Analysis and Controls of Time-Delay Systems with Perturbations: Theory and Application

Topic Information

Dear Colleagues,

The field of time-delay systems with perturbations is receiving a lot of interest in terms of modeling and analysis of practical systems. Specifically, the main focus is on performance analysis and controller design of time-delay systems across many fields of science and engineering. To include the uncertain properties of systems, systems with perturbances are used to present the variations and nonlinearities. Instability and poor performance will be caused by those main factors.

The focus of this Special Issue is addressing recent progress in both theoretical and practical developments on topics relating to the analysis, design, implementation, and technology of time-delay systems with perturbations. Topics that are invited for submission include (but are not limited to):

• Linear fractional perturbations;
• Switching signal designs and switched systems;
• Sampling and data-hold;
• Parallel distributed compensator and T–S fuzzy systems;
• Complex systems;
• Chaotic and hyperchaotic systems;
• Fractional-order dynamical systems;
• Networked control systems;
• Linear parameter varying systems;
• Mixed performances and controls;
• Fault diagnosis for time-delay systems;
• Case studies.

Prof. Dr. Chang-Hua Lien
Prof. Dr. Hamid Reza Karimi
Dr. Sundarapandian Vaidyanathan
Topic Editors

Keywords

Participating Journals

Journal Name	Impact Factor	CiteScore	Launched Year	First Decision (median)	APC
Fractal and Fractional fractalfract	5.4	3.6	2017	18.9 Days	CHF 2700
Mathematics mathematics	2.4	3.5	2013	16.9 Days	CHF 2600
Axioms axioms	2.0	-	2012	21.8 Days	CHF 2400
Symmetry symmetry	2.7	4.9	2009	16.2 Days	CHF 2400

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Published Papers (22 papers)

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19 pages, 431 KiB  

Open AccessArticle

A Discrete Integral Sliding Manifold for a Nonlinear System with Time Delay: An Event-Triggered Scheme

by Yassine Benyazid, Mohamed Fnadi and Ahmed Said Nouri

Mathematics 2023, 11(10), 2326; https://0-doi-org.brum.beds.ac.uk/10.3390/math11102326 - 16 May 2023

Cited by 1 | Viewed by 1096

Abstract

This paper presents a new approach to integral sliding mode control for discrete nonlinear systems with time delay. The approach is based on an event-triggered scheme and is applied to Takagi–Sugeno fuzzy models. In the first step, a new integral sliding function is [...] Read more.

This paper presents a new approach to integral sliding mode control for discrete nonlinear systems with time delay. The approach is based on an event-triggered scheme and is applied to Takagi–Sugeno fuzzy models. In the first step, a new integral sliding function is constructed, which avoids the limited assumptions of most existing fuzzy sliding mode control schemes. The design parameter matrices defining the sliding surface are obtained by solving linear matrix inequalities. In the second step, an event trigger-based integral sliding mode control protocol is developed to ensure the state trajectories of the Takagi–Sugeno fuzzy systems with time delays. Finally, the proposed strategies are evaluated through a simulation example to demonstrate their effectiveness. Full article

(This article belongs to the Topic Analysis and Controls of Time-Delay Systems with Perturbations: Theory and Application)

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17 pages, 541 KiB  

Open AccessArticle

A Switching Strategy for Stabilization of Discrete-Time Switched Positive Time-Varying Delay Systems with All Modes Being Unstable and Application to Uncertain Data

by Thanasak Mouktonglang, Kanyuta Poochinapan and Suriyon Yimnet

Axioms 2023, 12(5), 440; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms12050440 - 29 Apr 2023

Cited by 1 | Viewed by 842

Abstract

The stability problem of switched systems plays an essential role in the study of long-term behavior. In fact, systems containing both time delay and uncertainty terms may lead to performance degradation of those systems. Therefore, we are interested in the robust stability for [...] Read more.

The stability problem of switched systems plays an essential role in the study of long-term behavior. In fact, systems containing both time delay and uncertainty terms may lead to performance degradation of those systems. Therefore, we are interested in the robust stability for discrete-time switched positive time-varying delay systems with interval uncertainties in the case of all modes being unstable. Based on the proposed time-scheduled multiple co-positive Lyapunov–Krasovskii functional of each mode, new sufficient conditions for the global uniform asymptotic stability of the systems are derived. An effective time-dependent switching law utilized in this work is mode-dependent dwell time. In addition, the robust stability criteria in an asymptotic sense are formulated for the systems without time-varying delay. Compared with the existing related works, our results are less conservative and more general than some previous research. Finally, two numerical examples are provided to illustrate the effectiveness and correctness of the developed theoretical results. Full article

