Mathematics

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Preface to the Special Issue on “Identification, Knowledge Engineering and Digital Modeling for Adaptive and Intelligent Control”—Special Issue Book

by Natalia Bakhtadze

Mathematics 2023, 11(8), 1906; https://doi.org/10.3390/math11081906 - 18 Apr 2023

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Abstract

Starting our work on this Special Issue, we assumed that the research results presented here would reflect the solutions to various problems related to production management; however, the set of identified problems showed that their solutions could be useful for a wider range [...] Read more.

Starting our work on this Special Issue, we assumed that the research results presented here would reflect the solutions to various problems related to production management; however, the set of identified problems showed that their solutions could be useful for a wider range of applications [...] Full article

(This article belongs to the Special Issue Identification, Knowledge Engineering and Digital Modeling for Adaptive and Intelligent Control)

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16 pages, 4825 KiB

Open AccessArticle

Choice of Regularization Methods in Experiment Processing: Solving Inverse Problems of Thermal Conductivity

by Alexander Sokolov and Irina Nikulina

Mathematics 2022, 10(22), 4221; https://doi.org/10.3390/math10224221 - 11 Nov 2022

Cited by 1 | Viewed by 1096

Abstract

This work is aimed at numerical studies of inverse problems of experiment processing (identification of unknown parameters of mathematical models from experimental data) based on the balanced identification technology. Such problems are inverse in their nature and often turn out to be ill-posed. [...] Read more.

This work is aimed at numerical studies of inverse problems of experiment processing (identification of unknown parameters of mathematical models from experimental data) based on the balanced identification technology. Such problems are inverse in their nature and often turn out to be ill-posed. To solve them, various regularization methods are used, which differ in regularizing additions and methods for choosing the values of the regularization parameters. Balanced identification technology uses the cross-validation root-mean-square error to select the values of the regularization parameters. Its minimization leads to an optimally balanced solution, and the obtained value is used as a quantitative criterion for the correspondence of the model and the regularization method to the data. The approach is illustrated by the problem of identifying the heat-conduction coefficient on temperature. A mixed one-dimensional nonlinear heat conduction problem was chosen as a model. The one-dimensional problem was chosen based on the convenience of the graphical presentation of the results. The experimental data are synthetic data obtained on the basis of a known exact solution with added random errors. In total, nine problems (some original) were considered, differing in data sets and criteria for choosing solutions. This is the first time such a comprehensive study with error analysis has been carried out. Various estimates of the modeling errors are given and show a good agreement with the characteristics of the synthetic data errors. The effectiveness of the technology is confirmed by comparing numerical solutions with exact ones. Full article

(This article belongs to the Special Issue Identification, Knowledge Engineering and Digital Modeling for Adaptive and Intelligent Control)

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17 pages, 1190 KiB

Open AccessArticle

Identification of Quadratic Volterra Polynomials in the “Input–Output” Models of Nonlinear Systems

by Yury Voscoboynikov, Svetlana Solodusha, Evgeniia Markova, Ekaterina Antipina and Vasilisa Boeva

Mathematics 2022, 10(11), 1836; https://doi.org/10.3390/math10111836 - 26 May 2022

Cited by 3 | Viewed by 1613

Abstract

In this paper, we propose a new algorithm for constructing an integral model of a nonlinear dynamic system of the “input–output” type in the form of a quadratic segment of the Volterra integro-power series (polynomial). We consider nonparametric identification of models using physically [...] Read more.

