energies

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

Institute of Robotics and Machine Intelligence, Poznan University of Technology, 60965 Poznan, Poland;
dariusz.janiszewski@put.poznan.pl

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.

Keywords: motion control; variable-speed drives; automatic control; predictive control; sensorless
control; observers; Kalman filters; unscented Kalman filter; system modeling

Citation: Janiszewski, D. Sensorless

Model Predictive Control of
Permanent Magnet Synchronous
1. Introduction
Motors Using an Unscented Kalman
Filter. Energies 2024, 17, 2387. Permanent Magnet Synchronous Motor (PMSM) drives find extensive use in industrial
https://doi.org/10.3390/en17102387 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
Academic Editors: Emmanuel
applications where precision, responsiveness, and efficiency are paramount [1–4]. Discus-
Delaleau, Jean-Matthieu Bourgeot
sions often revolve around enhancing the accuracy, stability, and robustness of their control
and Ahmed Abu-Siada
methodologies. A widely used industrial control method, such as field-oriented control of
Received: 11 February 2024 PMSMs, heavily relies on an accurate rotor position [2]. However, the requirement for
Revised: 21 April 2024 high-precision sensors comes at a cost, often requiring enclosed housing, which limits
Accepted: 1 May 2024 their applicability in specialized environments. In demanding conditions involving severe
Published: 16 May 2024 pressure, humidity, temperature, and vibration, traditional mechanical encoders prove in-
adequate [5,6]. Systems called sensorless systems present a potential solution by eliminating
the necessity for mechanical sensors. This becomes particularly crucial during safety opera-
tions following sensor failure, especially in inaccessible environments [5]. Consequently,
Copyright: © 2024 by the author.
significant attention has been directed towards developing sensorless drives, utilizing the
Licensee MDPI, Basel, Switzerland.
This article is an open access article
inherent behavior of PMSMs as a kind of virtual position sensor. In general, the advantages
distributed under the terms and
of these drive systems encompass potentially reduced drive costs, fewer integrated parts, a
conditions of the Creative Commons
smaller size, a heightened reliability, and easier maintenance. However, their drawbacks
Attribution (CC BY) license (https:// primarily include a low load capacity, a limited speed accuracy, and instability in generating
creativecommons.org/licenses/by/ low-speed ranges [4,7].
4.0/).

Energies 2024, 17, 2387. https://doi.org/10.3390/en17102387 https://www.mdpi.com/journal/energies

Energies 2024, 17, 2387 2 of 19

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 empha-
sis 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 [4,15–17]. However, due to the inherent unreli-
ability 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 miti-
gates 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 statisti-
cal 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 [25–27]. 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 [28–30] 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.
Energies 2024, 17, 2387 3 of 19

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 , ib , ic (measured by sensors) translated into iα , i β and
desired voltages uα , u β , are defined as desired by the MPC scheme. For the values of
natural output (currents, speed and position), only the desired speed ωr∗ was chosen. The
presented algorithm requires knowledge of the state variables x̂.

Figure 1. Control system scheme.

The effectiveness of the proposed drive system has been verified in the scheme in
Figure 1.
Energies 2024, 17, 2387 4 of 19

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:
di
ud = Rs id + Ld d − pωr Lq iq , (1)
dt
diq
uq = Rs iq + Lq + pωr Ld id + pωr Ψm . (2)
dt
where ud , uq are dq axis voltages; id , iq are dq axis currents; Ld , Lq are dq axis inductances;
Rs is stator resistance; and Ψm 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:

3 
· p Ψm − Lq − Ld id · iq ,


Te = (3)
2
3
where p is the number of pole pairs, and the fraction 2 stems from frame conversion, i.e.,
perpendicular stator αβ over rotor dq.
The drive dynamics can be described as:

dωr
Te − Tload = J , (4)
dt
where Tload 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:
 
dωr p 3 Te
Ψm − Lq − Ld id · iq −



= . (5)
dt J 2 J

The otor position γ can be described by the derivative equation of the rotational speed:

dγ
= p · ωr . (6)
dt
Drawing inspiration from the concept of the dual Kalman filter described in refer-
ence [41], one or more parameters can be treated as estimation values. The load torque
Tl 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 Tl is invariable in a narrow interval holds:

