Mathematics

Editorial

Jump to: Research, Review

8 pages, 452 KiB

Open AccessEditorial

Preface to the Special Issue on “Control, Optimization, and Mathematical Modeling of Complex Systems”

by Mikhail Posypkin, Andrey Gorshenin and Vladimir Titarev

Mathematics 2022, 10(13), 2182; https://doi.org/10.3390/math10132182 - 23 Jun 2022

Viewed by 1729

Abstract

Complex systems have long been an integral part of modern life and can be encountered everywhere [...] Full article

(This article belongs to the Special Issue Control, Optimization, and Mathematical Modeling of Complex Systems)

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Research

Jump to: Editorial, Review

21 pages, 2223 KiB

Open AccessArticle

Statistical Feature Construction for Forecasting Accuracy Increase and Its Applications in Neural Network Based Analysis

by Andrey Gorshenin and Victor Kuzmin

Mathematics 2022, 10(4), 589; https://doi.org/10.3390/math10040589 - 14 Feb 2022

Cited by 15 | Viewed by 2461

Abstract

This paper presents a feature construction approach called Statistical Feature Construction (SFC) for time series prediction. Creation of new features is based on statistical characteristics of analyzed data series. First, the initial data are transformed into an array of short pseudo-stationary windows. For [...] Read more.

This paper presents a feature construction approach called Statistical Feature Construction (SFC) for time series prediction. Creation of new features is based on statistical characteristics of analyzed data series. First, the initial data are transformed into an array of short pseudo-stationary windows. For each window, a statistical model is created and characteristics of these models are later used as additional features for a single window or as time-dependent features for the entire time series. To demonstrate the effect of SFC, five plasma physics and six oceanographic time series were analyzed. For each window, unknown distribution parameters were estimated with the method of moving separation of finite normal mixtures. First four statistical moments of these mixtures for initial data and increments were used as additional data features. Multi-layer recurrent neural networks were trained to create short- and medium-term forecasts with a single window as input data; additional features were used to initialize the hidden state of recurrent layers. A hyperparameter grid-search was performed to compare fully-optimized neural networks for original and enriched data. A significant decrease in RMSE metric was observed with a median of

11.4 %

. There was no increase in RMSE metric in any of the analyzed time series. The experimental results have shown that SFC can be a valuable method for forecasting accuracy improvement. Full article

(This article belongs to the Special Issue Control, Optimization, and Mathematical Modeling of Complex Systems)

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28 pages, 13334 KiB

Open AccessArticle

Rock Segmentation in the Navigation Vision of the Planetary Rovers

by Boyu Kuang, Mariusz Wisniewski, Zeeshan A. Rana and Yifan Zhao

Mathematics 2021, 9(23), 3048; https://doi.org/10.3390/math9233048 - 27 Nov 2021

Cited by 23 | Viewed by 3852

Abstract

Visual navigation is an essential part of planetary rover autonomy. Rock segmentation emerged as an important interdisciplinary topic among image processing, robotics, and mathematical modeling. Rock segmentation is a challenging topic for rover autonomy because of the high computational consumption, real-time requirement, and [...] Read more.

Visual navigation is an essential part of planetary rover autonomy. Rock segmentation emerged as an important interdisciplinary topic among image processing, robotics, and mathematical modeling. Rock segmentation is a challenging topic for rover autonomy because of the high computational consumption, real-time requirement, and annotation difficulty. This research proposes a rock segmentation framework and a rock segmentation network (NI-U-Net++) to aid with the visual navigation of rovers. The framework consists of two stages: the pre-training process and the transfer-training process. The pre-training process applies the synthetic algorithm to generate the synthetic images; then, it uses the generated images to pre-train NI-U-Net++. The synthetic algorithm increases the size of the image dataset and provides pixel-level masks—both of which are challenges with machine learning tasks. The pre-training process accomplishes the state-of-the-art compared with the related studies, which achieved an accuracy, intersection over union (IoU), Dice score, and root mean squared error (RMSE) of 99.41%, 0.8991, 0.9459, and 0.0775, respectively. The transfer-training process fine-tunes the pre-trained NI-U-Net++ using the real-life images, which achieved an accuracy, IoU, Dice score, and RMSE of 99.58%, 0.7476, 0.8556, and 0.0557, respectively. Finally, the transfer-trained NI-U-Net++ is integrated into a planetary rover navigation vision and achieves a real-time performance of 32.57 frames per second (or the inference time is 0.0307 s per frame). The framework only manually annotates about 8% (183 images) of the 2250 images in the navigation vision, which is a labor-saving solution for rock segmentation tasks. The proposed rock segmentation framework and NI-U-Net++ improve the performance of the state-of-the-art models. The synthetic algorithm improves the process of creating valid data for the challenge of rock segmentation. All source codes, datasets, and trained models of this research are openly available in Cranfield Online Research Data (CORD). Full article