In this paper, we propose a new algorithm for constructing an integral model of a nonlinear dynamic system of the “input–output” type in the form of a quadratic segment of the Volterra integro-power series (polynomial). We consider nonparametric identification of models using physically realizable piecewise linear test signals in the time domain. The advantage of the presented approach is to obtain explicit formulas for calculating the transient responses (Volterra kernels), which determine the unique solution of the Volterra integral equations of the first kind with two variable integration limits. The numerical method proposed in the paper for solving the corresponding equations includes the use of smoothing splines. An important result is that the constructed identification algorithm has a low methodological error. Full article

(This article belongs to the Special Issue Identification, Knowledge Engineering and Digital Modeling for Adaptive and Intelligent Control)

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14 pages, 619 KiB

Open AccessArticle

Bayes Synthesis of Linear Nonstationary Stochastic Systems by Wavelet Canonical Expansions

by Igor Sinitsyn, Vladimir Sinitsyn, Eduard Korepanov and Tatyana Konashenkova

Mathematics 2022, 10(9), 1517; https://doi.org/10.3390/math10091517 - 2 May 2022

Cited by 3 | Viewed by 1351

Abstract

This article is devoted to analysis and optimization problems of stochastic systems based on wavelet canonical expansions. Basic new results: (i) for general Bayes criteria, a method of synthesized methodological support and a software tool for nonstationary normal (Gaussian) linear observable stochastic systems [...] Read more.

This article is devoted to analysis and optimization problems of stochastic systems based on wavelet canonical expansions. Basic new results: (i) for general Bayes criteria, a method of synthesized methodological support and a software tool for nonstationary normal (Gaussian) linear observable stochastic systems by Haar wavelet canonical expansions are presented; (ii) a method of synthesis of a linear optimal observable system for criterion of the maximal probability that a signal will not exceed a particular value in absolute magnitude is given. Applications: wavelet model building of essentially nonstationary stochastic processes and parameters calibration. Full article

(This article belongs to the Special Issue Identification, Knowledge Engineering and Digital Modeling for Adaptive and Intelligent Control)

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12 pages, 23920 KiB

Open AccessArticle

Differential Neural Network-Based Nonparametric Identification of Eye Response to Enforced Head Motion

by Isaac Chairez, Arthur Mukhamedov, Vladislav Prud, Olga Andrianova and Viktor Chertopolokhov

Mathematics 2022, 10(6), 855; https://doi.org/10.3390/math10060855 - 8 Mar 2022

Cited by 3 | Viewed by 2222

Abstract

Dynamic motion simulators cannot provide the same stimulation of sensory systems as real motion. Hence, only a subset of human senses should be targeted. For simulators providing vestibular stimulus, an automatic bodily function of vestibular–ocular reflex (VOR) can objectively measure how accurate motion [...] Read more.

Dynamic motion simulators cannot provide the same stimulation of sensory systems as real motion. Hence, only a subset of human senses should be targeted. For simulators providing vestibular stimulus, an automatic bodily function of vestibular–ocular reflex (VOR) can objectively measure how accurate motion simulation is. This requires a model of ocular response to enforced accelerations, an attempt to create which is shown in this paper. The proposed model corresponds to a single-layer spiking differential neural network with its activation functions are based on the dynamic Izhikevich model of neuron dynamics. An experiment is proposed to collect training data corresponding to controlled accelerated motions that produce VOR, measured using an eye-tracking system. The effectiveness of the proposed identification is demonstrated by comparing its performance with a traditional sigmoidal identifier. The proposed model based on dynamic representations of activation functions produces a more accurate approximation of foveal motion as the estimation of mean square error confirms. Full article

(This article belongs to the Special Issue Identification, Knowledge Engineering and Digital Modeling for Adaptive and Intelligent Control)

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13 pages, 607 KiB

Open AccessArticle

Identification of Linear Time-Invariant Systems with Dynamic Mode Decomposition

by Jan Heiland and Benjamin Unger

Mathematics 2022, 10(3), 418; https://doi.org/10.3390/math10030418 - 28 Jan 2022

Cited by 3 | Viewed by 2165

Abstract

Dynamic mode decomposition (DMD) is a popular data-driven framework to extract linear dynamics from complex high-dimensional systems. In this work, we study the system identification properties of DMD. We first show that DMD is invariant under linear transformations in the image of the [...] Read more.

