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Advances in Nonlinear Dynamical Systems and Control
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Advances in Nonlinear Dynamical Systems and Control

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "C1: Difference and Differential Equations".

Deadline for manuscript submissions: closed (31 January 2024) | Viewed by 9697

Special Issue Editor

Prof. Dr. Sundarapandian Vaidyanathan

E-Mail Website
Guest Editor
Research and Development Centre, Vel Tech University, Vel Nagar, Avadi, Chennai 600 062, Tamil Nadu, India
Interests: control systems; chaos theory; hyperchaos; dynamical systems; stability; sliding mode control; mathematical modelling; scientific computing

Special Issue Information

Dear Colleagues,

One of the main purposes of research in nonlinear control systems is to design various types of control techniques for the various nonlinear control models arising in real-world applications. Nonlinear control systems are often governed by nonlinear differential equations. A special type of nonlinear dynamical system is the family of chaotic systems, which are highly sensitive to changes in their initial conditions. Backstepping control, adaptive control, sliding mode control, fuzzy logic control, and fuzzy-based sliding mode control are some of the popular techniques used in the nonlinear control theory. Various mathematical techniques have been developed for analyzing the nonlinear differential equations such as Lyapunov stability theory, limit cycle theory, Poincaré maps, bifurcation theory, chaos theory, and describing functions.

Fractional-order differential equations and control systems have been also paid much attention to in the control literature. A fractional-order system is a dynamical system that can be modeled by a fractional differential equation containing derivatives of non-integer order. Fractional-order systems are highly useful in analyzing the anomalous behavior of nonlinear dynamical systems arising in physics, biology, electrochemistry, memristors, and chaotic systems.

Prof. Dr. Sundarapandian Vaidyanathan
Guest Editor

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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.

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Keywords

  • active and adaptive control for nonlinear systems
  • backstepping control and applications
  • chaotic systems and control
  • fractional-order dynamical systems and control
  • fuzzy logic control
  • geometric techniques for control
  • Lyapunov stability theory
  • memristors
  • nonlinear control systems
  • nonlinear dynamical systems
  • sliding mode control and applications
  • soft computing techniques and control

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Published Papers (6 papers)