(This article belongs to the Topic Analysis and Controls of Time-Delay Systems with Perturbations: Theory and Application)

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25 pages, 1728 KiB  

Open AccessArticle

Robust (Q,S,R)-γ-Dissipative and H₂ Performances for Switched Systems with Mixed Time Delays

by Ker-Wei Yu, Chang-Hua Lien, Hao-Chin Chang, Cheung-Chieh Ku, Sundarapandian Vaidyanathan, Cheng-Ruei Yu, Hung-Chih Li and Yi-You Hou

Mathematics 2023, 11(7), 1574; https://0-doi-org.brum.beds.ac.uk/10.3390/math11071574 - 23 Mar 2023

Viewed by 889

Abstract

This paper investigates the switching on switching rule and sampling input to guarantee (Q,S,R)-$\mathsf{\gamma }$-dissipative and $H_2$ performances for switched systems with mixed time delays. Synchronous switching can be used to overcome the difficulty in implementation of real-time switching for signals [...] Read more.

This paper investigates the switching on switching rule and sampling input to guarantee (Q,S,R)-$\mathsf{\gamma }$-dissipative and $H_2$ performances for switched systems with mixed time delays. Synchronous switching can be used to overcome the difficulty in implementation of real-time switching for signals and sampling. The design of sampling input under arbitrary switching for a switched system is also considered in this paper. A new proposal for a full matrix formulation approach and inequality are applied to achieve the main results. Finally, some numerical examples are illustrated to show the efficiency of the main contribution. Full article

(This article belongs to the Topic Analysis and Controls of Time-Delay Systems with Perturbations: Theory and Application)

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14 pages, 2195 KiB  

Open AccessArticle

Sequential Predictors for Uncertain Euler–Lagrange Systems with Large Transmission Delays

by Shaobo Shen, Liyue Fu, Aiguo Song and Tao Li

Fractal Fract. 2023, 7(3), 244; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract7030244 - 08 Mar 2023

Viewed by 933

Abstract

This paper investigates the state prediction problems for uncertain Euler–Lagrange systems with large time delays during data transmissions. A set of sequential predictors is proposed to estimate the actual real-time states of the systems by using the delayed information of measurements. The arbitrarily [...] Read more.

This paper investigates the state prediction problems for uncertain Euler–Lagrange systems with large time delays during data transmissions. A set of sequential predictors is proposed to estimate the actual real-time states of the systems by using the delayed information of measurements. The arbitrarily large delays are handled by applying adequate numbers of serial sub-predictors. Meanwhile, the novel prediction structure of each subsystem is designed to deal with nonlinearities and unknown dynamics in the systems. Then, the predictor design is extended to the case without using delayed velocity measurements by updating the structure of the first sub-predictor. Sufficient conditions for the design of predictor gains, ensuring the boundness of prediction errors, are obtained through Lyapunov–Krasovskii functionals. The effectiveness and robustness of the uncertainties of the proposed method are verified by comparative results in simulations. Full article

(This article belongs to the Topic Analysis and Controls of Time-Delay Systems with Perturbations: Theory and Application)

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32 pages, 6901 KiB  

Open AccessArticle

Neural Network-Based Adaptive Fractional-Order Backstepping Control of Uncertain Quadrotors with Unknown Input Delays

by Yi Yang and Haiyan H. Zhang

Fractal Fract. 2023, 7(3), 232; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract7030232 - 05 Mar 2023

Cited by 1 | Viewed by 1268

Abstract

Adaptive control is essential and effective for reliable quadrotor operations in the presence of uncertain modeling parameters and unknown time-delayed inputs. This paper presents an original radial basis function neural network-based adaptive fractional-order backstepping controller (RBF-ADFOBC). The nonlinearity of the time-delayed inputs is [...] Read more.

Adaptive control is essential and effective for reliable quadrotor operations in the presence of uncertain modeling parameters and unknown time-delayed inputs. This paper presents an original radial basis function neural network-based adaptive fractional-order backstepping controller (RBF-ADFOBC). The nonlinearity of the time-delayed inputs is eliminated by introducing an augmented state variable via Pade’s approximation method. For each subsystem in the quadrotor dynamics, a companioned second-order compensation system is developed. The candidate Lyapunov functions are then properly designed by incorporating the control errors, parameter uncertainties and estimation errors of the neural networks’ weight vectors. It is shown that the semi-globally uniformly ultimately boundedness of all the state variables and the estimation error of uncertain parameters can be guaranteed. In addition, the trajectory-tracking error of the state variables can be driven to an adjustable small neighborhood of origin by properly setting the selectable parameters. Numerical simulations reveal that the tracking performance of the proposed controller can be improved continuously as the fractional order increases to a specific positive value, and the controller with a negative order may demonstrate higher robustness to the modeling uncertainties. Favorably, the comparison to the other two previous controllers further reveals the superior tracking accuracy and robustness of the proposed RBF-ADFOBC controller. Full article

