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Parallel Computing Utilization in Nonlinear
Model Predictive Control of Permanent Magnet
Synchronous Motor
MICHAL KOZUBIK1,2 , Libor Vesely1,2, Eyke Aufderheide3, and Pavel Vaclavek1,2 (Senior
Member, IEEE)
1CEITEC - Central European Institute of Technology, Brno University of Technology, Purkynova 123, 612 00 Brno, Czech Republic
2Department of Control and Instrumentation, Brno University of Technology, Technicka 12, 616 00 Brno, Czech Republic
3Chair of Electrical Drive Systems, and Power Electronics, Technical University of Munich, Munich 80333, Germany
Corresponding author: Michal Kozubik (e-mail: michal.kozubik@ceitec.vutbr.cz).
The work was performed in the project A-IQ Ready: Artificial Intelligence using Quantum measured Information for real-time distributed
systems at the edge No 101096658/9A22002. The work was co-funded by grants of Ministry of Education, Youth and Sports of the Czech
Republic and Chips Joint Undertaking (Chips JU). The completion of this paper was made possible by the grant No. FEKT-S-23-8451 -
”Research on advanced methods and technologies in cybernetics, robotics, artificial intelligence, automation and measurement” financially
supported by the Internal science fund of Brno University of Technology. The work was supported by the infrastructure of RICAIP, which
has received funding from the European Union Horizon 2020 research and innovation programme under grant agreement No. 857306 and
from the Ministry of Education, Youth and Sports under OP RDE grant agreement No. CZ.02.1.01/0.0/0.0/17_043/0010085.
ABSTRACT Permanent Magnet Synchronous Motor (PMSM) drives are widely used for motion control
industrial applications and electrical vehicle powertrains, where they provide a good torque-to-weight ratio
and a high dynamical performance. With the increasing usage of these machines, the demands on exploiting
their abilities are also growing. Usual control techniques, such as field-oriented control (FOC), need some
workaround to achieve the requested behavior, e.g., field-weakening, while keeping the constraints on the
stator currents. Similarly, when applying the linear MPC, the linearization of the torque function and defined
constraints lead to a loss of essential information and sub-optimal performance. That is the reason why the
application of nonlinear theory is necessary. Nonlinear Predictive Control (NMPC) is a promising alternative
to linear control methods. This approach has a major drawback in its computational demands. This paper
presents a novel approach to the implementation of PMSMs’ NMPC. The proposed controller utilizes
the native parallelism of population-based optimization methods and the supreme performance of field-
programmable gate arrays to solve the nonlinear optimization problem in the time necessary for proper motor
control. The paper presents the verification of the algorithm’s behavior both in simulation and laboratory
experiments. The proposed controller’s behavior is compared to the standard control technique of FOC and
linear MPC. The achieved results prove the superior quality of control performed by NMPC in comparison
with FOC and LMPC. The controller was able to follow the Maximal Torque Per Ampere strategy without
any supplementary algorithm, altogether with constraint handling.
INDEX TERMS evolutionary algorithms, motor control, nonlinear control, parallel computing, predictive
control
I. INTRODUCTION
DUE to their beneficial features, such as high efficiency,
reliability, and performance factor, permanent magnet
synchronous motors (PMSM) are widely used in various
industries. All these fields, from classical ones, such as com-
puter numerical control (CNC) machinery and robotics, to
more delicate areas, e.g., electric cars, strive to use the maxi-
mum potential of the motor.
Currently, most PMSMs are controlled using a cascaded field-
oriented control (FOC) structure. [1] Although this has earned
its spurs in industrial applications, several significant limita-
tions lead to inferior performance in high-speed operations or
lower energy efficiency. The first of these is the non-existent
handling of essential motor control constraints. In FOC, limits
on voltage and currents are usually addressed by anti-windup
techniques. These are more of a workaround than a precise so-
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Kozubik, M. et al.: Parallel Computing Utilization in Nonlinear Model Predictive Control of Permanent Magnet Synchronous Motor
lution to a constrained control problem. As FOC does not in-
trinsically handle the problem of field-weakening, additional
algorithms are necessary to utilize controlled PMSM power.
On top of that, due to its cascade structure, the parameters of
several PI/PID controllers require careful tuning because they
are not clearly connected to the behavior of the motor itself.
Nonlinear Model Predictive Control (NMPC) is a promising
method of advanced control of PMSM drives [2] as well as
other AC drives [3] and other industrial applications [4]. Due
to its nonlinear nature, NMPC is able to overcome the afore-
mentioned drawbacks of FOC. However, finding the solution
to an optimization problem in every control iteration presents
significant computational demands, making it difficult for
practical use.
This is the reason why most proposed MPC algorithms for
PMSM control are limited to predictive current control while
keeping a cascaded control structure [5]–[9]. It is easy to
implement on commonly used hardware (HW), but this ap-
proach cannot fully exploit the abilities of MPC.
The most challenging task is full MPC implementation for AC
electrical drive control because of their nonlinear behavior
[10]. Approaches based on linear MPC, where a computa-
tionally efficient Quadratic Programming (QP) solution is
possible (especially when considering explicit MPC [11],
[12]), were published, often using improved linearization
techniques [13]–[15]. Another possibility is to simplify the
optimization problem by using a Finite Control Set (FCS)
[16]–[21], but this typically leads to problems with the current
and torque ripple caused by limited switching states and still
necessary compromises in switching frequency [22] because
of the necessary optimization computing time.
Allthough all these approaches do to some extent use the
advantages of MPC, they fail to maximize its potential. This
is either due to linearization or to not utilizing the multi-
variable control capability. All for the sake of simplifying
the optimization problem. Only a nonlinear MPC can do all
this. However, there is still the question of computational
complexity.
Researchers have recently been using different parallel com-
puting platforms to accelerate the necessary computations
in a large number of areas, such as machine learning [23],
optimization [24], [25] or motor control [26], [27].
This article builds on these results and proposes a novel
NMPC algorithm, that overcomes all mentioned simplifica-
tions by controlling the motor as a one nonlinear system. The
presented algorithm focuses on the root of the NMPC com-
putational burden, which is the solution to an optimization
problem. To speed up the execution, the proposed algorithm
combines the exceptional parallel computing power of field-
programmable gate array (FPGA) and the native parallelism
of the population-based optimization methods. As the algo-
rithm is built from the ground up, it allows more general
shapes of the cost function and possible constraints.
