Sensorless Model Predictive Control of Permanent Magnet Synchronous Motors Using an Unscented Kalman Filter

Abstract

This paper deals with the application of the Model Predictive Control (MPC) algorithm to the sensorless control of a Permanent Magnet Synchronous Motor (PMSM). The proposed estimation strategy, based on the unscented Kalman filter (UKF), uses only the measurement of the motor current for the online estimation of speed, rotor position and load torque. Information about the system state is fed into the MPC algorithm. The results verify the effectiveness and applicability of the proposed sensorless control technique. To demonstrate its real-world applicability, implementation in low-speed direct drive astronomy telescope mount systems is investigated. The outcomes of the implementation are thoroughly examined, leading to insightful conclusions drawn from the observed results. Through rigorous theoretical analysis and extensive simulation studies, this paper establishes a solid foundation for the proposed sensorless control technique. The results obtained from simulation studies and real-world applications underscore the efficacy and versatility of the proposed approach, offering valuable insights for the advancement of sensorless control strategies in motor applications. The main aim of this work is to demonstrate and validate the practical feasibility of combining two complex techniques, establishing that such an integration is not only possible but also effective in achieving the desired objectives.

1. Introduction

Permanent Magnet Synchronous Motor (PMSM) drives find extensive use in industrial processes owing to their inherently high torque-to-inertia ratio facilitated by their high flux density, compact size, and enhanced dynamic performance, making them ideal for applications where precision, responsiveness, and efficiency are paramount [1,2,3,4]. Discussions often revolve around enhancing the accuracy, stability, and robustness of their control methodologies. A widely used industrial control method, such as field-oriented control of PMSMs, heavily relies on an accurate rotor position [2]. However, the requirement for high-precision sensors comes at a cost, often requiring enclosed housing, which limits their applicability in specialized environments. In demanding conditions involving severe pressure, humidity, temperature, and vibration, traditional mechanical encoders prove inadequate [5,6]. Systems called sensorless systems present a potential solution by eliminating the necessity for mechanical sensors. This becomes particularly crucial during safety operations following sensor failure, especially in inaccessible environments [5]. Consequently, significant attention has been directed towards developing sensorless drives, utilizing the inherent behavior of PMSMs as a kind of virtual position sensor. In general, the advantages of these drive systems encompass potentially reduced drive costs, fewer integrated parts, a smaller size, a heightened reliability, and easier maintenance. However, their drawbacks primarily include a low load capacity, a limited speed accuracy, and instability in generating low-speed ranges [4,7].

In recent years, considerable research has focused on flux and shaft position estimation methods for drive systems in both induction motors and PMSMs, broadly categorized into two groups: those based on physical phenomena and those based on algorithmic methods, called observers. The exploration of these estimation methods reflects the growing emphasis on advancing motor control technologies, with a particular interest in developing robust and accurate techniques that leverage the understanding of physical motor behavior as well as sophisticated algorithmic approaches to ensure optimal performance and reliability in diverse industrial applications. Reference [8] outlines a primary method based on the physical non-linear behavior of magnetic circuits in induction motors. Additionally, the injection of high-frequency signals to emulate similar properties has also been applied [3,9]. The second major group involves algorithmic methods. The electromagnetic force (EMF) estimator has been extensively used to acquire speed and position information, as seen in [4,10,11]. Other proposed computational methods, such as those based on the basic sliding mode observer, are developed in [12,13]. Observers based on the Kalman filter (KF) [14] also find applications. For non-linear systems like PMSMs, extended Kalman filter (EKF)-based observers are prevalent [4,15,16,17]. However, due to the inherent unreliability and inaccuracy of the EKF for analytically undifferentiable non-linear circuits, an alternative approach using unscented transformations has emerged [18]. The unscented Kalman filter (UKF) stands out as an advanced filtering technique that significantly mitigates non-linearity issues associated with Gaussian noise and truncation errors, positioning itself as a viable and advantageous replacement for the extended Kalman filter (EKF). By employing a deterministic sampling approach, the UKF excels in capturing the true statistical characteristics of the underlying non-linear system dynamics [19,20]. This distinctive feature allows the UKF to outperform the EKF, especially in scenarios where the traditional filter may struggle with inaccuracies due to linearization, providing a more robust and accurate solution for state estimation in various engineering applications [4,21,22]. For certain complex non-linear systems, the resulting UKF more accurately estimates the true mean and covariance [23] in the scope of fast computations [24].

Model Predictive Control (MPC) is an advanced method in complex process control, frequently applied in chemical, pharmaceutical, petrochemical, and vehicle dynamics control systems. Its widespread adoption is attributed to its ability to optimize control inputs over a finite time horizon, considering the dynamic nature of the system and constraints imposed on the control variables. This forward-looking characteristic enables MPC to anticipate future system behavior and make proactive adjustments, making it particularly well suited for processes with intricate dynamics and stringent operational requirements [25,26,27]. The effectiveness of MPC lies not only in its ability to address complex control challenges but also in its integration with real-time optimization; its computational complexity and time demands limit applications to slower control cycle steps. MPC operates on an iterative control concept, leveraging a system model to predict future states within finite horizons, considering constraints and dynamic behaviors, requiring knowledge of the full system state—either measured or estimated. Notably, it has found successful applications in electromechanical systems in [28,29,30] and robotics in [31]. A neat and noteworthy summary of the possibilities of using MPC in motor control is presented in [30]. The recent literature [32] introduces MPC solutions for electric drive control, including current control in induction machines [16,33,34] and even in the domain of brushless DC motors with separate speed and current control [35]. Cascade MPC is commonly employed for speed control, enabling torque limitation to the maximum value while ensuring an optimal dynamic performance [32,34]. The solutions proposed in [36] and [37] advocate using the EKF as a state observer with a full state vector for MPC. Based on the comprehensive review of the literature presented in this article, it is evident that no comparable solutions or approaches have been documented or identified. A notable aspect worth highlighting is its implementation in a solution that closely resembles an industrial environment, contributing to its practical applicability and relevance to real-world scenarios.

