Continuous Control Set Predictive Control with Affine Registration Technique for Permanent Magnet Synchronous Motor Drive

Abstract

This article introduces an affine registration (AR) algorithm to improve the performance of ultra-local (UL) model-free predictive control (MFPC) methods. The UL model is extensively utilized in continuous control set (CCS)-MFPC to estimate lumped parameters. However, in these cases, variations in inductance during operation can significantly impact the input gain coefficient of the UL model. Little work has focused on this aspect, which makes the motivation and novelty of this paper. The AR technique, derived from image processing methodologies, is employed to establish a mapping between predicted states and actual states during the commissioning process. Subsequently, during actual operation, the resolved voltage reference is modified based on the AR matrix to compensate for errors in the UL model. The theoretical control model is rigorously derived, and experimental results are presented to validate the superior performance of the proposed method.

1. Introduction

Recently, continuous control set (CCS) model predictive control (MPC) has garnered considerable attention in the field of electrical drive systems [1,2,3]. Compared to finite control set (FCS)-MPC, CCS-MPC provides the flexibility to apply any reference voltage vector and synthesize it through pulse width modulation (PWM). This capability not only reduces current ripple but also enhances overall control performance [4,5]. However, despite these benefits, CCS-MPC still faces challenges, such as sensitivity to model parameters and vulnerability to external unknown disturbances [6,7,8,9].

To mitigate the challenges posed by parametric uncertainties, various approaches have been explored recently. These approaches can be broadly classified into model-based and model-free methods. Model-based methods typically involve disturbance compensation through observers, such as a Luenberger observer [10,11], sliding mode observer [12,13,14], and extended state observer [15,16]. The use of disturbance observers is widely recognized and applied. Conversely, with the growing trend towards data-driven control schemes in the scientific community, model-free approaches have gained prominence. These methods are increasingly acknowledged for their ease of implementation and independence from mathematical models.

Among model-free methods, the ultra-local (UL) model stands out as a popular and highly effective strategy. It simplifies PMSM systems into a first-order model that includes a lumped unknown term. This model, based on I/O data, facilitates predictive controller design, offering improved robustness against parameter variations and delivering superior performance. The literature [17] introduced a six-vector model-free predictive current control method based on the UL model to enhance the steady-state control performance of conventional MPC methods. In [18], the UL model with a predictive current control scheme was used for a doubly fed induction generator to address poor parameter robustness caused by complex machine parameters. The authors in [5] established a UL model for a reduced-order PMSM drive system, improving the dynamic response of CCS-MPC. In summary, model-based control methods rely on mathematical models to describe system dynamics. They offer high accuracy and predictability when the system model is well-defined, but inaccuracies or unmodeled dynamics can lead to suboptimal performance or instability, often necessitating the use of observers [19,20,21]. On the other hand, model-free control methods, such as reinforcement learning and adaptive control, can be simpler to implement initially, as they do not require detailed system models [22,23]. However, the training process may demand significant computational resources, making an online tracking model with a simple algorithm preferable.

Despite substantial research advancements, UL model-based MPC still suffers from variations in the input gain coefficient. The observer estimates only the lumped parameter, leaving the input gain coefficient constant within the control system. However, this coefficient typically includes the stator inductance, which varies with current fluctuations. Consequently, the predicted future states may be inaccurately estimated, affecting the voltage reference calculated by CCS-MPC. This undesired voltage input can lead to less effective control performance. To our knowledge, this issue has received limited attention to date.