To prove its superiority, the proposed algorithm control be-
havior is compared to different MPC based on linear theory,
and conventional control structure – FOC – in simulation and
laboratory experiments.
The organization of the paper is as follows: The following
chapter introduces the notation used in the article and the
mathematical model of the controlled motor. Then, the moti-
vation for the development of NMPC algorithms is presented.
Chapters IV and V focus on the control algorithm from the
theoretical and design perspective in the first mentioned,
followed by the description of the platform the algorithm was
tested on and its implementation. The sixth chapter presents
the results of both simulation and laboratory experiments. The
final chapter sums up the results of experiments and the whole
paper.
II. PRELIMINARIES
This chapter presents the mathematical notation used
throughout the whole article and the derivation of the PMSM
model used in the presented control algorithm.
A. MATHEMATICAL NOTATION
This paper adopts the following mathematical notation:
Vectors are denoted by small bold letters without italics, e.g.,
vector of system states x. An individual element of a
vector is referenced by index, e.g., xiis i-th element of a
vector x.
Matrices are denoted by bold capital letters and not in ital-
ics. For example, Arepresents a matrix. An individual
element of a matrix is referenced using row and column
indices like Ai,jdenoting an element of a matrix Aon
i-th row and j-th column. Similarly, a=Aidescribes
i-th row vector of matrix A.
Variables and functions are written in italics. For example,
xis a variable, and f(x)is a function of the variable x.
Constants are not italicized, e.g., Tsis a constant.
Considering different time domains, trepresents a continu-
ous time, and kis a discrete time. Whether the variable is in a
discrete or a continuous time domain is denoted in a subscript.
For example, id|tis a direct part of a current in a continuous
time domain, and id|kis its discrete equivalent.
B. MODEL DERIVATION
Let’s have a PMSM fed by the 3-phase 2-level voltage source
inverter (VSI). A common technique used in the field of motor
control is the transformation from phase abc-reference frame
into the dq-reference frame via Clarke’s and Park’s transfor-
mations. [28] The vector of systems states then consists of
the individual parts of the current (id,iq) and a mechanical
angular velocity (ωm)x=idiqωmTand input vector
is given by the voltage in direct (ud) and quadrature (uq) axis
u=uduqT.
Note that differential equations describing the electrical part
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Kozubik, M. et al.: Parallel Computing Utilization in Nonlinear Model Predictive Control of Permanent Magnet Synchronous Motor
of a motor
did|t
dt=−R
Ld
id|t+ Pp
Lq
Ld
iq|tωm|t+1
Ld
uq|t(1a)
diq|t
dt=−R
Lq
iq|t−Pp
Ld
Lq
id|tωm|t−PpΨPM
Lq
ωm|t+1
Lq
uq|t,
(1b)
where
Ris the stator winding resistance,
Ld,qis the direct/quadrature part of the rotor inductance,
Ppis the number of pole pairs,
ΨPM is the permanent magnet flux,
contain nonlinear terms given by the product of the system
states - iq|tand ωm|tin (1a) and id|tmultiplied by ωm|tin (1b).
These nonlinearities show the coupling between the direct and
quadrature parts of the current; also, they show how the rotor
speed affects the dynamics of the electrical part.
The torque generated by the motor Teis dependent on the
values of individual currents
Te|t= 1.5PpΨPMiq|t+ (Ld−Lq)id|tiq|t.(2)
The equation (2) serves as the basis for the differential equa-
tion describing the dynamics of the angular speed ωm. The
second term, describing reluctance torque [29], contains the
product of current parts, which are system states. Thus mak-
ing the equation nonlinear. The dynamics of the angular speed
is then given by the difference between the torque generated
by the motor and load torque TL
dωm|t
dt=1
JTe|t−TL|t
=1
J1.5PpΨPMiq|t+ (Ld−Lq)id|tiq|t−TL|t,
(3)
where Jis the moment of inertia.
For its proper function, the control algorithm needs to evalu-
ate the model in distinctive moments. Therefore, the discrete
equivalent of the model is necessary. It is obtained by the ap-
plication of Euler’s method of discretization on the equations
(1a), (1b) and (3), resulting in
id|k+1 =id|k+ (Ts/Ld)−Rid|k+ PpLqiq|kωm|k+ud|k
iq|k+1 =iq|k+ (Ts/Lq)−Riq|k−PpLdid|kωm|k+
+ (Ts/Lq)−PpΨPMωm|k+uq|k
ωm|k+1 =ωm|k+ (Ts/J)[1.5Pp(ΨPMiq|k+
+ (Ld−Lq)id|kiq|k)−TL|k].
(4)
III. MOTIVATION
This chapter consists of three parts. The first two cover the
problems of standard methods used to control PMSM. The
third section introduces why the proposed algorithm solves
the presented problems and how the paper’s main idea leads
to the successful usage of NMPC in PMSM control.
PI
ωm,r
iq,rPI
id,rPI
uq
ud
PMSM
Electrical
part
PMSM
Mechanical
part
iq
id
ωm
−
−
−
FIGURE 1: Cascaded scheme of the speed control
NMPC
Control
algorithm
ud
uqPMSM
ωm
ωm,r
iq
id
FIGURE 2: Proposed NMPC control scheme
A. FIELD-ORIENTED CONTROL
A common practice for the speed control of PMSMs is to
form a cascade control structure, as shown in Figure 1. This
structure does not address the problem of constraints, given
by the construction of drive, themselves but only their con-
sequences in the windup effect. That brings another problem,
as the saturation of the outer loop controller can increase an
integral part of the inner loop controller. [30]
In its default set-up, the FOC strategy requests the direct
component of the stator current idto be zero. [31] This request
reduces the efficiency of the controlled motor as it does not
use the reluctance part of the generated torque nor use the
field-weakening operation. To solve the problem, the control
scheme must be enhanced by another algorithm that finds the
optimal torque and converts it into the current requests.
On top of that, the motors state equations (1a), (1b) and (3) are
nonlinear, as shown earlier. However, the PI design is based
on linear theory; therefore, it has to rely on the linearization of
the system equations, which makes the controller ineffective
in high-speed operation.