(This article belongs to the Special Issue Control, Optimization, and Mathematical Modeling of Complex Systems)

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27 pages, 1842 KiB

Open AccessArticle

Fractional Dynamics of Stuxnet Virus Propagation in Industrial Control Systems

by Zaheer Masood, Muhammad Asif Zahoor Raja, Naveed Ishtiaq Chaudhary, Khalid Mehmood Cheema and Ahmad H. Milyani

Mathematics 2021, 9(17), 2160; https://doi.org/10.3390/math9172160 - 4 Sep 2021

Cited by 34 | Viewed by 3618

Abstract

The designed fractional order Stuxnet, the virus model, is analyzed to investigate the spread of the virus in the regime of isolated industrial networks environment by bridging the air-gap between the traditional and the critical control network infrastructures. Removable storage devices are commonly [...] Read more.

The designed fractional order Stuxnet, the virus model, is analyzed to investigate the spread of the virus in the regime of isolated industrial networks environment by bridging the air-gap between the traditional and the critical control network infrastructures. Removable storage devices are commonly used to exploit the vulnerability of individual nodes, as well as the associated networks, by transferring data and viruses in the isolated industrial control system. A mathematical model of an arbitrary order system is constructed and analyzed numerically to depict the control mechanism. A local and global stability analysis of the system is performed on the equilibrium points derived for the value of

α

= 1. To understand the depth of fractional model behavior, numerical simulations are carried out for the distinct order of the fractional derivative system, and the results show that fractional order models provide rich dynamics by means of fast transient and super-slow evolution of the model’s steady-state behavior, which are seldom perceived in integer-order counterparts. Full article

(This article belongs to the Special Issue Control, Optimization, and Mathematical Modeling of Complex Systems)

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21 pages, 4707 KiB

Open AccessArticle

Application of Mathematical Models to Assess the Impact of the COVID-19 Pandemic on Logistics Businesses and Recovery Solutions for Sustainable Development

by Han Khanh Nguyen

Mathematics 2021, 9(16), 1977; https://doi.org/10.3390/math9161977 - 18 Aug 2021

Cited by 8 | Viewed by 5846

Abstract

The logistics industry can be considered as the economic lifeline of each country because of its role in connecting production and business activities of enterprises and promoting socio-economic development between regions and countries. However, the COVID-19 pandemic, which began at the end of [...] Read more.

The logistics industry can be considered as the economic lifeline of each country because of its role in connecting production and business activities of enterprises and promoting socio-economic development between regions and countries. However, the COVID-19 pandemic, which began at the end of 2019, has seriously affected the global supply chain, causing heavy impacts on the logistics service sector. In this study, the authors used the Malmquist productivity index to assess the impact of the pandemic on logistics businesses in Vietnam. Moreover, the authors used a super-slack-based model to find strategic alliance partners for enterprises. The authors also used the Grey forecasting model to forecast the business situation for enterprises during the period 2021–2024, in order to provide the leaders of these enterprises with a complete picture of their partners as a solid basis for making decisions to implement alliances that will help logistics enterprises in Vietnam to develop sustainably. The results have found that the alliance between LO₇ and LO₁₀ is the most optimal, as this alliance can exploit freight in the opposite direction and reduce logistics costs, creating better competitiveness for businesses. Full article