(This article belongs to the Topic Analysis and Controls of Time-Delay Systems with Perturbations: Theory and Application)

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16 pages, 5402 KiB  

Open AccessArticle

Grey Wolf Optimizer-Based Optimal Controller Tuning Method for Unstable Cascade Processes with Time Delay

by Tufan Dogruer

Symmetry 2023, 15(1), 54; https://0-doi-org.brum.beds.ac.uk/10.3390/sym15010054 - 25 Dec 2022

Cited by 4 | Viewed by 1915

Abstract

This paper presents a controller design technique for cascade control systems based on the grey wolf optimization (GWO) algorithm. In the proposed control scheme, the proportional-integral-proportional-derivative (PI-PD) controller structure is used to control both the open-loop unstable process in the primary loop and [...] Read more.

This paper presents a controller design technique for cascade control systems based on the grey wolf optimization (GWO) algorithm. In the proposed control scheme, the proportional-integral-proportional-derivative (PI-PD) controller structure is used to control both the open-loop unstable process in the primary loop and the stable process in the secondary loop. To determine the optimal controller parameters, a new optimization algorithm is used in which the Euclidean distance function is used as a multi-objective function. A symmetry property of the Euclidean distance function is that its distance does not depend on the starting point and destination. The multi-objective function is designed according to system time response specifications such as settling time, overshoot, and steady-state error. Thus, the optimization algorithm allows the simultaneous determination of all controller parameters according to the desired output response. Three simulation studies are presented in the paper and the results are compared with studies using various methods based on internal model control, a modified Smith predictor, and a genetic algorithm. The simulation results reveal that the proposed method improves the performances of the systems in the control of cascade processes. Full article

(This article belongs to the Topic Analysis and Controls of Time-Delay Systems with Perturbations: Theory and Application)

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23 pages, 2421 KiB  

Open AccessArticle

Long-Term Bifurcation and Stochastic Optimal Control of a Triple-Delayed Ebola Virus Model with Vaccination and Quarantine Strategies

by Anwarud Din, Asad Khan and Yassine Sabbar

Fractal Fract. 2022, 6(10), 578; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract6100578 - 10 Oct 2022

Cited by 7 | Viewed by 1657

Abstract

Despite its high mortality rate of approximately $90\%$, the Ebola virus disease (EVD) has not received enough attention in terms of in-depth research. This illness has been responsible for over 40 years of epidemics throughout Central Africa. However, during 2014–2015, the [...] Read more.

Despite its high mortality rate of approximately $90\%$, the Ebola virus disease (EVD) has not received enough attention in terms of in-depth research. This illness has been responsible for over 40 years of epidemics throughout Central Africa. However, during 2014–2015, the Ebola-driven epidemic in West Africa became, and remains, the deadliest to date. Thus, Ebola has been declared one of the major public health issues. This paper aims at exploring the effects of external fluctuations on the prevalence of the Ebola virus. We begin by proposing a sophisticated biological system that takes into account vaccination and quarantine strategies as well as the effect of time lags. Due to some external perturbations, we extend our model to the probabilistic formulation with white noises. The perturbed model takes the form of a system of stochastic differential equations. Based on some non-standard analytical techniques, we demonstrate two main approach properties: intensity and elimination of Ebola virus. To better understand the impact of applied strategies, we deal with the stochastic control optimization approach by using some advanced theories. All of this theoretical arsenal has been numerically confirmed by employing some real statistical data of Ebola virus. Finally, we mention that this work could be a rich basis for further investigations aimed at understanding the complexity of Ebola virus propagation at pathophysiological and mathematics levels. Full article

(This article belongs to the Topic Analysis and Controls of Time-Delay Systems with Perturbations: Theory and Application)

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17 pages, 1140 KiB  

Open AccessArticle

Robust H_∞ Controller Design of Switched Delay Systems with Linear Fractional Perturbations by Synchronous Switching of Rule and Sampling Input

by Chang-Hua Lien, Hao-Chin Chang, Ker-Wei Yu, Hung-Chi Li and Yi-You Hou

Fractal Fract. 2022, 6(9), 479; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract6090479 - 29 Aug 2022

Cited by 3 | Viewed by 1231

Abstract

In this paper, we propose synchronous switching of rule and input to achieve H_∞ performance for an uncertain switched delay system with linear fractional perturbations. Our developed simple scheme utilizes the linear matrix inequality optimization problem to provide a feasible solution for [...] Read more.