d
T ≈ 0. (7)
dt l
Energies 2024, 17, 2387 5 of 19

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 Ts . 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:

x̂ k = Fk ( x̂ k−1 ) x̂ k−1 + Bk ( x̂ k−1 )uk , (8)

zk = Hk ( x̂ k ) x̂ k , (9)
For the synthesis of the unscented Kalman filter, state vector x̂ was chosen:
 T
x̂ = îd îq ω̂r γ̂ T̂l , (10)

with all natural state variables and estimated load torque T̂l . Considering the above, the
system matrix Fk has the form:
L
 
1 − Ts · RL s Ts · ωr Lq 0 0 0
d d
 − Ts · ωr Ld 1 − Ts · Rs − Ts · Ψm 0
 
 Lq Lq Lq 0 

Fk ( x̂ k ) = 
 0 T1 0 1 ,
0 − Ts · J  (11)
 
 0 0 Ts 1 0 
0 0 0 0 1

where
3 ph
i
T1 = Ts · Ψ f − Lq − Ld id .
2J
The output matrix Hk is a rotating matrix via the Clark/Parck transformation:
 
cos γ − sin γ 0 0 0
Hk ( x̂ k ) = , (12)
sin γ cos γ 0 0 0

and matrix Bk :
Ts · L1 cos γ 1
 
Ts · Ld sin γ
 − T · d1 sin γ Ts · 1
cos γ 
 s Lq Lq 
Bk ( x̂ k ) =  . (13)
 
 0 0 
 0 0 
0 0
The state matrices are simplified. The load torque Tl 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 Hk described as (12), Bk as (13), and Fk in the
form of (11), all of which are non-linear. Specifically, Hk and Bk incorporate trigonometric
functions, reflecting a dependency on state variables associated with two-axis rotation (αβ,
dq frames), while Fk 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 Px and
outputs Py , which characterizes the accuracy of the modeling process. This disruption
further complicates the characterization of variables in a non-linear space.
Energies 2024, 17, 2387 6 of 19

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 combi-
nation 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 x ka−1|k−1 can be defined as:

x ka−1|k−1 = [ x̂ kT−1|k−1 E⟨wkT ⟩ E⟨vkT ⟩ ] T (14)

which consists of state vector x̂k and expected noise terms wk and vk , with the definition of
each covariance given as:
 
P k −1| k −1 0 0
Pka−1|k−1 =  0 Qk 0 . (15)
0 0 Rk

where Qk is defined as the covariance of process noise and Rk is the covariance of observa-
tion (measurement) noise.
An important set of 2L + 1 sigma points, χk−1|k−1 , is derived from the augmented
state and covariances, where L is the dimension of the augmented state:

χ0k−1|k−1 = x ka−1|k−1 , (16)

q 
χik−1|k−1 = x ka−1|k−1 + ( L + λ)Pka−1|k−1 , (17)
i
for i = 1..L ,
q 
χik−1|k−1 = x ka−1|k−1 − ( L + λ)Pka−1|k−1 , (18)
i− L
for i = L + 1, . . . 2L ,
p 
The matrix square root ( L + λ)Pk should be calculated using numerically effi-
cient and stable methods such as the Cholesky decomposition, where the ( L + λ) coefficient
was chosen based on [44].
The sigma points χik|k−1 are propagated through the state-space transition function:

χik|k−1 = Fk χik−1|k−1 + Bk uk , i = 0..2L . (19)

Energies 2024, 17, 2387 7 of 19

The weighted sigma points χik|k−1 are recombined to produce the predicted state x̂ k|k−1
and covariance Pk|k−1 :
2L
x̂ k|k−1 = ∑ Wsi χik|k−1 , (20)
i =0
2L
P k | k −1 = ∑ Wci [χik|k−1 − x̂k|k−1 ][χik|k−1 − x̂k|k−1 ]T , (21)
i =0

where the weights Ws and Wc for the state and covariance [44] are given by:

