Improved Performance of the Permanent Magnet Synchronous Motor Sensorless Control System Based on Direct Torque Control Strategy and Sliding Mode Control Using Fractional Order and Fractal Dimension Calculus

Abstract

This article starts from the premise that one of the global control strategies of the Permanent Magnet Synchronous Motor (PMSM), namely the Direct Torque Control (DTC) control strategy, is characterized by the fact that the internal flux and torque control loop usually uses ON–OFF controllers with hysteresis, which offer easy implementation and very short response times, but the oscillations introduced by them must be cancelled by the external speed loop controller. Typically, this is a PI speed controller, whose performance is good around global operating points and for relatively small variations in external parameters and disturbances, caused in particular by load torque variation. Exploiting the advantages of the DTC strategy, this article presents a way to improve the performance of the sensorless control system (SCS) of the PMSM using the Proportional Integrator (PI), PI Equilibrium Optimizer Algorithm (EOA), Fractional Order (FO) PI, Tilt Integral Derivative (TID) and FO Lead–Lag under constant flux conditions. Sliding Mode Control (SMC) and FOSMC are proposed under conditions where the flux is variable. The performance indicators of the control system are the usual ones: response time, settling time, overshoot, steady-state error and speed ripple, plus another one given by the fractal dimension (FD) of the PMSM rotor speed signal, and the hypothesis that the FD of the controlled signal is higher when the control system performs better is verified. The article also presents the basic equations of the PMSM, based on which the synthesis of integer and fractional controllers, the synthesis of an observer for estimating the PMSM rotor speed, electromagnetic torque and stator flux are presented. The comparison of the performance for the proposed control systems and the demonstration of the parametric robustness are performed by numerical simulations in Matlab/Simulink using Simscape Electrical and Fractional-Order Modelling and Control (FOMCON). Real-time control based on an embedded system using a TMS320F28379D controller demonstrates the good performance of the PMSM-SCS based on the DTC strategy in a complete Hardware-In-the-Loop (HIL) implementation.

1. Introduction

Key applications for the Permanent Magnet Synchronous Motor (PMSM) include industrial applications, aerospace applications, electric drives and electric vehicles, computer peripherals and robotics. Common applications of the PMSM include AC drives, automated guided vehicles, heating, ventilation and air conditioning (HVAC), robots, servo drives and home appliances. The growing interest in the widespread use of PMSMs in a variety of applications can be explained both by the fact that PMSMs have a number of advantageous design features, and by the fact that PMSM control systems can be characterised by an extended range of control algorithms with increased flexibility and performance, ranging from their mathematical description to their implementation in embedded systems [1,2,3,4,5].

In terms of overall PMSM control strategies, two strategies can be distinguished, namely the Field-Oriented Control (FOC)-type strategy [6,7] and the Direct Torque Control (DTC)-type strategy [8,9]. The FOC-type strategy in the classical description contains two tandem control loops, the outer speed loop and the inner current loop, and the controllers are of the classical Proportional Integrator (PI) controller type. When using the DTC strategy, there are two tandem control loops, the outer one for speed control and the inner one for torque and flow control. As far as the controllers are concerned, in the classical description the speed controller is of the PI-type, while the flux and torque controllers are of the hysteresis ON–OFF type. At first glance, the advantages and disadvantages of using the FOC-type strategy vs. the DTC strategy in terms of control algorithm complexity vs. control algorithm computation times are clearly antagonistic. Obviously, the use of hysteresis ON–OFF controllers, which require very short computing times, will inherently introduce small oscillations that must be cancelled out by the outer speed loop controller. Typically, the classical PI-type controller for the outer speed control loop can be an acceptable compromise between the required computation time when carried out in an embedded system and the overall performance of the control system, represented by response time, settling time, overshoot, steady-state error and speed ripple.

In order to improve this global performance of PMSM control systems, given that PI-type controllers provide good performance around global operating points even for relatively small variations in external parameters and disturbances caused in particular by load torque variations, a number of more complex controllers of the following types can be used: adaptive [10], predictive [11], robust [12], fuzzy [13], and neural [14].

Although the performance of these controllers is superior to that of classical PI-type controllers, care must be taken when implementing them in real time in terms of the trade-off between the computing power of the embedded system and the overall cost of implementation.

A special type of controller can be obtained by using fractional elements. Thus, a number of advanced integer-order controllers can be described by fractional means, with the following improvements in control system performance [15,16,17,18,19,20].

A convergent approach to that represented by the control system synthesis is the use of velocity observers to estimate the PMSM rotor speed and possibly other control system characteristic parameters, in the case of the DTC strategy, torque and flux. The use of these observers ensures the sensorless nature of the control system and also contributes to the overall reliability of the PMSM control system through the use of sensors, particularly those using mechanical elements in motion [21,22,23,24].

In addition to the performance indicators described above, an additional one can be introduced, namely the fractal dimension (FD) of the controlled signal, in this case the FD of the PMSM speed signal. In the Papers [25,26], a hypothesis verified by numerical simulations is introduced, namely that more efficient control systems are also characterised by a higher value of the FD [27,28] of the controlled signal.

Taking into account these elements, the article starts from the PMSM control based on the DTC strategy, and shows the improvement in the control system performance in the case of constant reference flux by using PI-type fractional controllers, the optimisation of the controller parameters realized using a computational intelligence algorithm [29], particularly the Equilibrium Optimizer Algorithm (EOA)-type algorithm [30,31], the synthesis of a speed, torque and flux observer, and, in terms of the control system performance, the comparison between these control systems which uses both the usual performance indicators and the FD of the PMSM speed signal. In the case of a variable reference flux, the speed controller provides the reference for the torque controller, while a Sliding Mode Control (SMC) controller [32,33] is used to provide the flux reference in both the integer-order and fractional-order (FO) SMC variants [34].

The use of FO control algorithms, starting with PI and continuing with SMC, are presented in [35,36,37,38,39,40] and are applied to the control of generators, the control of DC-AC converters in microgrids and induction motors. Also, in [10], fractional calculus is used to improve the controller and realize an observer using a reference model. The main drawback is the high sensitivity to disturbances and model description. In these articles, numerical simulations are presented, but without real-time results. In [41], both numerical simulations and real-time implementation for the application of fractional-order control algorithms to PMSM control, using FOC control strategy, are presented.

It can be stated that the key elements discussed in this article are the DTC control strategy, and a PMSM observer for flux, load torque and rotor speed, together with the implementation of fractional calculus, to obtain an SMC or PI fractional-order controller. Roughly speaking, by using fractional calculus, more sophisticated PI and SMC type algorithms can be obtained, which will provide better specific performance of the control system for PMSM (response time, overshoot, settling error, torque, etc.).

This is based on the very definition of the fractional integro-differential equations used to obtain the above control laws, so that in addition to the tuning parameters of these control laws in the integer (non-fractional) case, additional parameters appear on the FOs of the elements of the tuning laws. It is clear that the presence of additional design parameters will lead to better results; in particular, finding the optimal values of these parameters is carried out using a computational intelligence algorithm of the EOA type.

The logic chain for achieving superior performance for the PMSM control system based on the DTC strategy proposed in this article can be clearly seen, together with the relative advantages and disadvantages, especially in terms of the cost of the physical controller in which the algorithms are implemented.

In fact, in most of the works presented in the literature, the comparison with the PI control law (a benchmark controller), both through numerical simulations and real-time implementation, is mainly used to obtain control laws that guarantee superior performance. Obviously, the existence of verified results on the performance of different control algorithms gives the control system designer a range of options, and a global techno-economic selection criterion should be used to select the control law.

