Fractional-order finite-time sliding mode control for uncertain teleoperated cyber-physical system with actuator fault

The stability of the teleoperated cyber-physical system with model uncertainty, external disturbance, and actuator fault is addressed in this study by using a suitable fractional-order sliding mode control (SMC) strategy. First, the sliding surface is designed to ensure the better tracking performance of the system. Second, the suggested control method combines SMC with an adaptive strategy to ensure that the system is stable in finite time. Third, neural network (NN) and fuzzy logic system (FLS) are used to estimate the model uncertainty, time-varying delay, external disturbance and unknown coefficient matrices of sliding mode surface, respectively. Finally, the advantages of the proposed control scheme are confirmed by the simulation example.

A fractional-order mathematical model for malaria and COVID-19 co-infection dynamics

This study proposes a fractional-order mathematical model for malaria and COVID-19 co-infection using the Atangana-Baleanu Derivative. We explain the various stages of the diseases together in humans and mosquitoes, and we also establish the existence and uniqueness of the fractional order co-infection model solution using the fixed point theorem. We conduct the qualitative analysis along with an epidemic indicator, the basic reproduction number R0 of this model. We investigate the global stability at the disease and endemic free equilibrium of the malaria-only, COVID-19-only, and co-infection models. We run different simulations of the fractional-order co-infection model using a two-step Lagrange interpolation polynomial approximate method with the aid of the Maple software package. The results reveal that reducing the risk of malaria and COVID-19 by taking preventive measures will reduce the risk factor for getting COVID-19 after contracting malaria and will also reduce the risk factor for getting malaria after contracting COVID-19 even to the point of extinction.

Bifurcations of a delayed fractional-order BAM neural network via new parameter perturbations

This paper makes a new breakthrough in deliberating the bifurcations of fractional-order bidirectional associative memory neural network (FOBAMNN). In the beginning, the corresponding bifurcation results are established according to self-regulating parameter, which is different from bifurcation outcomes available by using time delay as the bifurcation parameter, and greatly enriches the bifurcation results of continuous neural networks(NNs). The deived results manifest that a larger self-regulating parameter is more conducive to the stability of the system, which is consistent with the actual meaning of the self-regulating parameter representing the decay rate of activity. In addition to the innovation in the research object, this paper also has innovation in the procedure of calculating the bifurcation critical point. In the face of the quartic equation about the bifurcation parameters, this paper utilizes the methodology of implicit array to calculate the bifurcation critical point succinctly and effectively, which eschews the disadvantages of the conventional Ferrari approach, such as cumbersome formula and huge computational efforts. Our developed technique can be employed as a general method to solve the bifurcation point including the problem of dealing with the bifurcation critical point of delay. Ultimately, numerical experiments test the key theoretical fruits of this paper.

Novel results on asymptotic stability and synchronization of fractional-order memristive neural networks with time delays: The 0<δ≤1 case

This paper investigates the asymptotic stability and synchronization of fractional-order (FO) memristive neural networks with time delays. Based on the FO comparison principle and inverse Laplace transform method, the novel sufficient conditions for the asymptotic stability of a FO nonlinear system are given. Then, based on the above conclusions, the sufficient conditions for the asymptotic stability and synchronization of FO memristive neural networks with time delays are investigated. The results in this paper have a wider coverage of situations and are more practical than the previous related results. Finally, the validity of the results is checked by two examples.

Stability analysis and optimal control of a fractional-order generalized SEIR model for the COVID-19 pandemic

In view of the spread of corona virus disease 2019 (COVID-19), this paper proposes a fractional-order generalized SEIR model. The non-negativity of the solution of the model is discussed. Based on the established threshold R 0 , the existence of the disease-free equilibrium and endemic equilibrium is analyzed. Then, sufficient conditions are established to ensure the local asymptotic stability of the equilibria. The parameters of the model are identified based on the statistical data of COVID-19 cases. Furthermore, the validity of the model for describing the COVID-19 outbreak is verified. Meanwhile, the accuracy of the relevant theoretical results are also verified. Considering the relevant strategies of COVID-19 prevention and control, the fractional optimal control problem (FOCP) is proposed. Numerical schemes for Riemann-Liouville (R-L) fractional-order adjoint system with transversal conditions is presented. Based on the relevant statistical data, the corresponding FOCP is numerically solved, and the control effect of the COVID-19 outbreak under the optimal control strategy is discussed.

