}(-{\widehat{f}}_{ds}-{\widehat{d}}_3\left(t\right)+{\dot{i}}_{ds}^{*})+u_3\hfill \\ \hfill v_{qs}^{*}& =\frac{1}{b_4}(-{\widehat{f}}_{qs}-{\widehat{d}}_4\left(t\right)+{\dot{i}}_{qs}^{*})+u_4\hfill \\ \hfill T_e^{*}& =\frac{1}{a_1}(-{\widehat{d}}_5\left(t\right)+{\dot{\mathsf{\Omega }}}^{*})+u_5\hfill \\ \hfill v_{\beta s}^{*}& =\frac{1}{b_{\beta s}}(-{\widehat{f}}_{\beta s}-{\widehat{d}}_6\left(t\right)+{\dot{i}}_{\beta s}^{*})+u_6\hfill \end{array}\right.\hfill \end{array}$$

(39)

With

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & {\widehat{f}}_{dp}=c_1{\widehat{i}}_{dp}+c_2\widehat{\mathsf{\Omega }}{\widehat{i}}_{qp},{\widehat{f}}_{qp}=c_3{\widehat{i}}_{qp}-c_4\widehat{\mathsf{\Omega }}{\widehat{i}}_{dp}-c_5\widehat{\mathsf{\Omega }}\hfill \\ \hfill & {\widehat{f}}_{ds}=c_6{\widehat{i}}_{ds}+c_7\widehat{\mathsf{\Omega }}{\widehat{i}}_{qs},{\widehat{f}}_{qs}=c_8{\widehat{i}}_{qs}-c_9\widehat{\mathsf{\Omega }}{\widehat{i}}_{ds}-c_{10}\widehat{\mathsf{\Omega }}\hfill \\ \hfill & {\widehat{f}}_{\beta s}=c_{11}{\widehat{i}}_{\beta s}-c_{12}\widehat{\mathsf{\Omega }}cos\left(3\theta \right)\hfill \end{array}\end{array}$$

where ${\widehat{d}}_i\left(t\right)$ ($i=1,\dots ,6$) represents the estimated disturbances of NESO.

A Lyapunov function can be formulated in the following way:

$$V=\frac{1}{2}(\sum _{i=1}^6{s}_i^2)$$

(40)

Taking the time derivative of (40) yields

$$\begin{array}{cc}\hfill \dot{V}& ={\displaystyle \sum _{i=1}^6}(-\frac{k_i}{b_i}|s_i{|}^{{\alpha }_i}\mathrm{sign}(s_i)-\frac{m_i}{b_i}{s}_i)s_i\hfill \\ \hfill & =\left({\displaystyle \sum _{i=1}^6}(-A_i|s_i{|}^{{\alpha }_i}\left|s_i\right|-B_i{s}_i^2)\right)\le 0\hfill \end{array}$$

(41)

where $A_i=\frac{k_i}{b_i},B_i=\frac{m_i}{b_i}$. Achieving $\dot{V}\le 0$ stability requires satisfying the conditions $0<a_i<1$, $A_i>0$, and $B_i>0$.

5. Results Analysis

Experiments were conducted using MATLAB/Simulink to validate the structural diagram of the control strategy for the FPPMSM under healthy conditions and an OCF, as depicted in Figure 5 and Figure 6, respectively. The FPPMSM drive parameters are reported in

Table A1. Motor drive parameters [31].

Parameter	Value
$n_p$	2
$R_s$	$1.1\mathsf{\Omega }$
$L_{qp}$	8.32 mH
$L_{dp}$	6.54 mH
$L_{qs}$	1.68 mH
$L_{ds}$	1.78 mH
$L_{ls}$	1.35 mH
${\varphi }_{f1}$	0.512 Wb
${\varphi }_{f3}$	0.034 Wb
J	0.095 kg·m2
DC-link voltage	150 V
Switching frequency	10 kHz
Rated power	3 kW
Rated speed	1000 rpm
Maximum torque	55 Nm
Peak phase voltage	125 V
Peak phase current	21 A

. The proposed control algorithm for the motor was implemented on a digital signal processor (DSP) card (TMS320F379D), which produces PWM to control a five-phase voltage source inverter (VSI) that supplies the FPPMSM. The FPPMSM and VSI models were modeled in Simulink and simulated with a fixed step size of $1\times {10}^{-6}$ s. The generation of PWM is based on carrier pulse width modulation (CPWM), due to its easy and simple implementation for a five-phase motor spatially under an OCF. Therefore, the CPWM technique, dealing with modulation signals injected with zero-sequence signals, requires only a few mathematical calculations compared to other techniques. It uses the zero-sequence signal equation as follows [29]:

$$v_{no}\left(t\right)=-\frac{1}{2}(max(v_a^{*},v_b^{*},v_c^{*},v_d^{*},v_e^{*})+min(v_a^{*},v_b^{*},v_c^{*},v_d^{*},v_e^{*}))$$