The main contributions of this article are as follows:

• Synthesis and implementation of control algorithms, an observer for rotor speed, electromagnetic torque and stator flux, along with Software-In-the-Loop (SIL) implementation for performance comparison of the proposed PMSM-SCSs based on DTC strategy, using high-end controllers: PI, PI-EOA-EOA, FOPI, TID and FO-Lead-Lag in the constant flux case, and SMC and FOSMC in the variable-flux case;
• Calculation of PMSM speed signal FD for high-end controllers: PI, PI-EOA, FOPI, TID and FO-Lead-Lag for constant flux, and SMC and FOSMC for variable flux;
• Demonstration of the parametric robustness of the PMSM-SCS based on the DTC strategy by maintaining the control performance, even in the case of a 50% increase in the J parameter, which represents the combined inertia of rotor and load;
• Real-time Hardware-In-the-Loop (HIL) implementation in an embedded system of the proposed PMSM-SCS based on a DTC strategy in the case of constant flux and variable flux, using FOPI-, SMC- and FOSMC-type controllers.

The rest of the paper is presented and structured as follows: Section 2 presents the proposed PMSM-SCS using the DTC strategy, while the numerical simulations performed in Matlab/Simulink for the proposed PMSM-SCS using the DTC strategy in the case of constant flux and variable flux are presented in Section 3. The real-time implementation in the embedded system in the HIL stage and experimental setup are presented in Section 4, and some conclusions and suggestions for future research are presented in Section 5.

2. Proposed PMSM-SCS Based on DTC Strategy

The DTC-type control strategy, together with the FOC-type control strategy, are the most commonly used PMSM control strategies. Each of these control strategies has a number of advantages and disadvantages when compared directly with each other. Therefore, this section presents the DTC strategy based on the PMSM operating equations in the d-q rotating reference-frame coordinate system, and Figure 1 depicts the block diagram of the DTC strategy. The operating equations are given in Equations (1)–(4) [8,9,24].

$$\left[\begin{array}{c}{\psi }_d\\ {\psi }_q\end{array}\right]=\left[\begin{array}{cc}{L}_d& 0\\ 0& L_q\end{array}\right]\left[\begin{array}{c}{i}_d\\ i_q\end{array}\right]+\left[\begin{array}{c}{\lambda }_0\\ 0\end{array}\right]$$

(1)

$$\left[\begin{array}{c}{u}_d\\ u_q\end{array}\right]=R_s\left[\begin{array}{c}{i}_d\\ i_q\end{array}\right]+\left[\begin{array}{cc}{L}_d& 0\\ 0& L_q\end{array}\right]{\displaystyle \frac{d}{dt}}\left[\begin{array}{c}{i}_d\\ i_q\end{array}\right]+n_p\omega \left[\begin{array}{cc}0& -L_d\\ L_d& 0\end{array}\right]\left[\begin{array}{c}{i}_d\\ i_q\end{array}\right]+n_p\omega \left[\begin{array}{c}0\\ {\lambda }_0\end{array}\right]$$

(2)

$$T_e={\displaystyle \frac{3}{2}}{n}_p\left(\left(L_d-L_q\right)i_d{i}_q+{\lambda }_0{I}_q\right)$$

(3)

$$J\dot{\omega }=T_e-T_L-B\omega$$

(4)

In the above equations, the usual notations are used for the inductance L, the stator resistance R_s, the stator flux ψ, the currents i and the voltages u in the coordinate system of the d-q rotating reference frame, plus the mechanical angular speed of the PMSM rotor denoted by ω, the flux linkage denoted by λ₀, the viscous friction coefficient denoted by B, and the rotor inertia combined with the load inertia denoted by J. T_e is the electromagnetic torque, T_L is the load torque and the number of PMSM pole pairs is given by n_p.

The DTC strategy contains a two-loop control, where the outer loop implements speed control and the inner loop implements torque and flux control via two control loops, based on hysteresis ON-OFF controllers [8,9].

In the case of sensorless control, it is also necessary to synthesise an observer that provides the estimated value of the stator flux, the estimated value of the PMSM rotor speed and the load torque. The main feature of this DTC strategy is the use of quick-response controllers with reduced design simplicity for the inner loops, while classical PI-type controllers generally ensure good performance for the outer loop, maintaining a reduced complexity of real-time implementation.

Based on these aspects, this section presents implementations of control structures that provide improved performance without significantly burdening the hardware/software structure for real-time implementation. Therefore, these types of outer loop controllers are presented when the reference stator flux is constant. For cases where the reference stator flux is variable, a separate section presents the implementation of SMC and FOSMC-type control structures.

The implementation of a flux, speed and torque load observer is also presented in a separate section. In order to compare the performance for the proposed PMSM-SCS presented, the FD of the control signals, in this case the PMSM speed signal, is compared in addition to the classic comparison elements such as response time, settling time and PMSM speed-signal ripple. These can be summarized in a flowchart, as in Figure 1.

2.1. PMSM-SCS Based on DTC Strategy Using Constant Flux

In this section, as shown in Figure 2, the classic PI speed controller is replaced by optimally tuned PI controllers using an EOA-type algorithm and FO-type speed controllers: FOPI, TID, and FO-Lead-Lag. In all these cases, the stator reference flux is constant. Throughout this article, it is useful to present the PMSM operating equations in both the d-q rotating reference-frame coordinate system and the α-β stationary reference-frame coordinate system. For PI-type controller tuning, in addition to heuristic or classical Ziegler–Nichols tuning [42], superior control system results can be achieved by using computational intelligence algorithms, of which the EOA–type algorithm is presented in [30,31]. As a result of implementing this algorithm, the optimal parameter values for tuning the PI controller are achieved. EOA-type optimization algorithms have a high efficiency in obtaining optimal or near-optimal solutions in a small computational time for most of the studied problems, with high algorithmic robustness and low parametric sensitivity [30]. In Appendix A, some details about EOA are presented.

2.1.1. Fractional-Order Speed Controllers for the PMSM-SCS Based on DTC Strategy

For the implementation of the FO-type controllers proposed in Figure 1 (FOPI, TID and FO-Lead-Lag speed controllers), some notions of fractional integration and differentiation are briefly presented. Thus, the non-integer order operator applied for integration and differentiation is denoted by ${}_{a}D_t^{\alpha }$, where α is the fractional order, and the limits of the interval to which the operator is applied are denoted by a and t. The operator ${}_{a}D_t^{\alpha }$ is shown in the following equation [15,16,41]:

$${}_{a}D_t^{\alpha }=\left\{\begin{array}{c}{\displaystyle \frac{d^{\alpha }}{d{t}^{\alpha }}}\hspace{1em}\mathrm{Re}(\alpha )>0\\ 1\hspace{1em}\mathrm{Re}(\alpha )=0\\ {\displaystyle {\int }_a^t{(dt)}^{-\alpha }\hspace{1em}\mathrm{Re}(\alpha )<0}\end{array}\right.$$

(5)

The Riemann–Liouville definition of the operator ${}_{a}D_t^{\alpha }$ is given by the next equation [15,16]:

$${}_{a}D_t^{\alpha }f\left(t\right)={\displaystyle \frac{1}{\mathsf{\Gamma }\left(m-\alpha \right)}}{\left({\displaystyle \frac{d}{dt}}\right)}^m{\displaystyle \underset{\alpha }{\overset{t}{\int }}\frac{f\left(\tau \right)}{{\left(t-\tau \right)}^{\alpha -m+1}}d\tau }$$

(6)

where $m-1<\alpha <m,\hspace{0.33em}m\in N$; $\mathsf{\Gamma }(\cdot )$—Euler’s gamma function.

The Grünwald–Letnikov definition of the operator ${}_{a}D_t^{\alpha }$ is useful in practical applications of numerical representation [15,16]:

$${}_{a}D_t^{\alpha }f(t)=\underset{h\to 0}{\lim }{\displaystyle \frac{1}{h^{\alpha }}}{\displaystyle \sum _{j=0}^{\left[\frac{t-\alpha }{h}\right]}{\left(-1\right)}^j}\left(\begin{array}{c}\alpha \\ j\end{array}\right)f\left(t-jh\right)$$

(7)

where operator [·] provides the integer part.

