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1

Automated Linearization of a System of Nonlinear Ordinary Differential Equations

Mazakova Aigerim, Jomartova Sholpan, Wójcik Waldemar, Mazakov Talgat, Ziyatbekova Gulzat

International Journal of Electronics and Telecommunications

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2023

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Vol. 69, No. 4

655--600

EN

This paper investigates the possibility of automatically linearizing nonlinear models. Constructing a linearised model for a nonlinear system is quite labor-intensive and practically unrealistic when the dimension is greater than 3. Therefore, it is important to automate the process of linearisation of the original nonlinear model. Based on the application of computer algebra, a constructive algorithm for the linearisation of a system of non-linear ordinary differential equations was developed. A software was developed on MatLab. The effectiveness of the proposed algorithm has been demonstrated on applied problems: an unmanned aerial vehicle dynamics model and a twolink robot model. The obtained linearized models were then used to test the stability of the original models. In order to account for possible inaccuracies in the measurements of the technical parameters of the model, an interval linearized model is adopted. For such a model, the procedure for constructing the corresponding interval characteristic polynomial and the corresponding Hurwitz matrix is automated. On the basis of the analysis of the properties of the main minors of the Hurwitz matrix, the stability of the studied system was analyzed.

2

Zeroing the transfer matrix of the Roesser model of 2-D linear systems

Kaczorek Tadeusz

International Journal of Applied Mathematics and Computer Science

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2023

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Vol. 33, no. 4

513--519

EN

Controllability, observability and the transfer matrix of the discrete 2-D Roesser model are analyzed. It is shown that the controllability of the Roesser model is invariant under state feedbacks and the observability under output feedbacks. Sufficient conditions are established for the zeroing of the transfer matrix of the Roesser model.

3

Results on the controllability of Caputo’s fractional descriptor systems with constant delays

Beata Sikora

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2023

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Vol. 71, nr 4

art. no. e146287

EN

The paper investigates the controllability of fractional descriptor linear systems with constant delays in control. The Caputo fractional derivative is considered. Using the Drazin inverse and the Laplace transform, a formula for solving of the matrix state equation is obtained. New criteria of relative controllability for Caputo’s fractional descriptor systems are formulated and proved. Both constrained and unconstrained controls are considered. To emphasize the importance of the theoretical studies, an application to electrical circuits is presented as a practical example.

4

On the control of the final speed for a class of finite-dimensional linear systems: controllability and regulation

Rachik Mostafa, Khaloufi Issam, Benfatah Youssef, Boutayeb Hamza, Laarabi Hassan

Archives of Control Sciences

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2023

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Vol. 33, No. 3

497--525

EN

In this article, we extended the concept of controllability, traditionally used to control the final state of a system, to the exact control of its final speed. Inspired by Kalman’s theory, we have established some conditions to characterize the control that allows the system to reach a desired final speed exactly. When the assumptions ensuring speed-controllability are not met, we adopt a regulation strategy that involves determining the control law to make the system’s final speed approach as closely as possible to the predefined final speed, and this at a lower cost. The theoretical results obtained are illustrated through three examples.

5

On the computation of hierarchical control results for one-dimensional transmission line

Pinheiro de Carvalho Pitagoras, Pereira de Jesus Isaías, Pereira de Sá Neto Olimpio

Mathematica Applicanda

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2022

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Vol. 50, no. 1

107--128

EN

In this paper, motivated by a physics problem, we investigate some numerical and computational aspects for the problem of hierarchical controllability in one-dimensional wave equation in domains with a moving boundary. Some controls act in part of the boundary and define a strategy of equilibrium between them, considering a leader control and a follower. Thus, we introduced the concept of hierarchical control to solve the problem and mapped the Stackelberg Strategy between these controls. The numerical methods described here consist of a combination of the following: finite element method (FEM) for space approximation; finite difference method (FDM) for time discretization and fixed-point algorithm for the solution of the total discrete control problem. Data programming and computer simulations are performed in FreeFem++ and for a better presentation of the experiments we use Matlab.