Motivated by these observations, this paper introduces a novel affine registration (AR) algorithm-based CCS-MFPC method. The image registration method excels in aligning images from different data sets, enabling precise correction based on extensive data [24,25]. It is particularly advantageous in interdisciplinary fields for data mapping, ensuring accurate data fusion, analysis, and comparison. Several algorithms exist for image registration, with the AR algorithm being a subset of linear registration methods [26,27,28]. It accommodates not only translation and rotation but also deformations like scaling and shearing, making it more flexible and suitable for images undergoing different transformations. Compared to non-rigid registration, AR is less computationally intensive, leading to enhanced diagnostic accuracy and real-time applications. In the CCS-MFPC method, the real-time motor parameters are scattered around the ideal value. If some parameters are selected and plotted as an image, each sample can produce a slightly different graph, but with similar shapes, which can be considered as a deformation. In this context, the AR algorithm can be employed to minimize the distortion of predicted future states caused by these non-ideal factors [29,30]. The proposed solution is grounded in the philosophy that, during commissioning, the predicted future states are defined as raw points, while the sampled real states are recognized as deformed points. Both sets of points are depicted on the plot, representing two different images. The transformation from the raw image to the deformed image is resolved by the AR matrix and applied to adjust the voltage input during online operation. By employing the proposed method, the impact of input gain coefficient variations in the CCS-MFPC method can be minimized.

To clarify the main contributions of this paper, we highlight two key points. First, we propose an AR-based CCS-MFPC framework to address unknown nonlinear inductance variations in PMSM drive systems. Compared with existing CCS-MPC methods, this approach offers better representation of system uncertainties, further enhancing the generalization capability of the proposed design, making it more widely applicable to real-world systems. The second key contribution is the improvement in the estimation accuracy of the UL model by rectifying its input gain coefficient, which opens a promising area for future research wherever the UL model is employed. Finally, we demonstrate the merits of the proposed control methodology through a numerical example, comparing it with state-of-the-art CCS-MPC solutions for PMSM drive systems.

This article is organized as follows. In Section 2, the mathematical model of PMSM and ultra-local model are introduced, the AR transformation method is proposed, and the rectification process is discussed in detail. In Section 3, the modified control results are presented and analyzed. In Section 4, the proposed method is compared with the conventional CCS-MPC method experimentally, and the comparison results are discussed. Finally, Section 5 concludes this paper.

2. Materials and Methods

2.1. Conventional CCS-MPC Method

The dynamic model of an ideal surface-mounted PMSM can be described in d-q frame as follows:

$$\left\{\begin{array}{cc}{\displaystyle \frac{d}{dt}}{i}_d\hfill & =-{\displaystyle \frac{R_s}{L_d}}{i}_d+{\displaystyle \frac{{\omega }_e{L}_q}{L_d}}{i}_q+{\displaystyle \frac{u_d}{L_d}};\hfill \\ {\displaystyle \frac{d}{dt}}{i}_q\hfill & =-{\displaystyle \frac{R_s}{L_q}}{i}_q-{\displaystyle \frac{{\omega }_e{L}_d}{L_q}}{i}_d+{\displaystyle \frac{u_q}{L_q}}-{\displaystyle \frac{{\omega }_e{\psi }_f}{L_q}};\hfill \\ {\displaystyle \frac{d}{dt}}{\omega }_e\hfill & ={\displaystyle \frac{p}{J}}(T_e-T_L);\hfill \\ T_e\hfill & ={\displaystyle \frac{3}{2}}p{\psi }_f{i}_q,\hfill \end{array}\right.$$

(1)

where $i_d$ and $i_q$ represent the stator currents, $u_d$ and $u_q$ represent the stator voltages, $L_d$ and $L_q$ represent the stator inductances, $R_s$ represents the stator resistance, ${\omega }_e$ represents the electrical angular speed, ${\psi }_f$ represents the rotor exciting flux, p represents the pole pair number, J represents the inertia of the rotor, $T_e$ represents the electromagnetic torque, and $T_L$ represents the load torque.

To reduce the order of (1) and facilitate non-cascaded control between the speed loop and current loop, a new variable $e_{\omega }$ is introduced to replace the original state variable ${\omega }_e$:

$$e_w=({\eta }_m+{\displaystyle \frac{d}{dt}})({\omega }_e^{\ast }-{\omega }_e),$$

(2)

where ${\eta }_m$ is a user-defined coefficient of convergence and ${\omega }_e^{\ast }$ represents the speed reference.