Ws0 = λ
L+λ , Wc0 = λ
L+λ + (1 − α2 + β ) , (22)
1
Wsi = Wci = 2( L + λ )
, (23)

with:
λ = α2 ( L + κ ) (24)
where α, β, κ are noise distribution parameters, and λ 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 α, β, and κ
are conventionally set at 10−3 , 2, and 0, respectively. Introducing the κ factor provides an
opportunity to incorporate an additional degree of freedom during tuning, particularly
when non-Gaussian disturbance occurrences.
During correction, the sigma points χik|k−1 are projected through the observation func-
tion Hk :
Υik = Hk χik|k−1 , i = 0..2L . (25)

Based on weights Wsi and Wci from Equation (23) and the observation matrix Υik , it is
possible to obtain the following output signal:

2L
ẑk = ∑ Wsi Υik , (26)
i =0

and also the output covariance:

2L
Pzk zk = ∑ Wci [Υik − ẑk ][Υik − ẑk ]T . (27)
i =0

The solution of the classical form of the Kalman filter is adapted in the UKF. Correction
Kk depends directly on state covariances Pk|k−1 and the innovation of system covariances
Sk , so it is similar to (27):
Kk = P xk zk P− 1
zk zk , (28)
where P xk zk can be described as:

2L
P xk zk = ∑ Wci [χik|k−1 − x̂k|k−1 ][Υik − ẑk ]T . (29)
i =0

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

ỹ = zk − h( x̂ k|k−1 ) . (30)
k

Correction of state is achieved via:

x̂ k|k = x̂ k|k−1 + Kk (zk − ẑk ) . (31)

Energies 2024, 17, 2387 8 of 19

The adjusted covariance matrix Pk|k is a prediction of Pk|k−1 , corrected by weighted

values:
Pk|k = Pk|k−1 − Kk Pzk zk KkT . (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 accord-
ingly, 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 ten-
dency 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 hu-
man 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:

x k +1 = Ax k + Buk (33)
y = Cx k + Duk (34)
k

where:
x ∈ X ∈ Rn , y ∈ Y ∈ R p , u ∈ U ∈ Rm , (35)
Energies 2024, 17, 2387 9 of 19

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

k + N −1  
JN = ∑ x Tj Qx j + u Tj Ru j (36)
j=k

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

0 0 ··· 0
 

y
   H0  
0 C u0
 H1 H0 0 ··· 0 
 y   CA    u1 
1 H2 H1 H0 ··· 0
  
 .. = ..  x0 + 
  
 ..

 (37)

.
 
.   .. .. .. .. ..  . 
 
. . . . .
CA N −1
 
y u N −1
| N −1
{z } | {z } H N −1 H N −2 · · · · · · H0 | {z }
y ON
| {z } u
HN

written most simply as:

y = O N x 0 + H N u0 , (38)
where:
O0 = C, Ok = CAk , k = 1, 2, · · · , N − 1 (39)
H0 = D, Hk = CAk−1 B, k = 1, 2, · · · , 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:

N −1  T   h i
JN = ∑ wp − y
p
wp − y
p
= (w − y)T (w − y) .
N
(41)
p =0

The optimizer is a crucial part of the strategy, as it provides the control. The combina-
tion of (36) and (41) can be solved by the quadratic programming (QP) problem as:
 
JN = u T H TN H N u + 2 x0T O N H N − w T H N (42)

to minimize the value of the control signal u in each future predicted state N. The reference
speed signal is introduced into the w vector.
Since the problem depends on the current state x and the importance matrix Q intro-
duced into (42), it looks like:
 
JN = u T H TN QH N u + 2 x0T O N H N − w T H N . (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:

1
min JN → min u T Hu + fu , (44)
u 2

it is possible to introduce some constraints which state that:

Au ≤ b . (45)
Energies 2024, 17, 2387 10 of 19

Considering the maximum possible supply voltage:

umin ≤ uk ≤ umax (46)

and the suspected predicted state constraints:

x min ≤ x k ≤ x max , (47)

QP yields optimized results according to (44).