PL

W tym artykule, motywowanym problemem fizycznym, badamy pewne aspekty numeryczne i obliczeniowe problemu sterowalności hierarchicznej w jednowymiarowym równaniu falowym w domenach z ruchomą granicą. Niektóre sterowania są na brzegu i definiują strategię równowagi między nimi, biorąc pod uwagę sterowania lidera i naśladowcę. W związku z tym wprowadziliśmy koncepcję sterowań hierarchicznych w celu rozwiązania tego problemu i przyporządkowaliśmy strategię Stackelberga między tymi strategiami. Opisane tutaj metody numeryczne składają się z kombinacji następujących elementów: metoda elementów skończonych (MES) dla aproksymacji przestrzennej; metoda różnic skończonych (FDM) do dyskretyzacji czasu i algorytm stałoprzecinkowy do rozwiązania problemu całkowitego sterowania dyskretnego. Programowanie danych i symulacje komputerowe wykonujemy w FreeFem++, a dla lepszej prezentacji eksperymentów używamy Matlaba.

6

Some analysis problems of the linear systems

Kaczorek Tadeusz

Journal of Automation, Electronics and Electrical Engineering

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2022

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Vol. 4, No. 2

7--12

EN

New approaches to the transformations of the uncontrollable and unobservable matrices of linear systems to their canonical forms are proposed. It is shown that the uncontrollable pair (A,B) and unobservable pair (A,C) of linear systems can be transform to their controllable (𝐴̅ , 𝐵̅ ) and observable (𝐴̅ , 𝐶̅ ) canonical forms by suitable choice of nonsingular matrix M satisfying the condition 𝑀[𝐴 𝐵] = [𝐴̅ 𝐵̅ ] and [𝐴 𝐵]𝑀 = [𝐴̂ 𝐵̂ ], respectively. It is also shown that by suitable choice of the gain matrix K of the feedbacks of the derivative of the state vector it is possible to reduce the descriptor system to the standard one.

PL

Zaproponowano nowe podejścia do transformacji niesterowalnych i nieobserwowalnych macierzy układów liniowych do ich postaci kanonicznych. Wykazano, że niesterowalna para (A,B) i nieobserwowalna para (A,C) układów liniowych może być przekształcona do ich postaci kanonicznych sterowalnych i obserwowalnych prze odpowiedni dobór nieosobliwej macierzy M spełniającej warunki 𝑀[𝐴 𝐵] = [𝐴̅ 𝐵̅ ] i [𝐴 𝐵]𝑀 = [𝐴̂ 𝐵̂ ]. Pokazano,żeprzezodpowiednidobórmacierzyKsprzężeniazwrotnego od pochodnej wektora stanu jest możliwa redukcja układu deskryptorowegodoukładustandardowego.

7

Eigenvalues assignment in uncontrollable linear systems

Kaczorek Tadeusz

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2022

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Vol. 70, nr 6

art. no. e141987

EN

It is shown that in uncontrollable linear system x = Ax + Bu it is possible to assign arbitrarily the eigenvalues of the closed-loop system with state feedbacks u = Kx, K ∈ ℜn⨉m if rank [A B] = n. The design procedure consists in two steps. In the step 1 a nonsingular matrix M ∈ ℜn⨉m is chosen so that the pair (MA,MB) is controllable. In step 2 the feedback matrix K is chosen so that the closed-loop matrix Ac = A − BK has the desired eigenvalues. The procedure is illustrated by simple example.

8

Controllability of linear convex combination of linear discrete-time fractional systems

Kaczorek Tadeusz, Klamka Jerzy, Dzieliński Andrzej

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2022

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Vol. 70, nr 5

art. no. e143102

EN

In this paper the controllability properties of the convex linear combination of fractional, linear, discrete-time systems are characterized and investigated. The notions of linear convex combination and controllability in the context of fractional-order systems are recalled. Then, the controllability property of such a linear combination of discrete-time, linear fractional systems is proven. Further, the reduction of an infinite problem of transition matrix derivation is reduced to a finite one, which greatly simplifies the numerical burden of the controllability issue. Examples of controllable and uncontrollable, single-input, linear systems are presented. The possibility of extension of the considerations to multi-input systems is shown.