In this paper, we propose synchronous switching of rule and input to achieve H_∞ performance for an uncertain switched delay system with linear fractional perturbations. Our developed simple scheme utilizes the linear matrix inequality optimization problem to provide a feasible solution for the proposed results; if the optimization problem was feasible, our proposed robust H_∞ control could be designed. The feasibility of the optimization problem could be solved using the LMI toolbox of Matlab. In this paper, robust control with sampling is proposed to stabilize uncertain switching with interval time-varying delay and achieve H_∞ performance. Interval time-varying delay and sampling were considered instead of constant delay and pointwise sampling. A full-matrix formulation approach is presented to improve the conservativeness of our proposed results. Some numerical examples are demonstrated to show our main contributions. Full article

(This article belongs to the Topic Analysis and Controls of Time-Delay Systems with Perturbations: Theory and Application)

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16 pages, 1498 KiB  

Open AccessArticle

Sampled-Data Control for a Class of Singular Takagi-Sugeno Fuzzy Systems with Application in Truck-Trailer System

by Yongcheng Yang and Minjie Zheng

Symmetry 2022, 14(9), 1762; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14091762 - 24 Aug 2022

Cited by 3 | Viewed by 1497

Abstract

In order to solve the admissibility problem for a class of nonlinear singular systems with sampling, nonlinearity, and external disturbances, a sample-data control algorithm based on input delay methodology is proposed in this paper. Firstly, the system is converted to a time-delay system [...] Read more.

In order to solve the admissibility problem for a class of nonlinear singular systems with sampling, nonlinearity, and external disturbances, a sample-data control algorithm based on input delay methodology is proposed in this paper. Firstly, the system is converted to a time-delay system based on Takagi–Sugeno fuzzy models via an input delay approach, with many novel time-delay methods being used to deal with the sampled-data control problem. Secondly, both the upper and lower bounds of the sampling period are considered, which has a wider application scope. Thirdly, to obtain less conservative results, an appropriate Lyapunov–Krasovskii function, which involves several symmetric positive definite matrices, is established, and a relaxation variable is introduced by the method of reciprocally convex inequality. Then the conditions of admissibility are given, and the design method of the sampled-data controller is introduced. Finally, a truck-trailer example and two numerical examples are given to prove that the proposed approaches are valid and applicable. Full article

(This article belongs to the Topic Analysis and Controls of Time-Delay Systems with Perturbations: Theory and Application)

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21 pages, 1842 KiB  

Open AccessArticle

Tumour-Natural Killer and CD8⁺ T Cells Interaction Model with Delay

by Nor Aziran Awang, Normah Maan and Mohd Dasuki Sulain

Mathematics 2022, 10(13), 2193; https://0-doi-org.brum.beds.ac.uk/10.3390/math10132193 - 23 Jun 2022

Cited by 1 | Viewed by 1527

Abstract

The literature suggests that effective defence against tumour cells requires contributions from both Natural Killer (NK) cells and CD$8^{+}$ T cells. NK cells are spontaneously active against infected target cells, whereas CD$8^{+}$ T cells take some times to activate [...] Read more.

The literature suggests that effective defence against tumour cells requires contributions from both Natural Killer (NK) cells and CD$8^{+}$ T cells. NK cells are spontaneously active against infected target cells, whereas CD$8^{+}$ T cells take some times to activate cell called as cell-specific targeting, to kill the virus. The interaction between NK cells and tumour cells has produced the other CD$8^{+}$ T cell, called tumour-specific CD$8^{+}$ T cells. We illustrate the tumour–immune interaction through mathematical modelling by considering the cell cycle. The interaction of the cells is described by a system of delay differential equations, and the delay, $\tau$ represent time taken for tumour cell reside interphase. The stability analysis and the bifurcation behaviour of the system are analysed. We established the stability of the model by analysing the characteristic equation to produce a stability region. The stability region is split into two regions, tumour decay and tumour growth. By applying the Routh–Hurwitz Criteria, the analysis of the trivial and interior equilibrium point of the model provides conditions for stability and is illustrated in the stability map. Numerical simulation is carried out to show oscillations through Hopf Bifurcation, and stability switching is found for the delay system. The result also showed that the interaction of NK cells with tumour cells could suppress tumour cells since it can increase the population of CD$8^{+}$ T cells. This concluded that the inclusion of delay and immune responses (NK-CD$8^{+}$ T cells) into consideration gives us a deep insight into the tumour growth and helps us understand how their interactions contribute to kill tumour cells. Full article

