Mathematics

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18 pages, 511 KiB

Open AccessFeature PaperArticle

Adaptive Fault-Tolerant Tracking Control for Continuous-Time Interval Type-2 Fuzzy Systems

by Ming-Yang Qiao and Xiao-Heng Chang

Mathematics 2024, 12(23), 3682; https://doi.org/10.3390/math12233682 - 24 Nov 2024

Viewed by 275

Abstract

This paper investigated the tracking problem of mixed

H_{\infty}

and

L_{2} - L_{\infty}

adaptive fault-tolerant control (FTC) for continuous-time interval type-2 fuzzy systems (IT2FSs). For the membership function mismatch and uncertainty between the modules of the nonlinear system, the IT2 [...] Read more.

This paper investigated the tracking problem of mixed

H_{\infty}

and

L_{2} - L_{\infty}

adaptive fault-tolerant control (FTC) for continuous-time interval type-2 fuzzy systems (IT2FSs). For the membership function mismatch and uncertainty between the modules of the nonlinear system, the IT2 fuzzy model is applied to linearly approximate it. The observer can sensitively estimate the system state, and the adaptive fault estimation functions can estimate adaptively the fault signals, which enables the designed adaptive FTC scheme to ensure the asymptotic stability of the closed-loop control system and achieve the desired mixed

H_{\infty}

and

L_{2} - L_{\infty}

tracking performance. The designed adaptive control law can achieve the purpose of dynamic compensation for faults and disturbances, and the introduced lemmas further reduce the design conservatism by adjusting the slack parameters and matrices. Finally, a mass-spring-damping system is used to illustrate the effectiveness of the designed method. Full article

(This article belongs to the Special Issue Nonlinear Systems: Dynamics, Control, Optimization and Applications in Science and Engineering, 3rd Edition)

13 pages, 278 KiB

Open AccessArticle

An Improved Non-Monotonic Adaptive Trust Region Algorithm for Unconstrained Optimization

by Mingming Xu, Quanxin Zhu and Hongying Xiao

Mathematics 2024, 12(21), 3398; https://doi.org/10.3390/math12213398 - 30 Oct 2024

Viewed by 312

Abstract

The trust region method is an effective method for solving unconstrained optimization problems. Incorrectly updating the rules of the trust region radius will increase the number of iterations and affect the calculation efficiency. In order to obtain an effective radius for the trust [...] Read more.

The trust region method is an effective method for solving unconstrained optimization problems. Incorrectly updating the rules of the trust region radius will increase the number of iterations and affect the calculation efficiency. In order to obtain an effective radius for the trust region, an adaptive radius updating criterion is proposed based on the gradient of the current iteration point and the eigenvalue of the Hessian matrix which avoids calculating the inverse of the Hessian matrix during radius updating. This approach reduces the computation time and enhances the algorithm’s performance. On this basis, we apply adaptive radius and non-monotonic techniques to the trust region algorithm and propose an improved non-monotonic adaptive trust region algorithm. Under proper assumptions, the convergence of the algorithm is analyzed. Numerical experiments confirm that the suggested algorithm is effective. Full article

(This article belongs to the Special Issue Nonlinear Systems: Dynamics, Control, Optimization and Applications in Science and Engineering, 3rd Edition)

24 pages, 853 KiB

Open AccessArticle

Attack-Dependent Adaptive Event-Triggered Security Fuzzy Control for Nonlinear Networked Cascade Control Systems Under Deception Attacks

by Xi-Ming Liu, Xiao-Heng Chang and Li-Wei Hou

Mathematics 2024, 12(21), 3385; https://doi.org/10.3390/math12213385 - 29 Oct 2024

Viewed by 638

Abstract

This article investigates the issue of

H_{\infty}

security output feedback control for a nonlinear networked cascade control system with deception attacks. First, to further reduce the amount of communication data, reasonably schedule network resources, and alleviate the impact of multi-channel deception attacks, [...] Read more.