Building upon the insights gained from previous authors’ experience in sensorless control with PID controllers [4,38,39], the current research endeavors to elevate the precision of control systems through the integration of Model Predictive Control (MPC) for sensorless control of Permanent Magnet Synchronous Motors (PMSMs). A significant focus of this research lies in leveraging the state estimates obtained through the unscented Kalman filter (UKF) in conjunction with MPC. The seamless fusion of UKF-derived state information with the predictive capabilities of MPC holds promise for achieving a higher degree of accuracy in PMSM control.

This article covers the fundamental components of this research, including problem identification, a proposed solution supported by the literature, the theoretical framework, simulations, and a practical demonstration in a lab setting. This paper is divided into four sections: Following the introduction, the second section provides a detailed description of the considered control scheme, encompassing a basic mathematical model of the system, the fundamentals of the unscented Kalman filter, and Model Predictive Control, as well as a method for mutual exploitation. In the third section, the results, taking into account quality, are presented in two distinct steps: simulations and laboratory verification. Exemplary figures illustrating the operation of the proposed sensorless control structure are included. The document is then concluded in the final section.

2. Methods

The proposed cascade MPC is often used for speed control, which allows us to limit the torque (currents) to the maximum admissible value while providing optimal dynamic performance [32]. The opposite approach consists of introducing full Model Predictive Control. Controlled values are estimated by the unscented Kalman filter.

2.1. Proposed Sensorless Control Scheme

A block diagram of the proposed sensorless scheme is shown in Figure 1. Signals for the UKF, like stator currents $i_a$, $i_b$, $i_c$ (measured by sensors) translated into $i_{\alpha }$, $i_{\beta }$ and desired voltages $u_{\alpha }$, $u_{\beta }$, are defined as desired by the MPC scheme. For the values of natural output (currents, speed and position), only the desired speed ${\omega }_r^{*}$ was chosen. The presented algorithm requires knowledge of the state variables $\widehat{\underline{x}}$.

The effectiveness of the proposed drive system has been verified in the scheme in Figure 1.

2.2. Mathematical Model of a PMSM for MPC and the UKF

The process model plays a decisive role in the controller. In order to predict the future behavior of a process, we must have a model of how the process behaves. A model is also necessary for the expected signal estimation via the unscented Kalman filter. The chosen model must be able to capture the process dynamics to predict the future outputs based on estimated states with precision.

The mathematical model of the PMSM has three main parts: the electrical network, electromechanical torque production and the mechanical subsystem [40]. The stator of the PMSM and the induction motor are similar. The rotor consists of permanent magnets (modern rare-earth magnets with a high strength).

During investigations, some simplified assumptions are made: saturation is negligible, the inducted electromagnetic force is sinusoidal, eddy currents and hysteresis losses are negligible, there are no dynamical dependencies in the air gap, and there is no rotor cage. Under these assumptions, the rotor-oriented $dq$ electrical network equations of the PMSM can be described as:

$$u_d=R_s{i}_d+L_d\frac{d{i}_d}{dt}-p{\omega }_r{L}_q{i}_q,$$

(1)

$$u_q=R_s{i}_q+L_q\frac{d{i}_q}{dt}+p{\omega }_r{L}_d{i}_d+p{\omega }_r{\mathrm{\Psi }}_m.$$

(2)

where $u_d$, $u_q$ are $dq$ axis voltages; $i_d$, $i_q$ are $dq$ axis currents; $L_d$, $L_q$ are $dq$ axis inductances; $R_s$ is stator resistance; and ${\mathrm{\Psi }}_m$ is the magnetic flux produced by permanent magnets placed on the rotor.

The value of the produced electromagnetic torque is given by the following equation:

$$T_e=\frac{3}{2}·p\left[{\mathrm{\Psi }}_m-\left(L_q-L_d\right)i_d\right]·i_q,$$

(3)

where p is the number of pole pairs, and the fraction $\frac{3}{2}$ stems from frame conversion, i.e., perpendicular stator $\alpha \beta$ over rotor $dq$.

The drive dynamics can be described as:

$$T_e-T_{load}=J\frac{d{\omega }_r}{dt},$$

(4)

where $T_{load}$ is the load torque and J is the summary moment of inertia of the kinematic chain.

Based on (3) and (4), the movement equation is:

$$\frac{d{\omega }_r}{dt}=\frac{p}{J}\left[\frac{3}{2}\left({\mathrm{\Psi }}_m-\left(L_q-L_d\right)i_d\right)·i_q\right]-\frac{T_e}{J}.$$

(5)

The otor position $\gamma$ can be described by the derivative equation of the rotational speed:

$$\frac{d\gamma }{dt}=p·{\omega }_r.$$

(6)

Drawing inspiration from the concept of the dual Kalman filter described in reference [41], one or more parameters can be treated as estimation values. The load torque $T_l$ is considered as such a virtual parameter within the proposed framework, enabling its observation and estimation. This approach opens avenues for dynamic adaptation and refinement [4,39,42], as in this case, the load torque is not merely an unknown disturbance but is dynamically observed and estimated, enhancing the adaptability of the control system in response to varying operational conditions. The true assumption that the load torque $T_l$ is invariable in a narrow interval holds:

$$\frac{d}{dt}{T}_l\approx 0.$$

(7)

The discrete state space is obtained from the continuous state space by discretization of coefficient matrices with the first-order backward Euler method with step time $T_s$. The step time was chosen to be far beyond the Nyquist frequency based on system poles, and is equal to the PWM frequency of the power electronics part. This is a hardware-implementation-oriented assumption. The state-space model can be described as a classical discrete state-space model:

$${\underline{\widehat{x}}}_k={\mathbf{F}}_k\left({\widehat{\underline{x}}}_{k-1}\right){\widehat{\underline{x}}}_{k-1}+{\mathbf{B}}_k\left({\widehat{\underline{x}}}_{k-1}\right){\underline{u}}_k,$$

