A Finite-Set Integral Sliding Modes Predictive Control for a Permanent Magnet Synchronous Motor Drive System

Abstract

Finite-set model predictive control (FS-MPC) is an easy and intuitive control technique. However, parametric uncertainties reduce the accuracy of the prediction. Classical MPC requires many calculations; therefore, the calculation time generates a considerable time delay in the actuation. This delay deteriorates the performance of the system and generates a significant current ripple. This paper proposes a finite-set integral sliding modes predictive control (FS-ISMPC) for a permanent magnet synchronous motor (PMSM). The conventional decision function is replaced by an integral sliding cost function, which has several advantages, such as robustness to parameter uncertainties, and convergence in finite time. The proposed decision function does not require the inductance and resistance parameters of the motor. In addition, the proposal includes compensation for the calculation delay of the control vector. The proposed control strategy was compared with traditional predictive control with delay compensation using a real-time hardware-in-the-loop (HIL) simulation. The results obtained from the comparison indicated that the proposed controller has a lower THD and computational burden.

1. Introduction

The permanent magnet synchronous motor (PMSM) offers high efficiency, power density, and the ability to maintain constant power over a wide range of speeds. PMSM is widely used in industrial applications and electric vehicle traction systems. PMSM can be controlled using both trapezoidal and sinusoidal PWM techniques for speed and position. In [1], a genetic algorithm-based controller for speed and position is presented. The study compares the performance of trapezoidal and sinusoidal PWM commutation techniques, providing insights for optimizing motor control systems. The classic control methods used in PMSM drives are current vector control [2] and direct torque control [3]. However, traditional controllers have poor dynamic performance. Some of the problems include overshoot and torque fluctuations at low speeds. Finite-set model predictive control (FS-MPC) has gained high relevance in the last decade due to its implementation in digital systems. FSMPC does not require modulation, and nonlinear constraints can be easily incorporated [4].

Controllers based on FS-MPC for PMSM suffer from several drawbacks. The online control vector calculation time affects the performance of the controller [5]. Computation time includes filtering, sampling, and other factors, which generate a delay in the control signal. The state of the switches changes once the calculation is completed. Consequently, the measured currents present a ripple of significant amplitude [6,7]. Therefore, it is necessary to add compensation for the delay and improve the performance of the control system. The most widely used method for delay compensation is to increase the prediction and control horizon [8,9,10]. To increase the performance of the compensation, the control vector of the next sampling is calculated. This process is repeated due to the receding horizon principle. However, this compensation method depends on the accuracy of the model. On the other hand, parameter uncertainty is a challenge for FS-MPC, since parameters change due to PMSM conditions. The saturation effect of the magnetic field changes the inductance of the motor under load conditions. Temperature changes due to mechanical stress decrease the value of the constant flux linkage. In addition, the rising temperature increases the resistance of the winding [11]. Several solutions are presented in the literature to improve the robustness of the prediction. In practical applications, there is not only parametric uncertainty but also disturbances and nonlinearities. Therefore, disturbance estimation and compensation techniques are required [12,13,14]. In [15], a predictive current control based on a composite observer is proposed. The estimation method eliminates the influence of motor parameter variation and permanent magnet demagnetization on the current vector. The proposed observer is designed using sliding modes (SM) and Luenberger approaches. In [16], a moving horizon estimator is utilized; the effect of constant flux linkage is eliminated, and voltage errors caused by resistance and inductance are compensated.

The recent trend to solve the parameter mismatch has led to the study of model-free control (MFC) methods. In [17], a nonparametric predictive current control (NPCC) has been proposed that can predict future current behavior using the measured data instead of motor parameters. However, the performance of NPCC is deficient if two consecutive switching states are identical, causing a stagnation problem [18,19]. MFC can be performed using the ultra-local model (ULM) approach. The ULM technique can be implemented by utilizing the input and output measurement data of the system without prior knowledge of the system parameters. The mathematical model is replaced by a differential equation of order selected by the practitioner [20]. ULM has been successfully applied to MPCs for PMSM drives [21,22]. In [23], a model of extended affine ULM is built containing a two-order term based on affine arithmetic. The accuracy of the model is improved using extended affine ULM, even in the presence of nonlinear terms. ULM provides a detailed representation of the system behavior in a small region of operation, allowing for more accurate predictions and control strategies tailored to that specific region [24].