This paper focuses specifically on the dq axis current (id and iq ) limitation and maxi-
mum supply voltage (uα and u β ). 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 Q for the
optimization system. Taking into account the entire control vector:
 
x = i d i q ωr γ (48)

based on the proposed model with states defined as (10), Q can be a specified internal
importance factor: 
Q = diag ĩd ĩq ω̃r γ̃ . (49)
To align with the speed vector control scheme, prioritizing the speed variable ωr is
imperative, making its coefficient ω̃r pivotal and substantial. Given that vector control
employs cascade control alongside current controllers, ensuring non-zero values for ĩd and
ĩq is essential. In the scenario presented, where position control is not a consideration, the
associated value γ̃ should be set to zero.
An essential component of the MPC system, assuming a vector of control state vari-
ables x like (48), is the definition of an object in state-space form (A, B, C, 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), A, B, C, D take the form:
 Rs L 
1 − Ts · Ld Ts · ωr Lq 0 0
d
− Ts · ωr LLdq 1 − Ts · Rs
− Ts · ΨLmq 0 ,
 
A= Lq (50)


 0 T1 0 0 
0 0 Ts 1
where:
3 ph
i
T1 = Ts · Ψ f − Lq − Ld id ,
2J
Energies 2024, 17, 2387 11 of 19

Ts · L1 cos γ Ts · L1 sin γ
 
d d
 − Ts · L1q sin γ Ts · L1q cos γ 
 
B= . (51)
 0 0 
0 0
 
cos γ − sin γ 0 0
C= , (52)
sin γ cos γ 0 0
 
0 0
D= . (53)
0 0
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 pro-
vides 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 P0|0 , Q0 , and R0 . In
this instance, the initial matrices were determined based on references [4,47], with their
corresponding system parameters detailed in Table A1 (Appendix A), as:
n o
Q0 = diag 0.45 · 10−3 , 0.45 · 10−3 , 1.5 · 10−8 , 2.1 · 10−11 , 0.1 (54)

and n o
R0 = diag 0.45 · 10−3 , 0.45 · 10−3 (55)

which correlate with system and output noises, with P0|0 defined as a zero starting point:

P 0|0 = 0 , (56)

which is assessed throughout the UKF algorithm computation.

The set-speed ωr∗ 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.
Energies 2024, 17, 2387 12 of 19

10

-5

-10
0 0.05 0.1 0.15 0.2 0.25 0.3

-2

-4
0 0.05 0.1 0.15 0.2 0.25 0.3

0.5

-0.5

-1
0 0.05 0.1 0.15 0.2 0.25 0.3

Figure 2. Speed ωr , position γ and load torque Tload during desired speed change—simulation
investigations.

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 ω R∗ in a
finite time t = 0.5 s without static error (ωr∗ reachable). Measurable quality indexes like the
Integral Absolute Error and Integral Time Absolute Error were defined as:
Rt
I AE f ull = t end |ωr − ωr∗ |dt (57)
0
Rt
ITAEstart = t start t · |ωr − ωr∗ |dt (58)
0

where the experiment starting time t0 is 0 s, tend is 0.3 s, and tstart was chosen as the longest
acceptable settling time of desired speed during experiment, at 0.1 s.
Typical simulation results for Q = [1 1 30 0] with the best quality are presented in
Figures 2 and 3, with a focus on showing two issues: the trends in mechanical and electrical
quantities, respectively.

50

-50
0 0.05 0.1 0.15 0.2 0.25 0.3

10

-5

-10
0 0.05 0.1 0.15 0.2 0.25 0.3

10

0
0 0.05 0.1 0.15 0.2 0.25 0.3

Figure 3. αβ axis control voltages, dq axis current value, current modulus during desired speed
change —simulation investigations.
Energies 2024, 17, 2387 13 of 19

Table 1 presents the mentioned indexes in relation to the tuning process by changing
the Q vector.

Table 1. Influence of the tuning matrix Q on the control quality.

Horizon Q Stable ωr∗ Reachable IAEfull ITAEstart

7 [1 1 1 0] Y N 0.760 0.815
7 [1 1 2 0] Y N 0.727 0.797
7 [1 1 5 0] Y N 0.438 0.580
7 [1 1 10 0] Y N 0.262 0.338
7 [1 1 20 0] Y Y 0.177 0.257
7 [1 1 30 0] Y Y 0.147 0.226
7 [1 1 50 0] limit of stab. Y 0.123 0.204
7 [1 1 55 0] N N – –
3 [1 1 30 0] N N – –
5 [1 1 30 0] Y Y 0.172 0.244
7 [1 1 30 0] Y Y 0.147 0.226
9 [1 1 30 0] Y Y 0.136 0.217

3.2. Laboratory Verification

After the numerous simulation studies presented above, an experimental verification
of the proposed control method, following previous methods, was conducted. Implemen-
tation 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 syn-
chronous 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.