Synchronization and adaptive control for coupled fractional-order reaction-diffusion neural networks with multiple couplings

In this paper, two kinds of coupled fractional-order reaction-diffusion neural networks (CFORDNNs) with multiple state couplings or spatial diffusion couplings are proposed. By resorting to the Laplace transform and the properties of Mittag-Leffler functions, sufficient synchronization conditions are derived for the concerned network models. In addition, to guarantee the synchronization of these two networks, several appropriate adaptive control schemes are also developed. Ultimately, the validity of the devised adaptive strategies are verified by adopting some numerical examples with simulation results.

Mittag-Leffler stability of fractional-order quaternion-valued memristive neural networks with generalized piecewise constant argument

This paper studies the global Mittag-Leffler (M-L) stability problem for fractional-order quaternion-valued memristive neural networks (FQVMNNs) with generalized piecewise constant argument (GPCA). First, a novel lemma is established, which is used to investigate the dynamic behaviors of quaternion-valued memristive neural networks (QVMNNs). Second, by using the theories of differential inclusion, set-valued mapping, and Banach fixed point, several sufficient criteria are derived to ensure the existence and uniqueness (EU) of the solution and equilibrium point for the associated systems. Then, by constructing Lyapunov functions and employing some inequality techniques, a set of criteria are proposed to ensure the global M-L stability of the considered systems. The obtained results in this paper not only extends previous works, but also provides new algebraic criteria with a larger feasible range. Finally, two numerical examples are introduced to illustrate the effectiveness of the obtained results.

Chaotic attractors that exist only in fractional-order case

INTRODUCTION

Studying chaotic dynamics in fractional- and integer-order dynamical systems has let researchers understand and predict the mechanisms of related non-linear phenomena.

OBJECTIVES

Phase transitions between the fractional- and integer-order cases is one of the main problems that have been extensively examined by scientists, economists, and engineers. This paper reports the existence of chaotic attractors that exist only in the fractional-order case when using the specific selection of parameter values in a new hyperchaotic (Matouk's) system.

METHODS

This paper discusses stability analysis of the steady-state solutions, existence of hidden chaotic attractors and self-excited chaotic attractors. The results are supported by computing basin sets of attractions, bifurcation diagrams and the Lyapunov exponent spectrum. These tools verify the existence of chaotic dynamics in the fractional-order case; however, the corresponding integer-order counterpart exhibits quasi-periodic dynamics when using the same choice of initial conditions and parameter set. Projective synchronization via non-linear controllers is also achieved between drive and response states of the hidden chaotic attractors of the fractional Matouk's system.

RESULTS

Dynamical analysis and computer simulation results verify that the chaotic attractors exist only in the fractional-order case when using the specific selection of parameter values in the Matouk's hyperchaotic system.

CONCLUSIONS

An example of the existence of hidden and self-excited chaotic attractors that appears only in the fractional-order case is discussed. So, the obtained results give the first example that shows chaotic states are not necessarily transmitted between fractional- and integer-order dynamical systems when using a specific selection of parameter values. Chaos synchronization using the hidden attractors' manifolds provides new challenges in chaos-based applications to technology and industrial fields.

A PID controller for synchronization between master-slave neurons in fractional-order of neocortical network model

Modeling of the biological neurons is a way to understand the architecture of neural networks of the brain. A complex brain network includes the synchronization between some groups of neurons. The dynamic behavior of interactions between groups of slave-master neurons in the neocortical network is unpredictable and challenging. The purpose of synchronizing a neural interaction is to reduce the synchronization error between the chaotic slave-master neurons. This paper uses a proportional-integral-derivative (PID) controller to synchronize master-slave neurons in the fractional-order of the neocortical network model based on dendritic spike frequency adaptation (DSFA) uncertainties and unknown disturbance effects. The purpose of this article is in two parts: First, we implemented the effect of previous states of the neuron conditions by fractional-order of the differential equations in the neocortical network model. Second, by synchronizing the FO neocortical master-slave model by PID controller, we investigated the connection strength of the complex network in chaotic point of view. The optimized PID coefficients and fractional-order were calculated using root mean square error (RMSE) criteria to control the membrane voltage synchronization. The chaotic behavior of the system was evaluated by numerical techniques such as attractor analysis and time series diagrams. The optimal RMSE value for master-slave neurons occurred at fractional-orders 0.89. It is shown that the synchronization of master-slave neurons improves over time, and eventually they are fully synchronized while the controller error is reduced.