(This article belongs to the Special Issue Control, Optimization, and Mathematical Modeling of Complex Systems)

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22 pages, 7736 KiB

Open AccessArticle

3D Model Identification of a Soft Robotic Neck

by Fernando Quevedo, Jorge Muñoz, Juan Alejandro Castano Pena and Concepción A. Monje

Mathematics 2021, 9(14), 1652; https://doi.org/10.3390/math9141652 - 13 Jul 2021

Cited by 8 | Viewed by 3238

Abstract

Soft robotics is becoming an emerging solution to many of the problems in robotics, such as weight, cost and human interaction. In order to overcome such problems, bio-inspired designs have introduced new actuators, links and architectures. However, the complexity of the required models [...] Read more.

Soft robotics is becoming an emerging solution to many of the problems in robotics, such as weight, cost and human interaction. In order to overcome such problems, bio-inspired designs have introduced new actuators, links and architectures. However, the complexity of the required models for control has increased dramatically and geometrical model approaches, widely used to model rigid dynamics, are not enough to model these new hardware types. In this paper, different linear and non-linear models will be used to model a soft neck consisting of a central soft link actuated by three motor-driven tendons. By combining the force on the different tendons, the neck is able to perform a motion similar to that of a human neck. In order to simplify the modeling, first a system input–output redefinition is proposed, considering the neck pitch and roll angles as outputs and the tendon lengths as inputs. Later, two identification strategies are selected and adapted to our case: set membership, a data-driven, nonlinear and non-parametric identification strategy which needs no input redefinition; and Recursive least-squares (RLS), a widely recognized identification technique. The first method offers the possibility of modeling complex dynamics without specific knowledge of its mathematical representation. The selection of this method was done considering its possible extension to more complex dynamics and the fact that its impact in soft robotics is yet to be studied according to the current literature. On the other hand, RLS shows the implication of using a parametric and linear identification in a nonlinear plant, and also helps to evaluate the degree of nonlinearity of the system by comparing the different performances. In addition to these methods, a neural network identification is used for comparison purposes. The obtained results validate the modeling approaches proposed. Full article

(This article belongs to the Special Issue Control, Optimization, and Mathematical Modeling of Complex Systems)

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25 pages, 5640 KiB

Open AccessArticle

A New Approach of Soft Joint Based on a Cable-Driven Parallel Mechanism for Robotic Applications

by Luis Nagua, Carlos Relaño, Concepción A. Monje and Carlos Balaguer

Mathematics 2021, 9(13), 1468; https://doi.org/10.3390/math9131468 - 23 Jun 2021

Cited by 10 | Viewed by 3532

Abstract

A soft joint has been designed and modeled to perform as a robotic joint with 2 Degrees of Freedom (DOF) (inclination and orientation). The joint actuation is based on a Cable-Driven Parallel Mechanism (CDPM). To study its performance in more detail, a test [...] Read more.

A soft joint has been designed and modeled to perform as a robotic joint with 2 Degrees of Freedom (DOF) (inclination and orientation). The joint actuation is based on a Cable-Driven Parallel Mechanism (CDPM). To study its performance in more detail, a test platform has been developed using components that can be manufactured in a 3D printer using a flexible polymer. The mathematical model of the kinematics of the soft joint is developed, which includes a blocking mechanism and the morphology workspace. The model is validated using Finite Element Analysis (FEA) (CAD software). Experimental tests are performed to validate the inverse kinematic model and to show the potential use of the prototype in robotic platforms such as manipulators and humanoid robots. Full article

(This article belongs to the Special Issue Control, Optimization, and Mathematical Modeling of Complex Systems)

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14 pages, 2251 KiB

Open AccessArticle

Study of Synergistic Effects in Complex Stochastic Systems

by Gurami Tsitsiashvili

Mathematics 2021, 9(12), 1396; https://doi.org/10.3390/math9121396 - 16 Jun 2021

Cited by 2 | Viewed by 2036

Abstract

In this paper, a method for detecting synergistic effects of the interaction of elements in multi-element stochastic systems of separate redundancy, multi-server queuing, and statistical estimates of nonlinear recurrent relations parameters has been developed. The detected effects are quite strong and manifest themselves [...] Read more.