(This article belongs to the Topic Analysis and Controls of Time-Delay Systems with Perturbations: Theory and Application)

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13 pages, 2302 KiB  

Open AccessArticle

Adaptive Memoryless Sliding Mode Control of Uncertain Rössler Systems with Unknown Time Delays

by Jun-Juh Yan and Hang-Hong Kuo

Mathematics 2022, 10(11), 1885; https://0-doi-org.brum.beds.ac.uk/10.3390/math10111885 - 31 May 2022

Cited by 3 | Viewed by 1348

Abstract

In this paper, by adopting sliding mode control, an adaptive memoryless control scheme has been developed for uncertain Rössler chaotic systems with unknown time delays. Firstly, the proposed adaptive control can force the trajectories of controlled Rössler time-delayed chaotic systems into the specified [...] Read more.

In this paper, by adopting sliding mode control, an adaptive memoryless control scheme has been developed for uncertain Rössler chaotic systems with unknown time delays. Firstly, the proposed adaptive control can force the trajectories of controlled Rössler time-delayed chaotic systems into the specified sliding manifold. Then, the Riemann sum is introduced to analyze the stability of the equivalent dynamics in the sliding manifold. The control performance can be predicted even if the controlled systems have unmatched uncertainties and unknown time delays, which have not been well addressed in the literature. Numerical simulations are included to demonstrate the feasibility of the proposed scheme. Full article

(This article belongs to the Topic Analysis and Controls of Time-Delay Systems with Perturbations: Theory and Application)

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14 pages, 1158 KiB  

Open AccessEditor’s ChoiceArticle

State Estimation for Complex-Valued Inertial Neural Networks with Multiple Time Delays

by Yaning Yu and Ziye Zhang

Mathematics 2022, 10(10), 1725; https://0-doi-org.brum.beds.ac.uk/10.3390/math10101725 - 18 May 2022

Cited by 8 | Viewed by 1755

Abstract

In this paper, the problem of state estimation for complex-valued inertial neural networks with leakage, additive and distributed delays is considered. By means of the Lyapunov–Krasovskii functional method, the Jensen inequality, and the reciprocally convex approach, a delay-dependent criterion based on linear matrix [...] Read more.

In this paper, the problem of state estimation for complex-valued inertial neural networks with leakage, additive and distributed delays is considered. By means of the Lyapunov–Krasovskii functional method, the Jensen inequality, and the reciprocally convex approach, a delay-dependent criterion based on linear matrix inequalities (LMIs) is derived. At the same time, the network state is estimated by observing the output measurements to ensure the global asymptotic stability of the error system. Finally, two examples are given to verify the effectiveness of the proposed method. Full article

(This article belongs to the Topic Analysis and Controls of Time-Delay Systems with Perturbations: Theory and Application)

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13 pages, 266 KiB  

Open AccessArticle

A Note on the Reverse Order Law for g-Inverse of Operator Product

by Yingying Qin and Zhiping Xiong

Axioms 2022, 11(5), 226; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms11050226 - 12 May 2022

Viewed by 1467

Abstract

The generalized inverse has many important applications in aspects of the theoretical research of matrices and statistics. One of the core problems of the generalized inverse is finding the necessary and sufficient conditions of the reverse order laws for the generalized inverse of [...] Read more.

The generalized inverse has many important applications in aspects of the theoretical research of matrices and statistics. One of the core problems of the generalized inverse is finding the necessary and sufficient conditions of the reverse order laws for the generalized inverse of the operator product. In this paper, we study the reverse order law for the g-inverse of an operator product $T_1{T}_2{T}_3$ using the technique of matrix form of bounded linear operators. In particular, some necessary and sufficient conditions for the inclusion $T_3\left\{1\right\}{T}_2\left\{1\right\}{T}_1\left\{1\right\}\subseteq \left(T_1{T}_2{T}_3\right)\left\{1\right\}$ is presented. Moreover, some finite dimensional results are extended to infinite dimensional settings. Full article

(This article belongs to the Topic Analysis and Controls of Time-Delay Systems with Perturbations: Theory and Application)

17 pages, 1592 KiB  

Open AccessArticle

Observer-Based PID Control Strategy for the Stabilization of Delayed High Order Systems with up to Three Unstable Poles

by César Cruz-Díaz, Basilio del Muro-Cuéllar, Gonzalo Duchén-Sánchez, Juan Francisco Márquez-Rubio and Martín Velasco-Villa

Mathematics 2022, 10(9), 1399; https://0-doi-org.brum.beds.ac.uk/10.3390/math10091399 - 22 Apr 2022

Cited by 4 | Viewed by 1919

Abstract

In this paper, a new method to manage the stabilization and control problems of n-dimensional linear systems plus dead time, which includes one, two, or three unstable poles, is proposed. The control methodology proposed in this work is an Observer-based Proportional-Integral-Derivative (PID) [...] Read more.