9

Improving the controllability of the electro-hydraulic complex in emergency operating modes

Korenkova Tetyana, Kovalchuk Viktoriya

Przegląd Elektrotechniczny

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2021

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R. 97, nr 12

121--124

EN

The possibility of using reversible operating modes of hydraulic and electric machines in the problems of improving the controllability of an electrohydraulic complex is shown. The schemes of energy balance in pumping and turbine operating modes of hydraulic machines are given. A mathematical description of the four-quadrant characteristics of the pumping unit is proposed. The complete characteristics of the pump operating on the pipeline network with back pressure, when the direction of rotation of the pump electric motor was changed and there was reverse of the liquid in the hydraulic system, are obtained. A structural diagram of a system for active regulation of pumping unit parameters based on the use of hydro turbine units or reversible hydraulic machines is proposed.

PL

Pokazano możliwość użycia odwracalnego trybu pracy maszyn hydraulicznych i elektrycznych w polepszaniu sterowności zespołu elektrohydraulicznego. Podane zostały schematy równowagi energii w trybie pompowania i pracy turbinowej maszyny hydraulicznej. Dla jednostki pompowanie z czterokwadrantowym trybem pracy opracowano modele matematyczne. Otrzymano finalne charakterystyki działania pompy i sieci rurociągowej z ciśnieniem wstecznym w warunkach zmiany kierunku obrotów napędu elektrycznego pompy i cofania płynu w systemie hydraulicznym. Zaproponowany został diagram parametrów aktywnej regulacji jednostki pompowania, wykorzystujący jednostki turbin wodnych lub odwracalnych maszyn hydraulicznych.

10

Positive output controllability of linear discrete–time invariant systems

Ouyadri Mourad, Laabissi Mohamed, Achhab Mohammed Elarbi

Control and Cybernetics

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2021

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Vol. 50, No. 4

521--539

EN

This paper studies the output controllability of discrete linear time invariant systems (LTI) with non-negative input constraints. Some geometrical arguments and positive invariance concepts are used to derive the necessary and/or sufficient conditions for the positive output controllability of discrete LTI systems. The paper also provides several academic examples, which support the theoretical results.

11

Convex linear combination of the controllability pairs for linear systems

Kaczorek Tadeusz, Klamka Jerzy

Control and Cybernetics

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2021

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Vol. 50, No. 4

541--551

EN

The convex linear combination of the controllability pairs of linear continuous-time linear systems is defined and its properties are discussed. The main result is obtained using pure algebraic methods. In the illustrative examples different cases of linear convex combinations are analyzed.

12

On controllability of fractional positive continuous-time linear systems with delay

Sikora Beata, Matlok Natalia

Archives of Control Sciences

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2021

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Vol. 31, No. 1

29--51

EN

In the paper positive fractional continuous-time linear systems are considered. Positive fractional systems without delays and positive fractional systems with a single delay in control are studied. New criteria for approximate and exact controllability of systems without delays as well as a relative controllability criterion of systems with delay are established and proved. Numerical examples are presented for different controllability criteria. A practical application is proposed.

13

Application of maximum principle to optimization of production and storage costs

Popescu Liviu, Dimitrov Ramona

Archives of Control Sciences

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2021

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Vol. 31, No. 4

865--881

EN

A problem of optimization for production and storge costs is studied. The problem consists in manufacture of n types of products, with some given restrictions, so that the total production and storage costs are minimal. The mathematical model is built using the framework of driftless control affine systems. Controllability is studied using Lie geometric methods and the optimal solution is obtained with Pontryagin Maximum Principle. It is proved that the economical system is not controllable, in the sense that we can only produce a certain quantity of products. Finally, some numerical examples are given with graphical representation.

14

The practical feedback stabilization for evolution equations in Banach spaces

Hanen Damak

Acta Mechanica et Automatica

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2021

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Vol. 15, no. 2

58--65

EN

This paper investigates the notion of practical feedback stabilization of evolution equations satisfying some relaxed conditions in infinite-dimensional Banach spaces. Moreover, sufficient conditions are presented that guarantee practical stabilizability of uncertain systems based on Lyapunov functions. These results are applied to partial differential equations.

15

Controllability of multi-term time-fractional differential systems with state-dependent delay

Chaudhary Renu, Singh Vikram, Pandey D. N.