B. COMMON MPC APPROACH
The usual design of the MPC controller of PMSM draws on
a theory of linear MPC. The necessity of a linear model of
a controlled system causes the need for the linearization of
system equations at a specific operating point, which leads to
suboptimal control at different operating points. The problem
of constraints is addressed by linear MPC. However, the
linear approach needs linear constraints. That leads to their
approximation and, thus, sub-optimality again.
Moreover, the usual approach to linear MPC does not fully
omit the cascade structure. Therefore, the problems of struc-
ture are still present. On top of that, the focus on just one
controlled variable does not exploit the full potential of MPC
as multiple-input multiple-output (MIMO) controller.
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C. PRESENTED APPROACH
The presented algorithm solves all the mentioned drawbacks.
Firstly, the cascade structure is replaced by Figure 2. There-
fore, the presented algorithm fully utilizes the MIMO control
capabilities of NMPC and covers the mutual effect of all
system states.
Since the approach is nonlinear, there is no need for the
system linearization. In comparison with linear MPC, the pre-
sented algorithm is able to control the whole range of system
dynamics without suboptimalities in the solution. Same goes
for the constraints. Nonlinear optimization does not put the
requirements on the constraints present in the optimization
problem and system states and input can occur in their whole
ranges. And since there is no cascade with integrators, the
problem of mutually affecting windups cannot occur. The
windup cannot occur itself, due to the constraints handling
being directly imbued into to computation of control value.
IV. NMPC ALGORITHM
The main idea of MPC – either linear or nonlinear – is to use
the model of a controlled plant to predict its future behavior.
The predictive controller utilizes the predicted behavior to
optimize the control value. [32] To compute the control value,
the controller solves the optimization problem
min
u
C(x,u)
s. t. gi(x,u)=0,i= 1,...,m
hj(x,u)≤0,j= 1,...,p,
(5)
where the cost function C(·)represents demands on the con-
trol law. Equality constraints gi(·)define the trajectory of
the states and inputs. Inequality constraints hi(·)outline their
possible range.
The following sections cover the construction of the optimiza-
tion problem used in the presented NMPC algorithm. After
that, the optimization algorithm used is described.
A. INCREMENTAL FORM OF CONTROLLER
In its initial form, the NMPC controller would use the total
voltage value as a control value. There are some issues with
this approach. Firstly, the controller can compute the value of
voltage across its whole range. This leads to more aggressive
behavior of the controller with significant changes in voltage
resulting in oscillations of the current. Secondly, there is no
integral action in the controller, so the zero error with constant
reference is not guaranteed.
Both these issues are solved using the incremental form of the
controller. [33] This requires the change of the state vector x,
input vector uand the model of motor (4) itself. The voltage
becomes a state of the system
x=id|kiq|kωm|kud|kuq|kT(6)
and the input vector is composed of the voltage increments
u=∆ud|k∆uq|T.(7)
uq
ud
Udq|k
UMAX
FIGURE 3: Vector of voltage with box-limited increment
This change allows the introduction of the hard constraint put
on the change of voltage in the form of box constraint shown
in Figure 3.
This modification itself would result in an incremental form
of controller. However, it would prolong the horizon neces-
sary for the angular speed prediction. Modification of the
equations describing the change of currents eliminates the
prolonging.
The new model, expecting an immediate voltage change
id|k+1 =id|k+ (Ts/Ld)−Rid|k+ PpLqiq|kωm|k+
+ (Ts/Ld)(ud|k+ ∆ud|k)
iq|k+1 =iq|k+ (Ts/Lq)−Riq|k−PpLdid|kωm|k+
+ (Ts/Lq)−PpΨPMωm|k+uq|k+ ∆uq|k
ωm|k+1 =ωm|k+ (Ts/J)[1.5Pp(ΨPMiq|k+
+ (Ld−Lq)id|kiq|k)−TL|k]
ud|k+1 =ud|k+ ∆ud|k
uq|k+1 =uq|k+ ∆uq|k,
(8)
allows the controller to work in the incremental form while
not extending the necessary prediction horizon.
B. SPEED TRACKING OPERATION
The cost function penalizes the tracking error of angular
speed to achieve the tracking of the speed reference. On top of
that, there is a demand for the minimization of the consumed
energy. Thus, there is a penalization of the value of voltage
and current.
The overall construction of the electrical drive puts hard
constraints on the optimization problem, e.g., the maximal
voltage is limited by the DC source. In the dq-reference frame,
the voltage and current rated values limit their ℓ2-norm.
Given l= 0,...,N−1, where Nis the length of prediction
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Kozubik, M. et al.: Parallel Computing Utilization in Nonlinear Model Predictive Control of Permanent Magnet Synchronous Motor
horizon, the optimization problem
min
∆ud,q
N
X
i=1 wω(ωm,r|k−ωm|k+i)2+
+
N
X
i=1 widi2
d|k+i+ wiqi2
q|k+i+
+
N
X
i=1 wudu2
d|k+i+ wuqu2
q|k+i+
+
N
X
i=1 w∆ud(∆ud|k+i−1)2+
+
N
X
i=1 w∆uq(∆uq|k+i−1)2
(9a)
s. t. i2
d|k+l+1 +i2
q|k+l+1 −IMAX2≤0(9b)
u2
d|k+l+u2
q|k+l−UMAX2≤0(9c)
|∆ud|k+l| − ∆Ud,MAX ≤0(9d)
|∆uq|k+l| − ∆Uq,MAX ≤0(9e)
id|k+l+1 =id|k+l+ (Ts/Ld)−Rid|k+l+
+ (Ts/Ld)PpLqiq|k+lωm|k+l+
+ (Ts/Ld)ud|k+l+ ∆ud|k+l
iq|k+l+1 =iq|k+l+ (Ts/Lq)−Riq|k+l+
+ (Ts/Lq)−PpLdid|k+lωm|k+l+
+ (Ts/Lq)−PpΨPMωm|k+l+
+ (Ts/Lq)uq|k+l+ ∆uq|k+l
ωm|k+l+1 =ωm|k+l+
+ [(Ts/J) (1.5PpΨPM)] iq|k+l+
+ [(Ts/J)(1.5Pp∆L)] id|k+liq|k+l−
−(Ts/J)TL
ud|k+l+1 =ud|k+l+ ∆ud|k+l
uq|k+l+1 =uq|k+l+ ∆uq|k+l,
(9f)
where ∆L = Ld−Lq, ensures reference tracking if the
designer chooses the weighting coefficients accordingly. Cost
function (9a) guarantees the demands mentioned at the begin-
ning of this section. On top of that, it includes the penalization
of the voltage increment, which tunes the aggressivity of
the control. The inequality constraints of the optimization
problem (9b), (9c) deal with the aforementioned physical
limitations. The following inequality constraints (9d), (9e)
limit the increments of voltage in direct and quadrature axis
by the parameter ∆Ud,MAX,∆Uq,MAX respectively. The set
of equality constraints (9f) is the model of motor (4) used for
the prediction of state trajectories.
