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Archives of Control Sciences

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Archives of Control Sciences | 2021 | vol. 31 | No 4 |

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Abstract

Dynamics and control of discrete chaotic systems of fractional-order have received considerable attention over the last few years. So far, nonlinear control laws have been mainly used for stabilizing at zero the chaotic dynamics of fractional maps. This article provides a further contribution to such research field by presenting simple linear control laws for stabilizing three fractional chaotic maps in regard to their dynamics. Specifically, a one-dimensional linear control law and a scalar control law are proposed for stabilizing at the origin the chaotic dynamics of the Zeraoulia-Sprott rational map and the Ikeda map, respectively. Additionally, a two-dimensional linear control law is developed to stabilize the chaotic fractional flow map. All the results have been achieved by exploiting new theorems based on the Lyapunov method as well as on the properties of the Caputo h-difference operator. The relevant simulation findings are implemented to confirm the validity of the established linear control scheme.
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Authors and Affiliations

A. Othman Almatroud
1
Adel Ouannas
2
Giuseppe Grassi
3
Iqbal M. Batiha
4
Ahlem Gasri
5
M. Mossa Al-Sawalha
1

  1. Department of Mathematics, Faculty of Science, University of Ha'il, Ha'il 81451, Saudi Arabia
  2. Department of Mathematics and Computer Science, University of Larbi Ben M’hidi, Oum El Bouaghi 04000, Algeria
  3. Dipartimento Ingegneria Innovazione, Universita del Salento, 73100 Lecce, Italy
  4. Department of Mathematics, Faculty of Science and Information Technology, Irbid National University, Irbid, Jordan and Nonlinear Dynamics Research Center (NDRC), Ajman University, Ajman, UAE
  5. Department of Mathematics, University of Larbi Tebessi, Tebessa 12002, Algeria
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Abstract

Hybrid systems (HS) are roughly described as a set of discrete state transitions and continuous dynamics modeled by differential equations. Parametric HS may be constructed by having parameters on the differential equations, initial conditions, jump conditions, or a combination of the previous ones. In real applications, the best solution is obtained by a set of metrics functional over the set of solutions generated from a finite set of parameters. This paper examines the choice of parameters on delta-reachability bounded hybrid systems.We present an efficient model based on the tool pHL-MT to benchmark the HS solutions (based on dReach), and a non-parametric frontier analysis approach, relying on multidirectional efficiency analysis (MEA). Three numerical examples of epidemic models with variable growth infectivity are presented, namely: when the variable of infected individuals oscillates around some endemic (non-autonomous) equilibrium; when there is an asymptotically stable non-trivial attractor; and in the presence of bump functions.
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Authors and Affiliations

Eugénio Miguel Alexandre Rocha
1
Kelly Patricia Murillo
1

  1. Center for Research and Development in Mathematics and Applications, and Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal
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Abstract

The paper is concerned with the presentation and analysis of the Dynamic Matrix Control (DMC) model predictive control algorithm with the representation of the process input trajectories by parametrised sums of Laguerre functions. First the formulation of the DMCL (DMC with Laguerre functions) algorithm is presented. The algorithm differs from the standard DMC one in the formulation of the decision variables of the optimization problem – coefficients of approximations by the Laguerre functions instead of control input values are these variables. Then the DMCL algorithm is applied to two multivariable benchmark problems to investigate properties of the algorithm and to provide a concise comparison with the standard DMC one. The problems with difficult dynamics are selected, which usually leads to longer prediction and control horizons. Benefits from using Laguerre functions were shown, especially evident for smaller sampling intervals.
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Authors and Affiliations

Piotr Tatjewski
1

  1. Warsaw University of Technology, Nowowiejska15/19, 00-665 Warszawa, Poland
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Abstract

The paper is a newapproach to the Duhamel integral. It contains an overviewof formulas and applications of Duhamel’s integral as well as a number of new results on the Duhamel integral and principle. Basic definitions are recalled and formulas for Duhamel’s integral are derived via Laplace transformation and Leibniz integral rule. Applications of Duhamel’s integral for solving certain types of differential and integral equations are presented. Moreover, an interpretation of Duhamel’s formula in the theory of operator semigroups is given. Some applications of Duhamel’s formula in control systems analysis are discussed. The work is also devoted to the usage of Duhamel’s integral for differential equations with fractional order derivative.
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Authors and Affiliations

Michał Różański
1
Beata Sikora
1
Adrian Smuda
1
Roman Wituła
1

  1. Department of Applied Mathematics, Silesian University of Technology, Kaszubska 23, 44-100 Gliwice, Poland
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Abstract

In this work, we present optimal control formulation and numerical algorithm for fractional order discrete time singular system (DTSS) for fixed terminal state and fixed terminal time endpoint condition. The performance index (PI) is in quadratic form, and the system dynamics is in the sense of Riemann-Liouville fractional derivative (RLFD). A coordinate transformation is used to convert the fractional-order DTSS into its equivalent non-singular form, and then the optimal control problem (OCP) is formulated. The Hamiltonian technique is used to derive the necessary conditions. A solution algorithm is presented for solving the OCP. To validate the formulation and the solution algorithm, an example for fixed terminal state and fixed terminal time case is presented.
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Authors and Affiliations

Tirumalasetty Chiranjeevi
1
Raj Kumar Biswas
2
Ramesh Devarapalli
3
ORCID: ORCID
Naladi Ram Babu
2
Fausto Pedro García Márquez
4

  1. Department of Electrical Engineering, Rajkiya Engineering College Sonbhadra, U. P., India
  2. Department of Electrical Engineering, National Institute of Technology, Silchar, India
  3. Department of Electrical Engineering, BIT Sindri, Dhanbad 828123, Jharkhand, India
  4. Ingenium Research Group, University of Castilla-La Mancha, Spain
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Abstract

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.
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Authors and Affiliations

Liviu Popescu
1
Ramona Dimitrov
1

  1. University of Craiova, Faculty of Economics and Business Administration, Department of Statistics and Economic Informatics, Al. I. Cuza st., No. 13, Craiova 200585, Romania
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Abstract

Small bucket models with many short fictitious micro-periods ensure high-quality schedules in multi-level systems, i.e., with multiple stages or dependent demand. In such models, setup times longer than a single period are, however, more likely. This paper presents new mixedinteger programming models for the proportional lot-sizing and scheduling problem (PLSP) with setup operations overlapping multiple periods with variable capacity.
A new model is proposed that explicitly determines periods overlapped by each setup operation and the time spent on setup execution during each period. The model assumes that most periods have the same length; however, a few of them are shorter, and the time interval determined by two consecutive shorter periods is always longer than a single setup operation. The computational experiments showthat the newmodel requires a significantly smaller computation effort than known models.
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Authors and Affiliations

Waldemar Kaczmarczyk
1

  1. Department of Strategic Management, AGH University of Science and Technology, Al.Mickiewicza 30, 30-059, Kraków, Poland

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