In this paper, a method for detecting synergistic effects of the interaction of elements in multi-element stochastic systems of separate redundancy, multi-server queuing, and statistical estimates of nonlinear recurrent relations parameters has been developed. The detected effects are quite strong and manifest themselves even with rough estimates. This allows studying them with mathematical methods of relatively low complexity and thereby expand the set of possible applications. These methods are based on special techniques of the structural analysis of multi-element stochastic models in combination with majorant asymptotic estimates of their performance indicators. They allow moving to more accurate and rather economical numerical calculations, as they indicate in which direction it is most convenient to perform these calculations. Full article

(This article belongs to the Special Issue Control, Optimization, and Mathematical Modeling of Complex Systems)

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23 pages, 19676 KiB

Open AccessArticle

Modular and Self-Scalable Origami Robot: A First Approach

by Lisbeth Mena, Jorge Muñoz, Concepción A. Monje and Carlos Balaguer

Mathematics 2021, 9(12), 1324; https://doi.org/10.3390/math9121324 - 9 Jun 2021

Cited by 11 | Viewed by 5647

Abstract

This paper presents a proposal of a modular robot with origami structure. The proposal is based on a self-scalable and modular link made of soft parts. The kinematics of a single link and several links interconnected is studied and validated. Besides, the link [...] Read more.

This paper presents a proposal of a modular robot with origami structure. The proposal is based on a self-scalable and modular link made of soft parts. The kinematics of a single link and several links interconnected is studied and validated. Besides, the link has been prototyped, identified, and controlled in position. The experimental data show that the system meets the scalability requirements and that its response is totally reliable and robust. Full article

(This article belongs to the Special Issue Control, Optimization, and Mathematical Modeling of Complex Systems)

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17 pages, 954 KiB

Open AccessArticle

High-Performance Tracking for Proton Exchange Membrane Fuel Cell System PEMFC Using Model Predictive Control

by Mohamed Derbeli, Asma Charaabi, Oscar Barambones and Cristian Napole

Mathematics 2021, 9(11), 1158; https://doi.org/10.3390/math9111158 - 21 May 2021

Cited by 22 | Viewed by 3425

Abstract

Proton exchange membrane (PEM) fuel cell has recently attracted broad attention from many researchers due to its cleanliness, high efficiency and soundless operation. The obtention of high-performance output characteristics is required to overcome the market restrictions of the PEMFC technologies. Therefore, the main [...] Read more.

Proton exchange membrane (PEM) fuel cell has recently attracted broad attention from many researchers due to its cleanliness, high efficiency and soundless operation. The obtention of high-performance output characteristics is required to overcome the market restrictions of the PEMFC technologies. Therefore, the main aim of this work is to maintain the system operating point at an adequate and efficient power stage with high-performance tracking. To this end, a model predictive control (MPC) based on a global minimum cost function for a two-step horizon was designed and implemented in a boost converter integrated with a fuel cell system. An experimental comparative study has been investigated between the MPC and a PI controller to reveal the merits of the proposed technique. Comparative results have indicated that a reduction of

15.65 %

and

86.9 %

, respectively, in the overshoot and response time could be achieved using the suggested control structure. Full article

(This article belongs to the Special Issue Control, Optimization, and Mathematical Modeling of Complex Systems)

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21 pages, 1205 KiB

Open AccessFeature PaperArticle

Minimax Estimation in Regression under Sample Conformity Constraints

by Andrey Borisov

Mathematics 2021, 9(10), 1080; https://doi.org/10.3390/math9101080 - 11 May 2021

Cited by 3 | Viewed by 2067

Abstract

The paper is devoted to the guaranteeing estimation of parameters in the uncertain stochastic nonlinear regression. The loss function is the conditional mean square of the estimation error given the available observations. The distribution of regression parameters is partially unknown, and the uncertainty [...] Read more.

