Search for Articles:
Title / Keyword
Author / Affiliation / Email
Journal
Article Type
 
 
Advanced Search
 
Section
Special Issue
Volume
Issue
Number
Page
 
3.5
2.4
Journals
Mathematics
Special Issues
Dynamics and Control Theory with Applications
share announcement

Dynamics and Control Theory with Applications

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Dynamical Systems".

Deadline for manuscript submissions: closed (30 December 2023) | Viewed by 26973

Special Issue Editors

Prof. Dr. Yurii Vladimirovich Mitrishkin

E-Mail Website
Guest Editor
Faculty of Physics, Lomonosov Moscow State University, Moscow, Russia
Interests: identification; methods and systems of control of dynamical plants; plasma control in tokamaks
Dr. Nikolay M. Kartsev

E-Mail Website
Guest Editor
V.A. Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, 117997 Moscow, Russia
Interests: plasma control in tokamaks

Special Issue Information

Dear Colleagues,

In reality, especially in science and engineering, there are countless dynamic objects that are described by differential or difference equations and require automatic control, such as unstable dynamic objects. Such objects can be controlled either with the help of dynamic models (model-based design) or without them. In the first case, it is necessary to build mathematical models using the laws of physics, chemistry, biology, etc. (the first principal equations), or identification. Then, based on these models, either the known automatic control theory is used to control them, or a new theory is developed to create methods and systems mainly with feedback to control the models. The purpose of this is to apply the theory in practice. In the second case, control systems are tuned to an object without knowledge of its model by automatically tuning the controller, e.g., using reinforcement learning.

Topics include, but are not limited to, the following:

  1. Building mathematical models of complex dynamic objects using the laws of physics, chemistry, biology, etc.;
  2. Solving identification problems of dynamic objects to build adequate mathematical models suitable for the synthesis of control systems;
  3. Developing theory with formulation of new control problems and proving theorems for synthesis and analysis of control systems solving these problems;
  4. Developing cascade hierarchical control systems with different control laws at different control levels;
  5. Developing theory and control systems with artificial intelligence, using neural networks and machine learning, in particular reinforcement learning;
  6. Simulating synthesized control systems for dynamic objects on computers and in real time on test beds;
  7. Developing digital twins that provide information flows in both directions between the control system in real space and the control system in virtual space;
  8. Applying new control systems to real dynamic objects and obtaining new practical results.

Prof. Dr. Yurii Vladimirovich Mitrishkin
Dr. Nikolay M. Kartsev
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • mathematical dynamical models
  • identification
  • model-based design
  • control laws
  • feedback control systems
  • cascade hierarchical control systems
  • artificial intelligence for control
  • reinforcement learning for control
  • simulation
  • real-time test beds
  • digital twins
  • control of real dynamic objects

Published Papers (24 papers)

Download All Papers
Order results
Result details
Show export options expand_more Show export options expand_less
Select all
Export citation of selected articles as:

Research

Jump to: Other

38 pages, 2295 KiB  
Article
Synthesis Methodology for Discrete MIMO PID Controller with Loop Shaping on LTV Plant Model via Iterated LMI Restrictions
by Artem E. Konkov and Yuri V. Mitrishkin
Mathematics 2024, 12(6), 810; https://doi.org/10.3390/math12060810 - 9 Mar 2024
Viewed by 649
Abstract
This paper presents a methodology for synthesizing discrete MIMO PID controllers through iterative solutions of LMIs. It justifies the necessity of direct synthesis of discrete controllers in digital control systems. The proposed methodology allows for synthesizing an LTI controller for an LTV plant [...] Read more.
This paper presents a methodology for synthesizing discrete MIMO PID controllers through iterative solutions of LMIs. It justifies the necessity of direct synthesis of discrete controllers in digital control systems. The proposed methodology allows for synthesizing an LTI controller for an LTV plant model, ensuring the robust stability of the synthesized system. Robustness is further ensured by the small-gain theorem and a novel approach to loop shaping, enabling the use of arbitrary shape functions. As a result, this methodology provides a broad statement of the problem with performance criteria necessary for practical application in control systems. Numerical examples are used to illustrate the methodology, which is implemented using the MATLAB toolbox, freely available for use. Full article
(This article belongs to the Special Issue Dynamics and Control Theory with Applications)
►▼ Show Figures

Figure 1

20 pages, 6321 KiB  
Article
Design and Real-Time Implementation of a Cascaded Model Predictive Control Architecture for Unmanned Aerial Vehicles
by Patricio Borbolla-Burillo, David Sotelo, Michael Frye, Luis E. Garza-Castañón, Luis Juárez-Moreno and Carlos Sotelo
Mathematics 2024, 12(5), 739; https://doi.org/10.3390/math12050739 - 29 Feb 2024
Viewed by 765
Abstract
Modeling and control are challenging in unmanned aerial vehicles, especially in quadrotors where there exists high coupling between the position and the orientation dynamics. In simulations, conventional control strategies such as the use of a proportional–integral–derivative (PID) controller under different configurations are typically [...] Read more.
Modeling and control are challenging in unmanned aerial vehicles, especially in quadrotors where there exists high coupling between the position and the orientation dynamics. In simulations, conventional control strategies such as the use of a proportional–integral–derivative (PID) controller under different configurations are typically employed due to their simplicity and ease of design. However, linear assumptions have to be made, which turns into poor performance for practical applications on unmanned aerial vehicles (UAVs). This paper designs and implements a hierarchical cascaded model predictive control (MPC) for three-dimensional trajectory tracking using a quadrotor platform. The overall system consists of two stages: the mission server and the commander stabilizer. Different from existing works, the heavy computational burden is managed by decomposing the overall MPC strategy into two different schemes. The first scheme controls the translational displacements while the second scheme regulates the rotational movements of the quadrotor. For validation, the performance of the proposed controller is compared against that of a proportional–integral–velocity (PIV) controller taken from the literature. Here, real-world experiments for tracking helicoidal and lemniscate trajectories are implemented, while for regulation, an extreme wind disturbance is applied. The experimental results show that the proposed controller outperforms the PIV controller, presenting less signal effort fluctuations, especially in terms of rejecting external wind disturbances. Full article
(This article belongs to the Special Issue Dynamics and Control Theory with Applications)
►▼ Show Figures