Predefined performance-based model-free adaptive fractional-order fast terminal sliding-mode control of MIMO nonlinear systems

The purpose of this article is to tackle with the problem of data-driven robust control of multi-input multi-output discrete-time nonlinear plants under tracking error constraints and output perturbations. Thereby, based upon the concept of dynamic linearization, a novel predefined performance based model-free adaptive fractional-order fast terminal sliding-mode controller is proposed so that the tracking errors can converge and remain within a preassigned neighborhood. The presented approach does solely rely on the real-time input/output data of the process, and the transient response together with the steady-state manner of the errors can be arbitrarily predefined. In the meantime, the closed-loop behavior is investigated by mathematical analysis, and the efficiency of the method is validated through various simulation examples.

Is fractional-order chaos theory the new tool to model chaotic pandemics as Covid-19?

The deadly outbreak of the second wave of Covid-19, especially in worst hit lower-middle-income countries like India, and the drastic rise of another growing epidemic of Mucormycosis, call for an efficient mathematical tool to model pandemics, analyse their course of outbreak and help in adopting quicker control strategies to converge to an infection-free equilibrium. This review paper on prominent pandemics reveals that their dispersion is chaotic in nature having long-range memory effects and features which the existing integer-order models fail to capture. This paper thus puts forward the use of fractional-order (FO) chaos theory that has memory capacity and hereditary properties, as a potential tool to model the pandemics with more accuracy and closeness to their real physical dynamics. We investigate eight FO models of Bombay plague, Cancer and Covid-19 pandemics through phase portraits, time series, Lyapunov exponents and bifurcation analysis. FO controllers (FOCs) on the concepts of fuzzy logic, adaptive sliding mode and active backstepping control are designed to stabilise chaos. Also, FOCs based on adaptive sliding mode and active backstepping synchronisation are designed to synchronise a chaotic epidemic with a non-chaotic one, to mitigate the unpredictability due to chaos during transmission. It is found that severity and complexity of the models increase as the memory fades, indicating that FO can be used as a crucial parameter to analyse the progression of a pandemic. To sum it up, this paper will help researchers to have an overview of using fractional calculus in modelling pandemics more precisely and also to approximate, choose, stabilise and synchronise the chaos control parameter that will eliminate the extreme sensitivity and irregularity of the models.

Fractional-order delayed Ross-Macdonald model for malaria transmission

This paper proposes a novel fractional-order delayed Ross-Macdonald model for malaria transmission. This paper aims to systematically investigate the effect of both the incubation periods of Plasmodium and the order on the dynamic behavior of diseases. Utilizing inequality techniques, contraction mapping theory, fractional linear stability theorem, and bifurcation theory, several sufficient conditions for the existence and uniqueness of solutions, the local stability of the positive equilibrium point, and the existence of fractional-order Hopf bifurcation are obtained under different time delays cases. The results show that time delay can change the stability of system. System becomes unstable and generates a Hopf bifurcation when the delay increases to a certain value. Besides, the value of order influences the stability interval size. Thus, incubation periods and the order have a major effect on the dynamic behavior of the model. The effectiveness of the theoretical results is shown through numerical simulations.

Fractional-order discontinuous systems with indefinite LKFs: An application to fractional-order neural networks with time delays

In this article, we discuss bipartite fixed-time synchronization for fractional-order signed neural networks with discontinuous activation patterns. The Filippov multi-map is used to convert the fixed-time stability of the fractional-order general solution into the zero solution of the fractional-order differential inclusions. On the Caputo fractional-order derivative, Lyapunov-Krasovskii functional is proved to possess the indefinite fractional derivatives for fixed-time stability of fragmentary discontinuous systems. Furthermore, the fixed-time stability of the fractional-order discontinuous system is achieved as well as an estimate of the new settling time.. The discontinuous controller is designed for the delayed fractional-order discontinuous signed neural networks with antagonistic interactions and new conditions for permanent fixed-time synchronization of these networks with antagonistic interactions are also provided, as well as the settling time for permanent fixed-time synchronization. Two numerical simulation results are presented to demonstrate the effectiveness of the main results.