The paper is devoted to the guaranteeing estimation of parameters in the uncertain stochastic nonlinear regression. The loss function is the conditional mean square of the estimation error given the available observations. The distribution of regression parameters is partially unknown, and the uncertainty is described by a subset of probability distributions with a known compact domain. The essential feature is the usage of some additional constraints describing the conformity of the uncertain distribution to the realized observation sample. The paper contains various examples of the conformity indices. The estimation task is formulated as the minimax optimization problem, which, in turn, is solved in terms of saddle points. The paper presents the characterization of both the optimal estimator and the set of least favorable distributions. The saddle points are found via the solution to a dual finite-dimensional optimization problem, which is simpler than the initial minimax problem. The paper proposes a numerical mesh procedure of the solution to the dual optimization problem. The interconnection between the least favorable distributions under the conformity constraint, and their Pareto efficiency in the sense of a vector criterion is also indicated. The influence of various conformity constraints on the estimation performance is demonstrated by the illustrative numerical examples. Full article

(This article belongs to the Special Issue Control, Optimization, and Mathematical Modeling of Complex Systems)

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14 pages, 567 KiB

Open AccessArticle

Discrete Velocity Boltzmann Model for Quasi-Incompressible Hydrodynamics

by Oleg Ilyin

Mathematics 2021, 9(9), 993; https://doi.org/10.3390/math9090993 - 28 Apr 2021

Cited by 8 | Viewed by 2393

Abstract

In this paper, we consider the development of the two-dimensional discrete velocity Boltzmann model on a nine-velocity lattice. Compared to the conventional lattice Boltzmann approach for the present model, the collision rules for the interacting particles are formulated explicitly. The collisions are tailored [...] Read more.

In this paper, we consider the development of the two-dimensional discrete velocity Boltzmann model on a nine-velocity lattice. Compared to the conventional lattice Boltzmann approach for the present model, the collision rules for the interacting particles are formulated explicitly. The collisions are tailored in such a way that mass, momentum and energy are conserved and the H-theorem is fulfilled. By applying the Chapman–Enskog expansion, we show that the model recovers quasi-incompressible hydrodynamic equations for small Mach number limit and we derive the closed expression for the viscosity, depending on the collision cross-sections. In addition, the numerical implementation of the model with the on-lattice streaming and local collision step is proposed. As test problems, the shear wave decay and Taylor–Green vortex are considered, and a comparison of the numerical simulations with the analytical solutions is presented. Full article

(This article belongs to the Special Issue Control, Optimization, and Mathematical Modeling of Complex Systems)

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18 pages, 610 KiB

Open AccessArticle

Precise Trajectory Tracking Control of Ship Towing Systems via a Dynamical Tracking Target

by Ouxue Li and Yusheng Zhou

Mathematics 2021, 9(9), 974; https://doi.org/10.3390/math9090974 - 27 Apr 2021

Cited by 7 | Viewed by 2620

Abstract

This paper proposes a novel control strategy to address the precise trajectory tracking control problem of a ship towing system. At first, the kinematics and dynamics models of a ship towing system are established by introducing a passive steering angle and using its [...] Read more.

This paper proposes a novel control strategy to address the precise trajectory tracking control problem of a ship towing system. At first, the kinematics and dynamics models of a ship towing system are established by introducing a passive steering angle and using its structure relationship. Then, by using the motion law derived from its nonholonomic constraints, the relative curvature of the target trajectory curve is applied to design a dynamical tracking target. By applying the sliding mode control and inverse dynamic adaptive control methods, two appropriate robust torque controllers are designed via the dynamical tracking target, so that both the tugboat and the towed ship are able to track the desired path precisely. As we show, the proposed strategy has excellent agreement with the numerical simulation results. Full article

(This article belongs to the Special Issue Control, Optimization, and Mathematical Modeling of Complex Systems)

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13 pages, 415 KiB

Open AccessArticle

Qualitative Properties of Randomized Maximum Entropy Estimates of Probability Density Functions

by Yuri S. Popkov

Mathematics 2021, 9(5), 548; https://doi.org/10.3390/math9050548 - 5 Mar 2021

Cited by 2 | Viewed by 1857

Abstract

The problem of randomized maximum entropy estimation for the probability density function of random model parameters with real data and measurement noises was formulated. This estimation procedure maximizes an information entropy functional on a set of integral equalities depending on the real data [...] Read more.

