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1

Fractional optimal control approach to the diabetics model

Nadeem Muhammad, Habib Mustafa, Arif Ayesha, Asjad Muhammad Imran

Journal of Applied Mathematics and Computational Mechanics

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2024

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Vol. 23, nr 1

71-82

EN

Diabetes mellitus is one of the most critical diseases, affecting millions of people around the world. This work deals with the fractional optimal control of the dynamics of the population model on diabetes. This framework is based on the fractional order differential problems that describe the population before and after diabetes involving some health problems. We consider the Caputo derivatives for the study of the proposed model. The maximum principle of Pontryagin is utilized to derive the necessary conditions for the optimality of a dynamical system. Using a forward-backward sweep approach with the generalized Euler method accomplishes numerical solutions of formulated issues.

2

Degradation tolerant optimal control design for stochastic linear systems

Kanso Soha, Jha Mayank Shekhar, Theilliol Didier

International Journal of Applied Mathematics and Computer Science

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2024

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Vol. 34, no. 1

5--14

EN

Safety-critical and mission-critical systems are often sensitive to functional degradation at the system or component level. Such degradation dynamics are often dependent on system usage (or control input), and may lead to significant losses and a potential system failure. Therefore, it becomes imperative to develop control designs that are able to ensure system stability and performance whilst mitigating the effects of incipient degradation by modulating the control input appropriately. In this context, this paper proposes a novel approach based on an optimal control theory framework wherein the degradation state of the system is considered in the augmented system model and estimated using sensor measurements. Further, it is incorporated within the optimal control paradigm leading to a control law that results in deceleration of the degradation rate at the cost of system performance whilst ensuring system stability. To that end, the speed of degradation and the state of the system in discrete time are considered to develop a linear quadratic tracker (LQT) and regulator (LQR) over a finite horizon in a mathematically rigorous manner. Simulation studies are performed to assess the proposed approach.

3

Optimal docking problem of UAV at detected moving object

Nabadova Lamiya

Zeszyty Naukowe. Transport / Politechnika Śląska

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2023

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z. 120

205--214

EN

In the article, the problem of detecting a suspicious object in the control by unmanned air vehicle (UAV) and tracking it by reaching and changing its direction in the shortest period of time is explored. To solve this optimal control problem, it is considered that the flight of UAV is described with simple motion equations. In the beginning, known quantities are current coordinates and speed of UAV, equation of motion of detected suspicious object.

4

Internet digital content pricing and subscribers control

Mejjouali Sobhi, Tadj Lotfi

Operations Research and Decisions

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2023

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Vol. 33, No. 3

59--73

EN

We use optimal control theory to determine the optimal rate of change in the subscription fee and the optimal ratio of ad space to the total web page space for a web content provider. An optimal solution is obtained using the maximum principle approach and the model predictive control approach. Numerical experiments show that it is preferable to use the first approach when the planning horizon is short and the second approach when the planning horizon is long.

5

Mathematical model+ of a long-distance conveyor

Pihnastyi Oleh M., Khodusov Valery D., Kotova Anna V.

Mining Science

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2023

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Vol. 30

27--43

EN

The model of the multi-conveyor transport system of a conveyor type is given in the article. The conveyor is considered as a complex dynamic distributed system. The analytical expression which allows calculating linear density and material flow at any point of the transport route for a specific instant in time is obtained. The conveyor belt speed and the material flow from the accumulative bunker to the input of the conveyor are represented as given time functions. The decision analysis for the steady and transient periods of the transport system operation is accomplished. The estimated duration of the transient process is given. The model is of interest for the design of highly efficient flow control systems for long-ranged multi-conveyor transport systems of a conveyor type.

6

Optimal control problems without terminal constraints: The turnpike property with interior decay

Gugat Martin, Lazar Martin

International Journal of Applied Mathematics and Computer Science

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2023

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Vol. 33, no. 3

429--438

EN

We show a turnpike result for problems of optimal control with possibly nonlinear systems as well as pointwise-in-time state and control constraints. The objective functional is of integral type and contains a tracking term which penalizes the distance to a desired steady state. In the optimal control problem, only the initial state is prescribed. We assume that a cheap control condition holds that yields a bound for the optimal value of our optimal control problem in terms of the initial data. We show that the solutions to the optimal control problems on the time intervals [0, T] have a turnpike structure in the following sense: For large T the contribution to the objective functional that comes from the subinterval [T/2, T], i.e., from the second half of the time interval [0, T], is at most of the order 1/T. More generally, the result holds for subintervals of the form [r T,T], where r ∈ (0, 1/2) is a real number. Using this result inductively implies that the decay of the integral on such a subinterval in the objective function is faster than the reciprocal value of a power series in T with positive coefficients. Accordingly, the contribution to the objective value from the final part of the time interval decays rapidly with a growing time horizon. At the end of the paper we present examples for optimal control problems where our results are applicable.