(8)

$${\underline{z}}_k={\mathbf{H}}_k\left({\widehat{\underline{x}}}_k\right){\widehat{\underline{x}}}_k,$$

(9)

For the synthesis of the unscented Kalman filter, state vector $\widehat{x}$ was chosen:

$$\widehat{\underline{x}}={\left[{\widehat{i}}_d{\widehat{i}}_q{\widehat{\omega }}_r\widehat{\gamma }{\widehat{T}}_l\right]}^T,$$

(10)

with all natural state variables and estimated load torque ${\widehat{T}}_l$. Considering the above, the system matrix ${\mathbf{F}}_k$ has the form:

$${\mathbf{F}}_k\left({\widehat{\underline{x}}}_k\right)=\left[\begin{array}{ccccc}1-T_s·\frac{R_s}{L_d}& T_s·{\omega }_r\frac{L_q}{L_d}& 0& 0& 0\\ -T_s·{\omega }_r\frac{L_d}{L_q}& 1-T_s·\frac{R_s}{L_q}& -T_s·\frac{{\mathrm{\Psi }}_m}{L_q}& 0& 0\\ 0& T_1& 0& 0& -T_s·\frac{1}{J}\\ 0& 0& T_s& 1& 0\\ 0& 0& 0& 0& 1\end{array}\right],$$

(11)

where

$$T_1=T_s·\frac{3}{2}\frac{p}{J}\left[{\mathrm{\Psi }}_f-\left(L_q-L_d\right)i_d\right].$$

The output matrix ${\mathbf{H}}_k$ is a rotating matrix via the Clark/Parck transformation:

$${\mathbf{H}}_k\left({\widehat{\underline{x}}}_k\right)=\left[\begin{array}{ccccc}cos\gamma & -sin\gamma & 0& 0& 0\\ sin\gamma & cos\gamma & 0& 0& 0\end{array}\right],$$

(12)

and matrix ${\mathbf{B}}_k$:

$${\mathbf{B}}_k\left({\widehat{\underline{x}}}_k\right)=\left[\begin{array}{cc}{T}_s·\frac{1}{L_d}cos\gamma & T_s·\frac{1}{L_d}sin\gamma \\ -T_s·\frac{1}{L_q}sin\gamma & T_s·\frac{1}{L_q}cos\gamma \\ 0& 0\\ 0& 0\\ 0& 0\end{array}\right].$$

(13)

The state matrices are simplified. The load torque $T_l$ is treated as a known disturbance. The appearance of the model of decoupled $dq$ axes determines the method of control.

As evident from the proposed model analysis above, the system under consideration is non-linear, and this non-linearity is apparent in the state, input, and output matrices. The presented non-linear system involves matrices ${\mathbf{H}}_k$ described as (12), ${\mathbf{B}}_k$ as (13), and ${\mathbf{F}}_k$ in the form of (11), all of which are non-linear. Specifically, ${\mathbf{H}}_k$ and ${\mathbf{B}}_k$ incorporate trigonometric functions, reflecting a dependency on state variables associated with two-axis rotation ($\alpha \beta$, $dq$ frames), while ${\mathbf{F}}_k$ includes non-neglected state multiplications.

An issue may arise in accurately determining the current state based on both current and historical inputs. Additionally, in the presence of high non-linearities, the space of variable occurrences becomes disrupted, leading to non-linear variances for state ${\mathbf{P}}_x$ and outputs ${\mathbf{P}}_y$, which characterizes the accuracy of the modeling process. This disruption further complicates the characterization of variables in a non-linear space.

2.3. Unscented Kalman Filter

The primary challenge in estimating non-linear systems lies in determining the non-linear function of the state and output, as discussed by [19,43]. Additionally, addressing non-linearities in the probability distribution further complicates the task [18]. It seems that the non-linear transformation and the inclusion of Jacobians in the extended Kalman filter (EKF) may not accurately determine real covariances [18]. The unscented Kalman filter (UKF) emerges as an advancement over prior algorithms, presenting a novel solution to the estimation theory problem based on unscented transformations [18,19]. Unlike the EKF, the UKF simplifies the definition and approximation of the Gaussian distribution associated with each state vector variable, as opposed to relying on the approximation of non-linear function transformations [19]. Moreover, the EKF streamlines the algorithm by eliminating the need to calculate Jacobians at each working point. This streamlining is grounded in two key assumptions. Firstly, it involves determining the non-linear transformation of the function at work, rather than across the entire range of the probability density distribution function. Secondly, it revolves around identifying working points where this density aligns with the actual decomposition of the non-linear system. This filter, like its classical form, is based on two cycles: prediction and correction [14].

The prediction of the UKF can be used independently from the UKF update, in combination with a linear update. In this case, however, the estimation state vector of the value of disturbances is extended. Such a procedure makes it possible to estimate the state vector and its environment (with disturbances).

Based on [19], the defined augmented estimation vector ${\underline{x}}_{k-1|k-1}^a$ can be defined as:

$${\underline{x}}_{k-1|k-1}^a={\left[{\widehat{\underline{x}}}_{k-1|k-1}^{T}E\langle {\underline{w}}_k^T\rangle E\langle {\underline{v}}_k^T\rangle \right]}^T$$

(14)

which consists of state vector ${\widehat{x}}_k$ and expected noise terms $w_k$ and $v_k$, with the definition of each covariance given as:

$${\mathbf{P}}_{k-1|k-1}^a=\left[\begin{array}{ccccccc}& {\mathbf{P}}_{k-1|k-1}& & 0& & 0& \\ & 0& & {\mathbf{Q}}_k& & 0& \\ & 0& & 0& & {\mathbf{R}}_k\end{array}\right].$$

(15)

where ${\mathbf{Q}}_k$ is defined as the covariance of process noise and ${\mathbf{R}}_k$ is the covariance of observation (measurement) noise.