Figure 1

32 pages, 4578 KiB  
Article
Dimensionless Policies Based on the Buckingham π Theorem: Is This a Good Way to Generalize Numerical Results?
by Alexandre Girard
Mathematics 2024, 12(5), 709; https://doi.org/10.3390/math12050709 - 28 Feb 2024
Viewed by 575
Abstract
The answer to the question posed in the title is yes if the context (the list of variables defining the motion control problem) is dimensionally similar. This article explores the use of the Buckingham π theorem as a tool to encode the control [...] Read more.
The answer to the question posed in the title is yes if the context (the list of variables defining the motion control problem) is dimensionally similar. This article explores the use of the Buckingham π theorem as a tool to encode the control policies of physical systems into a more generic form of knowledge that can be reused in various situations. This approach can be interpreted as enforcing invariance to the scaling of the fundamental units in an algorithm learning a control policy. First, we show, by restating the solution to a motion control problem using dimensionless variables, that (1) the policy mapping involves a reduced number of parameters and (2) control policies generated numerically for a specific system can be transferred exactly to a subset of dimensionally similar systems by scaling the input and output variables appropriately. Those two generic theoretical results are then demonstrated, with numerically generated optimal controllers, for the classic motion control problem of swinging up a torque-limited inverted pendulum and positioning a vehicle in slippery conditions. We also discuss the concept of regime, a region in the space of context variables, that can help to relax the similarity condition. Furthermore, we discuss how applying dimensional scaling of the input and output of a context-specific black-box policy is equivalent to substituting new system parameters in an analytical equation under some conditions, using a linear quadratic regulator (LQR) and a computed torque controller as examples. It remains to be seen how practical this approach can be to generalize policies for more complex high-dimensional problems, but the early results show that it is a promising transfer learning tool for numerical approaches like dynamic programming and reinforcement learning. Full article
(This article belongs to the Special Issue Dynamics and Control Theory with Applications)
►▼ Show Figures

Figure 1

30 pages, 524 KiB  
Article
Nonlinear Optimal Control for Stochastic Dynamical Systems
by Manuel Lanchares and Wassim M. Haddad
Mathematics 2024, 12(5), 647; https://doi.org/10.3390/math12050647 - 22 Feb 2024
Viewed by 617
Abstract
This paper presents a comprehensive framework addressing optimal nonlinear analysis and feedback control synthesis for nonlinear stochastic dynamical systems. The focus lies on establishing connections between stochastic Lyapunov theory and stochastic Hamilton–Jacobi–Bellman theory within a unified perspective. We demonstrate that the closed-loop nonlinear [...] Read more.
This paper presents a comprehensive framework addressing optimal nonlinear analysis and feedback control synthesis for nonlinear stochastic dynamical systems. The focus lies on establishing connections between stochastic Lyapunov theory and stochastic Hamilton–Jacobi–Bellman theory within a unified perspective. We demonstrate that the closed-loop nonlinear system’s asymptotic stability in probability is ensured through a Lyapunov function, identified as the solution to the steady-state form of the stochastic Hamilton–Jacobi–Bellman equation. This dual assurance guarantees both stochastic stability and optimality. Additionally, optimal feedback controllers for affine nonlinear systems are developed using an inverse optimality framework tailored to the stochastic stabilization problem. Furthermore, the paper derives stability margins for optimal and inverse optimal stochastic feedback regulators. Gain, sector, and disk margin guarantees are established for nonlinear stochastic dynamical systems controlled by nonlinear optimal and inverse optimal Hamilton–Jacobi–Bellman controllers. Full article
(This article belongs to the Special Issue Dynamics and Control Theory with Applications)
►▼ Show Figures