7

Control and optimization of abstract continuous time evolution inclusions

Mokhtar-Kharroubi Hocine

Control and Cybernetics

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2023

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Vol. 52, No. 1

5--34

EN

Abstract controlled evolution inclusions are revisited in the Banach spaces setting. The existence of solution is established for each selected control. Then, the input–output (or, control-states) multimap is examined and the Lipschitz continuous well posedness is derived. The optimal control of such inclusions handled in terms of a Bolza problem is investigated by means of the so-called PF format of optimization. A strong duality is provided, the existence of an optimal pair is given and the system of optimalty is derived. A Fenchel duality is built and applied to optimal control of convex process of evolution. Finally, it will be shown how the general theory we provided can be applied to a wide class of controled integrodifferental inclusions.

8

Extended Identification-Based Predictive Control for adaptive impact mitigation

Graczykowski Cezary, Faraj Rami

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2023

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Vol. 71, nr 4

art. no. e145937

EN

The paper introduces Extended Identification-Based Predictive Control (EIPC), which is a novel control method developed for the problem of adaptive impact mitigation. The model-based approach utilizing the paradigm of Model Predictive Control is combined with sequential identification of selected system parameters and process disturbances. The elaborated method is implemented in the shock-absorber control system and tested under impact loading conditions. The presented numerical study proves the successful and efficient adaptation of the absorber to unknown excitation conditions as well as to unknown force and leakage disturbances appearing during the process. The EIPC is used for both semi-active and active control of the impact mitigation process, which are compared in detail. In addition, the influence of selected control parameters and disturbance identification on the efficiency of the impact absorption process is assessed. As a result, it can be concluded that an efficient and robust control method was developed and successfully applied to the problem of adaptive impact mitigation.

9

Deployment of a predictive-like optimal control law on a servo drive system using linear programming approach

Horla Dariusz, Pinczewski Piotr

Archives of Electrical Engineering

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2023

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Vol. 72, nr 4

1005--1016

EN

Current drive control systems tend to push control loops to the limits of their performance. One of the ways of doing so is to use advanced optimization algorithms, usually related to model-based off-line calculations, such as genetic algorithms, the particle swarmoptimisation or the others. There is, however, a simpler way, namely to use predictive control formalism and by formulation of a simple linear programming problem which is easy to solve using powerful solvers, without excessive computational burden, what is a reliable solution, as whenever the optimization problem has a feasible solution, a global minimizer can be efficiently found. This approach has been deployed for a servo drive system operated by a real-time sampled-data controller, verified between model-in-the-loop and hardwarein- the-loop configurations, for a range of prediction horizons, as an attractive alternative to classical quadratic programming-related formulation of predictive control task.

10

A boundary value problem for a non-linear difference equation

Lefebvre Mario

Archives of Control Sciences

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2023

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Vol. 33, No. 4

829--837

EN

A boundary value problem for a non-linear difference equation of order three is considered. We show that this equation can be interpreted as the equation satisfied by the value function in a stochastic optimal control problem. We thus obtain an expression for the solution of the non-linear difference equation that can be used to find an explicit solution to this equation. An example is presented.

11

An itae optimal sliding mode controller for systems with control signal and velocity limitations

Pietrala Mateusz, Leśniewski Piotr, Bartoszewicz Andrzej

Acta Mechanica et Automatica

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2023

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Vol. 17, no. 2

230--238

EN

In this paper, a sliding mode controller, which can be applied for second-order systems, is designed. Robustness to external disturbances, finite regulation time and a good system’s behaviour are required for a sliding mode controller. In order to achieve the first two of these three goals, a non-linear, time-varying switching curve is introduced. The representative point (state vector) belongs to this line from the very beginning of the control process, which results in elimination of the reaching phase. The stable sliding motion along the switching curve is provided. Natural limitations such as control signal and system’s velocity constraints will be taken into account. In order to satisfy them, the sliding line parameters will be properly selected. However, a good dynamical behaviour of the system has to be provided. In order to achieve that, the integral time absolute error (ITAE) quality index will be introduced and minimised. The simulation example will verify theoretical considerations.