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Research

22 pages, 308 KiB  
Article
New Stability Results for Periodic Solutions of Generalized Van der Pol Oscillator via Second Bogolyubov’s Theorem
by Safia Meftah, Fares Yazid, Tefaha Lejdel Ali, Keltoum Bouhali, Abdelkader Moumen, Tariq Alraqad and Hicham Saber
Mathematics 2023, 11(23), 4779; https://doi.org/10.3390/math11234779 - 27 Nov 2023
Viewed by 877
Abstract
A certain class of nonlinear differential equations representing a generalized Van der Pol oscillator is proposed in which we study the behavior of the existing solution. After using the appropriate variables, the first Levinson’s change converts the equations into a system with two [...] Read more.
A certain class of nonlinear differential equations representing a generalized Van der Pol oscillator is proposed in which we study the behavior of the existing solution. After using the appropriate variables, the first Levinson’s change converts the equations into a system with two equations, and the second converts these systems into a Lipschitzian system. Our main result is obtained by applying the Second Bogolubov’s Theorem. We established some integrals, which are used to compute the average function of this system and arrive at a new general condition for the existence of an asymptotically stable unique periodic solution. One of the well-known results regarding asymptotic stability appears, owing to the Second Bogolubov’s Theorem, and the advantage of this method is that it can be applied not only in the periodic dynamical systems, but also in non-almost periodic dynamical systems. Full article
(This article belongs to the Special Issue Advances in Nonlinear Dynamical Systems and Control)
18 pages, 332 KiB  
Article
Study of a Transmission Problem with Friction Law and Increasing Continuous Terms in a Thin Layer
by Yasmina Kadri, Aissa Benseghir, Salah Boulaaras, Hamid Benseridi and Mourad Dilmi
Mathematics 2023, 11(22), 4609; https://doi.org/10.3390/math11224609 - 10 Nov 2023
Viewed by 903
Abstract
The aim of this paper is to establish the asymptotic analysis of nonlinear boundary value problems. The non-stationary motion is given by the elastic constructive law. The contact is described with a version of Tresca’s law of friction. A variational formulation of the [...] Read more.
The aim of this paper is to establish the asymptotic analysis of nonlinear boundary value problems. The non-stationary motion is given by the elastic constructive law. The contact is described with a version of Tresca’s law of friction. A variational formulation of the model, in the form of a coupled system for the displacements and the nonlinear source terms, is derived. The existence of a unique weak solution of the model is established. We also give the problem in transpose form, and we demonstrate different estimates of the displacement and of the source term independently of the small parameter. The main corresponding convergence results are stated in the different theorems of the last section. Full article
(This article belongs to the Special Issue Advances in Nonlinear Dynamical Systems and Control)
16 pages, 444 KiB  
Article
Equivalent-Input-Disturbance Based Robust Control Design for Fuzzy Semi-Markovian Jump Systems via the Proportional-Integral Observer Approach
by Aravindh Dharmarajan, Parivallal Arumugam, Sakthivel Ramalingam and Kavikumar Ramasamy
Mathematics 2023, 11(11), 2543; https://doi.org/10.3390/math11112543 - 1 Jun 2023
Cited by 4 | Viewed by 1590
Abstract
This work focuses on the design of a unified control law, which enhances the accuracy of both the disturbance estimation and stabilization of nonlinear T-S fuzzy semi-Markovian jump systems. In detail, a proportional-integral observer based equivalent-input-disturbance (PIO-EID) approach is considered to model and [...] Read more.
This work focuses on the design of a unified control law, which enhances the accuracy of both the disturbance estimation and stabilization of nonlinear T-S fuzzy semi-Markovian jump systems. In detail, a proportional-integral observer based equivalent-input-disturbance (PIO-EID) approach is considered to model and develop the controller. The PIO approach includes a variable for relaxation in the system design along with an additional term for integration to improve the flexibility of the design and endurance of the system. The proposed stability criteria are formulated in the form of matrix inequalities using Lyapunov theory and depend on the sojourn time for robust control design. Final analyses are performed using MATLAB software with simulations to endorse the theoretical findings of this paper. Full article
(This article belongs to the Special Issue Advances in Nonlinear Dynamical Systems and Control)
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18 pages, 3812 KiB  
Article
A Novel Fault-Tolerant Super-Twisting Control Technique for Chaos Stabilization in Fractional-Order Arch MEMS Resonators
by Fawaz W. Alsaade and Mohammed S. Al-zahrani
Mathematics 2023, 11(10), 2276; https://doi.org/10.3390/math11102276 - 13 May 2023
Cited by 3 | Viewed by 1450
Abstract
With the increasing demand for high-performance controllers in micro- and nano-systems, it is crucial to account for the effects of unexpected faults in control inputs during the design process. To tackle this challenge, we present a new approach that leverages an estimator-based super-twisting [...] Read more.