(42)

The back-EMF of the motor exhibited a trapezoidal shape due to the presence of the third harmonic component, as shown in Figure 7. The fast Fourier transform analysis revealed that the third harmonic constituted $19.92\%$ of the back-EMF spectrum. To evaluate the effectiveness of the proposed control strategy, we assessed it in healthy and faulty conditions, comparing the results with PI and traditional SMC (TSMC) control strategies as benchmarks, since they are the most commonly used control methods for FPPMSMs in the literature. The TSMC used in the results combined two methodologies proposed in [12,13]. In [12], the authors advocated the use of a traditional sliding mode controller with the help of a PI controller in the control loop for the speed. In [13], the authors recommended the use of traditional sliding mode controllers in the control loops of the current. Additionally, to compare the performance of the MTO and MCL strategies, two performance metrics, namely torque ripples and speed fluctuations, were calculated as follows:

$$\begin{array}{c}{T}_{e_{ripple}}\%=\frac{(max\left(T_e\right) - min\left(T_e\right))}{mean\left(T_e\right)}\times 100,{\mathsf{\Omega }}_{fluc}\%=\frac{(max(\mathsf{\Omega }) - min(\mathsf{\Omega }\left)\right)}{mean\left(\mathsf{\Omega }\right)}\times 100\hfill \end{array}$$

(43)

5.1. Healthy Conditions

To illustrate the robustness of the proposed control (SMC with NESO) under normal operating conditions, we analyzed the effects of unmodeled dynamics and stator resistance uncertainty. The mismatched disturbances in (19) were modeled as follows: $d_1\left(t\right)=-\frac{\mathsf{\Delta }{R}_s}{L_{dp}}{i}_{dp}+{\delta }_{dp}$, $d_2\left(t\right)=-\frac{\mathsf{\Delta }{R}_s}{L_{qp}}{i}_{qp}+{\delta }_{qp}$, $d_3\left(t\right)=-\frac{\mathsf{\Delta }{R}_s}{L_{ds}}{i}_{ds}+{\delta }_{ds}$, and $d_4\left(t\right)=-\frac{\mathsf{\Delta }{R}_s}{L_{qs}}{i}_{qs}+{\delta }_{qs}$; where $\mathsf{\Delta }{R}_s=|R_s-R_{sn}|$; $R_{sn}$ denotes the nominal value of the stator resistance. ${\delta }_{dp}$, ${\delta }_{qp}$, ${\delta }_{ds}$, and ${\delta }_{qs}$ represent the unmodeled dynamics, which are characterized by sinusoidal forms accounting for dead-time effects [30].

Figure 8, Figure 9, Figure 10 and Figure 11 show the FPPMSM responses of the speed and torque under two scenarios involving step changes in load torque and reference speed. Figure 8 and Figure 9 present the first scenario, where the speed reference was fixed at 300 rpm, and the step load torque changed from 20 Nm to 40 Nm at time 1 s. At time 2 s, unmodeled dynamics were introduced as ${\delta }_{dp}=10sin\left(6\theta \right)$ and ${\delta }_{qp}=10cos\left(6\theta \right)$, and the stator resistance was increased by $+60\%R_s$, resulting in $1.6R_{sn}$. It is evident that all three methods could track the speed reference in the face of the step-increased load torque and mismatched disturbances. However, the TSMC achieved a shorter settling time, with no overshoot compared to the PI strategy. Nonetheless, the PI and TSMC exhibited noticeable ripples and fluctuations in the torque and speed responses at time 2 s, reflecting poor robustness in the presence of disturbances. On the other hand, the proposed control (SMC with NESO) achieved the shortest settling time and significantly minimized torque ripples and speed fluctuations compared to the traditional controllers. In the second scenario, shown in Figure 10 and Figure 11, the load torque was fixed at 40 Nm, and the step speed reference was changed from 100 rpm to 300 rpm and then back to 100 rpm at time 3 s. At time 2 s, the PMSM started operating under stator resistance uncertainty and unmodeled dynamics, as in the first scenario. It is clear that the SMC with NESO tracked the reference speed more accurately than the traditional controllers (PI and TSMC). Therefore, the effectiveness and robustness of the proposed control under healthy conditions were verified.

