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A special issue of Mathematical and Computational Applications (ISSN 2297-8747). This special issue belongs to the section "Engineering".

Deadline for manuscript submissions: closed (30 September 2018) | Viewed by 52488

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Prof. Dr. Guillermo Valencia-Palomo

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Tecnológico Nacional de México / Instituto Tecnológico de Hermosillo, Ave. Tecnológico y Periférico Poniente SN, Hermosillo 83170, Mexico
Interests: predictive control; optimization; LPV systems; fault detection and isolation
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Dr. Francisco Ronay López-Estrada

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Tecnológico Nacional de México / Instituto Tecnológico de Tuxtla Gutiérrez, TURIX-DYNAMICS Diagnosis and Control Group, Carretera Panamericana, Km 1080, Tuxtla Gutierrez 29050, Mexico
Interests: control applications; optimization; LMIs; Takagi–Sugeno; fault diagnosis
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Mathematical optimization is the selection of the best element in a set with respect to a given criterion. Optimization has become one of the most used tools in modern control theory to compute the control law, adjusting the controller parameters (tuning), model fitting, finding suitable conditions in order to fulfill a given closed-loop property among others. In the simplest case, optimization consist in maximize or minimize a function by systematically choosing input values from a valid input set and computing the function value. Nevertheless, real-world control systems need to comply with several conditions and constraints that has to be taken into account in the problem formulation which represent challenges in the application of the optimization algorithms.

In this Special Issue call, the aim is to offer a state-of-the-art of the most advanced optimization techniques (online and offline) and its applications in control engineering. Potential topics include (but not limited to):

Optimal control of nonlinear systems;
Optimal control of complex systems;
Predictive control;
Optimal observer design;
Principal component analysis;
Neuronal networks;
Numerical optimization;
Evolutionary optimization;
Constrained optimization;
Control systems.

Prof. Dr. Guillermo Valencia-Palomo
Dr. Francisco Ronay López-Estrada
Guest Editors

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Keywords

optimization
optimal control
optimal conditions

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Published Papers (12 papers)

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Research

11 pages, 625 KiB

Open AccessFeature PaperArticle

Modeling and Simulation of a Hydraulic Network for Leak Diagnosis

by José-Roberto Bermúdez, Francisco-Ronay López-Estrada, Gildas Besançon, Guillermo Valencia-Palomo, Lizeth Torres and Héctor-Ricardo Hernández

Math. Comput. Appl. 2018, 23(4), 70; https://doi.org/10.3390/mca23040070 - 6 Nov 2018

Cited by 17 | Viewed by 4396

Abstract

This work presents the modeling and simulation of a hydraulic network with four nodes and two branches that form a two-level water distribution system. It also proposes a distribution of hydraulic valves that allows emulating a leak using a valve and different network configurations, e.g., simple ducts, closed networks and branched networks. The network is modeled in the steady state considering turbulent flow. Numerical experiments are performed, and the results show that the proposed network is useful for the design of leakage diagnosis and control algorithms in different configurations and leakage scenarios. Full article

(This article belongs to the Special Issue Optimization in Control Applications)

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18 pages, 362 KiB

Open AccessArticle

Time Needed to Control an Epidemic with Restricted Resources in SIR Model with Short-Term Controlled Population: A Fixed Point Method for a Free Isoperimetric Optimal Control Problem

by Imane Abouelkheir, Fadwa El Kihal, Mostafa Rachik and Ilias Elmouki

Math. Comput. Appl. 2018, 23(4), 64; https://doi.org/10.3390/mca23040064 - 22 Oct 2018

Cited by 6 | Viewed by 3241

Abstract

In this paper, we attempt to determine the optimal duration of an anti-epidemic control strategy which targets susceptible people, under the isoperimetric condition that we could not control all individuals of this category due to restricted health resources. We state and prove the local and global stability conditions of free and endemic equilibria of a simple epidemic compartmental model devised in the form of four ordinary differential equations which describe the dynamics of susceptible-controlled-infected-removed populations and where it is taken into account that the controlled people cannot acquire long-lived immunity to move towards the removed compartment due to the temporary effect of the control parameter. Thereafter, we characterize the sought optimal control and we show the effectiveness of this limited control policy along with the research of the optimal duration that is needed to reduce the size of the infected population. The isoperimetric constraint is defined over a fixed horizon, while the objective function is defined over a free horizon present under a quadratic form in the payoff term. The complexity of this optimal control problem requires the execution of three numerical methods all combined together at the same time, namely, the forward–backward sweep method to generate the optimal state and control functions, the secant method adapted to the isoperimetric restriction, and, finally, the fixed point method to obtain the optimal final time. Full article