Note that the application of these definitions to the system presented in this article is under initial zero conditions [16,17], and the two definitions in Equations (6) and (7) become equivalent. Starting with PID control law, after applying the Laplace transform in the complex domain s, the following equation can be obtained [15]:

$$U(s)=K_{p}E(s)+K_i{\displaystyle \frac{1}{s}}E(s)+K_{d}sE(s)$$

(8)

From Equation (8), which represents the output of the controller in the complex domain s in the integer case, FO-type controllers can be further described in the fractional case, using the notions of fractional computation described above.

• FO-PI Speed Controller for PMSM-SCS Based on DTC Strategy

The most commonly used FO controllers are of the PI^λD^μ type. The description equation in the time domain is expressed as follows [15]:

$$u(t)=K_{p}e(t)+K_i{D}^{-\lambda }e(t)+K_d{D}^{\mu }e(t)$$

(9)

where e(t)—signal error.

Using the Laplace transform for the Equation (9) with zero initial conditions gives the next equation, which represents the FO-PI speed controller for the PMSM-SCS based on DTC strategy [15]:

$$G_c(s)=K_p+{\displaystyle \frac{K_i}{s^{\lambda }}}+K_d{s}^{\mu }$$

(10)

where the proportional factor is noted with K_p; the integral factor is noted with K_i; the integrator order (positive) is noted with λ; the differential factor is denoted by K_d; and the differentiator order is denoted by μ. For λ = μ = 1, the result is an integer-order PID controller.

• TID Speed Controller for PMSM-SCS Based on DTC Strategy

Another fractional controller similar to the PI^λD^μ controller is the TID controller, which can be expressed by the following transfer function for the PMSM-SCS, based on DTC strategy [15]:

$$G_c(s)={\displaystyle \frac{K_t}{s^{1/n}}}+{\displaystyle \frac{K_i}{s}}+K_{d}s$$

(11)

where the slope gain is denoted by K_t; the integration order of the slope term is noted with n; the gain of the integrator term is denoted by K_i; and the gain of the derivator term is denoted by K_d.

• FO-Lead-Lag Speed Controller for PMSM-SCS Based on DTC Strategy

The general form of the transfer function for a FO-Lead-Lag controller is expressed as follows [15]:

$$G_c(s)=K_c{\left({\displaystyle \frac{s+\frac{1}{\lambda }}{s+\frac{1}{x\lambda }}}\right)}^{\alpha }=K_c{x}^{\alpha }{\left({\displaystyle \frac{\lambda s+1}{x\lambda s+1}}\right)}^{\alpha },\hspace{0.33em}0<x<1$$

(12)

where the fractional order of the FO-Lead-Lag controller is denoted by λ.

Note that for α > 0, the FO-Lead-Lag controller has a lead effect, while for α < 0 the FO-Lead-Lag controller has a lag effect. In Equation (12), for $k^{\prime }=K_c{x}^{\alpha }$, the usual form of the FO-Lead-Lag speed controller is obtained [15]:

$$G_c(s)=k^{\prime }{\left({\displaystyle \frac{\lambda s+1}{x\lambda s+1}}\right)}^{\alpha }$$

(13)

Also, if $k^{\prime }=\alpha =1$, $\lambda =K_p/K_i$ and x has a high value (e.g., x > 10,000), the transfer function of the FO-Lead-Lag controller is transformed in the transfer function of the FO-PI controller. This allows greater flexibility in the use of the FO-Lead-Lag controller in a control loop and the possibility of obtaining higher performance from the PMSM-SCS, based on the DTC strategy.

2.1.2. Observer for Speed, Electromagnetic Torque, and Flux Estimations

The sensorless nature of the PMSM-SCS based on the DTC strategy is given by the fact that an observer is used to estimate rotor speed, electromagnetic torque and flux.

In this sense, a coordinate transformation between the d-q rotating reference frame and the α-β stationary reference frame for the quantities describing the PMSM operating equations is as follows [34]:

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

(14)

where f_α and f_β are the variables in the α-β stationary reference frame, and f_d and f_q are the variables in the d-q rotor reference frame.

In Equation (14), the coupling matrix uses the electrical angle θ, whose relation to the mechanical angle θ representing the position of the PMSM rotor is ${\theta }_e=n_p\theta$. Using relation (14), the following equations can be written in the α-β stationary reference-frame coordinate system to describe the operation of the PMSM.

$$\left[\begin{array}{c}{u}_{\alpha }\\ u_{\beta }\end{array}\right]=\left[\begin{array}{cc}R+p{L}_d& {\omega }_e\left(L_d-L_q\right)\\ -{\omega }_e\left(L_d-L_q\right)& R+p{L}_d\end{array}\right]\left[\begin{array}{c}{i}_{\alpha }\\ i_{\beta }\end{array}\right]+\left\{\left(L_d-L_q\right)\left({\omega }_e{i}_d-{\displaystyle \frac{d{i}_q}{dt}}\right)+{\omega }_e{\lambda }_0\right\}\left[\begin{array}{c}-\sin {\theta }_e\\ \cos {\theta }_e\end{array}\right]$$

(15)

The back-EMF voltages e_α and e_β are defined as follows [7]:

$$\left[\begin{array}{c}{e}_{\alpha }\\ e_{\beta }\end{array}\right]=\left\{\left(L_d-L_q\right)\left({\omega }_e{i}_d-{\displaystyle \frac{d{i}_q}{dt}}\right)+{\omega }_e{\psi }_f\right\}\left[\begin{array}{c}-\sin {\theta }_e\\ \cos {\theta }_e\end{array}\right]$$

(16)

The i_α and i_β current time variation may be derived from Equation (15), in the following form:

$$\left[\begin{array}{c}{\displaystyle \frac{d{i}_{\alpha }}{dt}}\\ {\displaystyle \frac{d{i}_{\beta }}{dt}}\end{array}\right]=\left[\begin{array}{cc}-{\displaystyle \frac{R}{L_d}}& -{\displaystyle \frac{{\omega }_e\left(L_d-L_q\right)}{L_d}}\\ {\displaystyle \frac{{\omega }_e\left(L_d-L_q\right)}{L_d}}& -{\displaystyle \frac{R}{L_d}}\end{array}\right]\left[\begin{array}{c}{i}_{\alpha }\\ i_{\beta }\end{array}\right]-\left[\begin{array}{cc}{\displaystyle \frac{1}{L_d}}& 0\\ 0& {\displaystyle \frac{1}{L_d}}\end{array}\right]\left[\begin{array}{c}{e}_{\alpha }\\ e_{\beta }\end{array}\right]+{\displaystyle \frac{1}{L_d}}\left[\begin{array}{c}{u}_{\alpha }\\ u_{\beta }\end{array}\right]$$

(17)

Similarly, the time variation of the stator flux in the α-β stationary reference frame can be written as follows [7]:

$$\left\{\begin{array}{l}{\displaystyle \frac{d{\psi }_{\alpha }}{dt}}=u_{\alpha }-R{i}_{\alpha }\\ {\displaystyle \frac{d{\psi }_{\beta }}{dt}}=u_{\beta }-R{i}_{\beta }\end{array}\right.$$

(18)

The expression for the electromagnetic torque T_e in the stationary reference frame α-β is expressed as follows [7]:

$$T_e={\displaystyle \frac{3n_p}{2}}\left({\psi }_{\alpha }{i}_{\beta }-{\psi }_{\beta }{i}_{\alpha }\right)$$

(19)

