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EN
The possibility and expediency of application of the principle of least action (PLA) in the optimum control of conditions of the electric power system (EPS) is shown. The loss of active power and electricity was taken as the criterion of optimality. The algorithm for determining the optimal condition of the EPS is presented. Its peculiarity is that, in accordance with the LPA, the ideal condition of the EPS is first determined, and then parametric restrictions are taken into account. The ideal condition of the system is characterized by the lowest possible losses, and taking into account active restrictions on the parameters transposes the EPS to the optimal condition. In the optimal condition of the EPS, the losses increase depending on the degree of heterogeneity of the system.
PL
Pokazano możliwość i celowość zastosowania zasady najmniejszego działania w optymalnej kontroli stanów systemu elektroenergetycznego. Utratę mocy czynnej i energii elektrycznej przyjęto za kryterium optymalności. Przedstawiono algorytm określania optymalnego stanu systemu elektroenergetycznego. Jego szczególną cechą jest to, że zgodnie z zasadą najmniejszego działania najpierw określa się idealny stan systemu elektroenergetycznego, a następnie uwzględnia się ograniczenia parametryczne. Stan idealny systemu charakteryzuje się najniższymi możliwymi stratami, a wzięcie pod uwagę aktywnych ograniczeń parametrów przenosi system elektroenergetyczny do stanu optymalnego. W optymalnym stanie systemu elektroenergetycznego straty rosną w zależności od stopnia niejednorodności układu.
EN
The fulfilment of the condition for the simultaneous achievement of the desired chemical composition and temperature of the metal is ensured by controlling the oxygen consumption and the position of the oxygen impeller lance. The method for solving Model Predictive Control with quadratic functionality in the presence of constraints is given. Implementation of the described solutions will contribute to increasing the proportion of scrap and reducing the melting period without changing of technological process.
PL
Spełnienie warunku jednoczesnego osiągnięcia pożądanego składu chemicznego i temperatury metalu jest zapewnione poprzez kontrolę zużycia tlenu i położenia palnika tlenowego. Zaprezentowano metodę rozwiązania Modelu Sterowania Predykcyjnego z funkcjonalnością kwadratową w obecności ograniczeń. Wdrożenie opisanego rozwiązania przyczyni się do zwiększenia udziału złomu i skrócenia czasu topnienia bez zmiany procesu technologicznego.
EN
The paper deals with an optimal control problem in a dynamical system described by a linear differential equation with the Caputo fractional derivative. The goal of control is to minimize a Bolza-type cost functional, which consists of two terms: the first one evaluates the state of the system at a fixed terminal time, and the second one is an integral evaluation of the control on the whole time interval. In order to solve this problem, we propose to reduce it to some auxiliary optimal control problem in a dynamical system described by a first-order ordinary differential equation. The reduction is based on the representation formula for solutions to linear fractional differential equations and is performed by some linear transformation, which is called the informational image of a position of the original system and can be treated as a special prediction of a motion of this system at the terminal time. A connection between the original and auxiliary problems is established for both open-loop and feedback (closed-loop) controls. The results obtained in the paper are illustrated by examples.
EN
This paper addresses the nonlinear Cucker-Smale optimal control problem under the interplay of memory effect. The aforementioned effect is included by employing the Caputo fractional derivative in the equation representing the velocity of agents. Sufficient conditions for the existence of solutions to the considered problem are proved and the analysis of some particular problems is illustrated by two numerical examples.
EN
In this paper we have studied the driftless control system on a Lie group which arises due to the invariance of Black-Scholes equation by conformal transformations. These type of studies are possible as Black-Scholes equation can be mapped to one dimensional free Schrödinger equation. In particular we have studied the controllability, optimal control of the resulting dynamics as well as stability aspects of this system. We have also found out the trajectories of the states of the system through two unconventional integrators along with conventional Runge-Kutta integrator.
6
Content available remote Optimal Control in a Model of Chemotherapy-induced Radiosensilisation
EN
In this work we consider a simple mathematical model of radiochemotherapy which includes a term responsible for radiosensilisation. We focus on finding theoretically optimal controls which maximise tumour cure probability for a finite, fixed therapeutic horizon. We prove that the optimal controls for both therapies are of 0-bang type, a result which is not altered by inclusion of the radiosensilisation term. By means of numerical simulations we show that optimal control offers a moderate increase in survival time over a sequential treatment. We then revisit in more detail a question of measuring the synergy between the therapies by means of isobolograms, a common experimental technique for measuring additivity of two treatments.