(a) (b)
Figure 4. The prototype astronomical mount with an 11” telescope: (a) laboratory setup, (b) one-axis
experimental setup with a variable moment of inertia.

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
Energies 2024, 17, 2387 14 of 19

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:
 
JN = u T H TN QH N u + 2 x0T O N H N − w T H N (59)

which is:
Q = diag{1 1 30 0}, (60)
with a maximum related weight equal to 30 for a speed ωr 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 |U | ≤ 48 V and the maximum value of currents
(state vector x elements) was |id | ≤ 8 A, |iq | ≤ 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 x̂ as a control state x.
The first part of this investigation was focused on controlling the system behavior via
reference speed ωr∗ changes and the possibilities of obtaining the initial position γ after the
algorithm was started.
Figure 5 shows the system’s response to the rapid change in speed ωr∗ . Additional
interesting signals (including actual control voltages u∗ ) are presented in Figure 6.

10

-5

-10
0 0.05 0.1 0.15 0.2 0.25 0.3

-2

-4
0 0.05 0.1 0.15 0.2 0.25 0.3

0.5

-0.5

-1
0 0.05 0.1 0.15 0.2 0.25 0.3

Figure 5. Speed ωr , position γ and load torque Tload during the desired speed change.

An initial point x̂0 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 γ 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 u∗ and motor
currents id , iq with modulus I are presented in Figure 8.
Energies 2024, 17, 2387 15 of 19

50

-50
0 0.05 0.1 0.15 0.2 0.25 0.3

10

-5

-10
0 0.05 0.1 0.15 0.2 0.25 0.3

10

0
0 0.05 0.1 0.15 0.2 0.25 0.3

Figure 6. αβ axis control voltages, dq axis current value, current modulus during the desired
speed change.

10

0
0.05 0.1 0.15 0.2 0.25

-2

-4
0.05 0.1 0.15 0.2 0.25

0
0.05 0.1 0.15 0.2 0.25

Figure 7. Speed ωr , position γ and load torque Tl during load acting.

20

10

-10

-20
0.05 0.1 0.15 0.2 0.25

1.5

0.5

0
0.05 0.1 0.15 0.2 0.25

1.5

0.5

0
0.05 0.1 0.15 0.2 0.25

Figure 8. αβ axis control voltages, dq axis current value, and current modulus during torque acting.
Energies 2024, 17, 2387 16 of 19

The proposed control algorithms are able to respond quickly to significant disturbances.
A fixed error in the controlled speed ωr 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 per-
manent disturbance, which could potentially explain the outcome. Based on the observed
values (e.g., T̂l ) 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 sensor-
less 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, demon-
strating 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, broad-
ening its applicability in diverse operational scenarios.
The demonstrated advantages of the MPC algorithm in high-dynamic systems un-
derscore 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.

Funding: This research received no external funding.

Data Availability Statement: Dataset available on request from the author.
Conflicts of Interest: The author declare no conflict of interest.

Abbreviations
The following abbreviations are used in this manuscript:

EKF Extended Kalman Filter

EMF Electromagnetic Force
IAE Integral Absolute Error
ITAE Integral Time Absolute Error
MPC Model Predictive Control
PMSM Permament Magnet Synchronous Motor
QP Quadratic Programming
UKF Unscented Kalman Filter
Energies 2024, 17, 2387 17 of 19

Appendix A
The laboratory setup, motor specifications, and crucial power electronics parameters
are detailed in Table A1 below.

Table A1. Laboratory setup parameters.