Figure 1

20 pages, 303 KiB  
Article
Spectral Decomposition of Gramians of Continuous Linear Systems in the Form of Hadamard Products
by Igor Yadykin
Mathematics 2024, 12(1), 36; https://doi.org/10.3390/math12010036 - 22 Dec 2023
Viewed by 629
Abstract
New possibilities of Gramian computation, by means of canonical transformations into diagonal, controllable, and observable canonical forms, are shown. Using such a technique, the Gramian matrices can be represented as products of the Hadamard matrices of multipliers and the matrices of the transformed [...] Read more.
New possibilities of Gramian computation, by means of canonical transformations into diagonal, controllable, and observable canonical forms, are shown. Using such a technique, the Gramian matrices can be represented as products of the Hadamard matrices of multipliers and the matrices of the transformed right-hand sides of Lyapunov equations. It is shown that these multiplier matrices are invariant under various canonical transformations of linear continuous systems. The modal Lyapunov equations for continuous SISO LTI systems in diagonal form are obtained, and their new solutions based on Hadamard decomposition are proposed. New algorithms for the element-by-element computation of Gramian matrices for stable, continuous MIMO LTI systems are developed. New algorithms for the computation of controllability Gramians in the form of Xiao matrices are developed for continuous SISO LTI systems, given by the equations of state in the controllable and observable canonical forms. The application of transformations to the canonical forms of controllability and observability allowed us to simplify the formulas of the spectral decompositions of the Gramians. In this paper, new spectral expansions in the form of Hadamard products for solutions to the algebraic and differential Sylvester equations of MIMO LTI systems are obtained, including spectral expansions of the finite and infinite cross - Gramians of continuous MIMO LTI systems. Recommendations on the use of the obtained results are given. Full article
(This article belongs to the Special Issue Dynamics and Control Theory with Applications)
26 pages, 6683 KiB  
Article
Modeling and Control of a DC-DC Buck–Boost Converter with Non-Linear Power Inductor Operating in Saturation Region Considering Electrical Losses
by Ernesto Molina-Santana, Felipe Gonzalez-Montañez, Jesus Ulises Liceaga-Castro, Victor Manuel Jimenez-Mondragon and Irma Siller-Alcala
Mathematics 2023, 11(22), 4617; https://doi.org/10.3390/math11224617 - 11 Nov 2023
Cited by 2 | Viewed by 1742
Abstract
The present work proposes a nonlinear model of a buck–boost DC-DC power converter considering the nonlinear magnetic characteristics of the power inductor and electrical losses of the system. The Euler–Lagrange formalism is used for formulating the proposed model. Previous research works have reported [...] Read more.
The present work proposes a nonlinear model of a buck–boost DC-DC power converter considering the nonlinear magnetic characteristics of the power inductor and electrical losses of the system. The Euler–Lagrange formalism is used for formulating the proposed model. Previous research works have reported mathematical models to describe power inductor dynamics. However, a gap in the literature remains regarding modeling this kind of element when it operates within power converters. Also, a linear-based controller scheme is proposed to regulate a non-ideal buck–boost DC-DC power converter. A methodology for tuning the proposed controller is presented, which considers the nonlinear model structure of the power converter, the linearization procedure based on an identification process, and a frequency domain analysis based on the approximated linear model. Finally, the tuned control scheme is tested on the nonlinear model of the power converter under several operational conditions showing excellent performance by effectively regulating the output voltage. The results are compared with those derived from alternative control strategies, and a better performance is generally obtained. Full article
(This article belongs to the Special Issue Dynamics and Control Theory with Applications)
►▼ Show Figures

Figure 1

15 pages, 2412 KiB  
Article
Robust Tracking as Constrained Optimization by Uncertain Dynamic Plant: Mirror Descent Method and ASG—Version of Integral Sliding Mode Control
by Alexander Nazin, Hussain Alazki and Alexander Poznyak
Mathematics 2023, 11(19), 4112; https://doi.org/10.3390/math11194112 - 28 Sep 2023
Cited by 2 | Viewed by 567
Abstract
A class of controlled objects is considered, the dynamics of which are determined by a vector system of ordinary differential equations with a partially known right-hand side. It is presumed that the state variables and their velocities can be measured. Designing a robust [...] Read more.
A class of controlled objects is considered, the dynamics of which are determined by a vector system of ordinary differential equations with a partially known right-hand side. It is presumed that the state variables and their velocities can be measured. Designing a robust tracking controller under some constraints to admissible state variables is the research goal. This construction, which extends the results for the average subgradient technique (ASG), and is an update of the subgradient descent technique (SDM) and integral sliding mode (ISM) approach, is realized by using the Legendre–Fenchel transform. A two-link robot manipulator with three revolute joints, powered by individual PMDC motors, is presented as an illustrative example of the suggested approach implementation. Full article
(This article belongs to the Special Issue Dynamics and Control Theory with Applications)
►▼ Show Figures

Figure 1

17 pages, 2128 KiB  
Article
Robust Sliding-Mode Control Design of DC-DC Zeta Converter Operating in Buck and Boost Modes
by Humam Al-Baidhani, Fabio Corti, Alberto Reatti and Marian K. Kazimierczuk
Mathematics 2023, 11(17), 3791; https://doi.org/10.3390/math11173791 - 4 Sep 2023
Cited by 1 | Viewed by 1029
Abstract
This paper presents a new nonlinear control scheme for a pulse-width modulated dc-dc Zeta converter operating in buck and boost modes. The averaged model of the dc-dc power converter is derived, based on which a robust control law is developed using a simplified [...] Read more.
This paper presents a new nonlinear control scheme for a pulse-width modulated dc-dc Zeta converter operating in buck and boost modes. The averaged model of the dc-dc power converter is derived, based on which a robust control law is developed using a simplified sliding-mode control technique. The existence and stability conditions are introduced to select proper controller gains that ensure fast output voltage convergence towards reference voltage. A detailed design procedure is provided to realize the control scheme using low-cost discrete components. The proposed control method handles large disturbances, accommodates the non-minimum phase property, and maintains regulated output voltage during step-up and step-down operation modes. The control system also maintains constant switching frequency, improves the transient response, and eliminates the steady-state error at the output voltage. A MATLAB/SIMULINK model is developed to simulate the closed-loop dc-dc Zeta converter in continuous conduction mode and investigate the tracking and regulation performance. The simulation results confirm the robustness and stability of the nonlinear controlled power converter under abrupt line and load variations. Full article
(This article belongs to the Special Issue Dynamics and Control Theory with Applications)
►▼ Show Figures