(This article belongs to the Special Issue Optimization in Control Applications)

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19 pages, 917 KiB

Open AccessArticle

Differential Evolution Algorithm for Multilevel Assignment Problem: A Case Study in Chicken Transportation

by Sasitorn Kaewman, Tassin Srivarapongse, Chalermchat Theeraviriya and Ganokgarn Jirasirilerd

Math. Comput. Appl. 2018, 23(4), 55; https://doi.org/10.3390/mca23040055 - 2 Oct 2018

Cited by 3 | Viewed by 3383

Abstract

This study aims to solve the real-world multistage assignment problem. The proposed problem is composed of two stages of assignment: (1) different types of trucks are assigned to chicken farms to transport young chickens to egg farms, and (2) chicken farms are assigned to egg farms. Assigning different trucks to the egg farms and different egg farms to the chicken farms generates different costs and consumes different resources. The distance and the idle space in the truck have to be minimized, while constraints such as the minimum number of chickens needed for all egg farms and the longest time that chickens can be in the truck remain. This makes the problem a special case of the multistage assignment (S-MSA) problem. A mathematical model representing the problem was developed and solved to optimality using Lingo v.11 optimization software. Lingo v.11 can solve to optimality only small- and medium-sized test instances. To solve large-sized test instances, the differential evolution (DE) algorithm was designed. An excellent decoding method was developed to increase the search performance of DE. The proposed algorithm was tested with three randomly generated datasets (small, medium, and large test instances) and one real case study. Each dataset is composed of 12 problems, therefore we tested with 37 instances, including the case study. The results show that for small- and medium-sized test instances, DE has 0.03% and 0.05% higher cost than Lingo v.11. For large test instances, DE has 3.52% lower cost than Lingo v.11. Lingo v.11 uses an average computation time of 5.8, 103, and 4320 s for small, medium and large test instances, while DE uses 0.86, 1.68, and 8.79 s, which is, at most, 491 times less than Lingo v.11. Therefore, the proposed heuristics are an effective algorithm that can find a good solution while using less computation time. Full article

(This article belongs to the Special Issue Optimization in Control Applications)

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28 pages, 10028 KiB

Open AccessArticle

Rapid Solution of Optimal Control Problems by a Functional Spreadsheet Paradigm: A Practical Method for the Non-Programmer

by Chahid Kamel Ghaddar

Math. Comput. Appl. 2018, 23(4), 54; https://doi.org/10.3390/mca23040054 - 28 Sep 2018

Cited by 1 | Viewed by 5022

Abstract

We devise a practical and systematic spreadsheet solution paradigm for general optimal control problems. The paradigm is based on an adaptation of a partial-parametrization direct solution method which preserves the original mathematical optimization statement, but transforms it into a simplified nonlinear programming problem (NLP) suitable for Excel NLP solver. A rapid solution strategy is implemented by a tiered arrangement of pure elementary calculus functions in conjunction with Excel NLP solver. With the aid of the calculus functions, a cost index and constraints are represented by equivalent formulas that fully encapsulate an underlining parametrized dynamical system. Excel NLP solver is then employed to minimize (or maximize) the cost index formula, by varying decision parameters, subject to the constraints formulas. The paradigm is demonstrated for several fixed and free-time nonlinear optimal control problems involving integral and implicit dynamic constraints with direct comparison to published results obtained by fundamentally different methods. Practically, applying the paradigm involves no more than defining a few formulas using basic Excel spreadsheet skills. Full article

(This article belongs to the Special Issue Optimization in Control Applications)

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14 pages, 383 KiB

Open AccessArticle

Optimal Control and Computational Method for the Resolution of Isoperimetric Problem in a Discrete-Time SIRS System

by Fadwa El Kihal, Imane Abouelkheir, Mostafa Rachik and Ilias Elmouki

Math. Comput. Appl. 2018, 23(4), 52; https://doi.org/10.3390/mca23040052 - 24 Sep 2018

Cited by 9 | Viewed by 3119

Abstract

We consider a discrete-time susceptible-infected-removed-susceptible “again” (SIRS) epidemic model, and we introduce an optimal control function to seek the best control policy for preventing the spread of an infection to the susceptible population. In addition, we define a new compartment, which models the dynamics of the number of controlled individuals and who are supposed not to be able to reach a long-term immunity due to the limited effect of control. Furthermore, we treat the resolution of this optimal control problem when there is a restriction on the number of susceptible people who have been controlled along the time of the control strategy. Further, we provide sufficient and necessary conditions for the existence of the sought optimal control, whose characterization is also given in accordance with an isoperimetric constraint. Finally, we present the numerical results obtained, using a computational method, which combines the secant method with discrete progressive-regressive schemes for the resolution of the discrete two-point boundary value problem. Full article