The problem of randomized maximum entropy estimation for the probability density function of random model parameters with real data and measurement noises was formulated. This estimation procedure maximizes an information entropy functional on a set of integral equalities depending on the real data set. The technique of the Gâteaux derivatives is developed to solve this problem in analytical form. The probability density function estimates depend on Lagrange multipliers, which are obtained by balancing the model’s output with real data. A global theorem for the implicit dependence of these Lagrange multipliers on the data sample’s length is established using the rotation of homotopic vector fields. A theorem for the asymptotic efficiency of randomized maximum entropy estimate in terms of stationary Lagrange multipliers is formulated and proved. The proposed method is illustrated on the problem of forecasting of the evolution of the thermokarst lake area in Western Siberia. Full article

(This article belongs to the Special Issue Control, Optimization, and Mathematical Modeling of Complex Systems)

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17 pages, 5718 KiB

Open AccessArticle

Sparse Grid Adaptive Interpolation in Problems of Modeling Dynamic Systems with Interval Parameters

by Alexander Yu Morozov, Andrey A. Zhuravlev and Dmitry L. Reviznikov

Mathematics 2021, 9(4), 298; https://doi.org/10.3390/math9040298 - 3 Feb 2021

Cited by 9 | Viewed by 2471

Abstract

The paper is concerned with the issues of modeling dynamic systems with interval parameters. In previous works, the authors proposed an adaptive interpolation algorithm for solving interval problems; the essence of the algorithm is the dynamic construction of a piecewise polynomial function that [...] Read more.

The paper is concerned with the issues of modeling dynamic systems with interval parameters. In previous works, the authors proposed an adaptive interpolation algorithm for solving interval problems; the essence of the algorithm is the dynamic construction of a piecewise polynomial function that interpolates the solution of the problem with a given accuracy. The main problem of applying the algorithm is related to the curse of dimension, i.e., exponential complexity relative to the number of interval uncertainties in parameters. The main objective of this work is to apply the previously proposed adaptive interpolation algorithm to dynamic systems with a large number of interval parameters. In order to reduce the computational complexity of the algorithm, the authors propose using adaptive sparse grids. This article introduces a novelty approach of applying sparse grids to problems with interval uncertainties. The efficiency of the proposed approach has been demonstrated on representative interval problems of nonlinear dynamics and computational materials science. Full article

(This article belongs to the Special Issue Control, Optimization, and Mathematical Modeling of Complex Systems)

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17 pages, 665 KiB

Open AccessArticle

Machine Learning Control Based on Approximation of Optimal Trajectories

by Askhat Diveev, Sergey Konstantinov, Elizaveta Shmalko and Ge Dong

Mathematics 2021, 9(3), 265; https://doi.org/10.3390/math9030265 - 29 Jan 2021

Cited by 15 | Viewed by 2793

Abstract

The paper is devoted to an emerging trend in control—a machine learning control. Despite the popularity of the idea of machine learning, there are various interpretations of this concept, and there is an urgent need for its strict mathematical formalization. An attempt to [...] Read more.

The paper is devoted to an emerging trend in control—a machine learning control. Despite the popularity of the idea of machine learning, there are various interpretations of this concept, and there is an urgent need for its strict mathematical formalization. An attempt to formalize the concept of machine learning is presented in this paper. The concepts of an unknown function, work area, training set are introduced, and a mathematical formulation of the machine learning problem is presented. Based on the presented formulation, the concept of machine learning control is considered. One of the problems of machine learning control is the general synthesis of control. It implies finding a control function that depends on the state of the object, which ensures the achievement of the control goal with the optimal value of the quality criterion from any initial state of some admissible region. Supervised and unsupervised approaches to solving a problem based on symbolic regression methods are considered. As a computational example, a problem of general synthesis of optimal control for a spacecraft landing on the surface of the Moon is considered as supervised machine learning control with a training set. Full article