PL
W tej pracy rozważamy prosty model radiochemioterapii, w którym uwzględniamy składnik odpowiedzialny za uwrażliwienie komórek na radioterapię indukowane przez podaną chemioterapię. Skupiamy się na znalezieniu teoretycznie optymalnych strategii leczenia, które maksymalizują prawdopodobieństwo wyleczenia guza (tumour cure probability) w zadanym, skończonym horyzoncie terapeutycznym. Dowodzimy, że optymalne sterowania są postaci 0-bang i składnik odpowiedzialny za uwrażliwienie nie ma wpływu na strukturę tych sterowań. Przeprowadzamy symulacje numeryczne by pokazać, że optymalne sterowanie oferuje nieznaczny wzrost przeżycia pacjentów w porównaniu do terapii sekwencyjnej. Następnie, by zmierzyć synergię pomiędzy chemio- i radioterapią, korzystamy z metody izobologramu, techniki stosowanej w eksperymentach do oceny addytywności tych terapii.
EN
In the paper, we consider an equilibrium problem for a 2D elastic body with a thin elastic inclusion crossing an external boundary. The elastic body has a defect which is characterized by a positive damage parameter. The presence of a defect means that the problem is formulated in a non-smooth domain. Non-linear boundary conditions at the defect faces are imposed to prevent the mutual penetration between the faces. Both variational and differential problem formulations are proposed, and existence of solutions is established. We study an asymptotics of solutions with respect to the damage parameter as well as with respect to a rigidity parameter of the inclusion. Identification problems for finding the damage parameter are investigated. To this end, existence of solutions of optimal control problems is proven.
EN
We consider a semilinear elliptic optimal control problem possibly subject to control and/or state constraints. Generalizing previous work, presented in Ahmad Ali, Deckelnick and Hinze (2016) we provide a condition which guarantees that a solution of the necessary first order conditions is a global minimum. A similiar result also holds at the discrete level where the corresponding condition can be evaluated explicitly. Our investigations are motivated by G¨unter Leugering, who raised the question whether the problem class considered in Ahmad Ali, Deckelnick and Hinze (2016) can be extended to the nonlinearity φ(s) = s|s|. We develop a corresponding analysis and present several numerical test examples demonstrating its usefulness in practice.
EN
In this paper, we give an overview of results for Cahn–Hilliard systems involving fractional operators that have recently been established by the authors of this note. We address problems concerning existence, uniqueness, and regularity of the solutions to the system equations, and we study optimal control problems for the systems. The well-posedness results are valid for a wide class of fractional operators of spectral type and for the typical double-well nonlinearities appearing in the Cahn–Hilliard system equations, namely the classical differentiable, the logarithmic, and the nondifferentiable double obstacle potentials. While this also applies to the existence of optimal controls in the related optimal control problems, the establishment of first-order necessary optimality conditions requires imposing much stronger assumptions on the admissible class of fractional operators. One main reason for this is the necessity of deriving suitable differentiability properties for the associated control-to-state mapping. Nevertheless, it turns out that also in the singular case of logarithmic potentials, the first-order necessary optimality conditions can be established under suitable assumptions, and a “deep quench” approximation, based on the results derived for logarithmic nonlinearities makes even the case of double obstacle potentials accessible.
EN
A problem of sparse optimal boundary control for a semilinear parabolic partial differential equation is considered, where pointwise bounds on the control and mixed pointwise control-state constraints are given. A standard quadratic objective functional is to be minimized that includes a Tikhonov regularization term and the L1-norm of the control accounting for the sparsity. Applying a recent linearization theorem, we derive first-order necessary optimality conditions in terms of a variational inequality under linearized mixed control state constraints. Based on this preliminary result, a Lagrange multiplier rule with bounded and measurable multipliers is derived and sparsity results on the optimal control are demonstrated.
EN
Optimal control problems governed by a transport equation are investigated that are motivated by optical flow problems. The control is given by the velocity field, corresponding to the optical flow, while the state corresponds to the brightness of image points. The problem is studied in the setting of spatially BVregular vector fields under very low regularity requirements. Existing stability results for the control-to-state operator are improved and based on this the existence of minimizers for several classes of optimal control problems is proved under mild assumptions on the admissible sets.