C. STEADY-STATE ERROR ANALYSIS
A common problem with MPC applications is a steady-state
error. Therefore, researchers have proposed methods of com-
pensation. [34], [35] Unfortunately, these approaches solve
the problem for the current control loop only.
The time derivative of the tracking control error in the speed
control loop (assuming constant reference)
eω=ωm,r−ωm
˙
eω= 0 −˙ωm
˙
eω=−(Ts/J) [(1.5Pp) (ΨPMiq+ (Ld−Lq)idiq)−TL]
(10)
depends on the stator’s current value and the load torque.
While solving the optimization problem, the control can af-
fect the value of the stator current, but it does not know the
load torque.
Thus, the generated torque, which should mitigate the track-
ing error, is different from the computed one. This mismatch
leads to a non-zero steady-state error. A possible solution is
to introduce information about the load torque to the control
algorithm, e.g., by measuring it or using the load torque
observer [36]. Another possible solution is to integrate the
speed control error and artificially change the requested speed
value.
Nevertheless, dealing with the load torque problem is out of
the scope of this paper, and in the experimental validation
of the presented algorithm, the load torque value held by the
dynamometer is directly injected into the control algorithm.
D. FIELD-WEAKENING OPERATION
The controller maintaining the direct part of the current id
zero is a common approach in motor control. [31] There-
fore, the weighting coefficient widis set high during the
controller design, which ensures the required behavior. On
the other hand, there are more advanced strategies utilizing
the permanent magnet’s field-weakening, thus generating the
proper reluctance torque that leads the motor to achieve a
higher angular speed. The negative direct part of the current is
necessary for correct reluctance torque generation because, in
the standard interior permanent magnet synchronous machine
(IPMSM), the inductance in the quadrature axis is greater than
in the direct axis. [37] A positive value of this current leads
to a lower achieved angular speed; hence, it is undesired.
One of the possible ways to meet the demands put on reluc-
tance torque is to replace the constant value of weighting co-
efficient widwith the function. Using the piecewise function
dependent on the value of current id
wid|k(id|k) = (wid,nif id|k≤0
wid,pif id|k>0,(11)
whose parameters satisfy the condition wid,p>> wid,n, meets
both demands put on the reluctance torque. The weighting
coefficient for the negative direct part of current wid,nmust be
set low enough to ensure the field-weakening. Nevertheless, it
cannot be set to zero because that would lead the algorithm to
perform the field-weakening operation even though it would
not be necessary.
Setting up the weighting factor wid|kthis way works as a
soft constraint put on the optimization problem. This soft
constraint was used because, during the experiments, the use
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Kozubik, M. et al.: Parallel Computing Utilization in Nonlinear Model Predictive Control of Permanent Magnet Synchronous Motor
of a hard constraint id|kturned out to be problematic, causing
numerical instabilities, mainly due to measurement noise.
E. POPULATION-BASED OPTIMIZATION
Population-based optimization uses a group of individuals,
or agents, distributed across the feasible set. These agents
represent the candidate for the solution to an optimization
problem. For the case of the proposed predictive controller,
each agent is represented by the vector of possible voltage
increments across the prediction horizon N
a=∆ud|k∆uq|k∆ud|k+1 · · · ∆uq|k+N−1.(12)
According to the strategy given by the optimization algo-
rithm, the agents move iteratively toward the real solution.
Because of the number and distribution of agents, population-
based optimization methods can be less prone to being stuck
in local optima. [38] Furthermore, these methods are easy to
parallelize and consequently reduce their computational time.
On the other hand, these methods are usually stochastic,
which can lead to unpredictable behavior. For this reason,
the presented NMPC algorithm uses as few stochastic parts
as possible. Therefore, the basis of the optimization in the
proposed MPC algorithm is the Differential Evolution algo-
rithm [39] with some tweaks. The most important one is the
movement of agents, which is described by the equation
aj=aj+ s(aBEST −aj).(13)
The changes are the following: Firstly, in every iteration,
every agent ajmoves toward the agent aBEST whose cost
function value is, in the given iteration, lowest; the algorithm
does not choose a random combination of agents. Secondly,
all agents always move by a specified fraction of the distance
s∈(0; 1) in every dimension, not only when a randomly
generated number exceeds the threshold. [40]
Figure 4 shows the result of the experiment performed to
outline the behavior of the agents during the optimization
step. Every line shows the trajectory of a different agent,
while every mark on the line shows the position in a specific
iteration. For practical reasons, the figure only shows the first
two dimensions of the 2N-space of voltage increments.
Yet, it still well documents how the agents’ movements
proceed during optimization. Since the trajectories are not
straight lines, it is clear that the agent with the continuous best
value of the objective function aBEST has changed in different
iterations.
Algorithm 1 shows the pseudocode of the controller’s func-
tion. Abbreviations NoA and NoI are parameters Number of
Agents and Number of Iterations. The Solve the constraints
section of the algorithm solves whether the agent is within the
area specified by the constraints. If not, the barrier functions
[41] are used to increase the value of the agent’s objective
function.
V. IMPLEMENTATION
The implementation of the NMPC control algorithm is car-
ried out on the National Instruments PXI rapid prototyping
−0.1−5·10−205·10−20.1
−0.1
−5·10−2
0
5·10−2
0.1
ud|k+0 (−)
uq|k+0 (−)
FIGURE 4: Trajectories of the agents in the first two dimen-
sions during the optimization step
Algorithm 1 Proposed controller algorithm
Require: x
Initialize A=aT
1· · · aT
NoAT
it ←1
while it ≤NoI do
j←1
while j≤NoA do
Evaluate model (9f) for (x,Aj)
Evaluate cost function (9a)
Solve constraints (9b), (9c), (9d), (9e)
end while
Find ABEST
Move agents according to (13)
end while
u←ABEST ,1ABEST ,2T
platform using the LabView software. The following section
describes the hardware resources used for the implementation
of the control algorithm.