Several robust control techniques exist in the field of AC drive systems. SM control is a special nonlinear control strategy, which considers robustness as a part of the design process. This control strategy can also be adapted to power electronics due to the switching nature of the control law. A stability analysis, using Lyapunov theory, is the main advantage [25]. However, sliding chattering is related to the robustness of the overall control system. In [26], an adaptive integral sliding mode predictive control is proposed to solve the chattering problem and improve the dynamics of the system. Recent research has presented a study of MPC using the SM cost function [27]. In [28], a different cost function using SM theory is proposed for a current controller of a grid-connected three-phase bidirectional power inverter. The sliding cost function presents a fast dynamic response and reduces computational burden. In [29], a finite-set sliding modes predictive current control (FS-SMPC) is applied to a PMSM.

Apart from the control technique, its testing and validation have critical importance. HIL is a real-time simulation technique. HIL verifies control algorithms and plant models, saving cost and time with high fidelity. PMSM and power converters have been successfully simulated in real time. Moreover, parameters can be changed during the simulation [30]. In [31], HIL implementation using digital signal processors (DSPs) and field programmable gate arrays (FPGA) for electric machine drives has been investigated. In addition, the drawbacks of both methods, such as complexity in development and verification, are presented. In [32], a high-fidelity real-time simulation using HIL is presented; the simulation is implemented on an FPGA platform; and HIL testing is validated against finite element analysis and experimental results.

In this paper, a robust model predictive control based on sliding mode theory is proposed for the current control of the PMSM drive system, instead of the traditional FS-MPC, which considers an error square cost function. The chattering effect is reduced by an integral sliding surface and an extended control set. To eliminate the delay due to calculation time, a one-step-ahead control horizon is implemented. The contributions of the paper are: (1) A robust current controller for PMSM is designed using an integral sliding cost function. (2) The computational burden of the proposed controller is low compared to traditional FS-MPC. (3) The designed controller is simple, and real-time simulation demonstrates the parameter robustness. This paper is organized as follows: In Section 2, the mathematical model of PMSM and traditional FS-MPC is introduced. In Section 3, the proposed FS-ISMPC strategy is described. Real-time simulation is demonstrated to verify the effectiveness of the proposed method in Section 4, which is followed by a discussion in Section 5. Finally, the paper is concluded in Section 6.

2. Mathematical Model of PMSM and FS-Strategies

2.1. The Mathematical Model of PMSM

In this study, the mathematical model of a PMSM in a dq synchronous reference frame is described as follows [33]:

$$\left\{\begin{array}{c}\frac{d{i}_d}{dt}=-\frac{R}{L_d}{i}_d+\frac{L_q}{L_d}{i}_q{\omega }_e+\frac{1}{L_d}{u}_d+f_d\\ \frac{d{i}_q}{dt}=-\frac{R}{L_q}{i}_q-\frac{L_d}{L_q}{i}_d{\omega }_e-\frac{{\omega }_e{\psi }_f}{L_q}+\frac{1}{L_q}{u}_q+f_q.\end{array}\right.$$

(1)

In this model, $u_d$ and $u_q$, represent the d and q axis voltages, respectively; $i_d$ and $i_q$, represent the d and q axis currents, respectively; $L_d$ and $L_q$ are the stator inductance; $R$ is the winding resistance; ${\psi }_f$ is the flux linkage of permanent magnets; ${\omega }_e$ is the electrical angular velocity; $f_d$ and $f_q$ are defined to represent the disturbance caused by the parameter variation. The parameters of the motor are mainly influenced by the temperature and magnetic saturation. Therefore, the deviation of the internal parameters is ranged in a finite scale. The mechanical motion equation and the electromagnetic torque of PMSM are written as follows:

$$\left\{\begin{array}{c}\frac{d{\omega }_m}{dt}=\frac{T_e}{J}-\frac{T_L}{J}-\frac{B}{J}{\omega }_m\\ T_e=\frac{3}{2}{n}_p\left[{\psi }_f+\left(L_d-L_q\right)i_d\right]i_q,\end{array}\right.$$

(2)

where $T_e$ and $T_L$, represent the electromagnetic torque and load torque, respectively; $B$ represents the viscous friction; ${\omega }_m$ is the mechanical angular velocity; $J$ is the rotor inertia; $n_p$ is the number of pole pairs. Moreover, for surface-mounted PMSM inductance $L_d$ is equal to $L_q$, therefore electromagnetic torque can be simplified as follows:

$$T_e=K_t{i}_q,$$

(3)

where $K_t$ is the torque constant, assumed to be equal to $(3/2)n_p{\psi }_f$.