An important set of $2L+1$ sigma points, ${\chi }_{k-1|k-1}$, is derived from the augmented state and covariances, where L is the dimension of the augmented state:

$$\begin{array}{ccc}\hfill {\chi }_{k-1|k-1}^0& =& {\underline{x}}_{k-1|k-1}^a,\hfill \end{array}$$

(16)

$$\begin{array}{ccc}\hfill {\chi }_{k-1|k-1}^i& =& {\underline{x}}_{k-1|k-1}^a+{\left(\sqrt{(L+\lambda ){\mathbf{P}}_{k-1|k-1}^a}\right)}_i,\hfill \\ \hfill \mathrm{for}i& =& 1..L,\hfill \end{array}$$

(17)

$$\begin{array}{ccc}\hfill {\chi }_{k-1|k-1}^i& =& {\underline{x}}_{k-1|k-1}^a-{\left(\sqrt{(L+\lambda ){\mathbf{P}}_{k-1|k-1}^a}\right)}_{i-L},\hfill \\ \hfill \mathrm{for}i& =& L+1,\cdots 2L,\hfill \end{array}$$

(18)

The matrix square root $\left(\sqrt{(L+\lambda ){\mathbf{P}}_k}\right)$ should be calculated using numerically efficient and stable methods such as the Cholesky decomposition, where the $(L+\lambda )$ coefficient was chosen based on [44].

The sigma points ${\chi }_{k|k-1}^i$ are propagated through the state-space transition function:

$${\chi }_{k|k-1}^i={\mathbf{F}}_k{\chi }_{k-1|k-1}^i+{\mathbf{B}}_k{\underline{u}}_k,i=0..2L.$$

(19)

The weighted sigma points ${\chi }_{k|k-1}^i$ are recombined to produce the predicted state ${\widehat{\underline{x}}}_{k|k-1}$ and covariance ${\mathbf{P}}_{k|k-1}$:

$${\widehat{\underline{x}}}_{k|k-1}=\sum _{i=0}^{2L}{W}_s^i{\chi }_{k|k-1}^i,$$

(20)

$${\mathbf{P}}_{k|k-1}=\sum _{i=0}^{2L}{W}_c^i[{\chi }_{k|k-1}^i-{\widehat{\underline{x}}}_{k|k-1}]{[{\chi }_{k|k-1}^i-{\widehat{\underline{x}}}_{k|k-1}]}^T,$$

(21)

where the weights $W_s$ and $W_c$ for the state and covariance [44] are given by:

$$\begin{array}{ccc}& W_s^0=\frac{\lambda }{L+\lambda },W_c^0=\frac{\lambda }{L+\lambda }+(1-{\alpha }^2+\beta ),& \end{array}$$

(22)

$$\begin{array}{ccc}& W_s^i=W_c^i=\frac{1}{2(L+\lambda )},& \hfill \end{array}$$

(23)

with:

$$\lambda ={\alpha }^2\left(L+\kappa \right)$$

(24)

where $\alpha$, $\beta$, $\kappa$ are noise distribution parameters, and $\lambda$ is chosen arbitrarily. Making informed choices during filter tuning, as emphasized in [19], proves beneficial. For most applications where disturbances adhere to Gaussian noise assumptions, typical values for $\alpha$, $\beta$, and $\kappa$ are conventionally set at ${10}^{-3}$, 2, and 0, respectively. Introducing the $\kappa$ factor provides an opportunity to incorporate an additional degree of freedom during tuning, particularly when non-Gaussian disturbance occurrences.

During correction, the sigma points ${\chi }_{k|k-1}^i$ are projected through the observation function ${\mathbf{H}}_k$:

$${{\rm Y}}_k^i={\mathbf{H}}_k{\chi }_{k|k-1}^i,i=0..2L.$$

(25)

Based on weights $W_s^i$ and $W_c^i$ from Equation (23) and the observation matrix ${{\rm Y}}_k^i$, it is possible to obtain the following output signal:

$${\widehat{\underline{z}}}_k=\sum _{i=0}^{2L}{W}_s^i{{\rm Y}}_k^i,$$

(26)

and also the output covariance:

$${\mathbf{P}}_{z_k{z}_k}=\sum _{i=0}^{2L}{W}_c^i[{{\rm Y}}_k^i-{\widehat{\underline{z}}}_k]{[{{\rm Y}}_k^i-{\widehat{\underline{z}}}_k]}^T.$$

(27)

The solution of the classical form of the Kalman filter is adapted in the UKF. Correction ${\mathbf{K}}_k$ depends directly on state covariances ${\mathbf{P}}_{k|k-1}$ and the innovation of system covariances ${\mathbf{S}}_k$, so it is similar to (27):

$${\mathbf{K}}_k={\mathbf{P}}_{x_k{z}_k}{\mathbf{P}}_{z_k{z}_k}^{-1},$$

(28)

where ${\mathbf{P}}_{x_k{z}_k}$ can be described as:

$${\mathbf{P}}_{x_k{z}_k}=\sum _{i=0}^{2L}{W}_c^i[{\chi }_{k|k-1}^i-{\widehat{\underline{x}}}_{k|k-1}]{[{{\rm Y}}_k^i-{\widehat{\underline{z}}}_k]}^T.$$

(29)

As in the classical Kalman filter, the residuum is output as:

$${\tilde{\underline{y}}}_k={\underline{z}}_k-h\left({\widehat{\underline{x}}}_{k|k-1}\right).$$

(30)

Correction of state is achieved via:

$${\widehat{\underline{x}}}_{k|k}={\widehat{\underline{x}}}_{k|k-1}+{\mathbf{K}}_k({\underline{z}}_k-{\widehat{\underline{z}}}_k).$$

(31)

The adjusted covariance matrix ${\mathbf{P}}_{k|k}$ is a prediction of ${\mathbf{P}}_{k|k-1}$, corrected by weighted values:

$${\mathbf{P}}_{k|k}={\mathbf{P}}_{k|k-1}-{\mathbf{K}}_k{\mathbf{P}}_{z_k{z}_k}{\mathbf{K}}_k^T.$$

(32)

The algorithm operates in a recursive manner, utilizing data from the current step as input for the subsequent step.