Figure 1

25 pages, 2423 KiB  
Article
Design of a Port-Hamiltonian Control for an Alt-Azimuth Liquid–Mirror Telescope
by Juan Cristobal Alcaraz Tapia, Carlos E. Castañeda, Héctor Vargas Rodriguez and P. Esquivel
Mathematics 2023, 11(16), 3443; https://doi.org/10.3390/math11163443 - 8 Aug 2023
Viewed by 793
Abstract
In this work, we design a control strategy to be applied in a port-Hamilton representation of a liquid-mirror telescope for an alt-azimuth configuration. Starting from a dynamical model for an alt-azimuth liquid-mirror telescope based on Lagrange mechanics, a transformation to the port-Hamilton form [...] Read more.
In this work, we design a control strategy to be applied in a port-Hamilton representation of a liquid-mirror telescope for an alt-azimuth configuration. Starting from a dynamical model for an alt-azimuth liquid-mirror telescope based on Lagrange mechanics, a transformation to the port-Hamilton form is made. Such a dynamical model is obtained by computing the kinetic and potential energy of the telescope and substituting them in the Euler–Lagrange equation of motion. Then, for the transformation to the port-Hamiltonian form, we obtain the relation between the Hamiltonian and the Lagrangian. The resulting open-loop model based on the Hamiltonian function is controlled using an extension of the interconnection and damping-assignment passivity-based control aiming for a robust and accurate steady behavior in the closed loop while tracking a star’s position. For comparison purposes, two different control strategies are applied to the Lagrangian model, inverse-dynamics control and sliding mode super-twisting control. Since the light is collected by the principal mirror of the telescope while tracking a star, we make a description of the liquid mirror’s behavior. The tracking star’s position is described as a function of the observer’s position and the star’s coordinates as well as the date of observation. The simulations’ results show that the port-Hamilton control has a good transitory and steady response as well as great accuracy competing with that of inverse-dynamics control but with greater robustness and no chattering drawback. Full article
(This article belongs to the Special Issue Dynamics and Control Theory with Applications)
►▼ Show Figures

Figure 1

19 pages, 6448 KiB  
Article
Optimal Control Design and Online Controller-Area-Network Bus Data Analysis for a Light Commercial Hybrid Electric Vehicle
by Aminu Babangida, Chiedozie Maduakolam Light Odazie and Péter Tamás Szemes
Mathematics 2023, 11(15), 3436; https://doi.org/10.3390/math11153436 - 7 Aug 2023
Cited by 1 | Viewed by 1601
Abstract
In this article, a hybrid powertrain for the Volkswagen (VW) Crafter is designed using the Model-In-The-Loop (MIL) method. An enhanced Proportional-Integral (PI) control technique based on integral cost functions is developed by carrying out a time-based simulation in MATLAB/Simulink software to realize the [...] Read more.
In this article, a hybrid powertrain for the Volkswagen (VW) Crafter is designed using the Model-In-The-Loop (MIL) method. An enhanced Proportional-Integral (PI) control technique based on integral cost functions is developed by carrying out a time-based simulation in MATLAB/Simulink software to realize the optimal fuel economy of the vehicle. Moreover, a comparative study is conducted between the vehicle’s hybrid and pure electric versions to assess the optimal battery energy consumption per unit distance traveled. Communication within our vehicles’ Electronic Control Units (ECUs) is facilitated by a message-based protocol called a Controller Area Network (CAN). Consequently, this paper presents an online CAN Bus data analysis using the Hardware-In-The-Loop (HIL) method. This method uses a standard frame, J1939 CAN protocol, implemented with Net CAN Plus 110 hardware. A graphical user interface is developed on a host Personal Computer (PC) using LabVIEW for decoding the acquired raw CAN data to physical values. The simulation results reveal that the proposed controller is promising and suitable for realizing optimal performance over the HIL method. Full article
(This article belongs to the Special Issue Dynamics and Control Theory with Applications)
►▼ Show Figures

Figure 1

20 pages, 7643 KiB  
Article
Sensorless Scheme for Permanent-Magnet Synchronous Motors Susceptible to Time-Varying Load Torques
by Christian Aldrete-Maldonado, Ramon Ramirez-Villalobos, Luis N. Coria and Corina Plata-Ante
Mathematics 2023, 11(14), 3066; https://doi.org/10.3390/math11143066 - 11 Jul 2023
Viewed by 1033
Abstract
This paper is devoted to designing a sensorless high-speed tracking control for surface-mount permanent-magnet synchronous motors, considering a time-varying load torque. This proposal consists of an extended-state observer interconnected with a PI-compensated controller, considering only the measurement of electrical variables for feedback. First, [...] Read more.
This paper is devoted to designing a sensorless high-speed tracking control for surface-mount permanent-magnet synchronous motors, considering a time-varying load torque. This proposal consists of an extended-state observer interconnected with a PI-compensated controller, considering only the measurement of electrical variables for feedback. First, to design the extended-state observer, a rotary coordinate model of the motor is extended in one state to estimate the load torque and the rotor’s position and speed. Later, the estimations are fedback to a PI-compensated controller to attenuate the time-varying load torques. Our proposed methodology aims to overcome a restriction regarding the solution of the Riccati equation respecting the Lipschitz condition for observer stability analysis. Therefore, a PI-compensated controller described as a closed-loop provides a sensorless scheme. Lyapunov stability analysis is applied to determine sufficient conditions to ensure that the states of the closed-loop system are ultimately bounded, which is one of our main contributions. The proposed observer-based controller scheme deals with unmeasured load torque fluctuations. Furthermore, we carry out high-precision emulations to provide testing scenarios of the permanent-magnet synchronous motor with some challenging load torque magnitudes and behaviors. Finally, we conduct experiments on the Technosoft® development platform to corroborate the feasibility of the proposed control scheme in a real-world scenario. Full article
(This article belongs to the Special Issue Dynamics and Control Theory with Applications)
►▼ Show Figures