12

Sterowanie optymalne mechanizmem płaskim identyfikowanym z danych

Pikuliński Maciej, Malczyk Paweł

Prace Naukowe Politechniki Warszawskiej. Elektronika

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2022

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z. 197, t. 2

191-200

PL

W tym artykule rozważamy modele płaskich układów wieloczłonowych (UW) o strukturze otwartych łańcuchów kinematycznych (OŁK), które są identyfikowane z danych za pomocą regresji średniokwadratowej z regularyzacją. Modele tej klasy posiadają na ogół niepewne współczynniki lub zależne od stanu układu wyrażenia, które nie są obecne w nominalnym UW. Synteza sterowania optymalnego wykorzystująca zidentyfikowane układu z zastosowaniem metody adjoint może charakteryzować się utratą jakości i brakiem odporności. Nadrzędnym celem pracy jest dyskusja rezultatów związanych z optymalnym sterowaniem UW o strukturze OŁK, kiedy model użyty do syntezy sterowania jest identyfikowany z danych.

EN

In this paper, we consider data-driven models of planar open-loop multi-rigid-body systems, which are identified by using regularized regression. Such models may possess uncertain coefficients or additional state-dependent terms, which are not present in the nominal systems. This may cause performance issues and lack of robustness when the adjoint-based optimal control is applied. The primary importance of this paper is to discuss the results of optimal control of fully actuated, open-loop multi-rigid-body systems, when a model is identified from data.

13

Minimum energy control of degenerate Cauchy problem with skew-Hermitian pencil

Benaoued Meriem, Bouagada Djillali

Journal of Applied Analysis

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2022

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Vol. 28, nr 2

251--261

EN

The present research paper deals with the effectiveness of the control of an infinite-dimensional degenerate Cauchy problem with variable operator coefficients, skew-Hermitian pencil and bounded input condition. This study explores the minimum energy control problem. The investigation follows a set of methods to examine the procedure for developing a new result to solve the problem. Indeed, by the use of decomposition transformation of the considered system and the application of the Gramian operator, the formula of the process for controlling the system with minimum energy is obtained. Afterwards, a procedure to compute the optimal input for minimizing the performance index is then proposed. In a nutshell, the obtained results indicate that optimal control for minimizing the performance index ensures the solution of the minimum energy control of an infinite-dimensional degenerate Cauchy problem.

14

Mathematical modeling and optimal control of the impact of rumors on the banking crisis

Tadmon Calvin, Njike-Tchaptchet Eric Rostand

Demonstratio Mathematica

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2022

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Vol. 55, nr 1

90--118

EN

The bank run phenomenon, mostly due to rumor spread about the financial health of given financial institutions, is prejudicious to the stability of financial systems. In this paper, by using the epidemiological approach, we propose a nonlinear model for describing the impact of rumor on the banking crisis spread. We establish conditions under which the crisis dies out or remains permanent. We also solve an optimal control problem focusing on the minimization, at the lowest cost, of the number of stressed banks, as well as the number of banks undergoing the restructuring process. Numerical simulations are performed to illustrate theoretical results obtained.

15

Structure of Optimal Control in Optimal Shaping of the Steel Arch

Mikulski Leszek, Jasińska Dorota, Dąbrowska Olga

Civil and Environmental Engineering Reports

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2022

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Vol. 32, no. 3

143--165

EN

The paper presents the problem of optimal shaping of the H-bar cross-section of a steel arch that ensures minimal mass. Nineteen combinations of nine basic load states are considered simultaneously in the problem formulation. The optimal shaping task is formulated as a control theory problem within the formal structure of the maximum Pontriagin’s principle. Since the ranges of constraint activity defining the control structure are a priori unknown and must be determined numerically, assuming the proper control structure plays a key role in the task solution. The main achievement of the present work is the determination of a solution of the multi-decision and multi-constraint optimization problem of the arch constituting a primary structural system of the existing building assuring the reduction of the structure mass up to 42%. In addition, the impact of the assumed state constraint value on the solution structure is examined.

16

Control of perishable inventory system with uncertain perishability process using neural networks and robust multicriteria optimization

Chołodowicz Ewelina, Orłowski Przemysław

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2022

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Vol. 70, nr 3

art. no. e141182

EN

The inventory systems are highly variable and uncertain due to market demand instability, increased environmental impact, and perishability processes. The reduction of waste and minimization of holding and shortage costs are the main topics studied within the inventory management area. The main difficulty is the variability of perishability and other processes that occurred in inventory systems and the solution for a trade-off between sufficient inventory level and waste of products. In this paper, the approach for resolving this trade-off is proposed. The presented approach assumes the application of a state-feedback neural network controller to generate the optimal quantity of orders considering an uncertain deterioration process and the FIFO issuing policy. The development of the control system is based on state-space close loop control along with neural networks. For modelling the perishability process Weibull distribution and FIFO policy are applied. For the optimization of the designed control system, the evolutionary NSGA-II algorithm is used. The robustness of the proposed approach is provided using the minimax decision rule. The worst-case scenario of an uncertain perishability process is considered. For assessing the proposed approach, simulation research is conducted for different variants of controller structure and model parameters. We perform extensive numerical simulations in which the assessment process of obtained solutions is conducted using hyper volume indicator and average absolute deviation between results obtained for the learning and testing set. The results indicate that the proposed approach can significantly improve the performance of the perishable inventory system and provides robustness for the uncertain changes in the perishability process.

