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1

A mathematical model for controlling a quadrotor UAV

Kowalik Rafał, Łusiak Tomasz, Novak Andrej

Transactions on Aerospace Research

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2021

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Nr 3 (264)

58--70

EN

Given the recent surge in interest in UAVs and their potential applications, a great deal of work has lately been done in the field of UAV control. However, UAVs belong to a class of nonlinear systems that are inherently difficult to control. In this study we devised a mathematical model for a PID (proportional integral derivative) control system, designed to control a quadrotor UAV so that it follows a predefined trajectory. After first describing quadrotor flight dynamics, we present the control model adopted in our system (developed in MATLAB Simulink). We then present simulated results for the use of the control system to move a quadrotor UAV to desired locations and along desired trajectories. Positive results of these simulation support the conclusion that a quadrotor UAV spatial orientation control system based on this model will successfully fulfil its task also in real conditions.

PL

Wobec rosnącego zainteresowania bezzałogowymi statkami powietrznymi (BSP) i ich potencjalnymi zastosowaniami, w ostatnim czasie wykonano wiele prac w dziedzinie ich lepszego sterowania. BSP jednak należą do klasy nielinearnych systemów, które są z natury trudne do sterowania. W niniejszej pracy opracowano model matematyczny dla systemu sterowania opartego na metodzie PID (proportional integral derivative), zaprojektowanego do sterowania czterowirnikowym BSP tak, aby podążał on po wcześniej zdefiniowanej trajektorii. Opisano dynamiki lotu czterowirnikowym BSP i zaprezentowano model kontroli zastosowany w naszym systemie (opracowany w programie MATLAB Simulink). Przedstawione są wyniki symulacji użycia systemu do sterowania czterowirnikowym BSP w celu jego przemieszczenia się do pożądanej lokalizacji oraz wzdłuż pożądanej trajektorii. Wyniki tych symulacji potwierdzają wniosek, że oparty na tym modelu system sterowania dla czterowirnikowego BSP może z powodzeniem spełnić swoje zadanie również w warunkach rzeczywistych.

2

Fractional Order Sliding Mode Speed Control of Feedback Linearized Induction Motor

Fetene Yeshiwas, Shibeshi Dereje

Power Electronics and Drives

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2020

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Vol. 5 (40)

109--122

EN

Induction machines have of late become the most popular workhorses in the industry replacing DC machines because of their advantages, such as reduced cost, reliability and the absence of commutators, which make them adapt to unfavourable conditions with lower maintenance requirement. However, higher-order models of AC machines, nonlinearities in model equations, uncertainties in parameters and load disturbances make induction motors difficult to control than that of DC motors. In this paper, a robust control strategy considering the recent advances in technology is proposed employing input-output feedback linearization for the exact decoupling of electromagnetic torque and rotor flux using fractional-order sliding mode controller (SMC) for the outer speed loop. The sliding mode observer is also designed to extract the rotor flux from the DC input voltage of the inverter and stator current measurements. Finally, a comparison between the commonly used proportional integral and derivative (PID) controller and the proposed fractional-order SMCs is made and the conclusion is reached.

3

Oscillatory criteria via linearization of half-linear second order delay differential equations

Baculikova Blanka, Dzurina Jozef

Opuscula Mathematica

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2020

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Vol. 40, no. 5

523--536

EN

In the paper, we study oscillation of the half-linear second order delay differential equations of the form [formula]. We introduce new monotonic properties of its nonoscillatory solutions and use them for linearization of considered equation which leads to new oscillatory criteria. The presented results essentially improve existing ones.

4

Transistor-based temperature measuring device

Hotra Oleksandra

Informatyka, Automatyka, Pomiary w Gospodarce i Ochronie Środowiska

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2020

|

T. 10, nr 2

4--7

EN

The schematic diagrams of the temperature measuring device based on transistor structures are presented in the paper. The temperature dependence of collector current without and with linearization of the conversion function is analysed. The linearization method based on compensation current formation is proposed. This allowed to reduce the temperature measurement error up to ± 0.006°C over the temperature ranges 40… 60°C and 60… 80°C and up to 0.08°C over the temperature range 10… 90°C.