Figure 1

14 pages, 846 KiB  
Article
Distributed Observers for State Omniscience with Stochastic Communication Noises
by Kairui Chen, Zhangmou Zhu, Xianxian Zeng and Junwei Wang
Mathematics 2023, 11(9), 1997; https://doi.org/10.3390/math11091997 - 23 Apr 2023
Cited by 1 | Viewed by 977
Abstract
The focus of this paper is on solving the state estimation problem for general continuous-time linear systems through the use of distributed networked observers. To better reflect the communication environment, stochastic noises are considered when observers exchange information. In the networked observers, each [...] Read more.
The focus of this paper is on solving the state estimation problem for general continuous-time linear systems through the use of distributed networked observers. To better reflect the communication environment, stochastic noises are considered when observers exchange information. In the networked observers, each local observer measures only part of the system output, and the state estimation can not be accomplished within a single observer. Then, all observers communicate through a pre-specified graph to make up information in the remaining system output. By solving a parametric algebraic Riccati equation (ARE), a simple method to calculate parameters in the observers is proposed. Furthermore, using the stability theory of stochastic differential equations, state omniscience is discussed in almost sure sense and in the mean square sense for the cases of state-dependent noises and non-state-dependent noises, respectively. It is shown that, for observable linear systems, the resulting observers work in a coordinated mode to reach state omniscience under the connected graph. Illustrative examples are provided to show the effectiveness of the distributed observers. Full article
(This article belongs to the Special Issue Dynamics and Control Theory with Applications)
►▼ Show Figures

Figure 1

15 pages, 1655 KiB  
Article
General Approach to the Evolving Plasma Equilibria with a Resistive Wall in Tokamaks
by Vladimir D. Pustovitov and Nikolay V. Chukashev
Mathematics 2023, 11(8), 1885; https://doi.org/10.3390/math11081885 - 16 Apr 2023
Viewed by 1040
Abstract
The dynamic problem of plasma equilibrium in a tokamak is considered taking into account the electromagnetic reaction of the vacuum vessel resistive wall. The currents induced in the wall during transient events contribute to the external magnetic field that determines the plasma shape [...] Read more.
The dynamic problem of plasma equilibrium in a tokamak is considered taking into account the electromagnetic reaction of the vacuum vessel resistive wall. The currents induced in the wall during transient events contribute to the external magnetic field that determines the plasma shape and position. Accordingly, the plasma geometry must evolve so that the inductive excitation of the wall current would properly compensate for the resistive losses. Simultaneous consideration of these factors presents the main difficulty of the description. It is performed in a general form using the Green’s function method that guarantees the mathematical accuracy of expressions for the magnetic fields from each source. At the same time, it is desirable to minimize the related complications, which is one of the goals here. The starting point is the standard solution of the external equilibrium problem given by integral relating the poloidal magnetic flux to the magnetic field at the plasma boundary. In the evolutionary problem, the additional equations for the plasma-wall electromagnetic coupling are transformed to an equation with a similar integral over the wall, but with either the time derivative of the poloidal magnetic flux or the wall current density in the integrand. The mentioned similarity allows to use the already developed techniques, which makes this formulation compact and convenient. It provides the basis for extension of the existing analytical theory of equilibrium to the case with non-circular plasma and wall. Full article
(This article belongs to the Special Issue Dynamics and Control Theory with Applications)
►▼ Show Figures

Figure 1

15 pages, 1814 KiB  
Article
Min–Max Dynamic Programming Control for Systems with Uncertain Mathematical Models via Differential Neural Network Bellman’s Function Approximation
by Alexander Poznyak, Sebastian Noriega-Marquez, Alejandra Hernandez-Sanchez, Mariana Ballesteros-Escamilla and Isaac Chairez
Mathematics 2023, 11(5), 1211; https://doi.org/10.3390/math11051211 - 1 Mar 2023
Cited by 2 | Viewed by 1481
Abstract
This research focuses on designing a min–max robust control based on a neural dynamic programming approach using a class of continuous differential neural networks (DNNs). The proposed controller solves the robust optimization of a proposed cost function that depends on the trajectories of [...] Read more.
This research focuses on designing a min–max robust control based on a neural dynamic programming approach using a class of continuous differential neural networks (DNNs). The proposed controller solves the robust optimization of a proposed cost function that depends on the trajectories of a system with an uncertain mathematical model satisfying a class of non-linear perturbed systems. The dynamic programming min–max formulation enables robust control concerning bounded modelling uncertainties and disturbances. The Hamilton–Jacobi–Bellman (HJB) equation’s value function, approximated by a DNN, permits to estimate the closed-loop formulation of the controller. The controller design is based on an estimated state trajectory with the worst possible uncertainties/perturbations that provide the degree of robustness using the proposed controller. The class of learning laws for the time-varying weights in the DNN is produced by studying the HJB partial differential equation. The controller uses the solution of the obtained learning laws and a time-varying Riccati equation. A recurrent algorithm based on the Kiefer–Wolfowitz method leads to adjusting the initial conditions for the weights to satisfy the final condition of the given cost function. The robust control suggested in this work is evaluated using a numerical example confirming the optimizing solution based on the DNN approximate for Bellman’s value function. Full article
(This article belongs to the Special Issue Dynamics and Control Theory with Applications)
►▼ Show Figures