In this paper, a new method to manage the stabilization and control problems of n-dimensional linear systems plus dead time, which includes one, two, or three unstable poles, is proposed. The control methodology proposed in this work is an Observer-based Proportional-Integral-Derivative (PID) strategy, where an observer and a PID controller are used to relocate the original unstable open-loop poles to stabilize the resultant closed-loop system. The observer provides an adequate estimation of the delayed-free variables and the PID uses the delay-free variables estimated by the proposed observer. Also, step-tracking is achieved in the overall control scheme. Necessary and sufficient conditions are presented to ensure closed-loop stability based on the open loop parameters of the system. The observer-based PID strategy considers five to seven constant parameters to obtain a stable closed-loop system. A general procedure to implement the proposed control strategy is presented and its performance is evaluated by means of numerical simulations. Full article

(This article belongs to the Topic Analysis and Controls of Time-Delay Systems with Perturbations: Theory and Application)

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24 pages, 743 KiB  

Open AccessArticle

Robust State Estimation for Uncertain Discrete Linear Systems with Delayed Measurements

by Zhijun Li, Minxing Sun, Qianwen Duan and Yao Mao

Mathematics 2022, 10(9), 1365; https://0-doi-org.brum.beds.ac.uk/10.3390/math10091365 - 19 Apr 2022

Cited by 3 | Viewed by 1764

Abstract

Measurement delays and model parametric uncertainties are meaningful issues in actual systems. Addressing the simultaneous existence of random model parametric uncertainties and constant measurement delay in the discrete-time linear systems, this study proposes a novel robust estimation method based on the combination of [...] Read more.

Measurement delays and model parametric uncertainties are meaningful issues in actual systems. Addressing the simultaneous existence of random model parametric uncertainties and constant measurement delay in the discrete-time linear systems, this study proposes a novel robust estimation method based on the combination of Kalman filter regularized least-squares (RLS) framework and state augmentation. The state augmentation method is elaborately designed, and the cost function is improved by considering the influence of modelling errors. A recursive program similar to the Kalman filter is derived. Meanwhile, the asymptotic stability conditions of the proposed estimator and the boundedness conditions of its error covariance are analyzed theoretically. Numerical simulation results show that the proposed method has a better processing capability for measurement delay and better robustness to model parametric uncertainties than the Kalman filter based on nominal parameters. Full article

(This article belongs to the Topic Analysis and Controls of Time-Delay Systems with Perturbations: Theory and Application)

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25 pages, 2213 KiB  

Open AccessArticle

Tuning of PID Control for the Double Integrator Plus Dead Time Model by Modified Real Dominant Pole and Performance Portrait Methods

by Mikulas Huba and Damir Vrancic

Mathematics 2022, 10(6), 971; https://0-doi-org.brum.beds.ac.uk/10.3390/math10060971 - 18 Mar 2022

Cited by 17 | Viewed by 5241

Abstract

The paper discusses the proportional-integral-derivative (PID) controller from the viewpoint of (a) the analytical tuning of the PID controller for the double integrator plus dead time (DIPDT) model and (b) the numerical tuning using the performance portrait method (PPM). In the first case, [...] Read more.

The paper discusses the proportional-integral-derivative (PID) controller from the viewpoint of (a) the analytical tuning of the PID controller for the double integrator plus dead time (DIPDT) model and (b) the numerical tuning using the performance portrait method (PPM). In the first case, the already published tuning with multiple real dominant pole, extended by integrated tuning procedures, which incorporate the inevitable low-pass filters by delay equivalences, is elaborated for modified sets of real poles. By considering several such modified sets of real poles, resulting in several new sets of controller parameters, the design can be better adapted to the requirements of the control tasks solved and to the limitations of the existing control loop hardware. In a noisy and uncertain environment, the balance between speed of setpoint and disturbance responses and acceptable excessive controller effort can thus be improved. The effectiveness of the analytical design can be evaluated using the numerical performance portrait method (PPM). For an already generated performance portrait (PP), it can offer a broad spectrum of controller settings that satisfy various design constraints. However, the results of the analytical design are still important as they facilitate the initial steps in the elaboration of the PPM and in explaining the nature of PID control. The developed controller tuning are compared using a new interpretation of PID controller as an extension of the stabilising PD controller by disturbance observer (DOB). The input disturbances reconstructed by DOB by evaluating the controller output of an integral process model in steady-state, can be estimated by a low-pass filter with a sufficiently long (integral) time constant. All analysed results are in full agreement with the proposed DOB interpretation, which furthermore contributes significantly to the explanation of the problems related to the optimal design of PID controllers. Full article