PL

W artykule zostały przedstawione schematy miernika temperatury opartego na strukturach tranzystorowych. Została przeanalizowana zależność prądu kolektora od temperatury bez i przy zastosowaniu linearyzacji funkcji przetwarzania. Zaproponowano metodę linearyzacji opartą na formowaniu prądu kompensacyjnego, która pozwoliła zmniejszyć błąd pomiaru temperatury do ± 0,006°C w zakresach temperatury 40… 60°C i 60… 80°C oraz do ±0,08°C w zakresie 10… 90°C.

5

Study of the admixtures effect on concrete creep using Bayesian Linear Regression

Zgheib E., Raphael W.

Archives of Civil Engineering

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2019

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Vol. 65, nr 3

127--140

EN

Admixtures are commonly used nowadays in the mix composition of concrete. These additions affect concrete properties and performance especially creep deformations. This paper shows the effect of admixtures on creep of concrete. In fact, creep deformations have prejudicial consequences on concrete behaviour; an incorrect or inaccurate prediction leads to undesirable consequences in structures. Therefore, an accurate estimation of these deformations is mandatory. Moreover, design codes do not consider admixtures’ effect while predicting creep deformations, thus it is necessary to develop models that predict accurately creep deformations and consider the effect of admixtures. Using a large experimental database coming from international laboratories and research centres, this study aims to update the Eurocode 2 creep model by considering the type and percentage of admixtures using Bayesian Linear Regression method. The effect of two types of admixtures is presented in this paper; the water reducer and silica fume.

6

Sterowanie zestawem artyleryjsko-rakietowym z wyko-rzystaniem zmodyfikowanego regulatora liniowo-kwadratowego

Szmidt P., Koruba Z., Gapiński D.

Zeszyty Naukowe Politechniki Rzeszowskiej. Mechanika

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2018

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z. 90 [298], nr 2

223--235

PL

W niniejszej pracy przedstawiono opracowanie układu sterowania zdalnie sterowanym zestawem artyleryjsko-rakietowym z wykorzystaniem regulatora LQR (liniowo-kwadratowego). Model zestawu wyprowadzono, opierając się na produkowanym w Polsce zestawie ZU 23-2MR. W pracy omówiono linearyzację modelu matematycznego zestawu za pomocą Jakobianów, przedstawiono strukturę i budowę układu regulacji oraz zaproponowano sposób niwelacji uchybu sterowania w stanie ustalonym bez wykorzystania dodatkowych członów całkujących, występujących w układzie regulacji. Badaniom poddano także wpływ oddziaływania zakłóceń pochodzących od strzelającej armaty na dokładność sterowania w trakcie śledzenia manewrującego celu. Przeanalizowano również wpływ niedokładności identyfikacji parametrów obiektu sterowania na pracę układu. Symulacje numeryczne przeprowadzono z wykorzystaniem środowiska Scilab, a najistotniejsze wyniki zostały przedstawione w postaci graficznej.

EN

This paper presents the development of a remote-controlled artillery-rocket control system using the LQR (linear-quadratic regulator). The system model was derived on the basis of the ZU 23-2MR set manufactured in Poland. The work discusses linearization of the mathematical model of the system with the help of Jacobians, presents the structure of the control system and proposes a method of minimizing the control error in the steady state, without the use of additional integrating elements occurring in the control system. The study also examined the impact of interference from a shooting cannon on the accuracy of control while tracking a maneuvering target. The influence of inaccurate identification of control object parameters on system operation was also analyzed. Numerical simulations were carried out using the Scilab environment, and the most important results were presented in the graphic form.

7

Contact between 3D beams with deformable circular cross-sections : numerical verification

Kawa O., Litewka P., Studziński R.