Figure 1

9 pages, 612 KiB  
Article
Algebraic Methods for Achieving Super-Resolution by Digital Antenna Arrays
by Boris A. Lagovsky and Evgeny Ya. Rubinovich
Mathematics 2023, 11(4), 1056; https://doi.org/10.3390/math11041056 - 20 Feb 2023
Cited by 2 | Viewed by 1095
Abstract
The actual modern problem of developing and improving measurement and observation systems (including robotic ones) is to increase the volume and quality of the information received. Increasing the angle resolution to values significantly exceeding the Rayleigh criterion, i.e. achieving super-resolution is one of [...] Read more.
The actual modern problem of developing and improving measurement and observation systems (including robotic ones) is to increase the volume and quality of the information received. Increasing the angle resolution to values significantly exceeding the Rayleigh criterion, i.e. achieving super-resolution is one of important ways to solve the problem. Angular super-resolution which makes it possible to detail images of research objects and their individual fragments, improves the quality of solutions to detection, recognition and identification problems, increases the range of such systems. In many papers methods developed by authors to achieve a super-resolution based on approximate solutions of inverse problems in the form of Fredholm integral equation of the first kind of convolution type called algebraic are presented. The methods used, as well as their varieties, make it possible to reduce solutions of inverse problems posed to solving sets of linear algebraic equations (SLAE). This paper presents results of further improvement of algebraic methods based on intelligent analysis of received signals. It is shown that their use in systems based on digital antenna arrays makes it possible to increase the achieved degree of exceeding the Rayleigh criterion. In the course of numerical experiments with a mathematical model, the stability of the solutions obtained and their adequacy were confirmed. The numerical results obtained open the following possibilities: (1) obtaining images of studied objects with a resolution exceeding the Rayleigh criterion by 4 to 10 times, (2) determining the angular coordinates of individual small-sized objects as part of multi-element complex objects (group targets), (3) clarifying boundaries of extended objects and their individual elements, (4) localizing individual bright objects on a smoothly inhomogeneous reflective background. Applying presented new methods does not require a significant computing power, what allows you to work in a real time mode using relatively simple and inexpensive computing devices. The ways of further improvement of presented algebraic methods for solving applied inverse problems are described. Full article
(This article belongs to the Special Issue Dynamics and Control Theory with Applications)
►▼ Show Figures

Figure 1

11 pages, 266 KiB  
Article
Dynamics of Evolutionary Differential Equations with Several Spatial Variables
by Alexei G. Kushner
Mathematics 2023, 11(2), 335; https://doi.org/10.3390/math11020335 - 9 Jan 2023
Cited by 3 | Viewed by 1053
Abstract
The article is devoted to a method for constructing exact and approximate solutions of evolutionary partial differential equations with several spatial variables. The method is based on the theory of completely integrable distributions. Examples of applying this method to calculating exact solutions of [...] Read more.
The article is devoted to a method for constructing exact and approximate solutions of evolutionary partial differential equations with several spatial variables. The method is based on the theory of completely integrable distributions. Examples of applying this method to calculating exact solutions of the generalized Kolmogorov–Petrovskii–Piskunov–Fishev equations with two space variables are given. Full article
(This article belongs to the Special Issue Dynamics and Control Theory with Applications)
18 pages, 12585 KiB  
Article
Methodology of Plasma Shape Reachability Area Estimation in D-Shaped Tokamaks
by Yuri V. Mitrishkin, Valerii I. Kruzhkov and Pavel S. Korenev
Mathematics 2022, 10(23), 4605; https://doi.org/10.3390/math10234605 - 5 Dec 2022
Cited by 1 | Viewed by 1231
Abstract
This paper suggests and develops a new methodology of estimation for a multivariable reachability region of a plasma separatrix shape on the divertor phase of a plasma discharge in D-shaped tokamaks. The methodology is applied to a spherical Globus-M/M2 tokamak, including the estimation [...] Read more.
This paper suggests and develops a new methodology of estimation for a multivariable reachability region of a plasma separatrix shape on the divertor phase of a plasma discharge in D-shaped tokamaks. The methodology is applied to a spherical Globus-M/M2 tokamak, including the estimation of a controllability region of a vertical unstable plasma position on the basis of the experimental data. An assessment of the controllability region and the reachability region of the plasma is important for the design of tokamak poloidal field coils and the synthesis of a plasma magnetic control system. When designing a D-shaped tokamak, it is necessary to avoid the small controllability region of the vertically unstable plasma, because such cases occur in practice at a restricted voltage on a horizon field coil. To make the estimations mentioned above robust, PID-controllers for vertical and horizontal plasma position control were designed using the Quantitative Feedback Theory approach, which stabilizes the system and provides satisfactory control indexes (stability margins, setting time, overshoot) during plasma discharges. The controllers were tested on a series of plasma models and nonlinear models of current inverters in auto-oscillation mode as actuators for plasma position control. The estimations were made on these models, taking into account limitations on control actions, i.e., voltages on poloidal field coils. This research is the first step in the design of the plasma shape feedback control system for the operation of the Globus-M2 spherical tokamak. The developed methodology may be used in the design of poloidal field coil systems in tokamak projects in order to avoid weak achievability and controllability regions in magnetic plasma control. It was found that there is a strong cross-influence from the PF-coils currents and the CC current on the plasma shape; hence, these coils should be used to control the plasma shape simultaneously. Full article
(This article belongs to the Special Issue Dynamics and Control Theory with Applications)
►▼ Show Figures