Based on (2), the ultra-local mathematical model of (1) yields [4,5]:

$$\dot{\mathit{x}}=\mathit{Au}+\mathit{g},$$

(3)

where

$$\begin{array}{c}\mathit{x}=\left[\begin{array}{c}{e}_{\omega }\\ i_d\end{array}\right],\mathit{A}=\left[\begin{array}{cc}\hfill -{\displaystyle \frac{3p^2{\psi }_f}{2L_{q}J}}& \hfill 0\\ \hfill 0& \hfill {\displaystyle \frac{1}{L_d}}\end{array}\right],\hfill \\ \mathit{u}=\left[\begin{array}{c}{u}_q\hfill \\ u_d\hfill \end{array}\right],\mathit{g}=\left[\begin{array}{c}{g}_{\omega }\hfill \\ g_d\hfill \end{array}\right],\hfill \\ \left\{\begin{array}{cc}{g}_w\hfill & ={\displaystyle \frac{3p^2{\psi }_f}{2J}}\left({\displaystyle \frac{R_s}{L_q}}{i}_q+{\displaystyle \frac{w_e{L}_d}{L_q}}{i}_d+{\displaystyle \frac{{\psi }_f}{L_q}}{w}_e-{\eta }_m{i}_q\right)+{\displaystyle \frac{{\eta }_{m}p}{J}}{T}_L;\hfill \\ g_d\hfill & ={\displaystyle \frac{w_e{L}_q}{L_d}}{i}_q-{\displaystyle \frac{R_s}{L_d}}{i}_d.\hfill \end{array}\right.\hfill \end{array}$$

(4)

The UL model is applied for estimating the lumped parameter $\mathit{g}$ [31].

$$\begin{array}{cc}\hfill \mathit{g}\left(k\right)=& -{\displaystyle \frac{3}{N_f^3}}\sum _{m=k-N_f+1}^k\left(\left(\left(N_f-2(m-1)\right)\mathit{x}(m-1)\right.\right.\hfill \\ \hfill & +\mathit{A}(m-1)T_s\left(N_f-(m-1)\right)\mathit{U}(m-1)\hfill \\ \hfill & \left.+\left(N_f-2m\right)\mathit{x}\left(m\right)+\mathit{A}m{T}_s\left(N_f-m\right)\mathit{U}\left(m\right)\right).\hfill \end{array}$$

(5)

where $N_f$ is the window sequence length and $T_s$ is the control cycle period.

In the CCS-MPC method, calculating the voltage reference requires both the incremental form of the system input $\Delta u(k+1)$ and the general form of the state variables $x(k+2)$. Therefore, it is necessary to unify them through an improved formula.

$$\left\{\begin{array}{c}\Delta x(k+1)=\Delta x\left(k\right)+T_{s}A\Delta u\left(k\right)+T_s\Delta g\left(k\right);\hfill \\ x(k+1)=\Delta x\left(k\right)+T_{s}A\Delta u\left(k\right)+T_s\Delta g\left(k\right)+x\left(k\right).\hfill \end{array}\right.$$

(6)

Let

$$X(k+1)=\left[\begin{array}{c}\Delta x(k+1)\\ x(k+1)\end{array}\right],U\left(k\right)=\Delta u\left(k\right).$$

(7)

Then, Equation (6) can be written as:

$$\mathit{X}(k+1)=\mathit{D}\mathit{X}\left(k\right)+\mathit{B}U\left(k\right)+\mathit{C}G\left(k\right),$$

(8)

where

$$\mathit{D}=\left[\begin{array}{cc}\mathit{I}& 0\\ \mathit{I}& \mathit{I}\end{array}\right],\mathit{B}=\left[\begin{array}{c}{T}_s\mathit{A}\\ T_s\mathit{A}\end{array}\right],\mathit{C}=\left[\begin{array}{c}{T}_s\mathit{I}\hfill \\ T_s\mathit{I}\hfill \end{array}\right],\mathit{I}=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right].$$

(9)

The voltage reference $u\left(k\right)$ is sent to the PWM generator at the starting instant of the kth control cycle. Therefore, it is $\Delta u(k+1)$ that needs to be determined during the kth control cycle. Accordingly, predicting the value of $x(k+2)$ is necessary to assist in computing the optimal $\Delta u(k+1)$ [3].