17

On the minimum energy compensation for linear time-varying disturbed systems

Magri El Mostafa, Amissi Chadi, Afifi Larbi, Lhous Mustapha

Archives of Control Sciences

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2022

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Vol. 32, No. 4

733--754

EN

We consider in this work a class of finite dimensional time-varying linear disturbed systems. The main objective of this work is to studied the optimal control which ensures the remediability of a disturbance of time-varying disturbed systems. The remediability concept consist to find a convenient control which bringing back the corresponding observation of disturbed system to the normal one at the final time. We give firstly some characterisations of compensation and in second party we find a control which annul the output of the system and we show also that the Hilbert Uniqueness Method can be used to solve the optimal control which ensure the remediability. A general approach was given to minimize the linear quadratic problem. Examples and numerical simulations are given.

18

Optimizing Reference Model for Disturbance Rejection Controller for 3-DoF Robot Manipulator

Hoang Duy, Manh Dung Do, Kim Thai Dinh, Le Hai Xuan, Pham Thanh Ngoc, Thai Nguyen Huu, Nguyen Long Mau

Annals of Computer Science and Information Systems

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2022

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Vol. 33

121--127

EN

In this article, an method is proposed combining optimal control for linear system and disturbances observer to control a 3 degree of freedom (3DoF) robot manipulator. By making the tracking error follow a given stable linear reference model through the observer, an optimal controller LQR will be designed to solve the optimization problem for the reference system, thereby leading to good control quality for the original system. The effectiveness of the method is shown through simulation results performed on Matlab/Simulink.

19

Optimal controller design for a birotor helicopter

Nekrouf Sadek, Chekroun Soufyane

Przegląd Elektrotechniczny

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2021

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R. 97, nr 11

93--96

EN

Robust control problem for a two degree of freedom (2-DOF) lab helicopter is investigated. The helicopter dynamics involves nonlinearity, uncertainties, and coupling. A new high performance bounded (HPB) linear quadratic regulator control law has been presented that extends classical LQR by providing faster settling times, eigenstructure to optimize its performance, and has much quicker computation times than classical LQR. The robust compensator is designed to restrain the effects of uncertainties, nonlinear properties, and disturbances. The simulation results on the 2-DOF lab helicopter demonstrate the effectiveness of the proposed control strategy.

PL

Zbadano problem solidnego sterowania helikopterem laboratoryjnym o dwóch stopniach swobody (2-DOF). Dynamika helikoptera obejmuje nieliniowość, niepewności i sprzężenie. Przedstawiono nowe prawo sterowania liniowym regulatorem kwadratowym o wysokiej wydajności (HPB), które rozszerza klasyczną LQR, zapewniając szybsze czasy ustalania, strukturę własną w celu optymalizacji jego działania i ma znacznie szybsze czasy obliczeń niż klasyczne LQR. Wytrzymały kompensator jest przeznaczony do ograniczania skutków niepewności, właściwości nieliniowych i zakłóceń. Wyniki symulacji na śmigłowcu laboratoryjnym 2-DOF pokazują skuteczność proponowanej strategii kontroli.

20

Optimal anti-surge control of gas pumping unit with gas turbine drive

Oliynyk Andriy P., Feshanych Lidiia I., Ushkalenko Irina M., Smolarz Andrzej, Kalimbetov Galim, Spabekova Marzhan

Przegląd Elektrotechniczny

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2021

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R. 97, nr 9

42--45

EN

The article is devoted to the determination of the optimal transient process in the real system of anti-surge control of a gas pumping unit with a gas turbine drive. The literature review on this topic was carried out. The problem of optimal control of the controlled object was set on the basis of the Mayer problem for the corresponding higher-order function, the optimal transition process was constructed and the optimal trajectory of the transition process was determined taking into account the obstacles. The conclusions based on the results of the research were formulated.

PL

Artykuł poświęcony jest wyznaczeniu optymalnego procesu przejściowego w rzeczywistym układzie regulacji przeciwzakłóceniowej gazowego zespołu pompowego z napędem turbiną gazową. Dokonano przeglądu literatury na ten temat. Postawiono problem optymalnego sterowania obiektem sterowanym na podstawie problemu Mayera dla odpowiedniej funkcji wyższego rzędu, skonstruowano optymalny proces przejściowy oraz wyznaczono optymalną trajektorię procesu przejściowego z uwzględnieniem przeszkód. Sformułowano wnioski oparte na wynikach badań.