Engineering Transactions

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2018

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Vol. 66, iss. 3

281--299

EN

In this paper a numerical analysis of contact between three-dimensional elastic beams with deformations at the contact zone is carried out. The authors propose a new model of beam-to-beam contact which is the continuation of ideas presented in [6, 7, 10]. The results of beam-to-beam contact analysis are compared with the ones for full 3D problem solved in the Abaqus/enviroment. The aim of the conducted numerical simulations was to select the most appropriate 3D model and to use it as a reference to verify the accuracy of the proposed beam-to-beam contact definition. The verifications were carried out for contact between beams with circular cross-sections. The obtained contact forces and the displacements of beams tips for different beams arrangements and boundary conditions showed a satisfactory correlation.

8

The linearization of mooring load distribution problem

Artyszuk J.

Zeszyty Naukowe Akademii Morskiej w Szczecinie

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2016

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nr 46 (118)

9--16

EN

This paper presents a novel step forward in finding the loads of particular mooring ropes that balance the steady environmental excitations for a ship staying at berth. The industrial static equilibrium method for a rough assessment of ship mooring safety is considered to be well-established. The static loads are directly related to the rope’s MBLs (minimum breaking loads) while applying a certain safety margin (usually 50%). The problem is reduced to a set of linear equations that may be solved analytically. The generality in terms of arbitrary horizontal and vertical angles of mooring ropes is preserved. All derivations are provided to enhance trust in the very simple yet absolutely accurate and fast linear solution. The accuracy is studied both analytically, throughout all the development stages, and finally by comparison to the exact numerical solution of the original nonlinear equilibrium equations for an exemplary mooring pattern. A discussion of selected effects in load distribution is also given. Using the approach presented, for instance, we can efficiently test mooring safety when any mooring rope of the set is accidentally broken.

9

Test bench and simulation research on hydrodynamic torque converter for different input signals

Wozniak M., Siczek K., Pawelski Z.

Journal of KONES

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2016

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Vol. 23, No. 4

561--568

EN

This paper presents the experimental research and simulating results of a ZM130 hydrodynamic torque converter under steady and unsteady working conditions with an external step signal and sinus on turbine’s shaft on a modified test bench. The mathematical model of torque converter was described by the one-dimensional flow theory with variable loss coefficients. The character of received equations from test bend showed by linearized transmittance functions with variable coefficients. For test bend object identification researches will be used an Identification Toolbox from a mathematical code in Matlab, which enables to create transfer functions as a linear model from measurements data as they got the highest correctness level with input signal (as they get the best correction level with input signal on the pump´s shaft). This mathematical model for dynamic loads will be verified by experimental tests. The aim of the carried out studies was determination the effect of sampling time decreasing on the quality of measured parameters. Results of present investigations were compared with those obtained earlier, but for the same input signals .It is possible to affirm that proposed theoretical model imitates the change of dynamic parameters, and the same the property of the torque converter. Researches led on the test bend with extortion input torque sinus signal on the turbine’s shaft confirmed correctness of this model.

10

Investigation of the dynamics of singular protected systems

Naumeyko I., Alja'afreh M.

ECONTECHMOD : An International Quarterly Journal on Economics of Technology and Modelling Processes

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2016

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Vol. 5, No 4

37--42

EN

The analysis of asymptotic representations of the systems protected from harmful influences is carried out. Various types of general models of the "manmachineenvironment" with protection are considered. Each of them adequately describes some of the practically important qualities of the object, and they all together describe the object in terms of its safe operation. The dynamic properties of complex ergonomic systems, presented in the form of systems of differential equations with a small parameter at the derivative are investigated The methods of reducing the impact on the person of harmful factors are theoretically substantiated. The dynamic protection response speed is considered to be significantly greater than the harmful factor production rate. Numerical solution of the general problem and the analytical solution for autonomous case is obtained for harmful effects. By using asymptotic the system of equations has been solved in closed form not only for autonomous case, but also for parameters smoothly changing in time. The estimates of the cost of protection was obtained for the various cost-functionals and for different functions in the right-hand side of the equation describing the dynamics of protection. To assess the accuracy of model calculations and for graphic representation of the results mathematical package MAXIMA is used.