In an obvious way, the relation between the stator flux and its projections in the α-β stationary reference frame and in the d-q rotating reference frame is as follows:

$${\psi }^2={\psi }_{\alpha }^2+{\psi }_{\beta }^2={\psi }_d^2+{\psi }_q^2$$

(20)

Starting from Equation (18), the expressions for the stator flux $\psi ={\left[\begin{array}{cc}{\psi }_{\alpha }& {\psi }_{\beta }\end{array}\right]}^T$ in the α-β stationary reference frame are expressed as follows:

$$\left\{\begin{array}{l}{\psi }_{\alpha }={\displaystyle \underset{0}{\overset{t}{\int }}\left(u_{\alpha }-R_s{i}_{\alpha }\right)dt}\\ {\psi }_{\beta }={\displaystyle \underset{0}{\overset{t}{\int }}\left(u_{\beta }-R_s{i}_{\beta }\right)dt}\end{array}\right.$$

(21)

$$\left\{\begin{array}{l}{\overline{\psi }}_s=\sqrt{{\widehat{\psi }}_{\alpha }^2+{\widehat{\psi }}_{\beta }^2}\\ \angle {\widehat{\psi }}_s=arctg{\displaystyle \frac{{\widehat{\psi }}_{\beta }}{{\widehat{\psi }}_{\alpha }}}\end{array}\right.$$

(22)

An observer based on Equation (23) is employed to estimate the stator flux expressed in the α-β stationary reference frame [23,24].

$$\left\{\begin{array}{l}{\dot{\widehat{\psi }}}_{\alpha }=-R_s{i}_{\alpha }+u_{\alpha }+k_1{\tilde{i}}_{\alpha }+K_{1SMO}sigmoid\hspace{0.17em}({\tilde{i}}_{\alpha })\\ {\dot{\widehat{\psi }}}_{\beta }=-R_s{i}_{\beta }+u_{\beta }+k_2{\tilde{i}}_{\beta }+K_{2SMO}sigmoid\hspace{0.17em}({\tilde{i}}_{\beta })\end{array}\right.$$

(23)

where ${\widehat{\psi }}_{\alpha }$ and ${\widehat{\psi }}_{\beta }$—estimated stator–flux variables in the α-β stationary frame; ${\tilde{i}}_{\alpha }$ and ${\tilde{i}}_{\beta }$—stator current errors in the α-β stationary frame if given in Equation (24); k₁, k₂ K_1SMO, and K_2SMO—projected factors (observer gains).

$$\left\{\begin{array}{l}{\tilde{i}}_{\alpha }=i_{\alpha }-{\widehat{i}}_{\alpha }\\ {\tilde{i}}_{\beta }=i_{\beta }-{\widehat{i}}_{\beta }\end{array}\right.$$

(24)

Equation (25) is utilized to estimate the ${\widehat{i}}_{\alpha }$ and ${\widehat{i}}_{\beta }$ stator currents in the α-β stationary reference frame [23,24].

$$\widehat{i}=T^{-1}\left(\widehat{\theta }\right)L^{-1}T\left(\widehat{\theta }\right)\widehat{\psi }+{\displaystyle \frac{{\lambda }_0}{L_d}}\left(\begin{array}{c}\cos \widehat{\theta }\\ -\sin \widehat{\theta }\end{array}\right)$$

(25)

where $\widehat{i}={\left[\begin{array}{cc}{\widehat{i}}_{\alpha }& {\widehat{i}}_{\beta }\end{array}\right]}^T$; $\widehat{\psi }={\left[\begin{array}{cc}{\widehat{\psi }}_{\alpha }& {\widehat{\psi }}_{\beta }\end{array}\right]}^T$; $L=\left[\begin{array}{cc}{L}_d& 0\\ 0& L_q\end{array}\right]$; and the transformation matrix $T\left(\widehat{\theta }\right)$ is given in Equation (14).

Equation (25), explained for the components α and β, can be written as

$$\left\{\begin{array}{l}{\widehat{i}}_{\alpha }{\displaystyle \frac{1}{L_d}}{\widehat{\psi }}_{\alpha }+{\displaystyle \frac{{\lambda }_0}{L_d}}\cos (\widehat{\theta })\\ {\widehat{i}}_{\beta }{\displaystyle \frac{1}{L_q}}{\widehat{\psi }}_{\beta }-{\displaystyle \frac{{\lambda }_0}{L_q}}\sin (\widehat{\theta })\end{array}\right.$$

(26)

The PMSM rotor angular position is estimated using the following Equation [23,24]:

$$\widehat{\theta }={\tan }^{-1}\left({\displaystyle \frac{{\widehat{\lambda }}_{\alpha \beta }}{{\widehat{\lambda }}_{\alpha \alpha }}}\right)$$

(27)

where

$$\left\{\begin{array}{l}{\widehat{\lambda }}_{\alpha \alpha }={\widehat{\psi }}_{\alpha }-L_d{i}_{\alpha }\\ {\widehat{\lambda }}_{\alpha \beta }={\widehat{\psi }}_{\beta }-L_q{i}_{\beta }\end{array}\right.$$

(28)

In Equation (23), which describes a Sliding Mode Observer (SMO), the sign function is substituted with the sigmoid function to achieve a smooth transition between +1 and –1. The definition of the sigmoid function is given by equation (29).

$$H(x)={\displaystyle \frac{2}{1+e^{-a(x-b)}}}-1$$

(29)

where a = 4 and b = 0 are positive constants chosen for this application.

The convergence of the observer is proved by choosing a Lyapunov function of the following form:

$$V={\displaystyle \frac{1}{2}}\left({\tilde{\psi }}^T{T}^{-1}{L}^{-1}T\tilde{\psi }\right)>0$$

(30)

in which $\tilde{\psi }=\psi -\widehat{\psi }$. After a few calculations, we can obtained $\dot{V}<0$, which demonstrates global asymptotic stability [23,24]. To estimate the electromagnetic torque value, the following equation is used, according to [23,24]:

$${\widehat{T}}_e={\displaystyle \frac{3}{2}}{n}_p\left({\widehat{\psi }}_{\alpha }{i}_{\alpha }-{\widehat{\psi }}_{\beta }{i}_{\beta }\right)$$

(31)

Similarly, by deriving Equation (27), the PMSM rotor speed estimate is expressed as follows:

$$\widehat{\omega }(k)={\displaystyle \frac{{\widehat{\lambda }}_{a\alpha }\left(k-1\right){\widehat{\lambda }}_{a\beta }\left(k\right)-{\widehat{\lambda }}_{a\beta }\left(k-1\right){\widehat{\lambda }}_{a\alpha }\left(k\right)}{T_s\left({\widehat{\lambda }}_{a\alpha }^2(k)+{\widehat{\lambda }}_{a\beta }^2(k)\right)}}$$

(32)

where the sampling period is denoted by T_s, and the current sampling step is noted with k.

2.2. PMSM-SCS Based on Variable-Flux DTC Strategy

If the reference stator flux is not constant, the DTC strategy is modified to use more complex controllers to obtain ψ_ref. Therefore, Figure 3 depicts the schema of the proposed PMSM-SCS, based on the DTC strategy using SMC and FOSMC variable-flux controllers.