Trajectory tracking control for electro-optical tracking system based on fractional-order sliding mode controller with super-twisting extended state observer

This paper investigates a trajectory tracking control scheme for electro-optical tracking systems subject to friction and other nonlinear disturbances. The proposed approach is based on a super-twisting extended state observer (ESO) and a fractional-order nonsingular terminal sliding mode (FONTSM) with a switching-type reaching law. The novel hybrid control scheme exhibits the following advantageous characteristics. First, the extended state observer injected with a super-twisting algorithm is capable of simultaneously estimating the friction and other nonlinear disturbances without detailed knowledge of the precise friction and disturbance models. Second, the FONTSM surface enhances the control accuracy and robustness, and provides a more flexible controller structure than its integer counterpart. Third, a novel switching-type reaching law is utilized to achieve fast response, constraining chattering in the system. Additionally, the finite-time convergence of the adopted ESO and the finite-time stability of the control system are demonstrated based on the rigorous Lyapunov criteria. Finally, the effectiveness of the proposed hybrid control scheme is demonstrated for an electro-optical tracking system via trajectory tracking experiments.

Intermittent control for finite-time synchronization of fractional-order complex networks

This paper is concerned with the finite-time synchronization problem for fractional-order complex dynamical networks (FCDNs) with intermittent control. Using the definition of Caputo's fractional derivative and the properties of Beta function, the Caputo fractional-order derivative of the power function is evaluated. A general fractional-order intermittent differential inequality is obtained with fewer additional constraints. Then, the criteria are established for the finite-time convergence of FCDNs under intermittent feedback control, intermittent adaptive control and intermittent pinning control indicate that the setting time is related to order of FCDNs and initial conditions. Finally, these theoretical results are illustrated by numerical examples.

Novel methods to global Mittag-Leffler stability of delayed fractional-order quaternion-valued neural networks

In this paper, a type of fractional-order quaternion-valued neural networks (FOQVNNs) with leakage and time-varying delays is established to simulate real-world situations, and the global Mittag-Leffler stability of the system is investigated by using the non-decomposition method. First, to avoid decomposing the system into two complex-valued systems or four real-valued systems, a new sign function for quaternion numbers is introduced based on the ones for real and complex numbers. And two novel lemmas for quaternion-valued sign function and Caputo fractional derivative are established in quaternion domain, which are used to investigate the stability of FOQVNNs. Second, a concise and flexible quaternion-valued state feedback controller is directly designed and a novel 1-norm Lyapunov function composed of the absolute values of real and imaginary parts is established. Then, based on the designed quaternion-valued state feedback controller and the proposed lemmas, some sufficient conditions are given to ensure the global Mittag-Leffler stability of the system. Finally, a numerical simulation is given to verify the theoretical results.

A fractional-order differential equation model of COVID-19 infection of epithelial cells

A novel coronavirus disease (COVID-19) appeared in Wuhan, China in December 2019 and spread around the world at a rapid pace, taking the form of pandemic. There was an urgent need to look for the remedy and control this deadly disease. A new strain of coronavirus called Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) was considered to be responsible for COVID-19. Novel coronavirus (SARS-CoV-2) belongs to the family of coronaviruses crowned with homotrimeric class 1 fusion spike protein (or S protein) on their surfaces. COVID-19 attacks primarily at our throat and lungs epithelial cells. In COVID-19, a stronger adaptive immune response against SARS-CoV-2 can lead to longer recovery time and leads to several complications. In this paper, we propose a mathematical model for examining the consequence of adaptive immune responses to the viral mutation to control disease transmission. We consider three populations, namely, the uninfected epithelial cells, infected cells, and the SARS-CoV-2 virus. We also take into account combination drug therapy on the dynamics of COVID-19 and its effect. We present a fractional-order model representing COVID-19/SARS-CoV-2 infection of epithelial cells. The main aim of our study is to explore the effect of adaptive immune response using fractional order operator to monitor the influence of memory on the cell-biological aspects. Also, we have studied the outcome of an antiviral drug on the system to obstruct the contact between epithelial cells and SARS-CoV-2 to restrict the COVID-19 disease. Numerical simulations have been done to illustrate our analytical findings.

A new model of Hopfield network with fractional-order neurons for parameter estimation

In this work, we study an application of fractional-order Hopfield neural networks for optimization problem solving. The proposed network was simulated using a semi-analytical method based on Adomian decomposition,, and it was applied to the on-line estimation of time-varying parameters of nonlinear dynamical systems. Through simulations, it was demonstrated how fractional-order neurons influence the convergence of the Hopfield network, improving the performance of the parameter identification process if compared with integer-order implementations. Two different approaches for computing fractional derivatives were considered and compared as a function of the fractional-order of the derivatives: the Caputo and the Caputo-Fabrizio definitions. Simulation results related to different benchmarks commonly adopted in the literature are reported to demonstrate the suitability of the proposed architecture in the field of on-line parameter estimation.