(This article belongs to the Special Issue Control, Optimization, and Mathematical Modeling of Complex Systems)

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16 pages, 353 KiB

Open AccessArticle

Automatic Convexity Deduction for Efficient Function’s Range Bounding

by Mikhail Posypkin and Oleg Khamisov

Mathematics 2021, 9(2), 134; https://doi.org/10.3390/math9020134 - 10 Jan 2021

Cited by 8 | Viewed by 2552

Abstract

Reliable bounding of a function’s range is essential for deterministic global optimization, approximation, locating roots of nonlinear equations, and several other computational mathematics areas. Despite years of extensive research in this direction, there is still room for improvement. The traditional and compelling approach [...] Read more.

Reliable bounding of a function’s range is essential for deterministic global optimization, approximation, locating roots of nonlinear equations, and several other computational mathematics areas. Despite years of extensive research in this direction, there is still room for improvement. The traditional and compelling approach to this problem is interval analysis. We show that accounting convexity/concavity can significantly tighten the bounds computed by interval analysis. To make our approach applicable to a broad range of functions, we also develop the techniques for handling nondifferentiable composite functions. Traditional ways to ensure the convexity fail in such cases. Experimental evaluation showed the remarkable potential of the proposed methods. Full article

(This article belongs to the Special Issue Control, Optimization, and Mathematical Modeling of Complex Systems)

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20 pages, 1658 KiB

Open AccessArticle

Facilitating Numerical Solutions of Inhomogeneous Continuous Time Markov Chains Using Ergodicity Bounds Obtained with Logarithmic Norm Method

by Alexander Zeifman, Yacov Satin, Ivan Kovalev, Rostislav Razumchik and Victor Korolev

Mathematics 2021, 9(1), 42; https://doi.org/10.3390/math9010042 - 27 Dec 2020

Cited by 17 | Viewed by 2675

Abstract

The problem considered is the computation of the (limiting) time-dependent performance characteristics of one-dimensional continuous-time Markov chains with discrete state space and time varying intensities. Numerical solution techniques can benefit from methods providing ergodicity bounds because the latter can indicate how to choose [...] Read more.

The problem considered is the computation of the (limiting) time-dependent performance characteristics of one-dimensional continuous-time Markov chains with discrete state space and time varying intensities. Numerical solution techniques can benefit from methods providing ergodicity bounds because the latter can indicate how to choose the position and the length of the “distant time interval” (in the periodic case) on which the solution has to be computed. They can also be helpful whenever the state space truncation is required. In this paper one such analytic method—the logarithmic norm method—is being reviewed. Its applicability is shown within the queueing theory context with three examples: the classical time-varying

M / M / 2

queue; the time-varying single-server Markovian system with bulk arrivals, queue skipping policy and catastrophes; and the time-varying Markovian bulk-arrival and bulk-service system with state-dependent control. In each case it is shown whether and how the bounds on the rate of convergence can be obtained. Numerical examples are provided. Full article

(This article belongs to the Special Issue Control, Optimization, and Mathematical Modeling of Complex Systems)

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18 pages, 634 KiB

Open AccessArticle

Fundamentals of Synthesized Optimal Control

by Askhat Diveev, Elizaveta Shmalko, Vladimir Serebrenny and Peter Zentay

Mathematics 2021, 9(1), 21; https://doi.org/10.3390/math9010021 - 23 Dec 2020

Cited by 29 | Viewed by 2845

Abstract

This paper presents a new formulation of the optimal control problem with uncertainty, in which an additive bounded function is considered as uncertainty. The purpose of the control is to ensure the achievement of terminal conditions with the optimal value of the quality [...] Read more.