Dynamic mode decomposition (DMD) is a popular data-driven framework to extract linear dynamics from complex high-dimensional systems. In this work, we study the system identification properties of DMD. We first show that DMD is invariant under linear transformations in the image of the data matrix. If, in addition, the data are constructed from a linear time-invariant system, then we prove that DMD can recover the original dynamics under mild conditions. If the linear dynamics are discretized with the Runge–Kutta method, then we further classify the error of the DMD approximation and detail that for one-stage Runge–Kutta methods; even the continuous dynamics can be recovered with DMD. A numerical example illustrates the theoretical findings. Full article

(This article belongs to the Special Issue Identification, Knowledge Engineering and Digital Modeling for Adaptive and Intelligent Control)

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23 pages, 1786 KiB

Open AccessArticle

Robust Stabilization via Super-Stable Systems Techniques

by Svetlana A. Krasnova, Yulia G. Kokunko, Victor A. Utkin and Anton V. Utkin

Mathematics 2022, 10(1), 98; https://doi.org/10.3390/math10010098 - 28 Dec 2021

Cited by 1 | Viewed by 1404

Abstract

In this paper, we propose a direct method for the synthesis of robust systems operating under parametric uncertainty of the control plant model. The developed robust control procedures are based on the assumption that the structural properties of the nominal system are conservated [...] Read more.

In this paper, we propose a direct method for the synthesis of robust systems operating under parametric uncertainty of the control plant model. The developed robust control procedures are based on the assumption that the structural properties of the nominal system are conservated over the entire range of parameter changes. The invariant-to-parametric-uncertainties transformation of the initial model to a regular form makes it possible to use the concept of super-stable systems for the synthesis of a stabilizing feedback. It is essential that the synthesis of super-stable systems is carried out not on the basis of assigning eigenvalues to the matrix of the close-loop system, but in terms of its elements. The proposed approach is applicable to a wide class of linear systems with parametric uncertainties and provides a given degree of stability. Full article

(This article belongs to the Special Issue Identification, Knowledge Engineering and Digital Modeling for Adaptive and Intelligent Control)

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23 pages, 4639 KiB

Open AccessArticle

New Identification Approach and Methods for Plasma Equilibrium Reconstruction in D-Shaped Tokamaks

by Yuri V. Mitrishkin, Pavel S. Korenev, Artem E. Konkov, Valerii I. Kruzhkov and Nicolai E. Ovsiannikov

Mathematics 2022, 10(1), 40; https://doi.org/10.3390/math10010040 - 23 Dec 2021

Cited by 7 | Viewed by 2839

Abstract

The paper deals with the identification of plasma equilibrium reconstruction in D-shaped tokamaks on the base of plasma external magnetic measurements. The methods of such identification are directed to increase their speed of response when plasma discharges are relatively short, like in the [...] Read more.

The paper deals with the identification of plasma equilibrium reconstruction in D-shaped tokamaks on the base of plasma external magnetic measurements. The methods of such identification are directed to increase their speed of response when plasma discharges are relatively short, like in the spherical Globus-M2 tokamak (Ioffe Inst., St. Petersburg, Russia). The new approach is first to apply to the plasma discharges data the off-line equilibrium reconstruction algorithm based on the Picard iterations, and obtain the gaps between the plasma boundary and the first wall, and the second is to apply new identification methods to the gap values, producing plasma shape models operating in real time. The inputs for on-line robust identification algorithms are the measurements of magnetic fluxes on magnetic loops, plasma current, and currents in the poloidal field coils measured by the Rogowski loops. The novel on-line high-performance identification algorithms are designed on the base of (i) full-order observer synthesized by linear matrix inequality (LMI) methodology, (ii) static matrix obtained by the least square technique, and (iii) deep neural network. The robust observer is constructed on the base of the LPV plant models which have the novelty that the state vector contains the gaps which are estimated by the observer, using input and output signals. The results of the simulation of the identification systems on the base of experimental data of the Globus-M2 tokamak are presented. Full article