2.4. Model Predictive Control

Model Predictive Control is an advanced control scheme where the control signal is obtained by solving an open-loop optimal multi-variable control problem across a predefined predictive horizon. A method commonly referred to as Receding Horizon Control is used, as discussed in [27]. The objective of this approach is to optimize and generate an optimal control sequence, considering predefined cost criteria and adhering to specified control and system constraints.

Philosophically, Model Predictive Control (MPC) reflects human decision-making processes by autonomously selecting control actions that are projected to produce the most favorable predicted outcomes or outputs within a finite time. This parallels the cognitive process in which humans anticipate future consequences and adjust their actions accordingly, emphasizing the proactive and forward-thinking nature of both MPC and human decision-making. In practical terms, it imposes constraints on planning, decision-making, and resource utilization, necessitating efficient utilization within the specified timeframe. Understanding the finite nature of the horizon is crucial for managing expectations, setting goals, and implementing strategies effectively. This approach aligns with the human tendency to consider future consequences when making decisions. In normal applications, a singular MPC-based controller assumes the pivotal role of overseeing the entire system, possessing a comprehensive understanding of system behavior and all computed control signals. This centralized control perspective allows MPC to proactively shape the system’s trajectory, adjusting inputs in real time to optimize performance and adhere to defined objectives. The dynamic and anticipatory nature of MPC underscores its similarity with human decision-making processes, emphasizing the importance of foresight and adaptability in achieving optimal outcomes in complex control scenarios.

2.4.1. Principle

One notable advantage of predictive control mentioned above lies in its inherent simplicity and intuitive concepts. The essence of Model Predictive Control (MPC) is rooted in the straightforward idea that the controller must anticipate the future behavior of the system. At the heart of MPC lies the model of the system, as elucidated above. The efficacy of the control law is intricately tied to the accuracy of predictions, making the model a pivotal component in steering the system toward optimal performance. By leveraging this predictive capability, MPC enables a proactive and adaptive control strategy, making it a valuable tool in scenarios where anticipating and responding to future states is crucial for achieving desired control outcomes.

In the literature, MPC is formulated in the state-space approach. Let the model of the plant to be controlled be described by the linear discrete-time state-space equations:

$$\begin{array}{ccc}\hfill {\underline{x}}_{k+1}& =& \mathbf{A}{\underline{x}}_k+\mathbf{B}{\underline{u}}_k\hfill \end{array}$$

(33)

$$\begin{array}{ccc}\hfill {\underline{y}}_k& =& \mathbf{C}{\underline{x}}_k+\mathbf{D}{\underline{u}}_k\hfill \end{array}$$

(34)

where:

$$\underline{x}\in \mathbf{X}\in {\mathbb{R}}^n,\underline{y}\in \mathbf{Y}\in {\mathbb{R}}^p,\underline{u}\in \mathbf{U}\in {\mathbb{R}}^m,$$

(35)

are the vectors of state, output and input, respectively. The cost function $J_N$ that has to be minimized in the receding horizon N generally takes the quadratic form:

$$J_N=\sum _{j=k}^{k+N-1}\left({\underline{x}}_j^{T}Q{\underline{x}}_j+{\underline{u}}_j^{T}R{\underline{u}}_j\right)$$

(36)

where Q and R are particular scaling parameters of sensitivity of the minimization of internal behavior $x_j$ (represented by the Q factor) and external $u_j$ behavior (represented by the R factor).

When we assume an N-step prediction, the system output behavior can be determined in the following N steps:

$$\underset{\mathbf{y}}{\underbrace{\left[\begin{array}{c}{\underline{y}}_0\\ {\underline{y}}_1\\ ⋮\\ {\underline{y}}_{N-1}\end{array}\right]}}=\underset{{\mathbf{O}}_N}{\underbrace{\left[\begin{array}{c}\mathbf{C}\\ \mathbf{CA}\\ ⋮\\ {\mathbf{CA}}^{N-1}\end{array}\right]}}{\underline{x}}_0+\underset{{\mathbf{H}}_N}{\underbrace{\left[\begin{array}{ccccc}{\mathbf{H}}_0& 0& 0& \cdots & 0\\ {\mathbf{H}}_1& {\mathbf{H}}_0& 0& \cdots & 0\\ {\mathbf{H}}_2& {\mathbf{H}}_1& {\mathbf{H}}_0& \cdots & 0\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ {\mathbf{H}}_{N-1}& {\mathbf{H}}_{N-2}& \cdots & \cdots & {\mathbf{H}}_0\end{array}\right]}}\underset{\mathbf{u}}{\underbrace{\left[\begin{array}{c}{\underline{u}}_0\\ {\underline{u}}_1\\ ⋮\\ {\underline{u}}_{N-1}\end{array}\right]}}$$

(37)

written most simply as:

$$\mathbf{y}={\mathbf{O}}_N{\underline{x}}_0+{\mathbf{H}}_N{\underline{u}}_0,$$

(38)

where:

$${\mathbf{O}}_0=\mathbf{C},{\mathbf{O}}_k={\mathbf{CA}}^k,k=1,2,\cdots ,N-1$$

(39)

$${\mathbf{H}}_0=\mathbf{D},{\mathbf{H}}_k={\mathbf{CA}}^{k-1}\mathbf{B},k=1,2,\cdots ,N-1$$