Figure 1

37 pages, 3243 KiB  
Article
Optimization of Robust LMI-Control Systems for Unstable Vertical Plasma Position in D-Shaped Tokamak
by Yuri Mitrishkin, Evgeniia Pavlova and Mikhail Khlebnikov
Mathematics 2022, 10(23), 4531; https://doi.org/10.3390/math10234531 - 30 Nov 2022
Viewed by 1231
Abstract
The paper is devoted to the synthesis, comparison, and optimization of robust LMI-control systems for the vertical plasma position in a D-shaped tokamak, specifically the T-15MD tokamak (Kurchatov Institute, Moscow, Russia). The novelty of this work is to find out the possibilities of [...] Read more.
The paper is devoted to the synthesis, comparison, and optimization of robust LMI-control systems for the vertical plasma position in a D-shaped tokamak, specifically the T-15MD tokamak (Kurchatov Institute, Moscow, Russia). The novelty of this work is to find out the possibilities of LMI robust control systems, according to the criteria of the robust stability radius and control power peak at the rejection of a minor disruption type disturbance and a reference step signal using a unique unstable first-order plasma model. The plant under control consists of the connected in series plasma model with additive disturbance containing plant uncertainties, horizontal field coil (HFC), and actuator model as a multiphase rectifier. A set of robust controllers was designed by Linear Matrix Inequalities (LMI) method with pole placement in the LMI regions, state H2/H∞ performance, and output signal performance. The LMI theorems of the paper are directed to design the robust controllers and study the systems with the aim of eliminating the gap between theory and practice. The main achievement of this work consists in the optimization of robust control systems of the unstable plant with uncertain disturbance on the set of LMI synthesis approaches. The control systems have original quality criteria, such as control power and robust stability radius. The best control system on the basis of two criteria, namely, Dα,r,ϑ control system provides stabilization of the vertical plasma position on the real-time digital control testbed. Full article
(This article belongs to the Special Issue Dynamics and Control Theory with Applications)
►▼ Show Figures

Figure 1

20 pages, 340 KiB  
Article
Controllability of Stochastic Delay Systems Driven by the Rosenblatt Process
by Barakah Almarri, Xingtao Wang and Ahmed M. Elshenhab
Mathematics 2022, 10(22), 4223; https://doi.org/10.3390/math10224223 - 11 Nov 2022
Viewed by 1052
Abstract
In this work, we consider dynamical systems of linear and nonlinear stochastic delay-differential equations driven by the Rosenblatt process. With the aid of the delayed matrix functions of these systems, we derive the controllability results as an application. By using a delay Gramian [...] Read more.
In this work, we consider dynamical systems of linear and nonlinear stochastic delay-differential equations driven by the Rosenblatt process. With the aid of the delayed matrix functions of these systems, we derive the controllability results as an application. By using a delay Gramian matrix, we provide sufficient and necessary criteria for the controllability of linear stochastic delay systems. In addition, by employing Krasnoselskii’s fixed point theorem, we present some necessary criteria for the controllability of nonlinear stochastic delay systems. Our results improve and extend some existing ones. Finally, an example is given to illustrate the main results. Full article
(This article belongs to the Special Issue Dynamics and Control Theory with Applications)
30 pages, 368 KiB  
Article
Approximate Controllability of Semilinear Stochastic Generalized Systems in Hilbert Spaces
by Zhaoqiang Ge
Mathematics 2022, 10(17), 3050; https://doi.org/10.3390/math10173050 - 24 Aug 2022
Cited by 1 | Viewed by 1071
Abstract
Approximate controllability of two types of nonlinear stochastic generalized systems is investigated in the sense of mild solution in Hilbert spaces. Firstly, the approximate controllability of semilinear stochastic generalized systems with control only acting on the drift terms is discussed by GE-evolution operator [...] Read more.
Approximate controllability of two types of nonlinear stochastic generalized systems is investigated in the sense of mild solution in Hilbert spaces. Firstly, the approximate controllability of semilinear stochastic generalized systems with control only acting on the drift terms is discussed by GE-evolution operator and Nussbaum fixed-point theorem. Secondly, the approximate controllability of semilinear stochastic systems with control acting on both drift and diffusion terms is handled by using GE-evolution operator and Banach fixed-point theorem. At last, two illustrative examples are given. Full article
(This article belongs to the Special Issue Dynamics and Control Theory with Applications)
24 pages, 11803 KiB  
Article
Synthesis of a Control System for a Waste Heat Boiler with Forced Circulation under Restrictions on Control Actions
by Anton V. Utkin, Victor A. Utkin and Svetlana A. Krasnova
Mathematics 2022, 10(14), 2397; https://doi.org/10.3390/math10142397 - 8 Jul 2022
Cited by 4 | Viewed by 1309
Abstract
As is often the case in practice, classical control theory does not fully provide sufficient tools for solving applied problems of a certain specific class. Such problems include automatic control of waste heat boilers where physical restrictions are imposed on controls, namely, they [...] Read more.
As is often the case in practice, classical control theory does not fully provide sufficient tools for solving applied problems of a certain specific class. Such problems include automatic control of waste heat boilers where physical restrictions are imposed on controls, namely, they must belong to the class of non-negative functions with restrictions on amplitude and growth rate. In the theoretical part of this paper, within the framework of the block approach, methods for the synthesis of general linear systems with one-sided restrictions on control actions and state variables were developed. Then, the developed algorithms were applied to the waste heat boiler control system under conditions of incomplete measurements of state variables and the action of parametric and external uncontrolled disturbances. The effectiveness of the proposed algorithms was confirmed by numerical simulation. Full article
(This article belongs to the Special Issue Dynamics and Control Theory with Applications)
►▼ Show Figures