2.2. Traditional FS-MPC Strategy

In power electronics, MPC can be implemented considering a finite number of control actions or switching states. FS-MPC is intuitive, the first step is obtaining the dynamic model of the system. According to the Euler equation and neglecting the perturbation terms, (1) can be written in discrete form as follows:

$$i_{dq}^p\left(k+1\right)=\mathit{A}\left(k\right)i_{dq}\left(k\right)+\mathit{B}{u}_{dq}\left(k\right)+\mathit{C}\left(k\right),$$

(4)

where $k$ and $k+1$ represent the value of the current moment and the next moment, respectively, $T_s$ is the sampling period, and written as follows:

$$\begin{array}{c}{i}_{dq}^p\left(k+1\right)=\left[\begin{array}{c}{i}_d^p\left(k+1\right)\\ i_q^p\left(k+1\right)\end{array}\right]; i_{dq}\left(k\right)=\left[\begin{array}{c}{i}_d\left(k\right)\\ i_q\left(k\right)\end{array}\right];\hfill \\ u_{dq}\left(k\right)=\left[\begin{array}{c}{u}_d\left(k\right)\\ u_q\left(k\right)\end{array}\right];\mathit{A}\left(k\right)=\left[\begin{array}{cc}1-\frac{T_{s}R}{L_d}& {\omega }_e\left(k\right)T_s\\ -{\omega }_e\left(k\right)T_s& 1-\frac{T_{s}R}{L_q}\end{array}\right];\hfill \\ \mathit{B}=\left[\begin{array}{cc}\frac{T_s}{L_d}& 0\\ 0& \frac{T_s}{L_q}\end{array}\right]; \mathit{C}\left(k\right)=\left[\begin{array}{c}0\\ -\frac{{\omega }_e\left(k\right){\psi }_f{T}_s}{L_q}\end{array}\right].\hfill \end{array}$$

The PMSM is driven by a two-level three-phase voltage source inverter. The switching states can be summarized in a single complex switching function given by:

$$\mathit{S}=\frac{2}{3}\left(S_a+S_b\mathit{a}+S_c{\mathit{a}}^2\right),$$

(5)

where $S_a$, $S_b$, and $S_c$ are the switching functions of the inverter, $V_{dc}$ is the DC source voltage and $\mathit{a}$ is a unitary vector equal to $e^{j2\pi /3}$. Considering all possible combinations, there are eight different voltage vectors, as depicted in Figure 1. To ensure that the output current follows a specified trajectory, a cost function is defined as follows [3]:

$$J_k={\left[i_d^{ref}-i_d^p\left(k+1\right)\right]}^2+{\left[i_q^{ref}-i_q^p\left(k+1\right)\right]}^2,$$

(6)

where $i_d^{ref}$ is the d axis reference current; $i_q^{ref}$ is the q axis reference current. When field-oriented control is adopted $i_d^{ref}$ = 0, and $i_q^{ref}$ is obtained from the speed controller.

Considering the digital delay in the control action, a one-step-ahead control horizon is used. The cost function (6) evaluates the error two sampling periods ahead. The traditional cost function with delay compensation is defined as follows [34]:

$$J_{k+1}={\left[i_d^{ref}-i_d^p\left(k+2\right)\right]}^2+{\left[i_q^{ref}-i_q^p\left(k+2\right)\right]}^2.$$

(7)

2.3. FS-SMPC Strategy

MPC is applied in a way that an optimal control sequence is given every sampling period. Moreover, the prediction is performed in an open loop. Therefore, stability is not guaranteed [35]. In sliding mode control theory, the stability of the closed-loop system is proved using a Lyapunov function [25].