$$\left\{\begin{array}{c}\mathit{X}(k+1)=\mathit{D}\mathit{X}\left(k\right)+\mathit{B}U\left(k\right)+\mathit{C}G\left(k\right);\hfill \\ \mathit{X}(k+2)=\mathit{D}\mathit{X}(k+1)+\mathit{B}U(k+1)+\mathit{C}G(k+1).\hfill \end{array}\right.$$

(10)

Let

$${\mathit{X}}_{\mathit{S}}=\left[\begin{array}{c}\mathit{x}(k+1)\\ \mathit{x}(k+2)\end{array}\right],{\mathit{U}}_{\mathit{S}}=\left[\begin{array}{c}\mathit{U}\left(\mathit{k}\right)\\ \mathit{U}(\mathit{k}+\mathbf{1})\end{array}\right],{\mathit{G}}_{\mathit{S}}=\left[\begin{array}{c}\mathit{G}\left(\mathit{k}\right)\\ \mathit{G}(\mathit{k}+\mathbf{1})\end{array}\right].$$

(11)

Equation (10) can be written as:

$${\mathit{X}}_{\mathit{S}}=\mathbf{\Omega }\mathit{X}\left(k\right)+\mathbf{\Gamma }{\mathit{U}}_{\mathit{S}}\left(k\right)+\mathbf{\Lambda }{\mathit{G}}_{\mathit{S}}\left(k\right).$$

(12)

where

$$\mathbf{\Omega }=\left[\begin{array}{c}\mathit{D}\\ {\mathit{D}}^2\end{array}\right],\mathbf{\Gamma }=\left[\begin{array}{cc}\mathit{B}& 0\\ \mathit{DB}& \mathit{B}\end{array}\right],\mathbf{\Lambda }=\left[\begin{array}{cc}\mathit{C}& 0\\ \mathit{DC}& \mathit{C}\end{array}\right].$$

(13)

Based on (12), the cost function can be designed as:

$$\mathit{J}={({\mathit{X}}_{\mathit{ref}}-{\mathit{X}}_S)}^T\mathit{Q}({\mathit{X}}_{\mathit{ref}}-{\mathit{X}}_S)+{\mathit{U}}_S^T\mathit{W}{\mathit{U}}_S.$$

(14)

where ${\mathit{X}}_{\mathit{ref}}$ is the state reference and $\mathit{Q}$ and $\mathit{W}$ are the weighting factors of tracking errors and control effort penalization, respectively.

According to the solution of the quadratic program, the optimal ${\mathit{U}}_S$ without constraints can be calculated to minimize $\mathit{J}$ as:

$${\displaystyle \frac{\partial \mathit{J}}{\partial {\mathit{U}}_S}}=0.$$

(15)

$${\mathit{U}}_S={\left({\mathrm{\Gamma }}^T\mathit{Q}\mathrm{\Gamma }+\mathit{W}\right)}^{-1}{\mathrm{\Gamma }}^T\mathit{Q}\left({\mathit{X}}_{\mathit{ref}}-\mathrm{\Omega }X\left(k\right)-\mathrm{\Lambda }G\left(k\right)\right).$$

(16)

2.2. Defect of Conventional Method

The conventional computation process for $\Delta u(k+1)$ mentioned above involves a significant inaccuracy, which can deteriorate the estimation result of $x(k+2)$, consequently leading to suboptimal $u(k+1)$ provided by the control system. The problem is that, as indicated in (3), no matter how accurate the $\mathit{g}$ is, the estimation result of the ultra-local model is still compromised if the input gain coefficient $\mathit{A}$ lacks precision. Unfortunately, $\mathit{A}$ contains the inductance $L_d$ and $L_q$, as can be seen in (4). It is well known that the current fluctuates as the motor rotates, and the changing saturation condition alters the inductance.

Taking the inaccuracy of $\mathrm{\Gamma }$ into account and assuming that $G_S\left(k\right)$ can be precisely calculated, the predicted future state is calculated as:

$${\widehat{X}}_S=\mathrm{\Omega }X\left(k\right)+\widehat{\mathrm{\Gamma }}{U}_S\left(k\right)+\mathrm{\Lambda }{G}_S\left(k\right),$$

(17)

where ${\widehat{X}}_S$ represents the predicted future state and $\widehat{\mathrm{\Gamma }}$ represents the input gain coefficient with constant value. In contrast, (12) describes the actual value.