Parameter Symbol Value

Stator phase resistance Rs 3.55 Ω
Stator inductance (d axis) Ld 17.16 µH
Stator inductance (q axis) Lq 17.16 µH
Permanent magnet flux
Ψf 2.45 Wb
linkage
Number of pole pairs p 12
Moment of inertia J 39.5 mkgm2
Inverter supply voltage UDC 48
Maximum inverter phase
Imax 25 A
current
Sampling time Ts 100 µs

References
1. Pellegrino, G.; Vagati, A.; Guglielmi, P.; Boazzo, B. Performance Comparison Between Surface-Mounted and Interior PM Motor
Drives for Electric Vehicle Application. IEEE Trans. Ind. Electron. 2012, 59, 803–811. [CrossRef]
2. Vas, P. Vector Control of AC Machines; Monographs in Electrical and Electronic Engineering; Oxford University Press: Oxford,
UK, 1990.
3. Wang, G.; Valla, M.; Solsona, J. Position Sensorless Permanent Magnet Synchronous Machine Drives—A Review. IEEE Trans. Ind.
Electron. 2020, 67, 5830–5842. [CrossRef]
4. Urbanski, K.; Janiszewski, D. Sensorless Control of the Permanent Magnet Synchronous Motor. Sensors 2019, 19, 3546. [CrossRef]
[PubMed]
5. Batzel, T.; Lee, K. Electric propulsion with the sensorless permanent magnet synchronous motor: Model and approach. IEEE
Trans. Energy Convers. 2005, 20, 818–825. [CrossRef]
6. Briz, F.; Degner, M.W. Rotor Position Estimation. IEEE Ind. Electron. Mag. 2011, 5, 24–36. [CrossRef]
7. Vas, P. Sensorless Vector and Direct Torque Control; Monographs in Electrical and Electronic Engineering; Oxford University Press:
Oxford, UK, 1998; Number 42.
8. Schroedl, M. An improved position estimator for sensorless controlled permanent magnet synchronous motors. In Proceedings
of the 4th European Conference on Power Electronics and Applications–EPE ’91, Firenze, Italy, 3–6 September 1991 ; Volume 3,
pp. 418–423.
9. Wang, S.; Yang, K.; Chen, K. An Improved Position-Sensorless Control Method at Low Speed for PMSM Based on High-Frequency
Signal Injection into a Rotating Reference Frame. IEEE Access 2019, 7, 86510–86521. [CrossRef]
10. Urbanski, K. Estimation of Back EMF for PMSM at Low Speed Range. MM Mod. Mach. Sci. J. 2015, 564–569. [CrossRef]
11. He, C.; Xu, S.; Yan, B.; Wang, Z.; Wang, M. A Fixed-Point Position Observation Algorithm and System-on-Chip Design Suitable
for Sensorless Control of High-Speed Permanent Magnet Synchronous Motor. Electronics 2023, 12, 3160. [CrossRef]
12. Woldegiorgis, A.T.; Ge, X.; Li, S.; Hassan, M. Extended Sliding Mode Disturbance Observer-Based Sensorless Control of IPMSM
for Medium and High-Speed Range Considering Railway Application. IEEE Access 2019, 7, 175302–175312. [CrossRef]
13. Ma, Z.; Zhang, X. FPGA Implementation of Sensorless Sliding Mode Observer with a Novel Rotation Direction Detection for
PMSM Drives. IEEE Access 2018, 6, 55528–55536. [CrossRef]
14. Kalman, R.E. A New Approach to Linear Filtering and Prediction Problems. Trans. ASME–J. Basic Eng. 1960, 82, 35–45. [CrossRef]
15. Taheri, A.; Ren, H.; Holakooie, M.H. Sensorless Loss Model Control of the Six-Phase Induction Motor in All Speed Range by
Extended Kalman Filter. IEEE Access 2020, 8, 118741–118750. [CrossRef]
16. Wang, Y.; Yu, H.; Che, Z.; Wang, Y.; Zeng, C. Extended State Observer-Based Predictive Speed Control for Permanent Magnet
Linear Synchronous Motor. Processes 2019, 7, 618. [CrossRef]
17. Dilys, J.; Stankevič, V.; Łuksza, K. Implementation of Extended Kalman Filter with Optimized Execution Time for Sensorless
Control of a PMSM Using ARM Cortex-M3 Microcontroller. Energies 2021, 14, 3491. [CrossRef]
18. Julier, S.J. The scaled unscented transformation. In Proceedings of the 2002 American Control Conference (IEEE Cat. No.CH37301),
Anchorage, AK, USA, 8–10 May 2002; Volume 6, pp. 4555–4559. [CrossRef]
19. Julier, S.J.; Uhlmann, J.K. Unscented filtering and nonlinear estimation. Proc. IEEE 2004, 92, 401–422. [CrossRef]
20. Yin, L.; Deng, Z.; Huo, B.; Xia, Y.; Li, C. Robust Derivative Unscented Kalman Filter Under Non-Gaussian Noise. IEEE Access
2018, 6, 33129–33136. [CrossRef]
21. Rosafalco, L.; Eftekhar Azam, S.; Manzoni, A.; Corigliano, A.; Mariani, S. Unscented Kalman Filter Empowered by Bayesian
Model Evidence for System Identification in Structural Dynamics. Comput. Sci. Math. Forum 2022, 2, 3. [CrossRef]
Energies 2024, 17, 2387 18 of 19