This article investigates the issue of

H_{\infty}

security output feedback control for a nonlinear networked cascade control system with deception attacks. First, to further reduce the amount of communication data, reasonably schedule network resources, and alleviate the impact of multi-channel deception attacks, an attack-dependent adaptive event-triggered mechanism is introduced into the primary network channel, and its adaptive triggered threshold can be adjusted according to the random attack probability. Secondly, the output dynamic quantization of the secondary network channel is considered. Then, a novel security cascade output feedback controller design framework based on the Takagi–Sugeno (T-S) fuzzy networked cascade control system under deception attacks is established. In addition, by introducing the Lyapunov–Krasovskii stability theory, the design conditions of the controller are given. Finally, the effectiveness and superiority of the proposed design strategies are verified by two simulation examples of power plant boiler–turbine system and power plant boiler power generation control system. Full article

(This article belongs to the Special Issue Nonlinear Systems: Dynamics, Control, Optimization and Applications in Science and Engineering, 3rd Edition)

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21 pages, 577 KiB

Open AccessArticle

Wave Speeds for a Time-Periodic Bistable Three-Species Lattice Competition System

by Chaohong Pan, Jiali Zhan and Hongyong Wang

Mathematics 2024, 12(20), 3304; https://doi.org/10.3390/math12203304 - 21 Oct 2024

Viewed by 695

Abstract

In this paper, we consider propagation direction (which can be used to predict which species will occupy the habitat or win the competition eventually) of a bistable wave for a three-species time-periodic lattice competition system with bistable nonlinearity, aiming to address an open [...] Read more.

In this paper, we consider propagation direction (which can be used to predict which species will occupy the habitat or win the competition eventually) of a bistable wave for a three-species time-periodic lattice competition system with bistable nonlinearity, aiming to address an open problem. As a first step, by transforming the competition system to a cooperative one, we study the asymptotic behavior for the bistable wave profile and then prove the uniqueness of the bistable wave speed. Secondly, we utilize comparison principle and build up two couples of upper and lower solutions to judge the sign of the bistable wave speed which partially provides the answer to the open problem. As an application, we reduce the time-periodic system to a space–time homogeneous system, we obtain the corresponding criteria and carry out numerical simulations to illustrate the availability of our results. Moreover, an interesting phenomenon we have found is that the two weak competitors can wipe out the strong competitor under some circumstances. Full article

(This article belongs to the Special Issue Nonlinear Systems: Dynamics, Control, Optimization and Applications in Science and Engineering, 3rd Edition)

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15 pages, 455 KiB

Open AccessFeature PaperArticle

Finite-Time Asynchronous H_∞ Control for Non-Homogeneous Hidden Semi-Markov Jump Systems

by Qian Wang, Xiaojun Zhang, Yu Shao and Kaibo Shi

Mathematics 2024, 12(19), 3036; https://doi.org/10.3390/math12193036 - 28 Sep 2024

Viewed by 521

Abstract

This article explores the finite-time control problem associated with a specific category of non-homogeneous hidden semi-Markov jump systems. Firstly, a hidden semi-Markov model is designed to characterize the asynchronous interactions that occur between the true system mode and the controller mode, and emission [...] Read more.

This article explores the finite-time control problem associated with a specific category of non-homogeneous hidden semi-Markov jump systems. Firstly, a hidden semi-Markov model is designed to characterize the asynchronous interactions that occur between the true system mode and the controller mode, and emission probabilities are used to establish relationships between system models and controller modes. Secondly, a novel piecewise homogeneous method is introduced to tackle the non-homogeneous issue by taking into account the time-dependent transition rates for the jump rules between different modes of the system. Thirdly, an asynchronous controller is developed by applying Lyapunov theory along with criteria for stochastic finite-time boundedness, ensuring the specified

H_{\infty}

performance level is maintained. Finally, the effectiveness of this method is verified through two simulation examples. Full article

(This article belongs to the Special Issue Nonlinear Systems: Dynamics, Control, Optimization and Applications in Science and Engineering, 3rd Edition)

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15 pages, 274 KiB

Open AccessArticle

The Dynamic Behavior of a Stochastic SEIRM Model of COVID-19 with Standard Incidence Rate

by Yuxiao Zhao, Hui Wang and Dongxu Wang

Mathematics 2024, 12(19), 2966; https://doi.org/10.3390/math12192966 - 24 Sep 2024

Viewed by 447

Abstract

This paper studies the dynamic behavior of a stochastic SEIRM model of COVID-19 with a standard incidence rate. The existence of global solutions for dynamic system models is proven by integrating stochastic process theory and the concept of stopping times, together with the [...] Read more.