2.2.1. SMC-Type Controller for PMSM-SCS Based on DTC Strategy Using Variable Flux

Equations (1)–(4) are rewritten in the following form [41]:

$$\left(\begin{array}{c}{\dot{i}}_d\\ {\dot{i}}_q\\ \dot{\omega }\end{array}\right)=\left(\begin{array}{ccc}-{\displaystyle \frac{R_s}{L}}& n_p\omega & 0\\ -n_p\omega & -{\displaystyle \frac{R_s}{L}}& -{\displaystyle \frac{n_p{\lambda }_0}{L}}\\ 0& {\displaystyle \frac{K_t}{J}}& -{\displaystyle \frac{B}{J}}\end{array}\right)\left(\begin{array}{c}{i}_d\\ i_q\\ \omega \end{array}\right)+\left(\begin{array}{c}{\displaystyle \frac{u_d}{L}}\\ {\displaystyle \frac{u_q}{L}}\\ -{\displaystyle \frac{T_L}{J}}\end{array}\right)$$

(33)

to synthesise an SMC-type controller for PMSM control, the state variables x₁ and x₂ are selected and described by the following equations:

$$x_1={\omega }_{ref}-\omega$$

(34)

where x₁—speed tracking error.

$$x_2={\dot{x}}_1={\displaystyle \frac{{\omega }_{ref}-\omega }{dt}}=-\dot{\omega }$$

(35)

The sliding surface S of the zero error variation is defined in Equation (36), and by deriving this equation we can obtain Equation (37).

$$S=c{x}_1+x_2$$

(36)

$$\dot{S}=c{x}_2+{\dot{x}}_2=c{x}_2-D{\dot{i}}_q$$

(37)

where the parameter c is positive and adjustable, and $D={\displaystyle \frac{3n_p{\lambda }_0}{2J}}$ is obtained from the Equations (1)–(3).

Equation (38) shows the condition for the surface S time evolution; this condition ensures the control of the response time of the PMSM-SCS based on DTC strategy.

$$\dot{S}=-\epsilon \mathrm{sgn}(S)-qS,\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\epsilon ,q>0$$

(38)

where the parameters ε and q are positive and the signum function is noted with sgn().

The above sgn function is replaced by the sigmoid function defined in Equation (29), and this is used in the synthesis of the SMC-type controller in order to reduce the chattering phenomenon characteristic of the structure of this type of controller. From the above equations, according to SMC specific methods, the next equation can be written as

$$i_{qref}(t)={\displaystyle \frac{1}{D}}{\displaystyle \underset{0}{\overset{t}{\int }}\left[c{x}_2+\epsilon H(S)+qS\right]dt}$$

(39)

From Equations (9–11), the following custom equations can be obtained for the reference values of the electromagnetic torque, the reference current i_qref and the reference value of the stator flux in the usual case, where L_d = L_q:

$$i_{qref}={\displaystyle \frac{T_{eref}}{1.5n_p{\lambda }_0}}$$

(40)

$${\psi }_{ref}={\lambda }_0^2+{\left({\displaystyle \frac{L_q{T}_{eref}}{1.5n_p{\lambda }_0}}\right)}^2$$

(41)

The stability property of the PMSM-SCS based on DTC strategy using the control law expressed through Equation (39) is demonstrated by selecting a Lyapunov function of the following form:

$$V={\displaystyle \frac{1}{2}}{S}^2$$

(42)

The following expression is obtained by calculating the derivative $\dot{V}$:

$$\dot{V}=S\dot{S}=S\left[-\epsilon H(S)-qS\right]=-\epsilon H(S)-q{S}^2$$

(43)

The stability property of the system is given by the negativity of the derivative $\dot{V}$.

2.2.2. FOSMC-Type Controller for PMSM-SCS Based on DTC Strategy Using Variable Flux

To improve the performance of PMSM-SCS based on DTC strategy, the synthesis of a FOSMC-type controller is proposed to obtain i_qref and then flux ψ_ref, based on Equations (40) and (41). For this purpose, the sliding surface S is selected as follows:

$$S=k_p{x}_1+k_d{D}^{\mu }{x}_1=k_p{x}_1+k_d{D}^{\mu -1}{x}_2$$

(44)

By calculating the derivative of the surface S, the next equation can be obtained:

$$\dot{S}=k_p{\dot{x}}_1+k_d{D}^{\mu +1}{x}_1=k_p{x}_2+k_d{D}^{\mu -1}{\dot{x}}_2$$

(45)

Based on the descriptive equations of the PMSM mathematical model, the following equation can be written:

$${\dot{x}}_2={\ddot{\omega }}_{ref}-1.5{\displaystyle \frac{n_p{\lambda }_0}{J}}{\dot{i}}_q+{\displaystyle \frac{1}{J}}{\dot{T}}_L+{\displaystyle \frac{B}{J}}\dot{\omega }$$

(46)

Substituting Equation (46) into Equation (45) gives

$$\dot{S}=k_p{x}_2+k_d{D}^{\mu -1}\left({\ddot{\omega }}_{ref}-1.5{\displaystyle \frac{n_p{\lambda }_0}{J}}{\dot{i}}_q+{\displaystyle \frac{1}{J}}{\dot{T}}_L+{\displaystyle \frac{B}{J}}\dot{\omega }\right)$$

(47)

According to the SMC methods for $\dot{S}=0$, the following equation can be obtained:

$$-\epsilon H(S)-qS-k_p{x}_2=k_d{D}^{\mu -1}\left({\ddot{\omega }}_{ref}-1.5{\displaystyle \frac{n_p{\lambda }_0}{J}}{\dot{i}}_q+{\displaystyle \frac{1}{J}}{\dot{T}}_L+{\displaystyle \frac{B}{J}}\dot{\omega }\right)$$

(48)

Applying the operator $D^{1-\mu }$ to Equation (48) to both members, the following equation can be written:

$$D^{1-\mu }\left(-\epsilon H(S)-qS-k_p{x}_2\right)=k_d\left({\ddot{\omega }}_{ref}-1.5{\displaystyle \frac{n_p{\lambda }_0}{J}}{\dot{i}}_q+{\displaystyle \frac{1}{J}}{\dot{T}}_L+{\displaystyle \frac{B}{J}}\dot{\omega }\right)$$

(49)

After processing Equation (49), the following equation can be obtained:

$${\displaystyle \frac{1}{k_d}}{D}^{1-\mu }\left(-\epsilon H(S)-qS-k_p{x}_2\right)={\ddot{\omega }}_{ref}-1.5{\displaystyle \frac{n_p{\lambda }_0}{J}}{\dot{i}}_q+{\displaystyle \frac{1}{J}}{\dot{T}}_L+{\displaystyle \frac{B}{J}}\dot{\omega }$$

(50)

By rewriting Equation (50), the next equation can be written as

$$1.5{\displaystyle \frac{n_p{\lambda }_0}{J}}{\dot{i}}_q={\ddot{\omega }}_{ref}+{\displaystyle \frac{1}{J}}{\dot{T}}_L+{\displaystyle \frac{B}{J}}\dot{\omega }-{\displaystyle \frac{1}{k_d}}{D}^{1-\mu }\left(-\epsilon \hspace{0.17em}H(S)-qS-k_p{x}_2\right)$$

(51)

The current equation i_qref is achieved from Equation (51), as follows:

$$i_{qref}(t)={\displaystyle \frac{1}{1.5\frac{n_p{\lambda }_0}{J}}}\cdot {\displaystyle \underset{0}{\overset{t}{\int }}\left[{\ddot{\omega }}_{ref}+\frac{1}{J}{\dot{T}}_L+\frac{B}{J}\dot{\omega }-\frac{1}{k_d}{D}^{1-\mu }\left(-\epsilon \hspace{0.17em}H(S)-qS-k_p{x}_2\right)\right]}dt$$

(52)

Then, as in the case of the integer-order SMC-type controller, the reference value of the flux ψ_ref is obtained by using Equations (40) and (41).

3. Numerical Simulations Based on Matlab/Simulink for the Proposed PMSM-SCS Using DTC Strategy

In this section, for the values of a PMSM given in

Table 1. Nominal PMSM parameters of the PMSM-SCS based on the DTC strategy.