(This article belongs to the Special Issue Optimization in Control Applications)

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30 pages, 477 KiB

Open AccessFeature PaperArticle

Optimal Strategies for Psoriasis Treatment

by Ellina Grigorieva and Evgenii Khailov

Math. Comput. Appl. 2018, 23(3), 45; https://doi.org/10.3390/mca23030045 - 4 Sep 2018

Cited by 4 | Viewed by 2816

Abstract

Within a given time interval we consider a nonlinear system of differential equations describing psoriasis treatment. Its phase variables define the concentrations of T-lymphocytes, keratinocytes and dendritic cells. Two scalar bounded controls are introduced into this system to reflect medication dosages aimed at suppressing interactions between T-lymphocytes and keratinocytes, and between T-lymphocytes and dendritic cells. For such a controlled system, a minimization problem of the concentration of keratinocytes at the terminal time is considered. For its analysis, the Pontryagin maximum principle is applied. As a result of this analysis, the properties of the optimal controls and their possible types are established. It is shown that each of these controls is either a bang-bang type on the entire time interval or (in addition to bang-bang type) contains a singular arc. The obtained analytical results are confirmed by numerical calculations using the software “BOCOP-2.0.5”. Their detailed analysis and the corresponding conclusions are presented. Full article

(This article belongs to the Special Issue Optimization in Control Applications)

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29 pages, 2012 KiB

Open AccessArticle

Cost-Effective Analysis of Control Strategies to Reduce the Prevalence of Cutaneous Leishmaniasis, Based on a Mathematical Model

by Dibyendu Biswas, Suman Dolai, Jahangir Chowdhury, Priti K. Roy and Ellina V. Grigorieva

Math. Comput. Appl. 2018, 23(3), 38; https://doi.org/10.3390/mca23030038 - 25 Jul 2018

Cited by 3 | Viewed by 4221

Abstract

Leishmaniasis is a neglected tropical vector-borne epidemic disease, and its transmission is a complex process. Zoonotic transmission to humans or animals occurs through the bites of female Phlebotominae sand flies. Here, reservoir is considered as a major source of endemic pathogen pool for disease outbreak, and the role of more than one reservoir animal becomes indispensable. To study the role of the reservoir animals on disease dynamics, a mathematical model was constructed consisting of susceptible and infected populations of humans and two types of reservoir (animal) and vector populations, respectively. Our aim is to prevent the disease by applying a control theoretic approach, when more than one type of reservoir animal exists in the region. We use drugs like sodium stibogluconate and meglumine antimoniate to control the disease for humans and spray insecticide to control the sand fly population. Similarly, drugs are applied for infected reservoir animals of Types A and B. We calculated the cost-effectiveness of all possible combinations of the intervention and control policies. One of our findings is that the most cost-effective case for Leishmania control is the spray of insecticides for infected sand fly vector. Alternate strategic cases were compared to address the critical shortcomings of single strategic cases, and a range of control strategies were estimated for effective control and economical benefit of the overall control strategy. Our findings provide the most innovative techniques available for application to the successful eradication of cutaneous leishmaniasis in the future. Full article

(This article belongs to the Special Issue Optimization in Control Applications)

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14 pages, 2541 KiB

Open AccessFeature PaperArticle

A Simple Spectral Observer

by Lizeth Torres, Javier Jiménez-Cabas, José Francisco Gómez-Aguilar and Pablo Pérez-Alcazar

Math. Comput. Appl. 2018, 23(2), 23; https://doi.org/10.3390/mca23020023 - 1 May 2018

Cited by 8 | Viewed by 4501

Abstract

The principal aim of a spectral observer is twofold: the reconstruction of a signal of time via state estimation and the decomposition of such a signal into the frequencies that make it up. A spectral observer can be catalogued as an online algorithm for time-frequency analysis because is a method that can compute on the fly the Fourier Transform (FT) of a signal, without having the entire signal available from the start. In this regard, this paper presents a novel spectral observer with an adjustable constant gain for reconstructing a given signal by means of the recursive identification of the coefficients of a Fourier series. The reconstruction or estimation of a signal in the context of this work means to find the coefficients of a linear combination of sines a cosines that fits a signal such that it can be reproduced. The design procedure of the spectral observer is presented along with the following applications: (1) the reconstruction of a simple periodical signal, (2) the approximation of both a square and a triangular signal, (3) the edge detection in signals by using the Fourier coefficients, (4) the fitting of the historical Bitcoin market data from 1 December 2014 to 8 January 2018 and (5) the estimation of a input force acting upon a Duffing oscillator. To round out this paper, we present a detailed discussion about the results of the applications as well as a comparative analysis of the proposed spectral observer vis-à-vis the Short Time Fourier Transform (STFT), which is a well-known method for time-frequency analysis. Full article