This paper studies the dynamic behavior of a stochastic SEIRM model of COVID-19 with a standard incidence rate. The existence of global solutions for dynamic system models is proven by integrating stochastic process theory and the concept of stopping times, together with the contradiction method. Moreover, we construct appropriate Lyapunov functions to analyze system stability and apply Dynkin’s formula and Fatou’s lemma to handle stopping times and expectations of stochastic processes. Notably, the extinction study provides mathematical proof that under the given system dynamics, the total population does not grow indefinitely but tends to stabilize over time. The properties of the diffusion matrix are harnessed to guarantee the system’s stationary distribution. Conclusively, numerical simulations confirm the model’s extinction outcomes. Full article

(This article belongs to the Special Issue Nonlinear Systems: Dynamics, Control, Optimization and Applications in Science and Engineering, 3rd Edition)

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17 pages, 1968 KiB

Open AccessArticle

Command-Filtered Nussbaum Design for Nonlinear Systems with Unknown Control Direction and Input Constraints

by Yuxuan Liu

Mathematics 2024, 12(14), 2167; https://doi.org/10.3390/math12142167 - 10 Jul 2024

Viewed by 718

Abstract

This paper studies the problem of adaptive fuzzy control based on command filtering for a class of nonlinear systems characterized by an input dead zone, input saturation, and unknown control direction. First, this paper proposes a novel equivalent transformation technique that simplifies the [...] Read more.

This paper studies the problem of adaptive fuzzy control based on command filtering for a class of nonlinear systems characterized by an input dead zone, input saturation, and unknown control direction. First, this paper proposes a novel equivalent transformation technique that simplifies the design complexity of multiple input constraints by converting the input dead zone and saturation nonlinearities into a unified functional form. Subsequently, a fuzzy logic system is utilized to handle the unknown nonlinear functions, and the command-filtering method is employed to address the issue of complexity explosion, while the Nussbaum function is utilized to resolve the challenge of an unknown control direction. Based on Lyapunov stability, it is proven that the tracking error converges to a small neighborhood around the origin, and all closed-loop signals are bounded. Finally, a numerical simulation result and an actual simulation result of a pendulum are presented to verify the feasibility and effectiveness of the proposed control strategy. Full article

(This article belongs to the Special Issue Nonlinear Systems: Dynamics, Control, Optimization and Applications in Science and Engineering, 3rd Edition)

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22 pages, 477 KiB

Open AccessArticle

Exponential Synchronization of Coupled Neural Networks with Hybrid Delays and Stochastic Distributed Delayed Impulses

by Gang Zhang, Yinfang Song and Xiaoyou Liu

Mathematics 2024, 12(13), 1995; https://doi.org/10.3390/math12131995 - 27 Jun 2024

Viewed by 660

Abstract

This paper is concerned with exponential synchronization for a class of coupled neural networks with hybrid delays and stochastic distributed delayed impulses. First of all, based on the average impulsive interval method, total probability formula and ergodic theory, several novel impulsive Halanay differential [...] Read more.

This paper is concerned with exponential synchronization for a class of coupled neural networks with hybrid delays and stochastic distributed delayed impulses. First of all, based on the average impulsive interval method, total probability formula and ergodic theory, several novel impulsive Halanay differential inequalities are established. Two types of stochastic impulses, i.e., stochastic distributed delayed impulses with dependent property and Markov property have been taken into account, respectively. Secondly, some criteria on exponential synchronization in the mean square of a class of coupled neural networks with stochastic distributed delayed impulses are acquired by combining the proposed lemmas and graph theory. The validity of the theoretical results is demonstrated by several numerical simulation examples. Full article

(This article belongs to the Special Issue Nonlinear Systems: Dynamics, Control, Optimization and Applications in Science and Engineering, 3rd Edition)

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15 pages, 592 KiB

Open AccessArticle

Stochastic Intermittent Control with Uncertainty

by Zhengqi Ma, Hongyin Jiang, Chun Li, Defei Zhang and Xiaoyou Liu

Mathematics 2024, 12(13), 1947; https://doi.org/10.3390/math12131947 - 23 Jun 2024

Viewed by 582

Abstract

In this article, we delve into the exponential stability of uncertainty systems characterized by stochastic differential equations driven by G-Brownian motion, where coefficient uncertainty exists. To stabilize the system when it is unstable, we consider incorporating a delayed stochastic term. By employing linear [...] Read more.