A. HARDWARE DESCRIPTION
The PXI platform contains the Real-Time (RT) controller, the
I/O module, and the FPGA. Figure 5 shows the implementa-
tion scheme with the utilization of named hardware resources.
The PXI RT controller is based on an 8-core processor with
a Linux RT operation system and manages the Peer-to-Peer
(P2P) streaming between FPGAs, data logging from FPGAs
via DMA channels, writes initial values to FPGA RAM
blocks, and a graphical user interface (GUI).
The I/O module provides combinations of the 16-bit analog-
to-digital converter (ADC) and digital I/O with a user-
programmable Kintex-7 325T FPGA for onboard signal pro-
cessing. The digital outputs drive a low-voltage power stage.
Analog voltages from Hall effect current transducers LEM
HY 5-P are applied to the analog inputs. The incremental
encoder with 1024 pulses per rotation is connected to the
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digital inputs. The core of the NMPC control algorithm
runs on the FPGA. The FPGA features a Kintex UltraScale
KU060.
B. ALGORITHM WORKFLOW
Figure 6 presents the primary sequence of the algorithm. This
sequence consists of eight essential steps. The first two steps
are to read the initial positions of agents from FPGA memory
blocks and the motor’s states from the I/O module. These two
steps run parallel.
The four independent parallel pipelines perform the two
following steps: compute the motor state prediction and
evaluate the cost function for a given agent position. A further
increase in the parallel pipelines would speed up the whole
algorithm. However, the presented implementation reached
the full hardware potential of the FPGA. Both steps necessary
for evaluating the cost function are performed in the for-loop.
The prediction horizon length Ndetermines the number of
for-loop iterations.
The next three steps are to find the best agent, move agents to
a new position according to (13), and write the new agents’
position to memory.
These five steps represent one iteration of the optimization
algorithm explained before. Thus, the NMPC algorithm runs
another for-loop, whose number iteration is determined by the
optimization algorithm’s settings. The final step is to send the
new stator voltage increments to the I/O module.
C. COMPUTATIONAL BURDEN
The length of prediction horizon Nand the choice of op-
timization parameters affect the computational time of the
NMPC algorithm. The proper field-weakening function re-
quires the minimal prediction horizon length N = 4. Com-
putational demands of the NMPC algorithm rise with the
increase of the prediction horizon; therefore, the presented
algorithm works with this minimum value.
Table 1 contains the optimization parameters. A solution to
the presented optimization lies within the polytope formed by
the original positions of agents in 2N-space. Therefore, the
Number of Agents depends on the length of the prediction
horizon. For N = 4, 8-simplex requires nine vertices [42],
which is the bare minimum for this choice. Since searching
the momentary best agents requires the number of agents in
the power of two, and a higher number of agents can span
more parts of the space, the algorithm works with 32 agents,
the maximal possible value enabled by HW.
For its impact on computational performance, fixed-point
arithmetic was used. Thus, all values are normalized in the
range of -1 to 1. Table 2 contains the norms of the necessary
states - voltage UN, current INand angular speed ΩN.
The number of iterations was determined in the simulation
experiment by calculating the values of the Karush-Kuhn-
Tucker conditions [41] for the computationally most complex
problems, e.g., field-weakening. Similarly, parameter swas
TABLE 1: Optimization parameters
Parameter Value
Number of Agents 32
Number of Iterations 30
s 0.3
TABLE 2: Parameters of IPMSM
Parameter Value Unit
R 0.38 Ω
Ld0.405 mH
Lq0.665 mH
ΨPM 0.02594 Wb
UDC 12 V
IR6 A
Pp 3 −
J 446 ·10−6kg m2
UN6.93 V
IN6 A
ΩN150 rad s−1
set according to simulation experiments.
Figure 7 shows the timing scheme of the implemented algo-
rithm with named parameters. The resulting computational
time allows a sampling time of 100 µs. This sampling time is
the time we aimed for as trade-off between the control of the
electrical and mechanical parts of the controlled motor.
VI. EXPERIMENTAL RESULTS
This chapter covers the experimental comparison of the pro-
posed algorithm with different control strategies in simulation
and laboratory test bench. The first three sections describe
the design of each control strategy – FOC, Linear Model
Predictive Control (LMPC), and the presented algorithm. The
following section compares named control strategies in the
simulation experiment performed in Matlab Simulink. The
final part presents the results of the laboratory experiment.
A. FOC
Figure 1 shows the basis of the tested control structure, which
was expanded by features enhancing the speed control perfor-
mance. To follow the Maximum Torque per Ampere (MTPA)
[43] strategy, the speed controller was changed to compute
the torque necessary for speed control and not the stator
current itself. The references for the direct and quadrature
were computed according to the mentioned strategy.
To minimize the oscillations of controlled values, the con-
trollers were tuned to hold 60 degrees phase margin. Each PI-
controller included back-calculation [44] part to overcome the
windup phenomenon. Dynamic decoupling [45] was used to
deal with the nonlinearities in current differential equations
(1a), (1b). Table 3 shows the parameters of each controller;
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+
+
PXI rapid prototyping platform
I/O module
Kintex-7 325T
FPGA
Kintex- UltraScale KU060
Real-Time Controller
Xeon 8-Core W-2245
DMA
DMA
P2P streaming
ω
iq
id
uq
ud
dq←abc
dq→abc SVM
Current
processing
Encoder
signal
processing
ϑ
DO
AIDI
DC Power
Source
Power
Stage
LEM
LEM
IPMSM
Incremental
Encoder
z−1
z−1
∆ud
∆uq
FIGURE 5: Implementation scheme
FPGA
Cost Fnc
Cost Fnc
Cost Fnc
Cost Fnc
Model
Model
Model
Model
N=4
Length of Prediction Horizon
it = 30
Number of Iterations
Find Best Agent
Move Agents’ Positions
Write Agents’ Poisitons
FPGA P2P
∆ud,∆uq
FPGA P2P
id,iq,ωm
Memory Read
Initial Agents’ Positions
FIGURE 6: Algorithm workflow
assuming the standard form of the controller Kpis the pro-
portional gain, Tiis the integral time constant and Ttis the
tracking time constant for anti-windup action.