11

A Novel Design Of An NTC Thermistor Linearization Circuit

Lukić J., Denić D.

Metrology and Measurement Systems

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2015

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Vol. 22, nr 3

351--362

EN

A novel design of a circuit used for NTC thermistor linearization is proposed. The novelty of the proposed design consists in a specific combination of two linearization circuits, a serial-parallel resistive voltage divider and a two-stage piecewise linear analog-to-digital converter. At the output of the first linearization circuit the quasi-linear voltage is obtained. To remove the residual voltage nonlinearity, the second linearization circuit, i.e., a two-stage piecewise linear analog-to-digital converter is employed. This circuit is composed of two flash analog-to-digital converters. The first analog-to-digital converter is piecewise linear and it is actually performing the linearization, while the second analog-to-digital converter is linear and it is performing the reduction of the quantization error introduced by the first converter. After the linearization is performed, the maximal absolute value of a difference between the measured and real temperatures is 0.014°C for the temperature range between - 25 and 75°C, and 0.001°C for the temperature range between 10 and 40°C.

12

Methods for assessing the operating parameters of motor vehicles

Różowicz S.

Logistyka

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2015

|

nr 4

8248--8253, CD 2

PL

W pracy przedstawiono trzy metody diagnozowania, które pozwalają nam uzyskać informacje o stanie technicznym poszczególnych elementów danego układu. We współczesnych pojazdach samochodowych steruje się nie tylko pracą układu wtryskowo-zapłonowego, lecz pracą całego silnika, dlatego zmniejszenie kosztów naprawy pojazdu można uzyskać poprzez zastosowanie jednej z metod diagnozowania.

13

Positivity and linearization of a class of nonlinear continuous-time systems by state feedbacks

Kaczorek T.

International Journal of Applied Mathematics and Computer Science

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2015

|

Vol. 25, no. 4

827--831

EN

The positivity and linearization of a class of nonlinear continuous-time system by nonlinear state feedbacks are addressed. Necessary and sufficient conditions for the positivity of the class of nonlinear systems are established. A method for linearization of nonlinear systems by nonlinear state feedbacks is presented. It is shown that by a suitable choice of the state feedback it is possible to obtain an asymptotically stable and controllable linear system, and if the closed-loop system is positive then it is unstable.

14

Dynamic balance research of protected systems

Naumeyko I., Alja'afreh M.

ECONTECHMOD : An International Quarterly Journal on Economics of Technology and Modelling Processes

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2015

|

Vol. 4, No 3

85-90

EN

The dynamic models of the complex ergatic objects' behavior, presented in the form of differentia equations and their systems were studied. The stability and other properties are researched. The methods of analysis and reduce of harmful factors and their impact on people were theoretically proved. The methods of analysis and critical points removal in dynamic models of hazards distribution are offered. The object of study is the system of the harmful external factors protection. Subject of research is the system of two nonlinear differential equations as a model of technical systems with protection. The object of protection is described by logistic equation. and defense system - by non-linear differential equation with a security functions of rather general form. This paper describes critical modes analysis and stationary states’ stability of protected systems with harmful influences. Numerical solution of general problem and also the analytical solution for the case of fixed expected harmful effects have been obtained. Various types of general models for "Man-machine-environment" systems were studied. Each of describes some kind of the practically important quality of object in an appropriate way. And All together they describe the object in terms of it’s safe operation. Their further detailing process results to either well-known, or some new subsystems’ models. Systems with "fast" protection at a relatively slow dynamics of the object were studied. This leads to the models with small parameter and asymptotic solutions of differentia equations. Some estimates for protection cost in different price-functional and for different functions in the right part of equation, which describes the dynamics of defense were obtained. For calculations, analysis and graphical representations some of mathematical packages was applied.