The space vector representation of (1) in the $\alpha \beta$ stationary reference frame is given by:

$$\frac{d{\mathit{i}}_s}{dt}=-\frac{R}{L_s}{\mathit{i}}_s-\frac{1}{L_s}\mathit{\xi }+\frac{1}{L_s}{\mathit{u}}_s+{\mathit{f}}_s,$$

(8)

where ${\mathit{u}}_s$ is the stator voltage, ${\mathit{f}}_s$ is the disturbance, ${\mathit{i}}_s$ is the stator current, ${\theta }_e$ is the electrical rotor position, $L_s=L_d=L_q$, and

$$\frac{d{\mathit{i}}_s}{dt}=\left[\begin{array}{c}\frac{d{i}_{\propto }}{dt}\\ \frac{d{i}_{\beta }}{dt}\end{array}\right];{\mathit{i}}_s=\left[\begin{array}{c}{i}_{\propto }\\ i_{\beta }\end{array}\right]; {\mathit{u}}_s=\left[\begin{array}{c}{u}_{\propto }\\ u_{\beta }\end{array}\right]; \mathit{\xi }=j{\omega }_e{\psi }_f{e}^{j{\theta }_e}.$$

A sliding surface is designed as the difference between the measured current and the current reference. The sliding surface can be written as follows:

$$\mathsf{\sigma }={\mathit{i}}_s-{\mathit{i}}_s^{ref}.$$

(9)

The high-frequency switching ripple is directly related to the stator current vector ${\mathit{i}}_s$ and the reference current ${\mathit{i}}_{ref}$ is a pure sinusoidal waveform. Taking the first-time derivative of the sliding surface, and considering (8), this yields the following:

$$\dot{\mathsf{\sigma }}\approx \frac{d{\mathit{i}}_{\mathit{s}}}{dt}.$$

(10)

This leads to the following cost function and optimization problem:

$$\begin{array}{c}{J}_{opt}=\min \left\{{\mathsf{\sigma }}^T\dot{\mathsf{\sigma }}\right\}<0\\ s.t. {\mathit{u}}_s\in V_{dc}\mathit{S}.\end{array}$$

(11)

The above expression can be simplified since $L_s$ is always positive and $R{\mathit{i}}_s\ll {\mathit{u}}_s$. The cost function (11) can be written in discrete form as follows [26,27]:

$$J_k^{sm}={\left({\mathit{i}}_s\left(k\right)-{\mathit{i}}_s^{ref}\left(k\right)\right)}^T\left(-\mathit{\xi }\left(k\right)+{\mathit{u}}_s\left(k\right)\right).$$

(12)

3. Proposed FS-ISMPC

3.1. Integral Sliding Mode Predictive Control

The integral sliding mode control algorithm is an attractive alternative. The order of the motion equation in integral sliding mode is equal to the order of the original system [26]. Therefore, the system robustness can be ensured by an overall response of the system starting from the initial time instance. Let ${\mathit{e}}_s={\mathit{i}}_s-{\mathit{i}}_s^{ref}$ be the tracking error vector, and $\eta$ be the control gain. Then, the integral sliding mode surface (ISMS) is adopted as follows [26]:

$$\mathsf{\sigma }={\mathit{e}}_s+\eta {\int }_0^t{\mathit{e}}_{s}d\tau .$$

(13)

Replacing (8) into the first-time derivative of (13), and neglecting the perturbation term, yields the following:

$$\dot{\mathsf{\sigma }}=-\frac{R}{L_s}{\mathit{i}}_s-\frac{1}{L_s}\mathit{\xi }+\frac{1}{L_s}{\mathit{u}}_s+\eta {\mathit{e}}_s.$$

(14)

ISMS is similar to a proportional-integral controller. In this work, the discrete integral term ${\mathit{e}}_i$ is expressed as [36]:

$${\mathit{e}}_i\left(k\right)={\mathit{e}}_s\left(k\right)+{\mathit{e}}_i\left(k-1\right).$$

(15)

The discrete-time version of (13) is given by the following equation:

$$\mathsf{\sigma }\left(k\right)={\mathit{e}}_s\left(k\right)+\eta {\mathit{e}}_i\left(k\right).$$

(16)

The cost function can be acquired by applying (11), and the following discrete expression is obtained using the following equation:

$$J_k^{ism}={\mathsf{\sigma }}^T\left(k\right)\left(-\frac{R}{L_s}{\mathit{i}}_s\left(k\right)-\frac{1}{L_s}\mathit{\xi }\left(k\right)+\frac{1}{L_s}{\mathit{u}}_s\left(k\right)+\eta {\mathit{e}}_s(k)\right).$$

(17)

From (17), it can be seen that the control input is optimally calculated based on the SM existence condition. Moreover, the same online optimization mechanism of typical FS-MPC is employed. The more negative the value of (17) is, the faster it reaches the sliding surface. This aspect is used in the cost function definition of the proposed predictive controller.