With the increasing demand for high-performance controllers in micro- and nano-systems, it is crucial to account for the effects of unexpected faults in control inputs during the design process. To tackle this challenge, we present a new approach that leverages an estimator-based super-twisting control technique that is capable of regulating chaos in fractional-order arch micro-electro-mechanical system (MEMS) resonators. We begin by studying the governing equation of a fractional-order arch MEMS resonator, followed by a thorough exploration of its chaotic properties. We then outline the design process for our novel control technique. The proposed technique takes into consideration the effects of uncertainty and faults in the control input by utilizing a finite time estimator and a super-twisting algorithm. The proposed technique addresses important challenges in the control of MEMS in real-world applications by providing fault tolerance, which enables the controller to withstand unexpected faults in the control input. We apply our controller to the fractional-order arch MEMS resonator, conducting numerical simulations. The numerical findings reveal that our proposed control technique is capable of stabilizing the system’s dynamics, even in the presence of a time-evolving fault in the control actuator. These results provide compelling evidence of the efficacy of our approach to control, despite the presence of an evolving fault. Full article
(This article belongs to the Special Issue Advances in Nonlinear Dynamical Systems and Control)
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24 pages, 1076 KiB  
Article
Finite-Time Synchronization of Quantized Markovian-Jump Time-Varying Delayed Neural Networks via an Event-Triggered Control Scheme under Actuator Saturation
by Saravanan Shanmugam, Rajarathinam Vadivel and Nallappan Gunasekaran
Mathematics 2023, 11(10), 2257; https://doi.org/10.3390/math11102257 - 11 May 2023
Cited by 13 | Viewed by 1743
Abstract
In this paper, we present a finite-time synchronization (FTS) for quantized Markovian-jump time-varying delayed neural networks (QMJTDNNs) via event-triggered control. The QMJTDNNs take into account the effects of quantization on the system dynamics and utilize a combination of FTS and event-triggered communication to [...] Read more.
In this paper, we present a finite-time synchronization (FTS) for quantized Markovian-jump time-varying delayed neural networks (QMJTDNNs) via event-triggered control. The QMJTDNNs take into account the effects of quantization on the system dynamics and utilize a combination of FTS and event-triggered communication to mitigate the effects of communication delays, quantization error, and efficient synchronization. We analyze the FTS and convergence properties of the proposed method and provide simulation results to demonstrate its effectiveness in synchronizing a network of QMJTDNNs. We introduce a new method to achieve the FTS of a system that has input constraints. The method involves the development of the Lyapunov–Krasovskii functional approach (LKF), novel integral inequality techniques, and some sufficient conditions, all of which are expressed as linear matrix inequalities (LMIs). Furthermore, the study presents the design of an event-triggered controller gain for a larger sampling interval. The effectiveness of the proposed method is demonstrated through numerical examples. Full article
(This article belongs to the Special Issue Advances in Nonlinear Dynamical Systems and Control)
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21 pages, 4612 KiB  
Article
Adaptive Backstepping Terminal Sliding Mode Control of Nonlinear System Using Fuzzy Neural Structure
by Xiaoyu Gong, Wen Fu, Xingao Bian and Juntao Fei
Mathematics 2023, 11(5), 1094; https://doi.org/10.3390/math11051094 - 22 Feb 2023
Cited by 7 | Viewed by 2147
Abstract
An adaptive backstepping terminal sliding mode control (ABTSMC) method based on a multiple−layer fuzzy neural network is proposed for a class of nonlinear systems with parameter variations and external disturbances in this study. The proposed neural network is utilized to estimate the nonlinear [...] Read more.
An adaptive backstepping terminal sliding mode control (ABTSMC) method based on a multiple−layer fuzzy neural network is proposed for a class of nonlinear systems with parameter variations and external disturbances in this study. The proposed neural network is utilized to estimate the nonlinear function to handle the unknown uncertainties of the system and reduce the switching term gain. It has a strong learning ability and high approximation accuracy due to the combination of a fuzzy neural network and recurrent neural network. The neural network parameters can be adaptively adjusted to optimal values through the adaptive laws derived from the Lyapunov theorem. To stabilize the control signal, the additional parameter adaptive law derived by the adaptive projection algorithm is used to estimate the control coefficient. The terminal sliding mode control (TSMC) is introduced on the basis of backstepping control, which can ensure that the tracking error converges in finite time. The simulation example is carried out on the DC–DC buck converter model to verify the effectiveness and superiority of the proposed control method. The contrasting results show that the ABTSMC−DHLRNN possesses higher steady−state accuracy and faster transient response. Full article
(This article belongs to the Special Issue Advances in Nonlinear Dynamical Systems and Control)
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