Journal of Applied Analysis

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2020

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Vol. 26, nr 2

241--255

EN

In this paper, controllability results for a class of multi-term time-fractional differential systems with state-dependent delay have been studied. The concept of fractional calculus, measure of noncompactness and Mönch fixed-point theorem has been implemented to obtain a new set of controllability results. Finally, an application is given to illustrate the obtained results.

16

Independence of the controllability and observability of resistances in linear electrical circuits

Kaczorek Tadeusz

Journal of Automation, Electronics and Electrical Engineering

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2019

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Vol. 1, No. 1

9--15

EN

Sufficient conditions are given under which the controllability and observability of linear electrical circuits is independent of their resistances. In some particular cases the observability depends only on the capacitances or inductances of the electrical circuits.

17

Semantics and Controllability of Time-Aware Business Processes

De Angelis Emanuele, Fioravanti Fabio, Meo Maria Chiara, Pettorossi Alberto, Proietti Maurizio

Fundamenta Informaticae

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2019

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Vol. 165, nr 3-4

205--244

EN

We present an operational semantics for time-aware business processes, that is, processes modeling the execution of business activities, whose durations are subject to linear constraints over the integers. We assume that some of the durations are controllable, that is, they can be determined by the organization that executes the process, while others are uncontrollable, that is, they are determined by the external world. Then, we consider controllability properties, which guarantee the completion of the execution of the process, satisfying the given duration constraints, independently of the values of the uncontrollable durations. Controllability properties are encoded by quantified reachability formulas, where the reachability predicate is recursively defined by means of constrained Horn clauses (CHCs). These clauses are automatically derived from the operational semantics of the process. Finally, we present two algorithms for solving the so called weak and strong controllability problems. Our algorithms reduce these problems to the verification of a set of quantified integer constraints, which are simpler than the original quantified reachability formulas, and can effectively be handled by state-of-the-art CHC solvers.

18

Assignability of numerical characteristics of time-varying systems

Babiarz A., Czornik A., Klamka J.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2019

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Vol. 67, nr 6

1007--1022

EN

The main aim of this article is to survey and discuss the existing state of art concerning the assignability by a feedback of numerical characteristics of linear continuous and discrete time-varying systems. Most of the results present necessary or sufficient conditions for different formulation of the Lyapunov spectrum assignability problem. These conditions are expressed in terms of various controllability types and optimalizability of the controlled systems and certain properties of the free system such as: regularity, diagonalizability, boundness away, integral separation and reducibility.

19

Organizational culture deformations conditions af intellectual work virtualization

Litvinenko O.

Zeszyty Naukowe Wyższej Szkoły Zarządzania Ochroną Pracy w Katowicach

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2017

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Nr 1(13)

66--78

EN

The study examines the effects of intellectual work virtualization on internal uncertainness of organizations using the entropy approach adopted from the information theory. The results are grouped using marginal cases marking limits of entropy value of virtualized organization linked to specific type of organizational culture.

PL

Artykuł bada wpływ wirtualizacji pracy intelektualnej na wewnętrzną niepewność organizacji stosujących podejście entropii przyjęte z teorii informacji. Wyniki zostały pogrupowane przy zastosowaniu marginalnych przypadków okreslających granice wartości entropii wirtualnych organizacji powiązanych ze specyficznym typem kultury organizacyjnej.

20

Methodology of simultaneous utilizationof case study and immersion durning teaching of managerial disciplines

Momot V.

Zeszyty Naukowe Wyższej Szkoły Zarządzania Ochroną Pracy w Katowicach

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2017

|

Nr 1(13)

111--122

EN

The study examines the effects of simultaneous utilization of immersion and case studies on the outcome of practical preparation of future professional raising their value for the labor market. The results are indicating that immersion is capable of amplifying the outcome of case studies by raising the level of focusing during the in-class work and independence of decision-making.

PL

Artykuł omawia wpływ równoległego wykorzystania immersji i studiów przypadku na praktyczne przygotowanie przyszłego pracownika podnosząc jego wartość na rynku pracy. Wyniki wskazują, że immersja może zwielokrotnić wyniki studiów przypadku poprzez podniesienie poziomu koncentracji w czasie pracy na zajęciach i wzmocnić niezależność w podejmowania decyzji.