(This article belongs to the Special Issue Identification, Knowledge Engineering and Digital Modeling for Adaptive and Intelligent Control)

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19 pages, 409 KiB

Open AccessArticle

On Spectral Decomposition of States and Gramians of Bilinear Dynamical Systems

by Alexey Iskakov and Igor Yadykin

Mathematics 2021, 9(24), 3288; https://doi.org/10.3390/math9243288 - 17 Dec 2021

Cited by 5 | Viewed by 2315

Abstract

The article proves that the state of a bilinear control system can be split uniquely into generalized modes corresponding to the eigenvalues of the dynamics matrix. It is also shown that the Gramians of controllability and observability of a bilinear system can be [...] Read more.

The article proves that the state of a bilinear control system can be split uniquely into generalized modes corresponding to the eigenvalues of the dynamics matrix. It is also shown that the Gramians of controllability and observability of a bilinear system can be divided into parts (sub-Gramians) that characterize the measure of these generalized modes and their interactions. Furthermore, the properties of sub-Gramians were investigated in relation to modal controllability and observability. We also propose an algorithm for computing the Gramians and sub-Gramians based on the element-wise computation of the solution matrix. Based on the proposed algorithm, a novel criterion for the existence of solutions to the generalized Lyapunov equation is proposed, which allows, in some cases, to expand the domain of guaranteed existence of a solution of bilinear equations. Examples are provided that illustrate the application and practical use of the considered spectral decompositions. Full article

(This article belongs to the Special Issue Identification, Knowledge Engineering and Digital Modeling for Adaptive and Intelligent Control)

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23 pages, 7251 KiB

Open AccessArticle

Maximum-Likelihood-Based Adaptive and Intelligent Computing for Nonlinear System Identification

by Hasnat Bin Tariq, Naveed Ishtiaq Chaudhary, Zeshan Aslam Khan, Muhammad Asif Zahoor Raja, Khalid Mehmood Cheema and Ahmad H. Milyani

Mathematics 2021, 9(24), 3199; https://doi.org/10.3390/math9243199 - 11 Dec 2021

Cited by 4 | Viewed by 2137

Abstract

Most real-time systems are nonlinear in nature, and their optimization is very difficult due to inherit stiffness and complex system representation. The computational intelligent algorithms of evolutionary computing paradigm (ECP) effectively solve various complex, nonlinear optimization problems. The differential evolution algorithm (DEA) is [...] Read more.

Most real-time systems are nonlinear in nature, and their optimization is very difficult due to inherit stiffness and complex system representation. The computational intelligent algorithms of evolutionary computing paradigm (ECP) effectively solve various complex, nonlinear optimization problems. The differential evolution algorithm (DEA) is one of the most important approaches in ECP, which outperforms other standard approaches in terms of accuracy and convergence performance. In this study, a novel application of a recently proposed variant of DEA, the so-called, maximum-likelihood-based, adaptive, differential evolution algorithm (ADEA), is investigated for the identification of nonlinear Hammerstein output error (HOE) systems that are widely used to model different nonlinear processes of engineering and applied sciences. The performance of the ADEA is evaluated by taking polynomial- and sigmoidal-type nonlinearities in two case studies of HOE systems. Moreover, the robustness of the proposed scheme is examined for different noise levels. Reliability and consistent accuracy are assessed through multiple independent trials of the scheme. The convergence, accuracy, robustness and reliability of the ADEA are carefully examined for HOE identification in comparison with the standard counterpart of the DEA. The ADEA achieves the fitness values of 1.43 × 10⁻⁸ and 3.46 × 10⁻⁹ for a population size of 80 and 100, respectively, in the HOE system identification problem of case study 1 for a 0.01 nose level, while the respective fitness values in the case of DEA are 1.43 × 10⁻⁶ and 3.46 × 10⁻⁷. The ADEA is more statistically consistent but less complex when compared to the DEA due to the extra operations involved in introducing the adaptiveness during the mutation and crossover. The current study may consider the approach of effective nonlinear system identification as a step further in developing ECP-based computational intelligence. Full article