22. Janiszewski, D. Load torque estimation in sensorless PMSM drive using Unscented Kalman Filter. In Proceedings of the 2011
IEEE International Symposium on Industrial Electronics, Gdansk, Poland, 27–30 June 2011; pp. 643–648. [CrossRef]
23. Gustafsson, F.; Hendeby, G. Some Relations Between Extended and Unscented Kalman Filters. Signal Process. IEEE Trans. 2012,
60, 545–555. [CrossRef]
24. Hendeby, G.; Karlsson, R.; Gustafsson, F. Particle Filtering : The Need for Speed. EURASIP J. Adv. Signal Process. 2010,
2010, 181403. [CrossRef]
25. Grüne, L.; Pannek, J. Nonlinear Model Predictive Control. In Nonlinear Model Predictive Control: Theory and Algorithms; Springer:
London, UK, 2011; pp. 43–66. [CrossRef]
26. Corriou, J.P. Process Control: Theory and Applications; Springer: London, UK, 2004; Chapter Model Predictive Control; pp. 575–615.
[CrossRef]
27. García, C.E.; Prett, D.M.; Morari, M. Model predictive control: Theory and practice–A survey. Automatica 1989, 25, 335–348.
[CrossRef]
28. Comarella, B.V.; Carletti, D.; Yahyaoui, I.; Encarnação, L.F. Theoretical and Experimental Comparative Analysis of Finite Control
Set Model Predictive Control Strategies. Electronics 2023, 12, 1482. [CrossRef]
29. Morel, F.; Lin-Shi, X.; Retif, J.M.; Allard, B.; Buttay, C. A Comparative Study of Predictive Current Control Schemes for a
Permanent-Magnet Synchronous Machine Drive. IEEE Trans. Ind. Electron. 2009, 56, 2715–2728. [CrossRef]
30. Peng, J.; Yao, M. Overview of Predictive Control Technology for Permanent Magnet Synchronous Motor Systems. Appl. Sci. 2023,
13, 6255. [CrossRef]
31. Zhang, A.; Lin, Z.; Wang, B.; Han, Z. Nonlinear Model Predictive Control of Single-Link Flexible-Joint Robot Using Recurrent
Neural Network and Differential Evolution Optimization. Electronics 2021, 10, 2426. [CrossRef]
32. Cortes, P.; Kazmierkowski, M.P.; Kennel, R.M.; Quevedo, D.E.; Rodriguez, J. Predictive Control in Power Electronics and Drives.
IEEE Trans. Ind. Electron. 2008, 55, 4312–4324. [CrossRef]
33. Rojas, C.A.; Yuz, J.I.; Silva, C.A.; Rodriguez, J. Comments on “Predictive Torque Control of Induction Machines Based on
State-Space Models ”. IEEE Trans. Ind. Electron. 2014, 61, 1635–1638. [CrossRef]
34. Cataldo, P.; Jara, W.; Riedemann, J.; Pesce, C.; Andrade, I.; Pena, R. A Predictive Current Control Strategy for a Medium-Voltage
Open-End Winding Machine Drive. Electronics 2023, 12, 1070. [CrossRef]
35. Darba, A.; De Belie, F.; D’haese, P.; Melkebeek, J.A. Improved Dynamic Behavior in BLDC Drives Using Model Predictive Speed
and Current Control. IEEE Trans. Ind. Electron. 2016, 63, 728–740. [CrossRef]
36. Li, J.; Zhang, L.; Niu, Y.; Ren, H. Model predictive control for extended Kalman filter based speed sensorless induction motor
drives. In Proceedings of the 2016 IEEE Applied Power Electronics Conference and Exposition (APEC), Long Beach, CA, USA,
20–24 March 2016; pp. 2770–2775. [CrossRef]
37. Soliman, A.I.; Farhan, A.; Abdelrahem, M.; Kennel, R. Enhanced Sensorless Model Predictive Control of Induction Motor Based