EN
In the complex RLC network, apart from the currents flows arising from the normal laws of Kirchhoff, other distributions of current, resulting from certain optimization criteria, may also be received. This paper is the development of research on distribution that meets the condition of the minimum energy losses within the network called energy-optimal distribution. Optimal distribution is not reachable itself, but in order to trigger it off, it is necessary to introduce the control system in current-dependent voltage sources vector, entered into a mesh set of a complex RLC network. For energy-optimal controlling, to obtain the control operator, the inversion of R(s) operator is required. It is the matrix operator and the dispersive operator (it depends on frequency). Inversion of such operators is inconvenient because it is algorithmically complicated. To avoid this the operator R(s) is replaced by the R’ operator which is a?matrix, but non-dispersive one (it does not depend on s). This type of control is called the suboptimal control. Therefore, it is important to make appropriate selection of the R’ operator and hence the suboptimal control. This article shows how to implement such control through the use of matrix operators of multiple differentiation or integration. The key aspect is the distribution of a single rational function H(s) in a series of ‘s’ or ‘s1’. The paper presents a new way of developing a given, stable rational transmittance with real coefficients in power series of ‘s/s1. The formulas to determine values of series coefficients (with ‘s/s1’) have been shown and the conditions for convergence of differential/integral operators given as series of ‘s/s1’ have been defined.
EN
Development of a method for calculating the optimal mode of conducting a train in terms of energy saving meet the safety requirements and schedules. The method of calculation must solve the assigned tasks without significant time spent on the calculation. To implement this method of calculation was used a simplified model of the train as a controlled system. The existing mathematical and algorithmic methods for solving isoperimetric problems of finding the optimal solution in the presence of restrictions on resources were the information base for methodology development. Scientific works of domestic and foreign scientists, professional periodicals, materials of scientific and practical conferences, methodical and normative materials, currently in force on Ukrainian Railways. The results of these studies were used to create simulators on the basis of computer technology for the training of locomotive drivers. The scientific novelty of the proposed calculation method consists in applying the simplified calculations of the status of the train as a controlled system, without the use of differential equations of motion that allows to significantly increase the speed of the calculations. This, in turn, will solve the problems of finding optimal control in real time, taking into account changing conditions during the movement of the train. The practical significance of the obtained results is the use of such a calculation method that does not require significant time for its implementation and can be used as a subsystem of the on-board train control system capable of performing calculations taking into account changes in the current train situation.
EN
This article presents the usage of a zero-sum differential game to control a nonlinear object, which, in the analysed problem, was a mathematical pendulum. The obtained control was optimal with regard to adopted quality indicator for the worst interference. The two-point boundary value problem was solved numerically by means of the Dircol software application. Numerical solutions, meeting all the necessary optimality conditions, were obtained for different values of the rough parameter and for different values of damping.
PL
W artykule wykorzystano grę różniczkową o sumie zerowej do sterowania obiektem nieliniowym, jakim w analizowanym problemie jest wahadło matematyczne. Uzyskano sterowanie optymalne ze względu na przyjęty wskaźnik jakości, przy najgorszym zakłóceniu. Dwupunktowy problem brzegowy został rozwiązany numerycznie przy wykorzystaniu programu Dircol. Rozwiązania numeryczne spełniające wszystkie warunki konieczne optymalności zostały otrzymane dla różnych wartości parametru szorstkości oraz przy różnych wartościach tłumienia.
EN
In the paper a construction of a control system for 2nd order, uncertain-parameter plant is discussed. The considered model of the plant is described by state space equation or by equivalent transfer function and it describes a huge class of real control plants, for example – electric drives or oriented PV systems. As a controller the digital proportional (P) controller was employed. The control system is going to be implemented at the microcontroller platform. Results are by the example depicted.
PL
Budynki wraz z rozwojem techniki oraz wzrostem wymagań dotyczących komfortu termicznego użytkowników zużywają coraz więcej energii. Obecnie dużo uwagi poświęca się zagadnieniom dotyczącym optymalizacji zużycia energii i ogólnej poprawy efektywności energetycznej budynków. W większości przypadków jest to wymóg określony w obowiązujących aktach prawnych. Zwiększa się również liczba obiektów, gdzie inwestor – świadomy korzyści wynikających z zastosowania systemów monitorujących zużycie energii – decyduje się na ich wykorzystanie bez konieczności spełnienia jakichkolwiek wymogów. Ważny jest aspekt zapewnienia komfortu użytkowego, ale też efektywne zużycie energii poprzez optymalne sterowanie pracy systemów technicznych przez system BMS.