This paper presents a new formulation of the optimal control problem with uncertainty, in which an additive bounded function is considered as uncertainty. The purpose of the control is to ensure the achievement of terminal conditions with the optimal value of the quality functional, while the uncertainty has a limited impact on the change in the value of the functional. The article introduces the concept of feasibility of the mathematical model of the object, which is associated with the contraction property of mappings if we consider the model of the object as a one-parameter mapping. It is shown that this property is sufficient for the development of stable practical systems. To find a solution to the stated problem, which would ensure the feasibility of the system, the synthesized optimal control method is proposed. This article formulates the theoretical foundations of the synthesized optimal control. The method consists in making the control object stable relative to some point in the state space and to control the object by changing the position of the equilibrium points. The article provides evidence that this approach is insensitive to the uncertainties of the mathematical model of the object. An example of the application of the method for optimal control of a group of robots is given. A comparison of the synthesized optimal control method with the direct method on the model without disturbances and with them is presented. Full article

(This article belongs to the Special Issue Control, Optimization, and Mathematical Modeling of Complex Systems)

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18 pages, 1164 KiB

Open AccessArticle

Optimal Control Problem Solution with Phase Constraints for Group of Robots by Pontryagin Maximum Principle and Evolutionary Algorithm

by Askhat Diveev, Elena Sofronova and Ivan Zelinka

Mathematics 2020, 8(12), 2105; https://doi.org/10.3390/math8122105 - 25 Nov 2020

Cited by 10 | Viewed by 2888

Abstract

A numerical method based on the Pontryagin maximum principle for solving an optimal control problem with static and dynamic phase constraints for a group of objects is considered. Dynamic phase constraints are introduced to avoid collisions between objects. Phase constraints are included in [...] Read more.

A numerical method based on the Pontryagin maximum principle for solving an optimal control problem with static and dynamic phase constraints for a group of objects is considered. Dynamic phase constraints are introduced to avoid collisions between objects. Phase constraints are included in the functional in the form of smooth penalty functions. Additional parameters for special control modes and the terminal time of the control process were introduced. The search for additional parameters and the initial conditions for the conjugate variables was performed by the modified self-organizing migrating algorithm. An example of using this approach to solve the optimal control problem for the oncoming movement of two mobile robots is given. Simulation and comparison with direct approach showed that the problem is multimodal, and it approves application of the evolutionary algorithm for its solution. Full article

(This article belongs to the Special Issue Control, Optimization, and Mathematical Modeling of Complex Systems)

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Review

Jump to: Editorial, Research

42 pages, 448 KiB

Open AccessReview

Review of the Latest Progress in Controllability of Stochastic Linear Systems and Stochastic GE-Evolution Operator

by Zhaoqiang Ge

Mathematics 2021, 9(24), 3240; https://doi.org/10.3390/math9243240 - 14 Dec 2021

Cited by 5 | Viewed by 2358

Abstract

According to the spatial dimension, equation type, and time sequence, the latest progress in controllability of stochastic linear systems and some unsolved problems are introduced. Firstly, the exact controllability of stochastic linear systems in finite dimensional spaces is discussed. Secondly, the exact, exact [...] Read more.

According to the spatial dimension, equation type, and time sequence, the latest progress in controllability of stochastic linear systems and some unsolved problems are introduced. Firstly, the exact controllability of stochastic linear systems in finite dimensional spaces is discussed. Secondly, the exact, exact null, approximate, approximate null, and partial approximate controllability of stochastic linear systems in infinite dimensional spaces are considered. Thirdly, the exact, exact null and impulse controllability of stochastic singular linear systems in finite dimensional spaces are investigated. Fourthly, the exact and approximate controllability of stochastic singular linear systems in infinite dimensional spaces are studied. At last, the controllability and observability for a type of time-varying stochastic singular linear systems are studied by using stochastic GE-evolution operator in the sense of mild solution in Banach spaces, some necessary and sufficient conditions are obtained, the dual principle is proved to be true, an example is given to illustrate the validity of the theoretical results obtained in this part, and a problem to be solved is introduced. The main purpose of this paper is to facilitate readers to fully understand the latest research results concerning the controllability of stochastic linear systems and the problems that need to be further studied, and attract more scholars to engage in this research. Full article

(This article belongs to the Special Issue Control, Optimization, and Mathematical Modeling of Complex Systems)

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