B. LMPC
The design of LMPC based on MPT3 solver [46] followed
the same guidelines, such as the structure in Figure 2, as the
presented NMPC algorithm. However, this required several
compromises. First, the solver requires the cost function in
the quadratic form. Thus, the weighting factor widwas set to
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30
4
640 ns
2560 ns 25 ns 200 ns 12.5 ns
12.5 ns
50 ns 83925 ns
83987.5 ns
12.5 ns
FPGA
P2P
Read
Memory
Read
Model
+
Cost
Find
Best
Agent
Move
Agents’
Positions
Write
Agents’
Positions
FPGA P2P Write
FIGURE 7: Timing scheme
TABLE 3: Controllers’ parameters
Angular speed controller
Ti7.14 ·10−3
Kp0.40
Tt1.79 ·10−3
Direct part of the current controller
Ti1.07 ·10−3
Kp1.36
Tt2.66 ·10−4
Quadrature part of the current controller
Ti1.75 ·10−3
Kp2.31
Tt4.39 ·10−4
constant value equal to wid,pin Table 4.
The solver is based on quadratic programming, and there
are also demands put on the constraints of the optimization
problem. They must be linear. Because of this demand, some
assumptions had to be made. First, in the model, inductances
were replaced by ˜
Ld=˜
Lq= (Ld+ Lq)/2, which eliminated
the reluctance torque from mechanical equation (3). This
change also affected the electrical equations, but it did not
linearize them because there was still coupling present. The
model was linearized by the second assumption ωm= 0.
Also, the circular limitations of the current and the volt-
age had to be linearized. They were replaced by polygons
inscribed to given circles. Because of the aforementioned
numerical stability reasons, the limitation id≤0could not be
imposed, and some positive value of the current was allowed.
C. NMPC – WEIGHTING COEFFICIENTS SELECTION
When choosing the weighting coefficients, it was necessary
to consider a critical factor resulting from the implementa-
tion in the normalized values. Each normalized value repre-
sents a different range of the state itself, e.g., 0.1represents
15 rad s−1and 0.6 A; thus, different values of the weighting
coefficients are needed to keep the error in the same order of
magnitude.
The correct choice of the speed monitoring weighting coeffi-
cient is the most important in terms of speed control function.
TABLE 4: Cost function parameters
Parameter Value
wω6.25 ·10−2
wiq7.50 ·10−6
wid,n1.25 ·10−6
wid,p8.00 ·10−3
wud6.00 ·10−8
wuq7.00 ·10−6
w∆ud5.00 ·10−5
w∆uq1.00 ·10−6
∆Ud,MAX 0.10
∆Uq,MAX 0.10
For this reason, this factor has been set to the highest value.
Furthermore, the quadrature component of the current iqis an
important aspect of the generated angular velocity. Setting the
weighting coefficient too high leads to slow control behavior.
On the other hand, setting it too low leads to oscillations and
an energy non-efficient operation.
The different settings of the weighting coefficient for positive
and negative values of the direct current component have been
explained above.
Finally, different values of the voltage increment weighting
coefficients affect how quickly the algorithm starts to perform
the field-weakening operation.
Table 4 shows all weighting coefficients and maximal voltage
increments used during the experiments.
D. EXPERIMENT DESCRIPTION
Either experiment, simulation, or laboratory was performed
with TGT3-0130 electric drive – the parameters are listed in
Table 2. The goal of the experiments was to observe which
control algorithm best tracks the angular speed waveform
while complying with given constraints.
There are three sectors in the waveform. First, there is a ramp
with a slope higher than the slope achievable by the controlled
motor. The increase of the requested angular speed ends with
a constant value greater than its base value, which requires
the field-weakening operation to be achieved. Then, the re-
quest falls to zero with a slope achievable by the controlled
motor. This section checks the controllers’ ability to keep the
stated constraints – the slope of the request for the current
constraints and the field-weakening for voltage constraints.
Since the drive is constructed as a servo drive, the second
portion of the angular speed request waveform consists of the
rapid changes in the rotation direction – a common operation
mode of the servo drive. There are alternating ramps in pos-
itive and negative directions. The ramps have two possible
slopes - one that controllers can achieve tracking of and the
other that they cannot. The last section is the same as the first
one, but it checks the controllers’ behavior in the opposing
direction.
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(a) Angular speed
0 0.2 0.4 0.6 0.8 1 1.2 1.4
−0.5
0
0.5
Time (s)
Angular Speed (−)
0 0.2 0.4 0.6 0.8 1 1.2 1.4
−0.5
0
0.5
Time (s)
Angular Speed (−)
0 0.2 0.4 0.6 0.8 1 1.2 1.4
−0.5
0
0.5
Time (s)
Angular Speed (−)
(b) Direct part of the current
0 0.2 0.4 0.6 0.8 1 1.2 1.4
−0.5
0
Time (s)
Current (−)
(c) Quadrature part of the current
0 0.2 0.4 0.6 0.8 1 1.2 1.4
−1
−0.5
0
0.5
1
Time (s)
Current (−)
(d) Direct part of voltage
0 0.2 0.4 0.6 0.8 1 1.2 1.4
−0.5
0
Time (s)
Voltage (−)
(e) Quadrature part of the voltage
0 0.2 0.4 0.6 0.8 1 1.2 1.4
−1
−0.5
0
0.5
1
Time (s)
Voltage (−)
FIGURE 8: Normalized results of the simulation experiment,
blue – FOC, red – NMPC, green – LMPC.
−1−0.5 0 0.5 1
−1
−0.5
0
0.5
1
Direct part of Current (−)
Quadrature part of Current (−)
FIGURE 9: Normalized currents in dq-reference frame, blue
– FOC, red – NMPC, green – LMPC, black - limit.
−1−0.5 0 0.5 1
−1
−0.5
0
0.5
1
Direct part of Voltage (−)
Quadrature part of Voltage (−)
FIGURE 10: Normalized voltages in dq-reference frame, blue
– FOC, red – NMPC, green – LMPC, black - limit.