3.2. Stability Analysis

The stability is compared by introducing a continuous approximation of the control input. The cost function (17) selects the amplitude and the sign of the optimal vector. In SM, the control input has two parts, an equivalent control input part, and a discontinuous control input part [37]. According to [25], the optimal voltage vector of FS-MPC can be expressed by its continuous set optimal voltage vector and the quantization error. In order to explain the robustness of the proposed controller, the required optimal voltage must be as follows:

$${\mathit{u}}_{opt}={\mathit{u}}_{eq}-L_s\mathit{\epsilon }sign\left(\mathsf{\sigma }\right),$$

(18)

where: ${\mathit{u}}_{opt}$ is the optimal voltage vector of FS-ISMPC, ${\mathit{u}}_{eq}$ is the equivalent control input, and $\mathit{\epsilon }$ is the quantization error. In the case of FS-MPC, the continuous-set voltage vector is obtained based on the deadbeat solution of (7). However, the equivalent control is added to cancel known terms. The control input in (17) presents the same property of cancelation. From (14), the equivalent control can be obtained by applying the invariance condition:

$${\mathit{u}}_{eq}=R{\mathit{i}}_s+\mathit{\xi }-{L_{s}k}_i{\mathit{e}}_s.$$

(19)

Consider the following Lyapunov function candidate:

$$V=\frac{1}{2}{\mathsf{\sigma }}^T\mathsf{\sigma }.$$

(20)

The system is stable if the variation of $V$ is decreasing:

$$\dot{V}={\mathsf{\sigma }}^T\dot{\mathsf{\sigma }}.$$

(21)

Then, using (8) and (21), the following expression is obtained:

$$\dot{V}={\mathsf{\sigma }}^T\left(-\mathit{\epsilon }sign\left(\mathsf{\sigma }\right)+{\mathit{f}}_s\right)<0.$$

(22)

According to the above analysis, the sufficient condition for (21) to decrease is $‖\mathit{\epsilon }‖>0$. This is met due to the negative value of the cost function, which is mentioned in (17). Under the condition $‖\mathit{\epsilon }‖>‖{\mathit{f}}_s‖$, the Equation (22) is negative and the surface (13) converges to zero in a finite time.

3.3. Delay Compensation

The discrete current predictive model of (8) can be written as follows:

$${\mathit{i}}_{\mathit{s}}^{\mathit{p}}\left(k+1\right)=\left(1-\frac{T_{s}R}{L_s}\right){\mathit{i}}_s\left(k\right)+\frac{1}{L_s}\left[-\mathit{\xi }\left(k\right)+{\mathit{u}}_s\left(k\right)\right].$$

(23)

Considering the digital delay in the control input calculation, one step ahead should be shifted. Assuming that $\mathit{\xi }\left(k+1\right)\approx \mathit{\xi }\left(k\right)$, (17) is modified as follows:

$$J_{k+1}^{ism}={\mathsf{\sigma }}^T\left(k+1\right)\left(-\frac{R}{L_s}{\mathit{i}}_s\left(k+1\right)-\frac{1}{L_s}\mathit{\xi }\left(k\right)+\frac{1}{L_s}{\mathit{u}}_s\left(k+1\right)+\eta {\mathit{e}}_s(k+1)\right).$$

(24)

Similarly, (16) is modified as the following:

$$\mathsf{\sigma }\left(k+1\right)={\mathit{e}}_s\left(k+1\right)+\eta \left[{\mathit{e}}_s\left(k+1\right)+{\mathit{e}}_i\left(k\right)\right].$$

(25)

The above expression requires of one-step ahead error vector. The reference current is pure sinusoidal. Assuming that ${\mathit{i}}_s^{ref}\left(k+1\right)\approx {\mathit{i}}_s^{ref}\left(k\right)$, one-step-ahead error can be calculated as follows:

$${\mathit{e}}_s\left(k+1\right)={\mathit{i}}_{\mathit{s}}^{\mathit{p}}\left(k+1\right)-{\mathit{i}}_s^{ref}\left(k\right).$$

(26)

According to [26], (24) can be simplified since the product $R{\mathit{i}}_s$ is negligible. Moreover, $L_s$ is always positive. The cost function (24) is modified as follows:

$$J_{k+1}^{ism}={\mathsf{\sigma }}^T\left(k+1\right)\left[-\mathit{\xi }\left(k\right)+{\mathit{u}}_s\left(k+1\right)+\eta {\mathit{e}}_s(k+1)\right].$$

(27)

3.4. Extended Control Set

FS-MPC utilizes two null vectors and six active vectors to control the system. The output of the controller is a switching pattern. Therefore, a modulation technique is not required. The transient performance of the controller is faster than a classical controller. However, the tracking current presents a large ripple. To improve the steady-state performance of the two-level power inverter, an additional six intermediate vectors are added to the existing set of voltage vectors, as shown in Figure 2.