15

Finite-time stability and stabilization of singular state-delay systems using improved estimation of a lower bound on a Lyapunov-like functional

Stojanovic S. B., Debeljkovic D. Lj., Antic D. S.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2015

|

Vol. 63, nr 2

479--487

EN

In this paper, the problems of finite-time stability and stabilization for a class of singular time-delay systems are studied. Using the Lyapunov-like functional (LLF) with (exponential or power) weighting function and a new estimation method for the lower bound on LLF, some sufficient stability conditions are introduced. It is shown that the weighting function significantly reduces the conservatism of the stability criteria in comparison to estimation of the lower bound on LLF without this function. To solve the finite-time stabilization problem, a stabilizing linear state controller is designed by exploiting the cone complementarity linearization algorithm. Two numerical examples are given to illustrate the effectiveness of the proposed method.

16

Analiza porównawcza metod wyznaczania transformacji linearyzujących nieliniowe równania stanu układu

Zawadzki A.

Przegląd Elektrotechniczny

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2014

|

R. 90, nr 4

28--33

PL

W pracy przedstawiono zastosowanie elementów teorii geometrii różniczkowej do konstrukcji transformacji przestrzeni stanu, linearyzującej dynamiczny układ nieliniowy. Omówiono metody wyznaczania transformacji linearyzujących nieliniowe równania stanu układu (wejście-stan układu nieliniowego) oraz przeprowadzono ich analizę porównawczą. Przedstawiono warunki, jakie musi spełniać układ nieliniowy aby możliwe było przeprowadzenie zabiegów linearyzujących.

EN

The paper presents an application of elements of the theory of differential geometry for building the state space transformation, which linearizes a nonlinear dynamic system. The methods of determining the transformations for linearizing the nonlinear state equations (input-state of nonlinear system) are discussed and comparative analysis of them is performed. The conditions to be met by a nonlinear system to be able to carry out linearizing operations are also presented.

17

Zastosowanie rachunku ułamkowego rzędu do modelowania pewnej klasy generatorów nieliniowych

Zawadzki A., Włodarczyk M.

Prace Naukowe Politechniki Śląskiej. Elektryka

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2014

|

z. 1

65--73

PL

W pracy przedstawiono metodę linearyzacji nieliniowego równania stanu z zastosowaniem geometrycznej transformacji zmiennych stanu. Do konstrukcji tej transformacji wykorzystano elementy algebry Liego. Podano odpowiednie definicje i twierdzenia, na podstawie których dokonano linearyzacji nieliniowego równania stanu drugiego rzędu, opisującego pewien nieliniowy generator. Następnie w zlinearyzowanym równaniu, zmieniając operator różniczkowy pierwszego rzędu na operator rzędu ułamkowego, uzyskano liniowy ciągły układ ułamkowego rzędu, który rozwiązano stosując macierzowe funkcje Mittag-Lefflera. Uzyskane wyniki w porównaniu z rozwiązaniem równania pierwszego rzędu przedstawiono na wykresach. Obliczenia wykonano dla rzędów mniejszych i większych od jednego, co pozwoliło na wyciągnięcie pewnych wniosków.

EN

Linearization method for nonlinear equation of state by using the geometric transformation of the state variables is shown in the paper. Some elements of Lie algebra was used for the construction of this transformation. All necesary definitions and principles that are required to linearize the nonlinear state equation describing a second-order non-linear generator is shown. Continuous linear fractional system is the result of change of the first order differential operator on the order of the fractional operator in the linearized equation. It was solved by using matrix Mittag-Leffler functions. A comparison of the results with the solution of equations of the first order is made. The results are shown in diagrams. The calculations were made for the lower and higher than first orders of equations.

18

Positivity and linearization of a class of nonlinear discrete-time systems by state-feedbacks

Kaczorek T.