As inferred from Equations (14)–(16), when the predicted $x(k+2)$ is inaccurate, the voltage reference $u(k+1)$ also deviates from the optimal value. Consequently, the control performance is deteriorated.

Based on (17), the suboptimal voltage reference in contrast to (16) should also be rewritten as:

$${\widehat{\mathit{U}}}_S={\left({\widehat{\mathrm{\Gamma }}}^T\mathit{Q}\widehat{\mathrm{\Gamma }}+\mathit{W}\right)}^{-1}{\widehat{\mathrm{\Gamma }}}^T\mathit{Q}\left({\mathit{X}}_{\mathit{ref}}-\mathrm{\Omega }X\left(k\right)-\mathrm{\Lambda }G\left(k\right)\right).$$

(18)

Introducing (17) into (18), the value of the predicted future state can be deduced:

$${\widehat{X}}_S={({\widehat{\mathrm{\Gamma }}}^T\mathit{Q}\widehat{\mathrm{\Gamma }}+\mathit{W})}^{-1}W(\mathrm{\Omega }X\left(k\right)+\mathrm{\Lambda }{G}_S\left(k\right)).$$

(19)

Under the effect of ${\widehat{\mathit{U}}}_S$, the real value of future state $X_S$ should be calculated by combining (12) and (18):

$${\widehat{X}}_S^{\ast }={({\widehat{\mathrm{\Gamma }}}^T\mathit{Q}\widehat{\mathrm{\Gamma }}+\mathit{W})}^{-1}(W+{\widehat{\mathrm{\Gamma }}}^T\mathit{Q}(\widehat{\mathrm{\Gamma }}-\mathrm{\Gamma }))(\mathrm{\Omega }X\left(k\right)+\mathrm{\Lambda }{G}_S\left(k\right)).$$

(20)

By comparing (19) and (20), it is observed that when $\widehat{\mathrm{\Gamma }}$ is equal to $\mathrm{\Gamma }$, the state variable can be predicted precisely. However, the value of inductance is misestimated during the real-time operation. And from (4), (9), and (13), it is shown that $\mathrm{\Gamma }$ contains the value of inductance. Therefore, the predicted ${\widehat{X}}_S$ is wrong. This can influence the calculated output ${\widehat{\mathit{U}}}_S$ and degrade the control performance.

2.3. Proposed Image Registration Technique-Based Predictive Control Solution

To this end, in this paper, we focus on an improved CCS-MPC solution combined with the image registration technique for PMSM drive systems. Firstly, the transformation relationship between the predicted states $\widehat{x}(k+2)$ and the actual sampled states $x(k+2)$ is established using image registration during the setup phase. Subsequently, the obtained transformation map is utilized to obtain the corrected voltage input $u(k+1)$ during real-time operation. This section begins by introducing the AR method as the selected image registration technique, renowned for its simplicity, versatility, and universality. Following that, the improved AR-based CCS-MPC solution is elaborated, demonstrating how to achieve a refined voltage reference with significantly enhanced accuracy.

2.3.1. Affine Registration Algorithm

The affine registration algorithm (ARA) technique is one type of polynomial transformation methods in the region of image registration. For two images depicted in ${\mathbb{R}}^2$, let $(x,y)\in \mathit{S}$ denote the point sets on the raw image, while $(X,Y)\in {\mathit{S}}^{\ast }$ denotes the point sets on the deformed image. The first-order AR transformation T can be expressed as:

$$\left[\begin{array}{c}X\hfill \\ Y\hfill \end{array}\right]=\left[\begin{array}{ccc}{T}_{x10}& T_{x01}& T_{x00}\\ T_{y10}& T_{y01}& T_{y00}\end{array}\right]\left[\begin{array}{c}x\hfill \\ y\hfill \\ 1\hfill \end{array}\right].$$

(21)