(This article belongs to the Topic Analysis and Controls of Time-Delay Systems with Perturbations: Theory and Application)

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51 pages, 554 KiB  

Open AccessArticle

Regularity of Fractional Heat Semigroup Associated with Schrödinger Operators

by Pengtao Li, Tao Qian, Zhiyong Wang and Chao Zhang

Fractal Fract. 2022, 6(2), 112; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract6020112 - 14 Feb 2022

Cited by 5 | Viewed by 1693

Abstract

Let $L=-\mathsf{\Delta }+V$ be a Schrödinger operator, where the potential V belongs to the reverse Hölder class. By the subordinative formula, we introduce the fractional heat semigroup [...] Read more.

Let $L=-\mathsf{\Delta }+V$ be a Schrödinger operator, where the potential V belongs to the reverse Hölder class. By the subordinative formula, we introduce the fractional heat semigroup ${\left\{e^{-t{L}^{\alpha }}\right\}}_{t>0}, 0<\alpha <1$, associated with L. By the aid of the fundamental solution of the heat equation: ${\partial }_{t}u+Lu={\partial }_{t}u-\mathsf{\Delta }u+Vu=0,$ we estimate the gradient and the time-fractional derivatives of the fractional heat kernel $K_{\alpha ,t}^L(·,·)$, respectively. This method is independent of the Fourier transform, and can be applied to the second-order differential operators whose heat kernels satisfy the Gaussian upper bounds. As an application, we establish a Carleson measure characterization of the Campanato-type space $BM{O}_L^{\gamma }\left({\mathbb{R}}^n\right)$ via the fractional heat semigroup ${\left\{e^{-t{L}^{\alpha }}\right\}}_{t>0}$. Full article

(This article belongs to the Topic Analysis and Controls of Time-Delay Systems with Perturbations: Theory and Application)

23 pages, 5550 KiB  

Open AccessArticle

Controller Design for Unstable Time-Delay Systems with Unknown Transfer Functions

by Hsun-Heng Tsai, Chyun-Chau Fuh, Jeng-Rong Ho, Chih-Kuang Lin and Pi-Cheng Tung

Mathematics 2022, 10(3), 431; https://0-doi-org.brum.beds.ac.uk/10.3390/math10030431 - 29 Jan 2022

Cited by 5 | Viewed by 2925

Abstract

This study developed a method for designing parallel two-degree-of-freedom proportional-integral-derivative controllers for unstable time-delay processes with unknown dynamic equations. First, a performance index accounting for both transient response performance and disturbance rejection was developed. To obtain useful data even if the output of [...] Read more.

This study developed a method for designing parallel two-degree-of-freedom proportional-integral-derivative controllers for unstable time-delay processes with unknown dynamic equations. First, a performance index accounting for both transient response performance and disturbance rejection was developed. To obtain useful data even if the output of the system exceeds the allowable range, an effective penalty function was included in the performance index. The N–M simplex method was used to iteratively determine the optimal controller parameters. The proposed approach has the following advantages: (1) it can be used regardless of the stability of the open-loop system; (2) the mathematical model and parameters of the process need not be known in advance; (3) it can be used for processes that include measurement noise; (4) it has good transient response performance and is also robust against external disturbances; and (5) it enables more efficient controller design and reduces costs. Full article

(This article belongs to the Topic Analysis and Controls of Time-Delay Systems with Perturbations: Theory and Application)

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16 pages, 455 KiB  

Open AccessArticle

Finite-Time Boundedness of Linear Uncertain Switched Positive Time-Varying Delay Systems with Finite-Time Unbounded Subsystems and Exogenous Disturbance

by Thanasak Mouktonglang and Suriyon Yimnet

Mathematics 2022, 10(1), 65; https://0-doi-org.brum.beds.ac.uk/10.3390/math10010065 - 25 Dec 2021

Cited by 2 | Viewed by 2302

Abstract

The problem of finite-time boundedness for a class of linear switched positive time-varying delay systems with interval uncertainties and exogenous disturbance is addressed. This characteristic research is that the studied systems include the finite-time bounded subsystems and finite-time unbounded subsystems. Both a slow [...] Read more.