In this article, we delve into the exponential stability of uncertainty systems characterized by stochastic differential equations driven by G-Brownian motion, where coefficient uncertainty exists. To stabilize the system when it is unstable, we consider incorporating a delayed stochastic term. By employing linear matrix inequalities (LMI) and Lyapunov–Krasovskii functions, we derive a sufficient condition for stabilization. Our findings demonstrate that an unstable system can be stabilized with a control interval within

(θ^{*}, 1)

. Some numerical examples are provided at the end to validate the correctness of our theoretical results. Full article

(This article belongs to the Special Issue Nonlinear Systems: Dynamics, Control, Optimization and Applications in Science and Engineering, 3rd Edition)

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14 pages, 387 KiB

Open AccessArticle

Imputation-Based Variable Selection Method for Block-Wise Missing Data When Integrating Multiple Longitudinal Studies

by Zhongzhe Ouyang, Lu Wang and Alzheimer’s Disease Neuroimaging Initiative

Mathematics 2024, 12(7), 951; https://doi.org/10.3390/math12070951 - 23 Mar 2024

Viewed by 1023

Abstract

When integrating data from multiple sources, a common challenge is block-wise missing. Most existing methods address this issue only in cross-sectional studies. In this paper, we propose a method for variable selection when combining datasets from multiple sources in longitudinal studies. To account [...] Read more.

When integrating data from multiple sources, a common challenge is block-wise missing. Most existing methods address this issue only in cross-sectional studies. In this paper, we propose a method for variable selection when combining datasets from multiple sources in longitudinal studies. To account for block-wise missing in covariates, we impute the missing values multiple times based on combinations of samples from different missing pattern and predictors from different data sources. We then use these imputed data to construct estimating equations, and aggregate the information across subjects and sources with the generalized method of moments. We employ the smoothly clipped absolute deviation penalty in variable selection and use the extended Bayesian Information Criterion criteria for tuning parameter selection. We establish the asymptotic properties of the proposed estimator, and demonstrate the superior performance of the proposed method through numerical experiments. Furthermore, we apply the proposed method in the Alzheimer’s Disease Neuroimaging Initiative study to identify sensitive early-stage biomarkers of Alzheimer’s Disease, which is crucial for early disease detection and personalized treatment. Full article

(This article belongs to the Special Issue Nonlinear Systems: Dynamics, Control, Optimization and Applications in Science and Engineering, 3rd Edition)

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13 pages, 690 KiB

Open AccessArticle

Fractional-Order Model-Free Adaptive Control with High Order Estimation

by Zhuo-Xuan Lv and Jian Liao

Mathematics 2024, 12(5), 784; https://doi.org/10.3390/math12050784 - 6 Mar 2024

Cited by 1 | Viewed by 972

Abstract

This paper concerns an improved model-free adaptive fractional-order control with a high-order pseudo-partial derivative for uncertain discrete-time nonlinear systems. Firstly, a new equivalent model is obtained by employing the Grünwald–Letnikov (G-L) fractional-order difference of the input in a compact-form dynamic linearization. Then, the [...] Read more.

This paper concerns an improved model-free adaptive fractional-order control with a high-order pseudo-partial derivative for uncertain discrete-time nonlinear systems. Firstly, a new equivalent model is obtained by employing the Grünwald–Letnikov (G-L) fractional-order difference of the input in a compact-form dynamic linearization. Then, the pseudo-partial derivative (PPD) is derived using a high-order estimation algorithm, which provides more PPD information than the previous time. A discrete-time model-free adaptive fractional-order controller is proposed, which utilizes more past input–output data information. The ultimate uniform boundedness of the tracking errors are demonstrated through formal analysis. Finally, the simulation results demonstrate the effectiveness of the proposed method. Full article

(This article belongs to the Special Issue Nonlinear Systems: Dynamics, Control, Optimization and Applications in Science and Engineering, 3rd Edition)

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12 pages, 511 KiB

Open AccessArticle

Synchronization for Reaction–Diffusion Switched Delayed Feedback Epidemic Systems via Impulsive Control

by Ruofeng Rao and Quanxin Zhu

Mathematics 2024, 12(3), 447; https://doi.org/10.3390/math12030447 - 30 Jan 2024

Cited by 3 | Viewed by 919

Abstract

Due to the facts that epidemic-related parameters vary significantly in different stages of infectious diseases and are relatively stable within the same stage, infectious disease models should be switch-type models. However, research on switch-type infectious disease models is scarce due to the complexity [...] Read more.