(This article belongs to the Special Issue Identification, Knowledge Engineering and Digital Modeling for Adaptive and Intelligent Control)

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16 pages, 643 KiB

Open AccessArticle

Analysis and Prediction of Electric Power System’s Stability Based on Virtual State Estimators

by Natalia Bakhtadze and Igor Yadikin

Mathematics 2021, 9(24), 3194; https://doi.org/10.3390/math9243194 - 10 Dec 2021

Cited by 4 | Viewed by 1977

Abstract

The stability of bilinear systems is investigated using spectral techniques such as selective modal analysis. Predictive models of bilinear systems based on inductive knowledge extracted by big data mining techniques are applied with associative search of statistical patterns. A method and an algorithm [...] Read more.

The stability of bilinear systems is investigated using spectral techniques such as selective modal analysis. Predictive models of bilinear systems based on inductive knowledge extracted by big data mining techniques are applied with associative search of statistical patterns. A method and an algorithm for the elementwise solution of the generalized matrix Lyapunov equation are developed for discrete bilinear systems. The method is based on calculating the sequence of values of a fixed element of the solution matrix, which depends on the product of the eigenvalues of the dynamics matrix of the linear part and the elements of the nonlinearity matrixes. A sufficient condition for the convergence of all sequences is obtained, which is also a BIBO (bounded input bounded output) systems stability condition for the bilinear system. Full article

(This article belongs to the Special Issue Identification, Knowledge Engineering and Digital Modeling for Adaptive and Intelligent Control)

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20 pages, 305 KiB

Open AccessArticle

Methods of Ensuring Invariance with Respect to External Disturbances: Overview and New Advances

by Aleksey Antipov, Svetlana Krasnova and Victor Utkin

Mathematics 2021, 9(23), 3140; https://doi.org/10.3390/math9233140 - 6 Dec 2021

Cited by 12 | Viewed by 1981

Abstract

In this paper, we carry out a demonstration and comparative analysis of known methods of the synthesis of various control laws ensuring the invariance of the output (controlled) variable with respect to external disturbances under various assumptions about their type and channels of [...] Read more.

In this paper, we carry out a demonstration and comparative analysis of known methods of the synthesis of various control laws ensuring the invariance of the output (controlled) variable with respect to external disturbances under various assumptions about their type and channels of acting on the control plant. Methods of the synthesis are presented on the example of a third-order nonlinear system with single input and single output (SISO-systems), dynamic feedback synthesis is presented at a descriptive level and the focus is on procedures of static feedback synthesis. For the systems in which the matching conditions are not satisfied, it is concluded that it is expedient to introduce smooth and bounded nonlinear local feedbacks. Within the framework of the block control principle, we developed an iterative procedure of synthesis of S-shaped sigmoid feedbacks for such systems. Nonlinear local feedbacks ensure stabilization of the output variable with the given accuracy and settling time as in a system with traditionally used linear local feedbacks with high gains. However, in contrast to it, sigmoid functions do not lead to a large overshoot of state variables and control actions. Full article

(This article belongs to the Special Issue Identification, Knowledge Engineering and Digital Modeling for Adaptive and Intelligent Control)

15 pages, 1237 KiB

Open AccessArticle

Optimal Stochastic Control in the Interception Problem of a Randomly Tacking Vehicle

by Andrey A. Galyaev, Pavel V. Lysenko and Evgeny Y. Rubinovich

Mathematics 2021, 9(19), 2386; https://doi.org/10.3390/math9192386 - 25 Sep 2021

Cited by 7 | Viewed by 1815

Abstract

This article considers the mathematical aspects of the problem of the optimal interception of a mobile search vehicle moving along random tacks on a given route and searching for a target, which travels parallel to this route. Interception begins when the probability of [...] Read more.