Parameter	Value	Unit
Stator resistance (Rs)	2.875	Ω
Inductances on the d-q rotating reference frame (Ld and Lq)	0.0085	H
Combined inertia of rotor and load (J)	0.0008	kg·m2
Combined viscous friction of rotor and load (B)	0.005	N·m·s/rad
Flux induced by the permanent magnets of the rotor in the stator phases (λ0)	0.175	Wb
PMSM pole pair number (nP)	4	-

, numerical simulations achieved in Matlab/Simulink are performed for the corresponding PMSM control systems according to the cases of DTC strategy controllers presented in Section 2.

Thus, in Section 3.1 the controller synthesis (PI, PI-EOA, FOPI, TID and FO-Lead-Lag) and numerical simulation for PMSM-SCS based on DTC strategy using constant flux are presented. In Section 3.2 the controller synthesis (SMC and FOSMC) and numerical simulation for PMSM-SCS based on the DTC strategy using variable flux are presented. The description of the fractal dimension and calculus for the speed signal controlled by the proposed controllers used in PMSM-SCS, based on the DTC strategy, are presented in Section 3.3.

3.1. Numerical Simulation for PMSM-SCS Based on DTC Strategy, Using Constant Flux

The software implementation carried out in Matlab/Simulink version 2021b for the PMSM-SCS is based on the DTC strategy in the case of constant flux. Thus, for the PI speed controller described in Equation (8), in the case of the sensorless PMSM control system based on the constant-flux DTC strategy, the tuning parameters obtained by empirical methods are K_p = 0.05 and K_i = 0.91. Using an EOA-type algorithm, the tuning parameters obtained are K_p = 0.03 and K_i = 0.66. The FOMCON toolbox in Matlab allows the tuning of an FOPI-type controller described by Equation (10), in the case of PMSM-SCS based on a constant-flux DTC strategy. The values obtained are the following: K_p = 6.2, K_i = 24, λ = 1.1, and K_d = μ = 0, and the FOPI speed-controller transfer function becomes

$$H_{FOPI}(s)={\displaystyle \frac{6.2s^{1.1}+24}{s^{1.1}}}$$

(53)

For the closed-loop study of the system consisting of the FOPI controller given in Equation (53) and the fixed part given by the PMSM carried out with the FOMCON toolbox from Matlab, the following fractional-order transfer function is obtained:

$$H_{FOPI\_CL}(s)={\displaystyle \frac{17.71s^{1.1}+68.58}{5.44\cdot {10}^{-8}{s}^{4.2}+2.58\cdot {10}^{-5}{s}^{3.2}+0.03s^{2.2}+18.47s^{1.1}+68.56}}$$

(54)

The stability of the closed-loop system of the PMSM-SCS based on the constant-flux DTC strategy and the PI speed controller is given by satisfying the condition given by $\left|\arg \left(eig\left(A\right)\right)\right|>q{\displaystyle \frac{\pi }{2}}$ [15], where $q\in (0,\hspace{0.33em}1)$ is the proportional order and eig(A) is the eigenvalue of the associated matrix A of the state-space representation for the system defined in Equation (54). The graphical representation is presented in Figure 4a, using the FOMCON toolbox, and the analysis of this figure shows that the system is stable. Figure 4b shows the step-response signal of the closed-loop system when using the FOPI speed controller. Figure 5 depicts the Bode diagram for the closed-loop system consisting of the FOPI controller given by Equation (53) in the case of PMSM-SCS based on the DTC strategy using the FOPI speed controller at constant flux. Thus, Figure 5a presents the Bode magnitude plot, and the Bode phase plot is presented in Figure 5b.

According to the Bode stability criterion, it is observed that the system is stable, with a phase stability reserve of about 90° and an amplitude stability reserve of about 20 dB. These values indicate that the system has a good reserve of stability in both phase and amplitude characteristic elements of a robust control system.

Based on the second FO-type controller of the PMSM-SCS based on the constant-flux DTC strategy, i.e., the TID controller described in Equation (33), whose tuning parameters are K_t = 0.035, K_i = 0.31, n = 10 and K_d = 0, the following transfer function is obtained:

$$H_{TID}(s)={\displaystyle \frac{0.035s^{0.9}+0.31}{s}}$$

(55)

For the FO-Lead-Lag speed controller shown in Equations (19) and (20), the following transfer function is obtained for the tuning parameters k’ = 300, x = 50, λ = 1.4, and α = 0.11:

$$H_{\mathrm{FO-Lead-Lag}}(s)={\displaystyle \frac{3.606\cdot {10}^8}{s^{2.1}+73730\hspace{0.33em}{s}^{1.1}+4.7\cdot {10}^6}}$$

(56)

For real-time implementation in an embedded system, it is necessary to achieve the equivalent transfer function of integer order of the transfer function of the FO-type speed controllers given in Equations (53), (55), and (56) in fractional form. For this, Appendix B presents the procedure to obtain these equivalent integer-order transfer functions using Oustaloup recursive filters [15].

Based on the nominal PMSM parameters in the case of the sensorless control system based on the DTC strategy using PI, PI-EOA, FOPI, TID, and FO-Lead-Lag speed controllers at constant flux, the numerical simulations are presented below. Plots of the following parameters of interest are presented for each sensorless control system based, on the DTC strategy using speed controllers: reference speed and estimated PMSM rotor speed, load and electromagnetic torque, stator currents and fluxes in the α-β stationary reference frame.

The load torque reference T_L is of 1 Nm and the speed reference signal ω_ref used is expressed as a sequence of steps: [400 600 900] rpm→[0 0.1 0.2]s. For the speed, electromagnetic torque and flux estimation observed are presented in Section 2.1.2; the tuning parameters are k₁ = 150, k₂ = 50, and K_1SMO = K_2SMO = 0.1.

Figure 6 shows the parameter evolution of the PMSM-SCS using the DTC strategy with PI speed controller at constant flux. Note that the response time is tr = 12 ms and the settling time is ts = 72 ms. The parameter evolution of the PMSM-SCS using the DTC strategy when using the PI-EOA speed controller is shown in Figure 7. In this case, the response time is tr = 9.2 ms and the settling time is ts = 30 ms. There is a 25% reduction in response time and a reduction in settling time of around 60%.

Using the FOPI speed controller for the PMSM-SCS with DTC strategy at constant flux, the evolution of the parameters is shown in Figure 8. This gives a response time of tr = 7.9 ms and a settling time of ts = 15 ms. There is a 34% reduction in response time and a 79% reduction in settling time. Using the TID speed controller for the PMSM-SCS with DTC strategy at constant flux, the evolution of the parameters is shown in Figure 9. A response time of tr = 5.9 ms and a settling time of ts = 14.5 ms are obtained. There is a 50% reduction in response time and an 80% reduction in settling time. Using the FO-Lead-Lag speed controller for the PMSM-SCS with DTC strategy at constant flux, the evolution of the parameters is shown in Figure 10. This gives a response time of tr = 3.2 ms and a settling time of ts = 9.1 ms. There is a 73% reduction in response time and an approximate 87% reduction in settling time.

In Figure 6, Figure 7, Figure 8, Figure 9 and Figure 10 (updated) the performance of the FO-type controllers is shown, having a gradually increasing complexity, and it can be observed that there is improvement in the main parameters of interest in this case: response time and stabilization time. Also, for each figure, the following magnitudes are presented: electromagnetic torque, stator currents and fluxes in the reference frame α-β. It can be observed that, in accordance with the performance improvement of the control systems in question for the proposed reference variations, at the moments of the occurrence of jumps in the reference, a slight increase in the electromagnetic torque appears, while the shapes of the currents and fluxes remain, sensitively, the same shape.

A comparison of the evolution of the PMSM rotor speed for the sensorless control system based on the DTC strategy using PI, PI-EOA, FOPI, TID and FO-Lead-Lag speed controllers at constant flux is shown in Figure 11. It can be seen that the use of FO-type controllers improves the performance of the control system, so that the best performance in the comparison shown is provided by FO-Lead-Lag speed controllers, followed by TID and FOPI speed controllers.