Figure 1 illustrates some fundamental functions achievable through first-order AR transformations. The transformation T can be a composition of several basic transformations to achieve a combined effect, and it can also be extended to higher degrees by using polynomial multiplication to accomplish more complex transformations. For example, in Equation (21), the degree $d=1$, so the basis monomials are 1, x, and y. Similarly, for $d=2$, the basis monomials are $1,x,y,xy,x^2$, and $y^2$. It is easily concluded that there are $(d+1)(d+2)/2$ basis monomials for a d-order AR transformation, which can be written in integral form:

$$\mathit{X}=\sum _{i=0}^d\sum _{j=0}^{d-i}{A}_{xij}{x}^i{y}^j,\mathit{Y}=\sum _{i=0}^d\sum _{j=0}^{d-i}{A}_{yij}{x}^i{y}^j.$$

(22)

Let ${\mathcal{P}}_{\mathit{S}}$ denote the subset of S, representing the reference points selected during the commissioning process, while ${\mathcal{P}}_{\mathit{S}}^{\ast }$ denotes the subset of S, representing the sampled real points. The calculation of AR transformation matrix T turns into the problem of minimizing the following cost function.

$$\epsilon \left(T\right)=\sum _i{\Vert {\mathcal{P}}_{\mathit{S}i}^{\ast }-T{\mathcal{P}}_{\mathit{S}i}\Vert }^2.$$

(23)

2.3.2. Improved CCS-MPC Solution

A set of points sampled from ${\widehat{X}}_S$ and ${\widehat{X}}_S^{\ast }$ can be depicted on the plot. The aforementioned AR technique can be used to resolve the transformation $T({\widehat{X}}_S,{\widehat{X}}_S^{\ast })$ from ${\widehat{X}}_S$ to ${\widehat{X}}_S^{\ast }$.

From (19) and (20), it is obtained that:

$${\widehat{X}}_S^{\ast }=W^{-1}(W+{\widehat{\mathrm{\Gamma }}}^T\mathit{Q}(\widehat{\mathrm{\Gamma }}-\mathrm{\Gamma })){\widehat{X}}_S.$$

(24)

By casting (24) into the image registration problem, the relationship between $T(\widehat{\mathrm{\Gamma }},\mathrm{\Gamma })$ and $T({\widehat{X}}_S,{\widehat{X}}_S^{\ast })$ can be derived:

$$T(\widehat{\mathrm{\Gamma }},\mathrm{\Gamma })=T_4{T}_3{T}_2{T}_{1}T({\widehat{X}}_S,{\widehat{X}}_S^{\ast }),$$

(25)

where

$$\begin{array}{c}{T}_1=T_3={\left[\begin{array}{ccc}1& 0& -1\\ 0& 1& -1\end{array}\right]}^T,T_2=\left[\begin{array}{ccc}1& 0& 0\\ 0& {\displaystyle \frac{{\left(2L_{q}J\right)}^2{k}_u}{{\left(3p^2{\psi }_f{T}_s\right)}^2{k}_{\omega }}}& 0\end{array}\right],\hfill \\ T_4=\left[\begin{array}{ccc}1& 0& 0\\ 0& -1& 0\end{array}\right],\hfill \end{array}$$

and $k_u$ and $k_{\omega }$ are the diagonal elements in $\mathit{Q}$ and $\mathit{W}$, respectively.

Therefore, during the real-time operation, the modified voltage reference can be obtained:

$${\widehat{\mathit{U}}}_S^{\ast }={({\left(T(\widehat{\mathrm{\Gamma }},\mathrm{\Gamma })\widehat{\mathrm{\Gamma }}\right)}^T\mathit{Q}T(\widehat{\mathrm{\Gamma }},\mathrm{\Gamma })\widehat{\mathrm{\Gamma }}+\mathit{W})}^{-1}({\widehat{\mathrm{\Gamma }}}^T\mathit{Q}\widehat{\mathrm{\Gamma }}+\mathit{W}){\widehat{\mathit{U}}}_S.$$

(26)