TABLE 5: Results of simulation experiment
Algorithm ISE tr[ms] ts[ms]
NMPC 128.93 5.49 12.03
LMPC 196.48 5.52 13.09
FOC 139.73 5.58 12.21
E. SIMULATION COMPARISON
For the simulation comparison, the Matlab (R2023b)
Simulink was used. The PMSM was simulated using the Sim-
scape library block. The simulation sampling time of 10 µs
was used to simulate the behavior of the motor between the
actions of the controller. FOC control structure utilizes most
of the built-in Simulink blocks. LMPC was implemented as
an s-function. In the case of NMPC, general-purpose com-
puting on a graphics processing unit was utilized, and the
algorithm was implemented on the Jetson Xavier platform as
written in [40].
The results of the experiment are in Figures 8, 9 and 10. Con-
sidering the angular speed, the subfigure 8a shows how badly
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Kozubik, M. et al.: Parallel Computing Utilization in Nonlinear Model Predictive Control of Permanent Magnet Synchronous Motor
the LMPC performed in the angular speed control. There is a
non-zero error for the constant value of request in the first and
the third sectors, showing that the controller did not perform
any form of field-weakening, unlike NMPC and FOC. On top
of that, the LMPC controller was not able to track the majority
of the ramp requests, and, when compared to the other two
strategies, it failed to meet the control demands.
The subfigure also contains two insets. The first of them –
black – shows how each controller deals with the rapid change
of rotation sense. NMPC controller achieved greater speed
and started tracking the reference before any other controller.
The red inset shows the settling on the constant value. With
NMPC controller, the angular speed settled on constant value
faster, but with a slight overshoot compared with FOC.
Considering tthat angular speed was the control objective, the
integral square error criterion (ISE)
ISE =
NS
X
i=1
(ωm,r|i−ωm|i)2,(14)
where NSis the number of samples, was used to measure the
overall performance of control strategies. Table 5 shows the
values of the criterion for individual controllers and highlights
the poor performance of the LMPC. Table also contains
commonly observed values – rise and settling times (tr,ts)
– for the initial ramp. These quantities not only underline
the poor performance of LMPC, but also show slightly better
dynamic properties of NMPC compared to FOC.
Currents in the time domain and dq-reference frame are
shown in Figures 8b, 8c and 9. The compromises necessary
for the LMPC implementation led to the controller’s inability
to control the idproperly. It was not able to perform the
field-weakening operation; on the contrary, the value of the
current was kept mostly positive, which resulted in lower
achieved angular speed. That did not occur for the other two
controllers. However, when the sense of rotation changes
rapidly, there are differences between FOC and NMPC. There
was only a slight decrease of idby NMPC in comparison with
FOC. Considering that both achieved the same speed in the
given moment, NMPC performed more effectively than FOC.
On top of that, FOC was not able to abide by the current
limitation, as shown in Figure 9. This was caused by the
fact that the speed controller does not operate with current
constraints in its design and relies on current regulators to
handle their potential violation.
Voltage limitations were met by all controllers, as shown
in Figure 10. Hence, there would be no problems for any
controller between expected and actual control value. Figures
8d and 8e show the voltages in the time domain. The visible
spikes present in the FOC’s voltages were mitigated in both
predictive controllers by the weighting factor put on the
voltage increments.
Dynamometer Control Panel
PMSM + Dynamometer
Power Stage
+
Current
Measurement
PXI
Power Supply
DC Load
FIGURE 11: Test bench
TABLE 6: Results of laboratory experiment
Algorithm ISE tr[ms] ts[ms]
NMPC 149.73 5.53 13.15
FOC 180.06 5.94 13.27
F. LABORATORY EXPERIMENT
Due to the poor performance of the LMPC controller in
the simulation experiment, the LMPC algorithm was omitted
from the laboratory experiment. Laboratory comparison was
performed on the test bench shown in Figure 11. The test
bench consists of the TGT3-0130 drive paired with the dy-
namometer. A programmable power supply feeds the motor
through the power stage. The PXI platform served to execute
control algorithms. A PC screen in the figure shows the user
interface used for the controllers’ parameters tuning.
Figure 12a shows the resulting speed for the tested con-
trollers. Even in the laboratory experiment, both controllers
were able to perform the field-weakening operation. How-
ever, settling on the requested speed was better in the case
of NMPC, as shown in the green inset. The black inset shows
the ability of controllers to handle the fast change of the speed
request. The predictive controller achieved higher angular
speeds and started to track the ramp sooner than the standard
control structure. That is caused by the essential difference
between the PI control and predictive control. The standard
control structure focuses solely on the actual value of the
control error, which started to reduce with the decrease of
the requested value. The last – red – inset shows the settling
on zero requested angular speed. As shown, the speed of the
motor controlled by NMPC settled sooner. This comparison is
summed up in Table 6, which shows that the value of the ISE
criterion of the FOC structure was 20 %higher that of NMPC,
thus showing the overall better performance of NMPC. Also
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Kozubik, M. et al.: Parallel Computing Utilization in Nonlinear Model Predictive Control of Permanent Magnet Synchronous Motor
(a) Angular speed
0 0.2 0.4 0.6 0.8 1 1.2 1.4
−0.5
0
0.5
Time (s)
Angular Speed (−)
00.2 0.4 0.6 0.8 1 1.2 1.4
−0.5
0
0.5
Time (s)
Angular Speed (−)
0 0.2 0.4 0.6 0.811.2 1.4
−0.5
0
0.5
Time (s)
Angular Speed (−)
0 0.2 0.4 0.6 0.811.2 1.4
−0.5
0
0.5
Time (s)
Angular Speed (−)
(b) Direct part of the current
0 0.2 0.4 0.6 0.8 1 1.2 1.4
−0.5
0
Time (s)
Current (−)
(c) Quadrature part of the current
0 0.2 0.4 0.6 0.8 1 1.2 1.4
−1
−0.5
0
0.5
1
Time (s)
Current (−)
(d) Direct part of voltage
0 0.2 0.4 0.6 0.8 1 1.2 1.4
−0.5
0
0.5
Time (s)
Voltage (−)
(e) Quadrature part of the voltage
0 0.2 0.4 0.6 0.8 1 1.2 1.4
−1
−0.5
0
0.5
1
Time (s)
Voltage (−)
FIGURE 12: Normalized results of the laboratory experi-
ment, blue – FOC, red – NMPC.