The cost function (27) indicates the direction of the current change. Therefore, the decision function selects a voltage vector with magnitude $2V_{dc}/3$. This can be solved by adding a soft constraint on the rate of voltage change. This type of functionality cannot be included in a traditional SM theory. An additional term can be included in the cost function. The resulting cost function is expressed as follows:

$$J_{k+1}^T=J_{k+1}^{ism}+\lambda ‖{\mathit{u}}_s\left(k+1\right)-{\mathit{u}}_s\left(k\right)‖.$$

(28)

where $J_{k+1}^T$ is the cost function with constraints, and $\lambda$ is the weighting factor. The voltage vectors in the αβ frame and the corresponding duty cycles are shown in

Table 1. Voltage vector extended control set.

Vector	Rectangular Form	Duty Cycle
		D0	D1	D2
${\overrightarrow{V}}_0$	$0$	0.5	0.5	0.5
${\overrightarrow{V}}_1$	$2V_{dc}/3$	1	0	0
${\overrightarrow{V}}_2$	$V_{dc}/3+j\sqrt{3}{V}_{dc}/3$	1	1	0
${\overrightarrow{V}}_3$	$-V_{dc}/3+j\sqrt{3}{V}_{dc}/3$	0	1	0
${\overrightarrow{V}}_4$	$-2V_{dc}/3$	0	1	1
${\overrightarrow{V}}_5$	$-V_{dc}/3-j\sqrt{3}{V}_{dc}/3$	0	0	1
${\overrightarrow{V}}_6$	$V_{dc}/3-j\sqrt{3}{V}_{dc}/3$	1	0	1
${\overrightarrow{V}}_7$	$V_{dc}/2+j\sqrt{3}{V}_{dc}/6$	1	0.5	0
${\overrightarrow{V}}_8$	$j\sqrt{3}{V}_{dc}/3$	0.5	1	0
${\overrightarrow{V}}_9$	$-V_{dc}/2+j\sqrt{3}{V}_{dc}/6$	0	1	0.5
${\overrightarrow{V}}_{10}$	$-V_{dc}/2-j\sqrt{3}{V}_{dc}/6$	0	0.5	1
${\overrightarrow{V}}_{11}$	$-j\sqrt{3}{V}_{dc}/3$	0.5	0	1
${\overrightarrow{V}}_{12}$	$V_{dc}/2-j\sqrt{3}{V}_{dc}/6$	1	0	0.5

. These voltage vectors are stored in a look-up table avoiding the necessity for an online modulation algorithm. The duty cycle is compared with a triangular carrier signal and the intersections define the switching instants. The combination of switching states is applied for a corresponding period. Figure 3 shows the switching pattern of each phase over a sampling period. The carrier signal limits the maximum switching frequency and alternates the zero vectors included in ${\overrightarrow{V}}_0$. However, the variable switching frequency is the main disadvantage of this control method. The overall flowchart of the proposed FS-ISMPC is depicted in Figure 4. The control diagram is shown in Figure 5.

4. Performance and Evaluation

4.1. Simulation Test

The proposed FS-ISMPC was compared to FS-PMC and FS-SMPC via simulations using MATLAB/Simulink models. The simulation parameters of the PMSM are listed in

Table 2. Parameters of the PMSM.

Parameter	Value
Rated torque	20 Nm
Rated current	24.5 A
Pole pair ($n_p$)	4
Stator Resistance ($R$)	0.203 Ω
Inductance ($L_s$)	2.1 mH
Inertia ($J$)	0.0048 kg∙m2
Permanent magnet flux (${\psi }_f$)	0.123 Wb
Viscous friction coefficient (B)	0.001 Nm∙s/rad

. The sampling period $T_s$ was set as 20 μs, the coefficient $\eta$ = 0.12, and $\lambda$ = 0.5. The PI speed controller parameters were set to $K_p$ = 2 and $K_i$ = 0.015.