EN
Along with the development of technology and the increase of thermal comfort requirements of users, buildings consume more and more energy. Currently, a lot of attention is devoted to issues related to the optimization of energy consumption and the general improvement of the energy efficiency of buildings. In most cases, this is a requirement set out in existing legal acts. The number of facilities is also growing, where an investor aware of the benefits of using energy monitoring systems decides to use them without having to meet any requirements. The aspect of providing comfort for users is as important as effective energy consumption provided by optimal process control of technical systems by the BMS.
17
Content available Transient flow in gas networks: Traveling waves
EN
In the context of gas transportation, analytical solutions are helpful for the understanding of the underlying dynamics governed by a system of partial differential equations. We derive traveling wave solutions for the one-dimensional isothermal Euler equations, where an affine linear compressibility factor is used to describe the correlation between density and pressure. We show that, for this compressibility factor model, traveling wave solutions blow up in finite time. We then extend our analysis to networks under appropriate coupling conditions and derive compatibility conditions for the network nodes such that the traveling waves can travel through the nodes. Our result allows us to obtain an explicit solution for a certain optimal boundary control problem for the pipeline flow.
EN
A deterministic mathematical model of the Middle East respiratory syndrome (MERS) disease is introduced. Medical masks, supportive care treatment and a government campaign about the importance of medical masks will be involved in the model as time dependent variables. The problem is formulated as an optimal control one to minimize the number of infected people and keep the intervention costs as low as possible. Assuming that all control variables are constant, we find a disease free equilibrium point and an endemic equilibrium point explicitly. The existence and local stability criteria of these equilibria depend on the basic reproduction number. A sensitivity analysis of the basic reproduction number with respect to control parameters tells us that the intervention on medical mask use and the campaign about the importance of medical masks are much more effective for reducing the basic reproduction number than supportive care intervention. Numerical experiments for optimal control problems are presented for three different scenarios, i.e., a scenario of different initial conditions for the human population, a scenario of different initial basic reproduction numbers and a scenario of different budget limitations. Under budget limitations, it is much better to implement the medical mask intervention in the field, rather than give supportive care to control the spread of the MERS disease in the endemic prevention scenario. On the other hand, the medical mask intervention should be implemented partially together with supportive care to obtain the lowest number of infected people, with the lowest cost in the endemic reduction scenario.
EN
The paper presents conceptual provisions for the construction of invariant multi-frequency piezoelectric resonance devices with a predictive reference model. The law of the optimal control of the system in real time is formulated, aimed at minimizing energy costs to ensure the trajectory of the system. The results of piezoelectric resonance system mathematical modelling in the conditions of temperature and vibration perturbations are presented.
PL
W artykule przedstawiono koncepcyjne zasady konstruowania niezmiennych wieloczęstotliwościowych piezoelektrycznych urządzeń z predykcyjnym modelem odniesienia. Zostało sformułowane prawo optymalnego sterowania systemem w czasie rzeczywistym, aby przy minimalnych stratach energii zapewnić trajektorię ruchu systemu. Przedstawiono wyniki matematycznego modelowania układu piezorezonansowego w warunkach zaburzeń temperatury i drgań.
PL
W artykule opisano sześć spośród dziewięciu przedstawionych metod wyznaczania bezpiecznej trajektorii statku w sytuacji mijania się na morzu ze spotkanymi statkami. Badaniom symulacyjnym, na przykładzie nawigacyjnej sytuacji mijania się z 18 spotkanymi statkami, poddano algorytmy: wieloetapowej gry pozycyjnej niekooperacyjnej i kooperacyjnej, wielokrokowej gry macierzowej niekooperacyjnej i kooperacyjnej, programowania dynamicznego z neuronowymi ograniczeniami stanu oraz programowania liniowego. Dla każdego z algorytmów porównano bezpieczne trajektorie statku w warunkach dobrej i ograniczonej widzialności na morzu.
EN
The paper describes six of the nine kinds of methods for determining safe trajectory of the ship passing the encountered ships at sea. Simulation research on the example of the navigation situation, the passing of 18 ships have encountered, were algorithms: multi-stage positional game non-cooperative and cooperative, multi-step matrix game non-cooperative and cooperative, dynamic programming with neural state constraints and linear programming. For each of the algorithms compared safe ship trajectories in conditions of good and restricted visibility at sea.
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