3.2. Numerical Simulation for PMSM-SCS Based on DTC Strategy Using Variable Flux

The software implementation carried out in Matlab/Simulink for the PMSM-SCS is based on the variable-flux DTC strategy. The Matlab/Simulink implementation presented is based on the SMC- and FOSMC-type controller description equations presented in Section 2.2. For the numerical simulation scenario performed in Matlab/Simulink using the SMC-type controller described in Section 2, the parameters in Equations (34)–(43) are as follows: ε = 300, q = 200, and c = 100. For the FOSMC-type controller described in Section 2, Equations (44)–(52), the parameters are as follows: k_p = 100, k_d = 1, and μ = 0.55.

Based the SMC-type controller for the PMSM-SCS with the DTC strategy at variable flux, the evolution of the parameters is shown in Figure 12. The response time is tr = 2.3 ms and the settling time is ts = 2.3 ms. There is an 80% reduction in response time and an approximate 96% reduction in settling time, compared to the original PI controller performance. Furthermore, using the FOSMC-type controller for the PMSM-SCS with the DTC strategy at variable flux, the evolution of the parameters is shown in Figure 13. The response time is tr = 2.2 ms and the settling time is ts = 2.2 ms. There is an 82% reduction in response time and an approximate 97% reduction in settling time compared to the initial performance of the PI controller.

In terms of parametric robustness, Figure 14 shows the parameter evolution of the PMSM-SCS using the DTC strategy with the FOSMC variable-flux controller, where the J parameter increases by 50% compared to the nominal value. Good dynamic performance is also observed in this case, in the sense that the overshoot is negligible and the steady-state error is less than 0.1%, just as in the case of the nominal value of the J parameter. Obviously, given the importance of the J parameter, which represents the combined inertia of the PMSM rotor and load, the response time increases to 4 ms and the stator currents increase by approximately 20%.

The flux space-vector trajectory for the PMSM-SCS based on the DTC strategy using the FOSMC in the case of the variable-flux controller is shown in Figure 15a, and the evolution of the flux reference for the FOSMC-type controller is shown in Figure 15b.

The comparison of the evolution of the PMSM rotor speed for the sensorless control system based on the DTC strategy using SMC and FOSMC variable-flux controllers is shown in Figure 16. Note that this comparison is made between the SMC- and FOSMC-type controllers and the FO-Lead-Lag controllers, which have the best performance of the group of controllers presented in the previous section. Thus, the best performance of the PMSM-SCS, based on the DTC strategy, is achieved using the FOSMC-type controller.

3.3. Description of the Fractal Dimension and Calculus for the Speed Signal Controlled by the Proposed Controllers Used in PMSM-SCS, Based on DTC Strategy

For comparison of the performance of the PMSM-SCS using the DTC strategy, in the case of all the controllers presented in the previous sections, the FD of the signal controlled by the control system, in this case the PMSM speed, is proposed as a performance indicator. According to [28], an FD value of the controlled signal is higher if the control system provides better performance. One of the most widely used methods for calculating the FD of a one-dimensional signal is the box-counting method [28]. For the PMSM with the parameters given in Table 1, the graphical representation of the FD of the speed signal for the PMSM-SCS based on the DTC strategy is shown in Figure 17. So, for the constant flux case, Figure 17a depicts the FD of the speed signal for the PI speed controller, Figure 17b depicts the FD of the speed signal for the PI-EOA speed controller, Figure 17c depicts the FD of the speed signal for the FOPI speed controller, Figure 17d depicts the FD of the speed signal for the TID speed controller, and Figure 17e depicts the FD of the speed signal for the FO-Lead-Lag speed controller. In addition, in the case of a variable flux, Figure 17f depicts the FD of the speed signal for the SMC-type controller and Figure 17g depicts the FD of the speed signal for the FOSMC-type controller. It can be seen that the FD of the speed signal increases according to the controllers, which bring performance improvements to the closed-loop PMSM-SCS, based on the DTC.

The usual performance of the PMSM-SCS based on the DTC strategy for the controllers shown in the previous sections is summarised in

Table 2. Performance of proposed controllers for PMSM-SCSs.

Type of Controller for PMSM-SCS		Response Time [ms]	Settling Time [ms]	Speed Ripple [rpm]	FD of PMSM Speed Signal
PI	constant flux	12	72	43.36	0.984540 +/− 0.051130
PI-EOA		9.2	30	37.17	0.984526 +/− 0.051321
FOPI		7.9	15	35.61	0.984599 +/− 0.051421
TID		5.9	14.5	34.38	0.986350 +/− 0.026742
FO-Lead-Lag		3.2	9.1	33.54	0.986710 +/− 0.026879
SMC	variable flux	2.3	2.3	32.79	0.987770 +/− 0.027187
FOSMC		2.2	2.2	32.21	0.987950 +/− 0.027261

. In addition to FD, time response and settling time, and speed ripple are shown [41]. Also, the usual performance indicators of a control system, namely overshoot and steady-state error, have not been included in Table 2, because for all the controller types presented, except the classic PI speed controller, the overshoot is negligible and the steady-state error is less than 0.1%. Improvements in the performance of the PMSM-SCS, based on the DTC strategy, are obtained when using a classical PI speed controller, when using an EOA-type algorithm and when using FO-type controllers, i.e., FOPI, TID, and FO-Lead-Lag speed controllers, at constant flux. In the case of variable flux, the SMC controller and the FO controller, i.e., the FOSMC controller, provide the best performance of the PMSM-SCS, based on the DTC strategy.

4. Real-Time Implementation and Experimental Set-Up

Section 3 introduced a very important step in the development of a prototype, namely the numerical simulation step. At the numerical simulation stage, generally referred to as Software-In-the-Loop (SIL), both the mathematical model of the controlled process and the mathematical model of the controller run on a host PC, and the execution of the closed-loop simulation program does not overlap with the so-called real-time execution, where there is a correlation between the evolution of the real signals and the speed of display in the monitoring-and-control software application. At this stage, the controller parameters are adjusted and the performance of the analysed control systems is compared in a safe operating environment. After this stage, there is the option of going through the Processor-In-the-Loop (PIL) stage, where the driven process is still simulated but the controller is real, but the final stage of developing a prototype controller is the Hardware-In-the-Loop (HIL) stage, where the driven process is real and the controller is also implemented in an embedded system [42].

The PMSM-SCS based on the DTC strategy presented in this article uses the Matlab/Simulink development environment, which uses the Motor Control Blockset (MCB) and Embedded Coder Support Package (ECSP) toolboxes suitable for Texas Instruments C2000 microcontrollers for real-time implementation [43,44]. The MCB toolbox provides the ability to implement the proposed control structures in the SIL stage through specialised blocks for real-time implementation in the HIL stage. The ECSP toolbox allows the corresponding code of the PMSM control algorithm to be generated and executed, based on the DTC control strategy. The generated code runs on C2000 microcontrollers, specifically the TMS320F28379D microcontroller [45].

Real-time software blocks include enhanced pulse-width modulator (ePWM) control, serial peripheral interface (SPI) serial communication control, host PC and embedded-system communication blocks, digital motor control (DMC) specific blocks, and the IQMath library for optimising code for real-time computation through fixed-point implementation, while maintaining high accuracy. Key features of the TMS320F28379D microcontroller include a 32-bit floating-point Control Law Accelerator (CLA) module that performs mathematical computations concurrent with the main processor on a 32-bit fixed-point architecture. We also recall that the main processor has a clock frequency of 200 MHz, 1 MB of flash memory, 16-bit digital-to-analogue converters (DACs), serial communication ports and a general-purpose input/output (GPIO) subsystem [42]. To interface with the TelcoMotion DT4260 PMSM (TelcoMotion, Taichung, Taiwan), a 3-phase smart gate module driver with current shunt amplifiers and SPI-type BOOSTXL-DRV8305 (Texas Instruments, Dallas, TX, USA) is used, which communicates with the LAUNCHXL-F28379D development board on which the TMS320F28379D main controller is integrated (Figure 18). Following the real-time implementation of the PMSM-SCS based on the DTC strategy, a series of speed reference-signal variations imposed by the real-time monitoring and control software interface were implemented in Simulink on the PC host.