Figure 6 shows the tracking current comparisons to evaluate the integral term in the cost function. It can be observed that FS-ISMPC presents the smallest steady-state tracking error. In this simulation, the parameter values of the PMSM were not deviated. The rotor speed was 1000 rpm and the load torque was 16 N∙m. In terms of the current ripple in the αβ frame, the extended control set was reflected in the current quality. Therefore, the proposed method has an apparent improvement in steady-state performance.

To verify the robustness of the proposed current controller, a surface plot comparison between FS-MPC, FS-SMPC, and FS-ISMPC is presented in Figure 7. Two different cases are included in this plot: the static current tracking root mean square error (RMSE) by changing $L_s$ and ${\psi }_f$ in the PMSM model. The simulated results show that the sensitivity to parameter variations is similar for both FS-MPC and FS-SMPC, whereas the RMSE of the proposed method is not significant. Compared with the above parameters, the resistance value presents a negligible impact on the performance of all three methods. On the other hand, it is well known that variations in the system parameter values can affect the performance of classical FS-MPC in terms of RMSE.

Table 3. Comparison between existing strategies.

Item	[10]	[28]	Proposed FS-ISMPC
Modulator	No	No	No
Cost function	${‖\mathit{e}‖}^2$	${\mathsf{\sigma }}^T\dot{\mathsf{\sigma }}$ $\mathsf{\sigma }=\mathit{e}$	${\mathsf{\sigma }}^T\dot{\mathsf{\sigma }}$ $\mathsf{\sigma }=\mathit{e}+\eta \int \mathit{e}dt$
Multi-objective optimization	Yes	Yes	Yes
Coefficient/WF	No	No	$\lambda , \eta$
Robustness	Low	Moderate	High
Constraint handling	Yes	Yes	Yes

compares and summarizes the proposed and existing strategies based on SM. It can be seen that the similarity between all three methods is that no modulator is required. Meanwhile, the variable switching frequency is the main disadvantage of the three strategies. However, the actuation constraint of the proposed strategy forces the switching frequency to remain in the vicinity of its maximum frequency. The proposed FS-ISMPC can be considered as an optimized discrete SMC control strategy with a multi-objective optimization and constraint-handling capability. Moreover, the proposed strategy requires weighting factors (WF) and tunable coefficients, which can be tuned by using heuristic methods. A large value of $\eta$ in the predictive surface generates a larger transient time in the current response, as depicted in Figure 8. A large value of $\lambda$ in the cost function decreases the robustness.

4.2. HIL Simulation

Discrete-time equations of PMSM were required for the real-time simulation. The difference equations were programmed using fixed-point in LabVIEW predefined DSP blocks. The PMSM model and proposed FS-ISMPC were implemented on an NI cRIO-9067 using a 40 MHz clock. In the case of the proposed FS-ISMPC, six PWM gate drive signals were generated by a digital I/O module NI 9401. An analog input module NI 9291 was used for stator current acquisition. In addition, an isolated frequency input module NI 9326 was used for encoder signal acquisition. On the other hand, the PMSM model required a DAC module NI 9262 to generate the stator current. The encoder signal was generated by a digital I/O module NI 9401. The schematic diagram of overall system interconnections is shown in Figure 9.

Table 4. Hardware resources employed in control strategies.

Resources	Total	FS-MPC [10]	FS-SMPC [28]	Proposed
Total slides	13,300	4945 37.2%	3618 27.2%	4913 36.9%
Slide registers	106,400	11,225 10.5%	9024 8.5%	11,181 10.5%
Slide LUTs	53,200	12,824 24.1%	9568 18%	12,644 23.8%
Block RAMs	140	4 2.9%	4 2.9%	4 2.9%
DSP48s	220	108 49.1%	8 3.6%	90 40.9%

compares the FPGA resource usage for the control strategies. The most demanding strategy in terms of resources is the classical FS-MPC due to the use of the mathematical model of PMSM in the optimization stage. The proposal presents a significant usage of available DSP48 blocks in comparison with FS-SMPC. The integral term requires arithmetic calculations based on adders/multipliers and accumulators. The calculation time of each strategy is listed in

Table 5. Computational burden comparison.

Strategy	Clock Cycles	Execution Time
FS-MPC [10]	212	5.30 μs
FS-SMPC [28]	137	3.43 μs
Proposed	194	4.85 μs

. The computational burden of the proposed FS-ISMPC is slightly lower than the classical FS-MPC.