Due to the facts that epidemic-related parameters vary significantly in different stages of infectious diseases and are relatively stable within the same stage, infectious disease models should be switch-type models. However, research on switch-type infectious disease models is scarce due to the complexity and intricate design of switching rules. This scarcity has motivated the writing of this paper. By assuming that switching instants and impulse times occur at different moments, this paper proposes switch rules suitable for impulse control and derives synchronization criteria for reaction–diffusion switch-type infectious disease systems under impulse control. The effectiveness of this method is validated through numerical simulations. It is important to mention that, based on the information available to us, this paper is currently the sole study focusing on switch-type reaction–diffusion models for infectious diseases. Full article

(This article belongs to the Special Issue Nonlinear Systems: Dynamics, Control, Optimization and Applications in Science and Engineering, 3rd Edition)

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19 pages, 475 KiB

Open AccessArticle

Generalized Halanay Inequalities and Asymptotic Behavior of Nonautonomous Neural Networks with Infinite Delays

by Dehao Ruan and Yao Lu

Mathematics 2024, 12(1), 155; https://doi.org/10.3390/math12010155 - 3 Jan 2024

Viewed by 904

Abstract

This paper focuses on the asymptotic behavior of nonautonomous neural networks with delays. We establish criteria for analyzing the asymptotic behavior of nonautonomous recurrent neural networks with delays by means of constructing some new generalized Halanay inequalities. We do not require to constructi [...] Read more.

This paper focuses on the asymptotic behavior of nonautonomous neural networks with delays. We establish criteria for analyzing the asymptotic behavior of nonautonomous recurrent neural networks with delays by means of constructing some new generalized Halanay inequalities. We do not require to constructi any complicated Lyapunov function and our results improve some existing works. Lastly, we provide some illustrative examples to demonstrate the effectiveness of the obtained results. Full article

(This article belongs to the Special Issue Nonlinear Systems: Dynamics, Control, Optimization and Applications in Science and Engineering, 3rd Edition)

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17 pages, 3234 KiB

Open AccessArticle

Optimal Control of SLBRS with Recovery Rates

by Xiangqing Zhao and Wanmei Hou

Mathematics 2024, 12(1), 132; https://doi.org/10.3390/math12010132 - 30 Dec 2023

Viewed by 912

Abstract

In the information age, frequent information exchange has provided a breeding ground for the spread of computer viruses. The significant losses caused by computer virus attacks have long rung the alarm for information security. From academia to businesses, and even to government, everyone [...] Read more.

In the information age, frequent information exchange has provided a breeding ground for the spread of computer viruses. The significant losses caused by computer virus attacks have long rung the alarm for information security. From academia to businesses, and even to government, everyone remains highly vigilant about information security. Researchers have put forward various approaches to combat computer viruses, involving innovative efforts in both the hardware and software aspects, as well as theoretical innovation and practical exploration. This article is dedicated to theoretical exploration, specifically investigating the stability of a computer virus model, known as SLBRS, from the perspective of optimal control. Firstly, a control system is introduced with the aim of minimizing the costs related to network detoxification and diminishing the percentage of computers impacted by the virus. Secondly, we employ the Pontryagin maximum principle to analyze the optimality of a control strategy for the proposed system. Thirdly, we validate the effectiveness of our theoretical analysis through numerical simulation. In conclusion, both theoretical analysis and numerical simulation reveal that the utilization of optimal control analysis to stabilize the SLBRS has been demonstrated to be advantageous in restoring contaminated computer network environments. Full article

(This article belongs to the Special Issue Nonlinear Systems: Dynamics, Control, Optimization and Applications in Science and Engineering, 3rd Edition)

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24 pages, 506 KiB

Open AccessArticle

Discounted Risk-Sensitive Optimal Control of Switching Diffusions: Viscosity Solution and Numerical Approximation

by Xianggang Lu and Lin Sun

Mathematics 2024, 12(1), 38; https://doi.org/10.3390/math12010038 - 22 Dec 2023

Viewed by 822

Abstract

This work considers the infinite horizon discounted risk-sensitive optimal control problem for the switching diffusions with a compact control space and controlled through the drift; thus, the the generator of the switching diffusions also depends on the controls. Note that the running cost [...] Read more.

This work considers the infinite horizon discounted risk-sensitive optimal control problem for the switching diffusions with a compact control space and controlled through the drift; thus, the the generator of the switching diffusions also depends on the controls. Note that the running cost of interest can be unbounded, so a decent estimation on the value function is obtained, under suitable conditions. To solve such a risk-sensitive optimal control problem, we adopt the viscosity solution methods and propose a numerical approximation scheme. We can verify that the value function of the optimal control problem solves the optimality equation as the unique viscosity solution. The optimality equation is also called the Hamilton–Jacobi–Bellman (HJB) equation, which is a second-order partial differential equation (PDE). Since, the explicit solutions to such PDEs are usually difficult to obtain, the finite difference approximation scheme is derived to approximate the value function. As a byproduct, the

ϵ

-optimal control of finite difference type is also obtained. Full article

(This article belongs to the Special Issue Nonlinear Systems: Dynamics, Control, Optimization and Applications in Science and Engineering, 3rd Edition)

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