(40)

Based on the above and the definition of optimization cost in (36), these can be compared with the desired state of the system:

$$J_N=\sum _{p=0}^{N-1}{\left({\underline{w}}_p-{\underline{y}}_p\right)}^T\left({\underline{w}}_p-{\underline{y}}_p\right)={\left[{\left(\mathbf{w}-\mathbf{y}\right)}^T\left(\mathbf{w}-\mathbf{y}\right)\right]}_N.$$

(41)

The optimizer is a crucial part of the strategy, as it provides the control. The combination of (36) and (41) can be solved by the quadratic programming (QP) problem as:

$$J_N={\mathbf{u}}^T{\mathbf{H}}_N^T{\mathbf{H}}_N\mathbf{u}+2\left({\underline{x}}_0^T{\mathbf{O}}_N{\mathbf{H}}_N-{\mathbf{w}}^T{\mathbf{H}}_N\right)$$

(42)

to minimize the value of the control signal $\mathbf{u}$ in each future predicted state N. The reference speed signal is introduced into the $\mathbf{w}$ vector.

Since the problem depends on the current state $\underline{x}$ and the importance matrix $\mathbf{Q}$ introduced into (42), it looks like:

$$J_N={\mathbf{u}}^T{\mathbf{H}}_N^T\mathbf{Q}{\mathbf{H}}_N\mathbf{u}+2\left({\underline{x}}_0^T{\mathbf{O}}_N{\mathbf{H}}_N-{\mathbf{w}}^T{\mathbf{H}}_N\right).$$

(43)

The solution of the MPC problem requires the online solution of a QP at each time step.

2.4.2. Constraints

Based on finding the min of cost function (42), which has the form:

$$min{J}_N\to \underset{\mathbf{u}}{min}\frac{1}{2}{\mathbf{u}}^T\mathbf{H}\mathbf{u}+\mathbf{f}\mathbf{u},$$

(44)

it is possible to introduce some constraints which state that:

$$\mathbf{A}\mathbf{u}\le \mathbf{b}.$$

(45)

Considering the maximum possible supply voltage:

$${\underline{u}}_{min}\le {\underline{u}}_k\le {\underline{u}}_{max}$$

(46)

and the suspected predicted state constraints:

$${\underline{x}}_{min}\le {\underline{x}}_k\le {\underline{x}}_{max},$$

(47)

QP yields optimized results according to (44).

This paper focuses specifically on the $dq$ axis current ($i_d$ and $i_q$) limitation and maximum supply voltage ($u_{\alpha }$ and $u_{\beta }$). Other limitations seem not to apply here.

The presented optimization approach can now be identified and simplified to address the QP problem with linear inequality constraints [45]. While the optimization problem exhibits the desirable property of convexity, the presence of constraints complicates the situation. Despite the convex nature of the problem, these constraints hinder the derivation of a straightforward, explicit solution formula. Regarding performance, there are a few methods for solving the QP problem, like interior-point, active set, and sequential qp methods, which possess unique characteristics when applied to MPC. In this work, the decision was made to utilize the active set algorithm with iterative state values [46].

3. Results

To achieve optimal performance and efficiency in control systems, diverse control techniques are employed to regulate parameters such as the speed, torque, and rotor position. This investigation has presented a comprehensive categorization of speed control methods. Among these methods, the vector control technique emerges as a prominent approach for governing the speed of PMSMs. Furthermore, the vector control technique is commonly known as field-oriented control (FOC) [2]. Model Predictive Control is one of the most practical advanced control techniques in industrial applications.

However, the MPC approach requires the selection of the control matrix $\mathbf{Q}$ for the optimization system. Taking into account the entire control vector:

$$\underline{x}=\left[i_d{i}_q{\omega }_r\gamma \right]$$

(48)

based on the proposed model with states defined as (10), $\mathbf{Q}$ can be a specified internal importance factor:

$$\mathbf{Q}=diag\left\{{\tilde{i}}_d{\tilde{i}}_q{\tilde{\omega }}_r\tilde{\gamma }\right\}.$$

(49)

To align with the speed vector control scheme, prioritizing the speed variable ${\omega }_r$ is imperative, making its coefficient ${\tilde{\omega }}_r$ pivotal and substantial. Given that vector control employs cascade control alongside current controllers, ensuring non-zero values for ${\tilde{i}}_d$ and ${\tilde{i}}_q$ is essential. In the scenario presented, where position control is not a consideration, the associated value $\tilde{\gamma }$ should be set to zero.

An essential component of the MPC system, assuming a vector of control state variables $\underline{x}$ like (48), is the definition of an object in state-space form ($\mathbf{A}$, $\mathbf{B}$, $\mathbf{C}$, $\mathbf{D}$) as the relations (33) and (34). Considering the model of an object described in Section 2.2, with state matrices described by (11), (12) and (13), $\mathbf{A}$, $\mathbf{B}$, $\mathbf{C}$, $\mathbf{D}$ take the form:

$$\mathbf{A}=\left[\begin{array}{cccc}1-T_s·\frac{R_s}{L_d}& T_s·{\omega }_r\frac{L_q}{L_d}& 0& 0\\ -T_s·{\omega }_r\frac{L_d}{L_q}& 1-T_s·\frac{R_s}{L_q}& -T_s·\frac{{\mathrm{\Psi }}_m}{L_q}& 0\\ 0& T_1& 0& 0\\ 0& 0& T_s& 1\end{array}\right],$$

(50)

where:

$$T_1=T_s·\frac{3}{2}\frac{p}{J}\left[{\mathrm{\Psi }}_f-\left(L_q-L_d\right)i_d\right],$$

$$\mathbf{B}=\left[\begin{array}{cc}{T}_s·\frac{1}{L_d}cos\gamma & T_s·\frac{1}{L_d}sin\gamma \\ -T_s·\frac{1}{L_q}sin\gamma & T_s·\frac{1}{L_q}cos\gamma \\ 0& 0\\ 0& 0\end{array}\right].$$

(51)

$$\mathbf{C}=\left[\begin{array}{cccc}cos\gamma & -sin\gamma & 0& 0\\ sin\gamma & cos\gamma & 0& 0\end{array}\right],$$

(52)

$$\mathbf{D}=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right].$$

(53)

Regarding performance, there are a few methods for solving the QP problem, like interior-point, active set, sequential qp methods, which possess unique characteristics when applied to MPC. In this work, the decision was made to utilize the active set algorithm with iterative state values.