The comparison of conventional and proposed methods is demonstrated in Figure 2. In the conventional method, the derived inaccurate voltage reference is directly used for voltage reference, as the blue path illustrates. In contrast, the proposed method calculates the AR transformation matrix in advance. And in the operational process, the voltage reference is modified. The red path presents the improved calculation method for the voltage reference. The principle of the proposed method is also explained by the plot as shown in Figure 3. For simplification, only the q-axis component of $X_S$, which is $e_{\omega }$, is depicted for demonstration. As can be seen, the ideal relationship between $\mathrm{\Omega }X\left(k\right)+\mathrm{\Lambda }G\left(k\right)$ and $X_{Sq}^{\ast }$ is illustrated by black circles, which corresponds to (12). Caused by the miscalculation of $\mathrm{\Gamma }$, the actual relationship deduced by the conventional method is depicted by blue circles, which correspond to (19). The AR transformation can rectify the image from a blue circle set to red circle set, which is much closer to a black circle set. As a result, ${\widehat{X}}_{Sq}^{\ast }$ is much closer to the ideal value $X_{Sq}$.

It is worth mentioning that, since all the dedication is based on the QP solution (14) and (15), the voltage constraints are not considered in this paper. When the calculated reference voltage exceeds the inverter limits, a simple method, as shown in Figure 4, will be used for handling it. And we will carefully monitor the experiments to make sure that such a situation rarely happens in order to promise the viability of our theory.

3. Results

In the last section, the principle and design process of the proposed method were demonstrated. In this section, the dynamic performances of the proposed method will be validated by experiments. The model-based CCS-MPC (4) and conventional model-free CCS-MPC (16) are set as candidates to show the difference in performance.

The parameters of the PMSM are given in

Table 1. Pmsm Parameters.

Parameter	Value
Stator resistance $R_s$	0.43 Ohm
Stator inductance $L_s$	6.8 mH
Rotor flux linkage ${\psi }_f$	0.127 Wb
Rated current I	6.0 A
Rated voltage U	300 V
Rated speed $\omega$	3000 rpm
Pole pair number P	4
Total rotor inertia J	0.3 g· m2
${\eta }_m$	800
$\tau$	0.4

, and the test bench is shown in Figure 5. The load motor is controlled by a user-defined servo inverter, which is set to the direct torque control mode in the experiments. The test motor is controlled by an integrated control system, which consolidates the control board, three-phase inverter bridge, speed sensor and current sensor altogether. It facilitates data transmission through an Ethernet cable connected to a computer. In the following tests, the steady and dynamic performances are comprehensively discussed.

The effectiveness of the AR transformation is demonstrated through a comparison between the conventional model-free method (16) and the proposed method (26). The study is conducted at 1500 rpm with no load torque. The discrepancy between predicted and actual $e_{\omega }$ is illustrated in Figure 6a,b, where the actual image is depicted in blue. The predicted images from the conventional and proposed methods are shown in red and green, respectively. It is evident in Figure 6a that the points of the actual image are scattered around the line defined by Equation (23), while the points of the predicted image are deformed due to the influence of inductance variation. After applying the AR transformation, the modified points in green align much better with the actual points. As shown in Figure 6c,d, the estimation error of $e_{\omega }$ in the proposed method is smaller than that of the conventional method. Consequently, as can be seen in Figure 6e,f, the $e_{\omega }$ tracking error of the improved method is smaller than that of the conventional method. It is noted that the static speed tracking errors of both methods are not entirely satisfactory. This is because the tracking objective of the reduced-order model is $e_{\omega }$, as defined in (2), rather than ${\omega }_e$.

4. Discussion

Figure 7 illustrates the dynamic performances under load torque disturbance at 3000 rpm speed operation. The designed test procedure involves the motor initially running at a speed of 3000 rpm with zero load torque. Subsequently, the reference load torque steps from 0 to half of the full load torque, achieved by positively adjusting the $i_q^{\ast }$ of the load motor. The waveforms are recorded until the motor returns to a steady state. Comparative dynamic performances are summarized in

Table 2. Dynamic performance comparison of 3 controllers under torque disturbance.

Control Scheme	MPC (4)		MFPC (16)		Proposed (27)
Operating Condition	600 rpm	3000 rpm	600 rpm	3000 rpm	600 rpm	3000 rpm
Speed settling time (ms)	19.0	19.5	3.4	3.5	2.8	2.8
Speed drop (rpm)	26.5	27.3	25.1	25.3	23.4	23.2
$i_q$ overshoot (A)	0.86	0.84	2.51	2.74	2.18	2.43
THD (%)	5.84	11.05	4.52	5.76	4.32	4.55

.