Logistyka

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2014

|

nr 6

5078-5083

EN

The positivity and linearization of a class of nonlinear discrete-time system by nonlinear state-feedbacks are addressed. Necessary and sufficient conditions for the positivity of the class of nonlinear systems are established. A method for linearization of the nonlinear systems by nonlinear state-feedbacks is presented. It is shown that by suitable choice of state-feedbacks it is possible to obtain asymptotically stable and controllable linear system.

PL

Praca jest poświęcona dodatniości i linearyzacji nieliniowych układów dyskretnych za pomocą nieliniowych sprzężeń zwrotnych od wektora stanu. Sformułowano warunki konieczne i wystarczające dodatniości tej klasy układów nieliniowych. Podano metodę linearyzacji tych układów za pomocą nieliniowych sprzężeń zwrotnych od wektora stanu. Wykazano, że przez odpowiedni dobór nieliniowych sprzężeń zwrotnych można uzyskać układ liniowy sterowalny i stabilny asymptotycznie.

19

Filtracja Kalmana w technice na przykładzie urządzenia SST

Zając M.

Zeszyty Naukowe Wyższej Szkoły Informatyki

|

2013

|

Vol. 12, Nr 1

5--20

PL

W artykule podjęto próbę opisu filtracji Kalmana zarówno dla zagadnień liniowych jak i nieliniowych. Omówiono linearyzację dla zagadnień nieliniowych, wykorzystano zgromadzone informacje do budowy praktycznej filtra Kalmana. Zbadano możliwość wykorzystania założeń teoretycznych w praktycznej realizacji dotyczącej urządzenia pomiarowego i badania właściwości ferromagnetyka przy wykorzystaniu SST.

EN

It is an attempt to describe the Kalman filter both for linear and non-linear. The linearization for nonlinear phenomenon was descussed, and information were used to build practical Kalman filter.

20

Solution of the Kalman filtering problem in control and modeling of a double inverted pendulum with rolling friction

Olejnik P., Awrejcewicz J., Niełaczny M.

Pomiary Automatyka Robotyka

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2013

|

R. 17, nr 1

63-70

EN

The problem of control and dynamical modeling of a unicycle-cyclist system treated as a double inverted pendulum with rolling friction and vibrating in the plane that is perpendicular to the direction of movement is studied. The object of analysis consists of two basic parts: the wheel and the double pendulum. The equations of motion have been derived using the Lagrange equation of the second kind. The kinematic excitation has been applied to the cyclist. The aim of control is to maintain the unicyclecyclist system in an unstable equilibrium around given angular position. Control moment of force has been applied to the wheel in a numerical procedure performed in Simulink. Kalman filtering problem has been solved basing on the feedback control system and functions included in Control System Toolbox of MATLAB. The proposed approach could be extended in future to solve some dynamical problem of transverse vibrations.

PL

Praca podejmuje rozwiązanie zagadnienia kontroli i modelowania dynamicznego układu monocykl-rowerzysta rozpatrzonego jako podwójne wahadło odwrócone z tarciem tocznym drgające w płaszczyźnie prostopadłej do kierunku ruchu. Analizowany obiekt składa się z dwóch podstawowych części, tzn. koła i podwójnego wahadła. Równania ruchu układu mechanicznego, w którym wymuszenie kinematyczne przyłożono do masy reprezentującej ciało rowerzysty wyprowadzono stosując równania Lagrange’a drugiego rodzaju. Zadaniem kontroli było utrzymanie modelu układu monocykl-rowerzysta w niestabilnym położeniu równowagi w pobliżu zadanej wartości kąta obrotu. W procedurze numerycznej zrealizowanej w Simulinku moment siły kontroli przyłożono do koła napędzającego układ modelowy. Zagadnienie doboru filtru Kalmana rozwiązano w układzie sterowania ze sprzężeniem zwrotnym przy użyciu wbudowanych funkcji numerycznych wchodzących w skład biblioteki Control System Toolbox pakietu MATLAB. Zaproponowane podejście można rozszerzyć w przyszłości w celu rozwiązania podobnego problemu dynamicznego uwzględniającego drgania poprzeczne.