3.1. Preliminary Simulation Research

Evaluating control algorithms through simulations has emerged as a highly effective strategy for assessing the adequacy of motor controller designs. This approach not only provides a comprehensive understanding of the algorithm’s performance but also significantly reduces the time and costs associated with the development process before transitioning to laboratory hardware testing. Simulations offer a controlled environment where various scenarios and operating conditions can be systematically examined. During this stage, a simulation model of a typical direct-drive environment was constructed. The core of simulations were performed using the MATLAB environment with the following toolboxes: Control System, Optimalization with set of QP problem solvers. As part of the verification process, a comparable scheme was employed, mirroring the setup used in the tests of the sensorless system with a PID controller as documented in [4,38]. The first element of the research was tuning at the appropriate observer level and selecting the parameters so that the estimation result is sufficiently close to that of another measurement method, assuming possible deviations.

During the utilization of the model, several simplifications were implemented. For instance, the mechanical component, including the intricate friction mechanism, was streamlined into a single-mass system with minimal friction. Additionally, the power electronics converter system, featuring a PWM modulator, was substituted with a unit delay and a conversion factor derived from the supply voltage. Similarly, the concurrent delay within the control system was replaced with a unit delay. A crucial aspect involves tuning the Kalman filter by selecting initial values for (15), such as ${\mathbf{P}}_{0|0}$, ${\mathbf{Q}}_0$, and ${\mathbf{R}}_0$. In this instance, the initial matrices were determined based on references [4,47], with their corresponding system parameters detailed in

Table A1. Laboratory setup parameters.

Parameter	Symbol	Value
Stator phase resistance	$R_s$	$3.55$$\mathrm{\Omega }$
Stator inductance (d axis)	$L_d$	$17.16\mu$H
Stator inductance (q axis)	$L_q$	$17.16\mu$H
Permanent magnet flux linkage	${\mathrm{\Psi }}_f$	$2.45$ Wb
Number of pole pairs	p	12
Moment of inertia	J	$39.5$ mkgm2
Inverter supply voltage	$U_{DC}$	48
Maximum inverter phase current	$I_{max}$	$25$ A
Sampling time	$T_s$	$100\mu$s

(Appendix A), as:

$${\mathbf{Q}}_0=diag\left\{0.45·{10}^{-3},0.45·{10}^{-3},1.5·{10}^{-8},2.1·{10}^{-11},0.1\right\}$$

(54)

and

$${\mathbf{R}}_0=diag\left\{0.45·{10}^{-3},0.45·{10}^{-3}\right\}$$

(55)

which correlate with system and output noises, with ${\mathbf{P}}_{0|0}$ defined as a zero starting point:

$${\mathbf{P}}_{0|0}=\mathbf{0},$$

(56)

which is assessed throughout the UKF algorithm computation.

The set-speed ${\omega }_r^{*}$ test was used for simulation tests, identical to subsequent laboratory tests as in Figure 2. The speed command has crucial elements, including the initialization of zero initial values, a sudden spike in the speed command, and an abrupt change in the direction of speed.

During a series of simulations, it is worth comparing the entire group qualitatively and quantitatively. It was decided to use qualitative indicators regarding the stability of the closed control loop (Stable) and the ability to reach a constant desired signal ${\omega }_R^{*}$ in a finite time $t=0.5$ s without static error (${\omega }_r^{*}$ reachable). Measurable quality indexes like the Integral Absolute Error and Integral Time Absolute Error were defined as:

$$\begin{array}{ccc}& IA{E}_{full}={\int }_{t_0}^{t_{end}}|{\omega }_r-{\omega }_r^{*}|dt& \end{array}$$

(57)

$$\begin{array}{cc}& ITA{E}_{start}={\int }_{t_0}^{t_{start}}t·|{\omega }_r-{\omega }_r^{*}|dt\end{array}$$

(58)

where the experiment starting time $t_0$ is $0$ s, $t_{end}$ is $0.3$ s, and $t_{start}$ was chosen as the longest acceptable settling time of desired speed during experiment, at $0.1$ s.

Typical simulation results for $\mathbf{Q}=\left[11300\right]$ with the best quality are presented in Figure 2 and Figure 3, with a focus on showing two issues: the trends in mechanical and electrical quantities, respectively.

Table 1. Influence of the tuning matrix $\mathbf{Q}$ on the control quality.

Horizon	Q	Stable	${\mathit{\omega }}_{\mathit{r}}^{*}$ Reachable	IAEfull	ITAEstart
7	$\left[1110\right]$	Y	N	0.760	0.815
7	$\left[1120\right]$	Y	N	0.727	0.797
7	$\left[1150\right]$	Y	N	0.438	0.580
7	$\left[11100\right]$	Y	N	0.262	0.338
7	$\left[11200\right]$	Y	Y	0.177	0.257
7	$\left[\mathbf{1}\mathbf{1}\mathbf{30}\mathbf{0}\right]$	Y	Y	0.147	0.226
7	$\left[11500\right]$	limit of stab.	Y	0.123	0.204
7	$\left[11550\right]$	N	N	–	–
3	$\left[11300\right]$	N	N	–	–
5	$\left[11300\right]$	Y	Y	0.172	0.244
7	$\left[11300\right]$	Y	Y	0.147	0.226
9	$\left[11300\right]$	Y	Y	0.136	0.217

presents the mentioned indexes in relation to the tuning process by changing the $\mathbf{Q}$ vector.