4.3. Real-Time Simulation Results

To validate the effectiveness of the proposed strategy, the classical FS-MPC, FS-SMPC, and the proposed FS-ISMPC were compared. The PMSM model implemented in HIL was (8), whereas the controllers equations implemented were (7) for FS-MPC, (12) for FS-SMPC, and (28) for proposed FS-ISMPC. A speed reference step change was applied, and the rotor speed was changed from 500 to 1000 rpm, as can be observed in Figure 10, the dynamic response is very similar for all three strategies. The settling time is about 30 ms, which is a fast response for PMSM. The performance of all three control strategies is shown in Figure 11, with a step change in load torque $T_L$ from 8 Nm to 16 Nm, while maintaining a constant reference speed of 1000 rpm. In this case, the response of the three strategies was similar, and the setting time was 50 ms. Additionally, according to the load step test, the rotor speed undershoot was about 4 rpm. The torque ripple was compared using the RMS value. According to the torque fluctuation analysis, the value of torque fluctuation of the FSMPC was 0.380 Nm, the value of FS-SMPC was 0.618 Nm, and the value of torque fluctuation of the proposed FS-ISMPC was 0.342 Nm.

Figure 12 illustrates the total harmonic distortion (THD) of $i_{\alpha }$ under different load torque values. The THD was tested using nominal parameter values at 1000 rpm. The FS-MPC and the proposed FS-ISMPC present similar values when the load torque is greater than 50%. The THD of the three strategies increased when the load torque was low. The full-load THD of FS-MPC was 2.8%. In the case of FS-SMPC, it was 3.48%. Finally, the THD of the proposed FS-ISMPC was 2.1%. Figure 13 shows the current tracking RMSE to evaluate the FS-ISMPC robustness under parameter mismatches using HIL simulation. The RMSE is tested at 1000 rpm and 8 N.m. As shown, the proposed FS-ISMPC exhibits less sensitivity to parameter variations compared to the FS-SMPC. Figure 14 shows the current harmonic spectrum using the classical FS-MPC, FS-SMPC, and the proposed FS-ISMPC. The experimental data are acquired by the oscilloscope and are processed in the FFT module of MATLAB. The THD was calculated in MATLAB following the methodology proposed in [38]. In the case of the proposal, a visible harmonic at 50 kHz exists due to the use of carrier signal and the extended control set.

5. Discussion

The HIL simulation results indicate that the amplitude of the torque ripple depends on the sampling time. Notably, these findings corroborate with previous research, as evidenced by agreement with [10]. In the case of the proposed controller, a larger sampling time can decrease the effects of calculation delay; however, it also increases the amplitude of torque ripple. Increasing the sampling time could decrease the computational burden and potentially eliminate the necessity for compensation Equations (23) and (25).

The effects of parameter variation are negligible due to the uncertainty considered in this work. The robustness of the controller depends on the sliding surface, which aligns with [27,28]. However, in practice, there may be conditions that deviate the parameters outside the bounds, decreasing prediction accuracy. On the other hand, the tracking current error is diminished due to integral action on the sliding surface. The RMSE results indicate that the proposed controller exhibits improved performance similar to that reported by [26].

A system stability test similar to [27] was not directly performed. However, the analogy proposed in (18) suggests that the robustness of the controller depends on the quantization error; the larger the error, the greater the robustness.

6. Conclusions and Future Works

In this paper, an FS-ISMPC is presented for controlling a PMSM. The cost function is replaced by an SM existence condition. The stability and robustness of the proposed FSISMPC are theoretically proven, but also via MATLAB and real-time HIL simulations. The inductance and resistance values are not required in the cost function since the integral term reduces the RMSE. The proposed controller offers remarkable accuracy, robustness, low computational burden, and easy tuning for implementation. The THD of the proposal is lower than classical FS-MPC, even at low load torque. According to the torque fluctuation analysis, the value of the torque fluctuation of the FS-MPC is 0.380 Nm, while the value of torque fluctuation of the FS-ISMPC is 0.342 Nm; consequently, the torque fluctuation is reduced by 11%.

In future works, it might be possible to study alternative methods to enhance rejection for more complex loads, such as time-varying torque. As an alternative, a fixed-switching method can be included to increase global efficiency. Additionally, another type of nonlinear sliding surface should be explored as a case study. Also, the implementation of the online system identification approach may enhance the robustness of the proposed current controller.