5.2. Faulty Conditions

Figure 12, Figure 13 and Figure 14 show the FPPMSM responses of the speed, torque, and currents under the influence of an OCF in phase “a”. This fault event occurred at time 1 s, while the motor was operating at a load torque of 20 Nm and a speed reference of 200 rpm. Fault detection occurred at time 1.5 s, prompting an immediate transition to the proposed FTC diagram under faulty conditions using the MCL control strategy. It is clear that the oscillations and ripples in speed and torque responses using the proposed control were significantly reduced compared to the traditional controllers. Figure 14 illustrates the notable irregularities in the currents between times 1 s and 1.5 s before fault detection. Following fault detection, the currents exhibited disparities in magnitude, with phases “d” and “c” increasing by 26.31% and phases “e” and “b” increasing by 46.78%.

To evaluate the MCL and MTO control criteria using controllers based on PI, TSMC, and SMC with NESO, the FPPMSM was operated with four phases under a load torque of 40 Nm and a speed reference of 300 rpm. As shown in Figure 15 and Figure 16, the torque ripple peak was larger for the MTO strategy compared to the MCL strategy. Meanwhile, the proposed control (SMC with NESO) exhibited minimal and almost identical torque ripples for each control criterion (see Figure 17). In addition, as noted in Figure 18, the MTO strategy kept all phase currents equal in magnitude and increased by 38.2% compared to normal operation with five phases. According to (43),

Table 1. Comparison of MTO and MCL using PI, TSMC, and SMC with NESO.

Controllers	Strategy	${\mathit{T}}_{{\mathit{e}}_{\mathit{ripple}}}\%$	${\mathbf{\Omega }}_{\mathit{fluc}}\%$
PI	MTO	$37.3153$	$1.9202$
	MCL	$30.8831$	$1.5769$
TSMC	MTO	$19.4586$	$0.9958$
	MCL	$16.0544$	$0.814$
SMC+NESO	MTO	$1.9396$	$0.0118$
	MCL	$1.8087$	$0.0094$

presents the calculation of torque ripples and speed fluctuations for the MTO and MCL strategies based on the traditional controllers and the proposed control. It is evident that the SMC with NESO had the lowest ripples and fluctuations, which were approximately identical for both strategies.

To validate the robustness of the proposed control under faulty operating conditions, we investigated the impact of parameter uncertainties, specifically by varying the stator resistance. The mismatched disturbances in (19) and (20) were modeled as follows: $d_1\left(t\right)=-\frac{\mathsf{\Delta }{R}_s}{L_{dp}}{i}_{dp}$, $d_2\left(t\right)=-\frac{\mathsf{\Delta }{R}_s}{L_{qp}}{i}_{qp}$, and $d_6\left(t\right)=-\frac{\mathsf{\Delta }{R}_s}{L_{ls}}{i}_{\beta s}$. Two control scenarios were investigated: one employed the proposed control based on the NESO that incorporated the HTF, and the other used an ESO without the HTF. The load torque and speed reference were set to 20 Nm and 300 rpm, respectively, and the stator resistance was adjusted to +60% of $R_s$ at time 0.5 s. Figure 19 and Figure 20 show the FFT spectra of the phase “d” stator current in the steady state. As can be seen, the proposed control using the HTF had a lower THD of 24.08% compared to the control without HTF, which had a THD of 35.75%. Figure 21 and Figure 22 illustrate the speed and torque responses. It is evident that when the stator resistance changed at time 0.5 s, the control without HTF showed more pronounced ripples in speed and torque compared to the proposed control with HTF. Nevertheless, when the speed reference was set to 100 rpm at time 2 s, the ripples in both scenarios were nearly identical.

6. Conclusions

This paper proposed a FTC strategy for a FPPMSM affected by the third-order harmonic component of back-EMF under a single-phase OCF. The strategy used SMC with NESO based on the HTF, and the Lyapunov theory proved its stability. Additionally, the paper presented optimal current references using a field-oriented control for torque decoupling and minimum copper losses under healthy conditions, along with two criteria for optimal current references under faulty conditions: minimum copper losses (MCL) and maximum torque output (MTO). The paper showed that the proposed control had similar low torque ripples for each criterion, unlike TSMC and PI controllers. Moreover, the paper verified that the proposed control improved the tracking accuracy and robustness to disturbances under both healthy and faulty conditions, significantly reducing torque ripples and speed fluctuations under a fault.