This article considers the mathematical aspects of the problem of the optimal interception of a mobile search vehicle moving along random tacks on a given route and searching for a target, which travels parallel to this route. Interception begins when the probability of the target being detected by the search vehicle exceeds a certain threshold value. Interception was carried out by a controlled vehicle (defender) protecting the target. An analytical estimation of this detection probability is proposed. The interception problem was formulated as an optimal stochastic control problem, which was transformed to a deterministic optimization problem. As a result, the optimal control law of the defender was found, and the optimal interception time was estimated. The deterministic problem is a simplified version of the problem whose optimal solution provides a suboptimal solution to the stochastic problem. The obtained control law was compared with classic guidance methods. All the results were obtained analytically and validated with a computer simulation. Full article

(This article belongs to the Special Issue Identification, Knowledge Engineering and Digital Modeling for Adaptive and Intelligent Control)

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14 pages, 4706 KiB

Open AccessArticle

Multi-Output Soft Sensor with a Multivariate Filter That Predicts Errors Applied to an Industrial Reactive Distillation Process

by Vladimir Klimchenko, Andrei Torgashov, Yuri A. W. Shardt and Fan Yang

Mathematics 2021, 9(16), 1947; https://doi.org/10.3390/math9161947 - 15 Aug 2021

Cited by 4 | Viewed by 1921

Abstract

The paper deals with the problem of developing a multi-output soft sensor for the industrial reactive distillation process of methyl tert-butyl ether production. Unlike the existing soft sensor approaches, this paper proposes using a soft sensor with filters to predict model errors, which [...] Read more.

The paper deals with the problem of developing a multi-output soft sensor for the industrial reactive distillation process of methyl tert-butyl ether production. Unlike the existing soft sensor approaches, this paper proposes using a soft sensor with filters to predict model errors, which are then taken into account as corrections in the final predictions of outputs. The decomposition of the problem of optimal estimation of time delays is proposed for each input of the soft sensor. Using the proposed approach to predict the concentrations of methyl sec-butyl ether, methanol, and the sum of dimers and trimers of isobutylene in the output product in a reactive distillation column was shown to improve the results by 32%, 67%, and 9.5%, respectively. Full article

(This article belongs to the Special Issue Identification, Knowledge Engineering and Digital Modeling for Adaptive and Intelligent Control)

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13 pages, 818 KiB

Open AccessArticle

Models of Strategic Decision-Making under Informational Control

by Dmitry Novikov

Mathematics 2021, 9(16), 1889; https://doi.org/10.3390/math9161889 - 9 Aug 2021

Cited by 4 | Viewed by 2090

Abstract

A general complex model is considered for collective dynamical strategic decision-making with explicitly interconnected factors reflecting both psychic (internal state) and behavioral (external-action, result of activity) components of agents’ activity under the given environmental and control factors. This model unifies and generalizes approaches [...] Read more.

A general complex model is considered for collective dynamical strategic decision-making with explicitly interconnected factors reflecting both psychic (internal state) and behavioral (external-action, result of activity) components of agents’ activity under the given environmental and control factors. This model unifies and generalizes approaches of game theory, social psychology, theories of multi-agent systems, and control in organizational systems by simultaneous consideration of both internal and external parameters of the agents. Two special models (of informational control and informational confrontation) contain formal results on controllability and properties of equilibriums. Interpretations of a general model are conformity (threshold behavior), consensus, cognitive dissonance, and other effects with applications to production systems, multi-agent systems, crowd behavior, online social networks, and voting in small and large groups. Full article

(This article belongs to the Special Issue Identification, Knowledge Engineering and Digital Modeling for Adaptive and Intelligent Control)

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