Figure 1

19 pages, 312 KiB  
Article
Spectral Decompositions of Gramians of Continuous Stationary Systems Given by Equations of State in Canonical Forms
by Igor Yadykin
Mathematics 2022, 10(13), 2339; https://doi.org/10.3390/math10132339 - 4 Jul 2022
Cited by 3 | Viewed by 1230
Abstract
The application of transformations of the state equations of continuous linear and bilinear systems to the canonical form of controllability allows one to simplify the computation of Gramians of these systems. In this paper, we develop the method and obtain algorithms for computation [...] Read more.
The application of transformations of the state equations of continuous linear and bilinear systems to the canonical form of controllability allows one to simplify the computation of Gramians of these systems. In this paper, we develop the method and obtain algorithms for computation of the controllability and observability Gramians of continuous linear and bilinear stationary systems with many inputs and one output, based on the method of spectral expansion of the Gramians and the iterative method for computing the bilinear systems Gramians. An important feature of the concept is the idea of separability of the Gramians expansion: separate computation of the scalar and matrix parts of the spectral Gramian expansion reduces the sub-Gramian matrices computation to calculation of numerical sequences of their elements. For the continuous linear systems with one output the method and the algorithm of the spectral decomposition of the controllability Gramian are developed in the form of Xiao matrices. Analytical expressions for the diagonal elements of the Gramian matrices are obtained, and by making use of which the rest of the elements can be calculated. For continuous linear systems with many outputs the spectral decompositions of the Gramians in the form of generalized Xiao matrices are obtained, which allows us to significantly reduce the number of calculations. The obtained results are generalized for continuous bilinear systems with one output. Iterative spectral algorithms for computation of elements of Gramians of these systems are proposed. Examples are given that illustrate theoretical results. Full article
(This article belongs to the Special Issue Dynamics and Control Theory with Applications)
21 pages, 1993 KiB  
Article
The New Second-Order Sliding Mode Control Algorithm
by Sergey Kochetkov, Svetlana A. Krasnova and Victor A. Utkin
Mathematics 2022, 10(13), 2214; https://doi.org/10.3390/math10132214 - 24 Jun 2022
Cited by 6 | Viewed by 1787
Abstract
A new class of regulators on the basis of the second-order sliding mode control is proposed. For the second-order system with smooth disturbances, special feedback is chosen with a discontinuous component and a radical function component. The synthesized control law provides a transient [...] Read more.
A new class of regulators on the basis of the second-order sliding mode control is proposed. For the second-order system with smooth disturbances, special feedback is chosen with a discontinuous component and a radical function component. The synthesized control law provides a transient oscillatory process with decaying amplitudes, which converge to zero in finite time. In contrast to existing algorithms, the condition of homogeneity of the closed-loop system differential equations is omitted. In comparison to the “twisting”-algorithm, which is well known, designed feedback provides an invariance property with respect to smooth external perturbation with less relay amplitude. With the help of a special Lyapunov function, the convergence proof is considered by using the averaging approach. It is shown that the average oscillation period convergence speed is strictly negative, and the estimation of the convergence time is presented. The simulation results of the designed control law for the one link robot-manipulator are presented, which shows less steady-state oscillations in comparison to existing approaches. Full article
(This article belongs to the Special Issue Dynamics and Control Theory with Applications)
►▼ Show Figures

Figure 1

Other

Jump to: Research

4 pages, 2281 KiB  
Obituary
In Memory of Prof. Dr. Yuri V. Mitrishkin (1946–2024)
by Artem E. Konkov, Pavel S. Korenev, Valerii I. Kruzhkov and Evgeniia A. Pavlova
Mathematics 2024, 12(7), 990; https://doi.org/10.3390/math12070990 - 26 Mar 2024
Viewed by 354
Abstract
On 22 January 2024, at the age of 78, Yuri Vladimirovich Mitrishkin (Figure 1), an outstanding scientist and remarkable educator with half a century of experience, passed away unexpectedly [...] Full article
(This article belongs to the Special Issue Dynamics and Control Theory with Applications)
►▼ Show Figures

Figure 1

Show export options expand_more Show export options expand_less
Select all
Export citation of selected articles as:
 
Displaying articles 1-24
Back to TopTop