on Extended Kalman Filter. In Proceedings of the 2019 IEEE Conference on Power Electronics and Renewable Energy (CPERE),
Aswan, Egypt, 23–25 October 2019; pp. 309–313. [CrossRef]
38. Janiszewski, D. Extended Kalman Filter Based Speed Sensorless PMSM Control with Load Reconstruction. In Proceedings of the
IECON 2006-32nd Annual Conference IEEE Industrial Electronics, Paris, France, 6–10 November 2006; pp. 1465–1468. [CrossRef]
39. Janiszewski, D. Disturbance estimation for sensorless PMSM drive with Unscented Kalman Filter. In Proceedings of the 12th
IEEE Int Advanced Motion Control (AMC) Workshop, Sarajevo, Bosnia and Herzegovina, 25–27 March 2012; pp. 1–7. [CrossRef]
40. Pillay, P.; Krishnan, R. Modeling, simulation, and analysis of permanent-magnet motor drives. I. The permanent-magnet
synchronous motor drive. IEEE Trans. Ind. Appl. 1989, 25, 265–273. [CrossRef]
41. Wan, E.A.; Van Der Merwe, R.; Nelson, A.T. Dual Estimation and the Unscented Transformation. In Proceedings of the 12th
International Conference on Neural Information Processing Systems, Cambridge, MA, USA, 29 November–4 December 1999;
pp. 666–672.
42. Goodwin, G.C.; Sin, K.S. Adaptive Filtering Prediction and Control; Dover Books on Electrical Engineering; Dover Publications:
Mineola, NY, USA, 2009.
43. Onat, A. A Novel and Computationally Efficient Joint Unscented Kalman Filtering Scheme for Parameter Estimation of a Class of
Nonlinear Systems. IEEE Access 2019, 7, 31634–31655. [CrossRef]
44. Wan, E.A.; Van Der Merwe, R. The unscented Kalman filter for nonlinear estimation. In Proceedings of the IEEE 2000 Adaptive
Systems for Signal Processing, Communications, and Control Symposium (Cat. No.00EX373), Lake Louise, AB, Canada, 4 October
2000; pp. 153–158. [CrossRef]
45. Gould, N.; Toint, P.L. Preprocessing for quadratic programming. Math. Program. 2004, 100, 95–132. [CrossRef]
46. Bartlett, R.; Wachter, A.; Biegler, L. Active set vs. interior point strategies for model predictive control. In Proceedings of the 2000
American Control Conference, ACC (IEEE Cat. No.00CH36334), Chicago, IL, USA, 28–30 June 2000; Volume 6, pp. 4229–4233.
[CrossRef]
Energies 2024, 17, 2387 19 of 19

47. Janiszewski, D. EKF estimation of mechanical quantities for drive with PM Synchronous Motor. In Proceedings of the International
Conference on Power Electronics and Intelligent Control for Energy Conservation (PELINCEC 2005), Warszawa, Poland, 16–19
October 2005; p. 2.
48. Kozlowski, K.; Pazderski, D.; Krysiak, B.; Jedwabny, T.; Piasek, J.; Kozlowski, S.; Brock, S.; Janiszewski, D.; Nowopolski, K. High
Precision Automated Astronomical Mount. In Proceedings of the Automation 2019; Szewczyk, R., Zieliński, C., Kaliczyńska, M.,
Eds.; Springer: Cham, Switzerland, 2020; pp. 299–315. [CrossRef]

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual
author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to
people or property resulting from any ideas, methods, instructions or products referred to in the content.