In the following, the real-time testing of the proposed PMSM-SCS based on the DTC strategy is presented in the constant-flux case and in the variable-flux case, as presented in Section 3, where numerical simulations were performed. Real-time performance was obtained by implementing the constant-flux PMSM control structure shown in Figure 2 using PI-EOA and FOPI controllers, and the variable-flux PMSM control structure shown in Figure 3, using SMC- and FOSMC-type controllers.

Note that the controller structures presented in the Matlab/Simulink numerical simulations are preserved in the real-time implementation, but the 1/s integrators expressed in the continuous domain are equated to integrators in the discrete domain by using a bilinear Tustin transform.

4.1. Real-Time PMSM-SCS Based on DTC Strategy Using Constant Flux

Using the Simulink model for monitoring the real-time signals from the LAUNCHXL-F28379D development board, a series of experimental tests were carried out by applying acceleration and deceleration steps to the PMSM rotor speed reference. The graphs obtained for the main parameters of interest of the sensorless PMSM control system based on the DTC strategy are presented below.

For example, Figure 19 shows the real-time rotor speed evolution for PMSM-SCS based on the DTC strategy at constant flux. Figure 19a shows the real-time speed response when using the PI-EOA controller and Figure 19b shows the real-time speed response when using the FOPI controller. It can be seen that the performance of the PMSM-SCS based on the DTC strategy with the PI-EOA controller is very good in real-time implementation. Small inherent differences with the numerical simulations are observed, but, from a practical point of view, we can say that the control system offers good performance even for applications where high accuracy is required; in this sense, an argument is that the steady-state value error is less than 0.2%. In addition, the settling time is 70 ms and the overshoot is less than 2%, which is good performance for real-time systems. Performance is enhanced by the use of the FOPI controller, which achieves 1% overshoot and a settling time of 60 ms.

The real-time evolution of the stator currents i_a and i_b is shown in Figure 20a for the case of using the PI-EOA controller, and in Figure 20b for the case of using the FOPI controller. A reduction in the current values during the transient regime is observed when using the FOPI controller in the PMSM control structure. In addition, the following graphs show the real-time evolution of some parameters of interest to the control system. Thus, Figure 21a shows the real-time evolution of the flux reference and flux feedback (estimated flux given by the flux observer); the real-time evolution of the electromagnetic torque is shown in Figure 21b, and the detail of the real-time evolution of the PMSM rotor position is shown in Figure 21c.

The graphical representation of the FD of the real-time speed signal for the PMSM-SCS based on the DTC strategy in the case of constant flux is depicted in Figure 22. Figure 22a depicts the FD of the real-time speed signal in the case of using the PI-EOA controller, and Figure 22b depicts the FD of the real-time speed signal in the case of using the FOPI controller.

4.2. Real-Time PMSM-SCS Based on DTC Strategy Using Variable Flux

Similarly to what was presented in the previous subsection, in this subsection we present the real-time experimental tests carried out for the case of the sensorless PMSM control system based on the DTC strategy, where the flow is variable and is provided by the two controllers of type SMC and FOSMC. By imposing the same series of steps on the reference speed of the PMSM rotor, the graphs obtained for the parameters of interest are presented, to highlight the control performance.

The real-time speed evolution for PMSM sensorless control based on DTC strategy in the case of variable flux using the SMC-type controller is shown in Figure 23a and in the case of using the FOSMC-type controller is shown in Figure 23b. In this case, using integer- and fractional-order SMC-type controllers, it is observed that the overshoot is practically zero and the settling time is 34.5 ms and 33.9 ms, respectively.

The real-time evolution of the flux reference and estimated flux for the PMSM-SCS based on the DTC strategy using the FOSMC-type controller is shown in Figure 24.

The real-time evolution of the stator currents i_a and i_b is shown in Figure 25a for the case where the SMC-type controller is used, and in Figure 25b for the case where the FOSMC-type controller is used.

The graphical representation of the FD of the real-time speed signal for the PMSM-SCS based on the DTC strategy in the case of variable flux is shown in Figure 26. Thus, Figure 26a shows the FD of the real-time speed signal when using the SMC-type controller, and Figure 26b shows the FD of the real-time speed signal when using the FOSMC-type controller.

The results of the experimental tests carried out in real time are summarized in

Table 3. Performance of the proposed controllers for PMSM-SCS in real-time implementation.

Type of Controller for PMSM Sensorless Real-Time Control System		Response Time [ms]	Settling Time [ms]	Speed Ripple [rpm]	FD of PMSM Speed Signal
PI-EOA	constant flux	18.2	70	16.76	0.96900 +/− 0.065622
FOPI		25.8	60	15.83	0.97465 +/− 0.060998
SMC	variable flux	34.5	34.5	14.95	0.98009 +/− 0.060727
FOSMC		33.9	33.9	10.61	0.98533 +/− 0.031485

, both for using a constant flux and for using a variable flux for the sensorless PMSM control system, based on the DTC strategy. The superior performance of SMC-type controllers compared to classical PI-type controllers is observed, as well as the preservation of this hierarchy when using fractional calculus. It should be noted that in real-time, PI-type controllers have better performance in terms of response time, but not in terms of overshoot, settling time, speed-signal ripple and FD of the controlled signal. It should also be noted that the ripple of the speed signal is lower in real time than in numerical simulations, because the number of samples is 1000 in the first case and 10,000 in the second. Regarding the FD of the speed signal, the hypothesis that a better performing controller implies a higher FD value is verified. It should be noted that, in the case of real-time implementations, if the embedded system’s computing power is insufficient, it is possible that complex control algorithms that perform better in numerical simulations than classical PI-type algorithms may give opposite results. In the case presented in this article, the development platform offers superior computational power, so that a biunivocity between the performance obtained in numerical simulations and the real-time implementation was achieved. In this case, FO-type controllers offer improvements over integer-type controllers, with the best performance offered by FOSMC-type controllers.

5. Conclusions

Using notions of fractional calculus, this article has presented several integer-order or FO controllers that have been integrated into the structure of the PMSM-SCS, based on the DTC strategy, to improve performance. Thus, starting from PI-type controllers, the features of the control system was improved by using the EOA-type algorithm for optimal tuning of controller parameters and by using FO-type controllers: FOPI, TID, and FO-Lead-Lag speed controllers. These types of controllers, embedded in the DTC strategy, are based on a constant reference flux for the flux control loop. When the reference flux is variable, the features of the control system are improved by using an integer-order SMC-type controller and the improved fractional FOSMC-type controller. The performance indicators of the PMSM-SCS based on the DTC strategy are the usual ones: response time, settling time, overshoot, steady-state error, speed ripple, and FD of the PMSM rotor speed signal.

The parametric robustness property of the control system based on the DTC strategy is also demonstrated by the fact that good control performance is obtained by increasing the J parameter, which represents the combined inertia of the PMSM rotor and the load.

The deployment of the software application in embedded system for a real-time HIL stage follows the stages presented in the controller synthesis and numerical simulations for the SIL stage, and good control performance of the PMSM based on the DTC strategy is obtained for the real-time control system.

Future work will build on the development of controller vendor libraries to implement specific machine-learning capabilities in Simulink to design and implement PMSM-SCS that deliver superior performance.