(This article belongs to the Special Issue Optimization in Control Applications)

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15 pages, 811 KiB

Open AccessFeature PaperArticle

Solution of Optimal Harvesting Problem by Finite Difference Approximations of Size-Structured Population Model

by Johanna Pyy, Anssi Ahtikoski, Alexander Lapin and Erkki Laitinen

Math. Comput. Appl. 2018, 23(2), 22; https://doi.org/10.3390/mca23020022 - 26 Apr 2018

Cited by 3 | Viewed by 3389

Abstract

We solve numerically a forest management optimization problem governed by a nonlinear partial differential equation (PDE), which is a size-structured population model. The formulated problem is supplemented with a natural constraint for a solution to be non-negative. PDE is approximated by an explicit or implicit in time finite difference scheme, whereas the cost function is taken from the very beginning in the finite-dimensional form used in practice. We prove the stability of the constructed nonlinear finite difference schemes on the set of non-negative vectors and the solvability of the formulated discrete optimal control problems. The gradient information is derived by constructing discrete adjoint state equations. The projected gradient method is used for finding the extremal points. The results of numerical testing for several real problems show good agreement with the known results and confirm the theoretical statements. Full article

(This article belongs to the Special Issue Optimization in Control Applications)

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28 pages, 1824 KiB

Open AccessArticle

Optimal Control Analysis of a Mathematical Model for Breast Cancer

by Segun Isaac Oke, Maba Boniface Matadi and Sibusiso Southwell Xulu

Math. Comput. Appl. 2018, 23(2), 21; https://doi.org/10.3390/mca23020021 - 24 Apr 2018

Cited by 54 | Viewed by 8000

Abstract

In this paper, a mathematical model of breast cancer governed by a system of ordinary differential equations in the presence of chemotherapy treatment and ketogenic diet is discussed. Several comprehensive mathematical analyses were carried out using a variety of analytical methods to study the stability of the breast cancer model. Also, sufficient conditions on parameter values to ensure cancer persistence in the absence of anti-cancer drugs, ketogenic diet, and cancer emission when anti-cancer drugs, immune-booster, and ketogenic diet are included were established. Furthermore, optimal control theory is applied to discover the optimal drug adjustment as an input control of the system therapies in order to minimize the number of cancerous cells by considering different controlled combinations of administering the chemotherapy agent and ketogenic diet using the popular Pontryagin’s maximum principle. Numerical simulations are presented to validate our theoretical results. Full article

(This article belongs to the Special Issue Optimization in Control Applications)

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23 pages, 5706 KiB

Open AccessArticle

Novel Spreadsheet Direct Method for Optimal Control Problems

by Chahid Kamel Ghaddar

Math. Comput. Appl. 2018, 23(1), 6; https://doi.org/10.3390/mca23010006 - 25 Jan 2018

Cited by 3 | Viewed by 5360

Abstract

We devise a simple yet highly effective technique for solving general optimal control problems in Excel spreadsheets. The technique exploits Excel’s native nonlinear programming (NLP) Solver Command, in conjunction with two calculus worksheet functions, namely, an initial value problem solver and a discrete data integrator, in a direct solution paradigm adapted to the spreadsheet. The technique is tested on several highly nonlinear constrained multivariable control problems with remarkable results in terms of reliability, consistency with pseudo-spectral reported answers, and computing times in the order of seconds. The technique requires no more than defining a few analogous formulas to the problem mathematical equations using basic spreadsheet operations, and no programming skills are needed. It introduces an alternative, simpler tool for solving optimal control problems in social and natural science disciplines. Full article

(This article belongs to the Special Issue Optimization in Control Applications)

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15 pages, 612 KiB

Open AccessArticle

Solution of Fuzzy Differential Equations Using Fuzzy Sumudu Transforms

by Raheleh Jafari and Sina Razvarz

Math. Comput. Appl. 2018, 23(1), 5; https://doi.org/10.3390/mca23010005 - 17 Jan 2018

Cited by 16 | Viewed by 3872

Abstract

The uncertain nonlinear systems can be modeled with fuzzy differential equations (FDEs) and the solutions of these equations are applied to analyze many engineering problems. However, it is very difficult to obtain solutions of FDEs. In this paper, the solutions of FDEs are approximated by utilizing the fuzzy Sumudu transform (FST) method. Significant theorems are suggested in order to explain the properties of FST. The proposed method is validated with three real examples. Full article

(This article belongs to the Special Issue Optimization in Control Applications)

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