−1−0.5 0 0.5 1
−1
−0.5
0
0.5
1
Direct part of Current (−)
Quadrature part of Current (−)
FIGURE 13: Normalized currents in dq-reference frame, blue
– FOC, red – NMPC, black - limit, black dashed - MTPA
curve.
−1−0.5 0 0.5 1
−1
−0.5
0
0.5
1
Direct part of Voltage (−)
Quadrature part of Voltage (−)
FIGURE 14: Normalized voltages in dq-reference frame, blue
– FOC, red – NMPC, black - limit.
rise and settling times show the edge NMPC has over FOC,
with the rise time of FOC being larger by 10 %.
Considering the direct part of the current, there are visible
decreases in the middle section of the waveform of the re-
quested speed for the NMPC controller. These agree with the
results of simulation 8b. FOC focused solely on generating
the maximal torque, which is also visible on the waveform of
iqshown in Figure 12c. The figure shows, how the control
action performed by FOC resulted in larger values of iq,
especially in the middle section, which resulted in larger
overshoots and slower reaction to the change of speed request
in comparison with NMPC. In general, both controllers were
able to keep the current within stated limits. For FOC, there
is a slight overshoot visible in Figure 13, but smaller than
in the simulation experiment. The figure also shows, how
both controllers utilized the MTPA control strategy, when
necessary.
Both controllers also held the constraints of voltage as shown
in Figure 14. There are visible differences between the volt-
ages generated by individual controllers. One of them occurs
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Kozubik, M. et al.: Parallel Computing Utilization in Nonlinear Model Predictive Control of Permanent Magnet Synchronous Motor
during the initial ramp; since there was voltage increment
limitation, NMPC voltage followed the ramp increase in com-
parison with the step change generated by FOC. After leaving
the field-weakening region, FOC aggressively changed the
value of id, which resulted in peaks in both udand uq. This
behavior might be required; however, as the previous results
have shown, there was no significant effect on the quality of
speed control in comparison with NMPC with a smoother
approach. Similarly, for the middle section, where there are
rapid changes in the rotation direction, less aggressive control
voltage generated by NMPC leads to a better quality of speed
control.
VII. CONCLUSION
Nonlinear Model Predictive Control is a promising alternative
to traditional control approaches in the control of electrical
motors. Its foundations in nonlinear optimization deal with
all complications where the solution typically requires a form
of a workaround in a control based on linear theory. The
major drawback of NMPC is its dependency on the solu-
tion of nonlinear optimization problems, which present large
computational demands. Together with the necessity of short
sampling times, using NMPC for motor control is hard to
achieve.
This paper addresses this major issue of NMPC by present-
ing a novel approach to the control algorithm design. The
combination of the exquisite parallel computational power
of well-known platforms, such as FPGA and GPU, which
are becoming more widely used in embedded devices, and
the population-based optimization algorithm resulted in the
computation of the control value in sampling times demanded
by motor control.
Every aspect of the controller design is thoroughly covered in
the specific chapter of the paper. The implementation, which
considers the parallel computing capabilities of FPGA and
its resources, is covered in the fifth chapter. The final chapter
deals with the comparison of the presented algorithm in both
simulation and laboratory experiments that tested its real-time
performance.
The comparison of the presented NMPC algorithm with its
linear counterpart and commonly used FOC cascade structure
has shown the superb quality of NMPC in motor control. Its
nonlinear background helped the controller in the tracking of
the requested value of angular speed. The controller was able
to deal with circular constraints, achieve the field-weakening
operation, and operate on the MTPA curve without any need
for additional algorithms.
The results obtained prove the initial assumption that a com-
bination of algorithms specifically designed for parallel com-
puting platforms enables the use of computationally complex
concepts such as NMPC to control systems requiring short
sampling times. The results thus open up new avenues in
research. First, due to the development of parallel comput-
ing in microprocessors, it is possible to implement NMPC
on simpler devices. Furthermore, it is possible to construct
algorithms on other parallel methods and exploit their prop-
erties. Finally, more profound exploration of the paralleliza-
tion of algorithms allows the use of otherwise non-parallel
algorithms but with different beneficial properties such as
speed of convergence, e.g., Newton’s method, in parallelized
computation.
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MICHAL KOZUBIK received the M.Sc. degree in
cybernetics from the Brno University of Technol-
ogy, Brno, Czech Republic, in 2019, where he is
currently working towards the Ph.D. degree
His research interests are nonlinear model pre-
dictive control of synchronous motors, parallel
computing, and mathematical optimization. He is
currently working as a research assistant at the
Central European Institute of Technology, Brno,
Czechia.
LIBOR VESELY received the M.Sc. and Ph.D. de-
grees in cybernetics from the Brno University of
Technology, Brno, Czech Republic, in 2006 and
2012, respectively.
His research interests include speed sensorless
control of permanent-magnet synchronous motors,
state estimation, system modeling, and parameters
estimation. He is currently a Junior Researcher
with the Central European Institute of Technology,
Brno University of Technology.
EYKE AUFDERHEIDE (birth name Liegmann) re-
ceived the B.Sc. and M.Sc. degrees in electrical
power engineering from RWTH Aachen Univer-
sity, Aachen, Germany, in 2013 and 2016, respec-
tively. Currently, he is pursuing the Dr.-Ing. degree
at the Technical University of Munich, Munich,
Germany.
His interests are model predictive control of elec-
trical drive systems and the real-time implementa-
tion of control algorithms on FPGAs and embed-
ded systems.
PAVEL VACLAVEK (M’04-SM’12) received MSc.
and Ph.D. degrees in cybernetics from the Brno
University of Technology, Brno, Czech Republic
in 1993 and 2001, respectively. He received also
MSc. degree in industrial management in 1998
from Brno University of Technology.
His research interests include advanced control
of electrical drives, model predictive control, sys-
tem modeling and parameters estimation. He is a
research Cybernetics and Robotics group leader
and Industrial Cybernetics research programme coordinator at the CEITEC
- Central European Institute of Technology, Brno University of Technology.
14 VOLUME XX, 2024
This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and
content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2024.3456432
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/