As observed, the model-based method exhibits the longest settling time of 19.5 ms, along with the largest overshoot of 27.3 rpm. In contrast, the two model-free methods demonstrate relatively shorter settling times and smaller overshoots. It can be inferred that parameter mismatch in experiments adversely affects the performance of the fixed-model CCS-MPC.

For the two model-free methods, their dynamic performances are nearly identical, and the speed recovers very rapidly. The settling times are 3.5 ms and 2.8 ms, and the overshoots are 25.3 rpm and 23.2 rpm. The proposed method demonstrates superior performance compared to the conventional method. From the $i_q$ plots, it is observed that when the load torque is free, the model-based method exhibits a large steady-state error. After the $i_q$ steps, the steady-state error diminishes. This phenomenon occurs because a large $i_q$ saturates the steel, thereby reducing the variation amplitude of inductance.

Figure 7 also illustrates the steady-state performances of the three control schemes after load torque disturbance. The THD values of the three methods are 11.05%, 5.76%, and 4.55%, respectively. The fundamental frequency of the phase current is 200 Hz. The MPC has the largest THD. The ultra-local model-based methods have a smaller THD than the fixed-model-based method mainly because of the data-driven observer. The fixed-model method inaccurately estimates the parameters, so the performances are also non-ideal. Additionally, it is evident that parameter mismatch also has an impact on the steady-state phase current waveforms. The fundamental amplitude of MPC is obviously smaller than that of the MFPC methods.

The performance of the three candidates at low-speed operation (600 rpm) was also tested, as is shown in Figure 8. The dynamic performances are similar to those of the high-speed experiments. However, the steady-state performances of all methods are better than their performances in low-speed tests. The THD of MPC decreases from 11.05% to 5.84%, and the THD of conventional MFPC decreases from 5.76% to 4.52%. It is speculated that when the motor rotates slowly, the change in $\mathit{g}$ computed at adjacent control instants is smaller. As a result, the parameter fluctuation in the control algorithm is smaller, and the performances are more satisfactory.

In addition, the start-up and braking performances of the proposed method are also compared with the other two strategies. As illustrated in Figure 9, the motor is accelerated from 0 to 3000 rpm. The MPC method, though maintaining a stable speed loop, exhibits the most limited waveform, with significant noise and spikes. The conventional MFPC method shows improved performance with fewer spikes, but it still falls short in terms of smoothness. The proposed method demonstrates superior performance by synchronizing q-axis current and speed control, leading to a highly responsive current waveform with minimal spikes. This method ensures precise and efficient motor operation, making it the most reliable choice among the three.

In Figure 10, the motor is braked from 3000 rpm to 1000 rpm. As can be seen, while the MPC method keeps the speed loop stable, it suffers from the most erratic waveform, with fluctuations and disturbances. The proposed and most effective method has a rapid and clean current response with minimal disturbances. To sum up, compared to the other two methods, the proposed method ensures precise deceleration and optimal motor performance, making it the preferred choice for the tested operations.

5. Conclusions

In this paper, an affine registration algorithm has been coupled with the continuous control set model-free predictive control method to improve its robustness. The conventional ultra-local model has a problem with its input gain coefficient when it is used for motor control application. The input gain coefficient involves inductance which is hard to observe in real-time operation. To this end, the AR method is introduced to rectify the output of the control system. The mathematical principle of the improvement process for the proposed method is presented in Section 2, where the mathematical model of PMSM and the ultra-local model are introduced, the AR transformation method is proposed, and the rectification process is discussed in detail. The rectification result of the proposed method is presented and analyzed in Section 3. In Section 4, the proposed method is compared with the conventional CCS-MPC method experimentally, and the steady-state performances as well as the dynamic performances under high-speed and low-speed conditions are thoroughly discussed. Experimental results show that the proposed method demonstrates excellent performance in both steady-state and dynamic operations. Finally, Section 2 concludes this paper.