3.2. Laboratory Verification

After the numerous simulation studies presented above, an experimental verification of the proposed control method, following previous methods, was conducted. Implementation in real astronomy telescope mount systems was investigated [48]. The base axis of the laboratory stand, presented in Figure 4, consists of a surface-mounted magnet synchronous motor, supplied by a three-phase MOSFET power inverter controlled by a Texas Instruments Sitara AM335x and an Altera FPGA Cyclone IV board. The proposed control algorithm is divided into a few parts: hardware maintenance, estimation (UKF), and control (MPC). All matrix linear algebra operations were implemented using dlib++ libraries like dlib/matrix, dlib/vector and dlib/rectangle. The entire MPC algorithm was implemented based on dlib/control and dlib/optimization.

The MPC algorithm that was developed was tested in extensive simulations under various conditions both prior to and during its implementation. The real results are only partly presented below to demonstrate its operation. The advantages of the sensorless UKF with a coordinated PI current and speed controllers were considered during the tuning of the observer. From the collection of sets, one conditional example was chosen for the number of predicted states $N=5$. Equation (42) was extended into:

$$J_N={\mathbf{u}}^T{\mathbf{H}}_N^T\mathbf{Q}{\mathbf{H}}_N\mathbf{u}+2\left({\underline{x}}_0^T{\mathbf{O}}_N{\mathbf{H}}_N-{\mathbf{w}}^T{\mathbf{H}}_N\right)$$

(59)

which is:

$$\mathbf{Q}=diag\left\{11300\right\},$$

(60)

with a maximum related weight equal to 30 for a speed ${\omega }_r$ control surrounded by ones based on numerous simulation studies. Originally, the prediction horizon was set to $N=7$, but after the first implementation, it turned out that the processor’s computational efficiency was too low and this number was reduced to $N=5$. The impact of this change was tested in simulation investigations and the values of quality factors were added to Table 1 above. An important point of this investigation under laboratory setup constraints setting the maximum value of the control signal was $\left|U\right|\le 48$ V and the maximum value of currents (state vector $\underline{x}$ elements) was $|i_d|\le 8$ A, $|i_q|\le 8$ A in a circular shape.

The state observer based on the UKF was tuned similar to a classical coordinate system [22], with any special tuning. This proved to be an important indication of the possibility of using the estimated state $\widehat{\underline{x}}$ as a control state $\underline{x}$.

The first part of this investigation was focused on controlling the system behavior via reference speed ${\omega }_r^{*}$ changes and the possibilities of obtaining the initial position $\gamma$ after the algorithm was started.

Figure 5 shows the system’s response to the rapid change in speed ${\omega }_r^{*}$. Additional interesting signals (including actual control voltages ${\underline{u}}^{*}$) are presented in Figure 6.

An initial point ${\widehat{x}}_0$ can be found by applying any voltage to the drive based on the first predictive step. The unscented Kalman filter in that form can determine the proper actual position of $\gamma$ after a few steps. This process does not interfere with the MPC algorithm.

Additional investigations concern the external load torque disturbances and the kick type additional force. The properties are presented in Figure 7. A fast dynamic reaction can be observed but a steady-state error remains. The actual control voltages ${\underline{u}}^{*}$ and motor currents $i_d$, $i_q$ with modulus I are presented in Figure 8.

The proposed control algorithms are able to respond quickly to significant disturbances. A fixed error in the controlled speed ${\omega }_r$ appeared. In some cases, where speed is at a standstill, some systematic fixed errors are encountered, and the source is static disturbances like observed rough friction. The proposed solution and presented results can be compared with other works, including [4,10,13,17]. The quality of the obtained estimates and the quality of control are similar, with the same level of estimation errors and control times. The resulting performance, particularly in terms of noise from system state estimation methods, is comparable.

The model considered during the control design phase failed to account for the permanent disturbance, which could potentially explain the outcome. Based on the observed values (e.g., ${\widehat{T}}_l$) from the UKF, it is possible to cancel all these errors, which could be a subject for further research.

4. Conclusions

This paper stands out for its significant contribution in introducing an original sensorless control scheme for permanent magnet drives, emphasizing the utilization of Model Predictive Control (MPC) to attain an exceptionally dynamic performance. Notably, the controller’s robust performance is showcased across a spectrum of operating points, demonstrating its effectiveness even under challenging conditions such as zero speed.

A noteworthy result is the application of the $dq$ model for control signal prediction, which engenders a control strategy reminiscent of vector control, as discussed by Vas [2]. This strategic alignment adds a layer of versatility to the proposed control scheme, broadening its applicability in diverse operational scenarios.

The demonstrated advantages of the MPC algorithm in high-dynamic systems underscore its suitability for situations characterized by rapid speed changes. This paper not only presents a successful design of an MPC-based control system incorporating an unscented Kalman filter but also highlights ongoing efforts aimed at further reducing the control errors and refining dynamical behavior. Moreover, there are plans to implement the proposed algorithm in an industrial drive system with constrained computing hardware, emphasizing the practicality and scalability of the proposed approach.

While the discussed topics have garnered considerable attention, there remains ample space for additional research and development in this field. Exploring the integration of predictive control with other established control methods not only holds promise for advancing the current understanding but also opens up avenues for innovative applications and improvements in control strategies for diverse systems. This paper thus serves as a springboard for continued exploration and investigations into the evolving landscape of advanced control methodologies in the realm of sensorless permanent magnet drives.