The problem of finite-time boundedness for a class of linear switched positive time-varying delay systems with interval uncertainties and exogenous disturbance is addressed. This characteristic research is that the studied systems include the finite-time bounded subsystems and finite-time unbounded subsystems. Both a slow mode-dependent average dwell time and a fast mode-dependent average dwell time switching techniques are utilized reasonably. And by applying a copositive Lyapunov-Krasovskii functional, novel delay-dependent sufficient criteria are derived to guarantee such systems to be finite-time bounded concerning the given parameters and designed switching signal. Furthermore, new finite-time boundedness criteria of the systems without interval uncertainties are also obtained. Finally, the efficiency of the theoretical results is presented in two illustrative examples. Full article

(This article belongs to the Topic Analysis and Controls of Time-Delay Systems with Perturbations: Theory and Application)

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19 pages, 309 KiB  

Open AccessArticle

Sobolev Regularity of Multilinear Fractional Maximal Operators on Infinite Connected Graphs

by Suying Liu and Feng Liu

Mathematics 2021, 9(22), 2883; https://0-doi-org.brum.beds.ac.uk/10.3390/math9222883 - 12 Nov 2021

Viewed by 1238

Abstract

Let G be an infinite connected graph. We introduce two kinds of multilinear fractional maximal operators on G. By assuming that the graph G satisfies certain geometric conditions, we establish the bounds for the above operators on the endpoint Sobolev spaces and [...] Read more.

Let G be an infinite connected graph. We introduce two kinds of multilinear fractional maximal operators on G. By assuming that the graph G satisfies certain geometric conditions, we establish the bounds for the above operators on the endpoint Sobolev spaces and Hajłasz–Sobolev spaces on G. Full article

(This article belongs to the Topic Analysis and Controls of Time-Delay Systems with Perturbations: Theory and Application)

24 pages, 1889 KiB  

Open AccessArticle

Reachable Set and Robust Mixed Performance of Uncertain Discrete Systems with Interval Time-Varying Delay and Linear Fractional Perturbations

by Chang-Hua Lien, Hao-Chin Chang, Ker-Wei Yu, Hung-Chi Li and Yi-You Hou

Mathematics 2021, 9(21), 2763; https://0-doi-org.brum.beds.ac.uk/10.3390/math9212763 - 30 Oct 2021

Cited by 1 | Viewed by 1390

Abstract

In this paper, the mixed performance and reachable set of uncertain discrete systems with slow variation interval time-varying delay are considered. The original uncertain discrete systems with interval time-varying delay are first transformed into a switched system. Then, the proposed improved results are [...] Read more.

In this paper, the mixed performance and reachable set of uncertain discrete systems with slow variation interval time-varying delay are considered. The original uncertain discrete systems with interval time-varying delay are first transformed into a switched system. Then, the proposed improved results are used to guarantee the stability and reachable set of the uncertain system with slow variation interval time-varying delay. The mixed performance (H₂/H_∞) can be derived in the same formulation simultaneously. The design scheme of robust switched control is also developed in this paper. The gains of the controller can be designed and switched to achieve stabilization and mixed performance of the system according to the delay value. Some comparisons with published results are made to show the main contribution of the proposed approach. Finally, some numerical examples are illustrated to show the main results. Full article

(This article belongs to the Topic Analysis and Controls of Time-Delay Systems with Perturbations: Theory and Application)

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16 pages, 270 KiB  

Open AccessArticle

On a Boundary Value Problem of Hybrid Functional Differential Inclusion with Nonlocal Integral Condition

by Ahmed M. A. El-Sayed, Wagdy G. El-Sayed and Somyya S. Amrajaa

Mathematics 2021, 9(21), 2667; https://0-doi-org.brum.beds.ac.uk/10.3390/math9212667 - 21 Oct 2021

Viewed by 1261

Abstract

In this work, we present a boundary value problem of hybrid functional differential inclusion with nonlocal condition. The boundary conditions of integral and infinite points will be deduced. The existence of solutions and its maximal and minimal will be proved. A sufficient condition [...] Read more.

In this work, we present a boundary value problem of hybrid functional differential inclusion with nonlocal condition. The boundary conditions of integral and infinite points will be deduced. The existence of solutions and its maximal and minimal will be proved. A sufficient condition for uniqueness of the solution is given. The continuous dependence of the unique solution will be studied. Full article

(This article belongs to the Topic Analysis and Controls of Time-Delay Systems with Perturbations: Theory and Application)

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