Passive filter design for fractional-order quaternion-valued neural networks with neutral delays and external disturbance

The problem on passive filter design for fractional-order quaternion-valued neural networks (FOQVNNs) with neutral delays and external disturbance is considered in this paper. Without separating the FOQVNNs into two complex-valued neural networks (CVNNs) or the FOQVNNs into four real-valued neural networks (RVNNs), by constructing Lyapunov-Krasovskii functional and using inequality technique, the delay-independent and delay-dependent sufficient conditions presented as linear matrix inequality (LMI) to confirm the augmented filtering dynamic system to be stable and passive with an expected dissipation are derived. One numerical example with simulations is furnished to pledge the feasibility for the obtained theory results.

A fractional-order SIRD model with time-dependent memory indexes for encompassing the multi-fractional characteristics of the COVID-19

COVID-19 is a novel coronavirus affecting all the world since December last year. Up to date, the spread of the outbreak continues to complicate our lives, and therefore, several research efforts from many scientific areas are proposed. Among them, mathematical models are an excellent way to understand and predict the epidemic outbreaks evolution to some extent. Due to the COVID-19 may be modeled as a non-Markovian process that follows power-law scaling features, we present a fractional-order SIRD (Susceptible-Infected-Recovered-Dead) model based on the Caputo derivative for incorporating the memory effects (long and short) in the outbreak progress. Additionally, we analyze the experimental time series of 23 countries using fractal formalism. Like previous works, we identify that the COVID-19 evolution shows various power-law exponents (no just a single one) and share some universality among geographical regions. Hence, we incorporate numerous memory indexes in the proposed model, i.e., distinct fractional-orders defined by a time-dependent function that permits us to set specific memory contributions during the evolution. This allows controlling the memory effects of more early states, e.g., before and after a quarantine decree, which could be less relevant than the contribution of more recent ones on the current state of the SIRD system. We also prove our model with Italy's real data from the Center for Systems Science and Engineering (CSSE) at Johns Hopkins University.

Complex dynamics in susceptible-infected models for COVID-19 with multi-drug resistance

Nowadays, exploring complex dynamic of epidemic models becomes a focal point for research after the outbreak of COVID-19 pandemic which has no vaccine or fully approved drug treatment up till now. Hence, complex dynamics in a susceptible-infected (SI) model for COVID-19 with multi-drug resistance (MDR) and its fractional-order counterpart are investigated. Existence of positive solution in fractional-order model is discussed. Local stability based on the fractional Routh-Hurwitz (FRH) conditions is considered. Also, new FRH conditions are introduced and proved for the fractional case (0,2]. All these FRH conditions are also applied to discuss local stability of the multi-drug resistance steady states. Chaotic attractors are also found in this model for both integer-order and fractional-order cases. Numerical tools such as Lyapunov exponents, Lyapunov spectrum and bifurcation diagrams are employed to confirm existence of these complex dynamics. This study helps to understand complex behaviors and predict spread of severe infectious diseases such as COVID-19.

Analysis and implementation of fractional-order chaotic system with standard components

This paper is devoted to the problem of uncertainty in fractional-order Chaotic systems implemented by means of standard electronic components. The fractional order element (FOE) is typically substituted by one complex impedance network containing a huge number of discrete resistors and capacitors. In order to balance the complexity and accuracy of the circuit, a sparse optimization based parameter selection method is proposed. The random error and the uncertainty of system implementation are analyzed through numerical simulations. The effectiveness of the method is verified by numerical and circuit simulations, tested experimentally with electronic circuit implementations. The simulations and experiments show that the proposed method reduces the order of circuit systems and finds a minimum number for the combination of commercially available standard components.

Stability analysis and optimal control of a fractional-order model for African swine fever

In this paper, a basic fractional-order model is proposed to describe the transmission of African swine fever. Two cases are considered: constant control and optimal control. In the former case, the existence and uniqueness of positive solution is proved firstly; then the basic reproduction number and the sufficient conditions for the stability of two equilibriums are obtained by using the next generation matrix method and Lyapunov LaSalle's invariance principle. In the latter case, optimal control is considered. By using the Hamiltonian function and Pontryagin's maximum principle, the optimal control formula is obtained. In addition, some examples and numerical simulations (based on Adama-Bashforth-Moulton predictor-corrector method) are performed to verify the theoretical results. At last, we present some brief discussion and conclusion.

A fractional-order SEIHDR model for COVID-19 with inter-city networked coupling effects

In the end of 2019, a new type of coronavirus first appeared in Wuhan. Through the real-data of COVID-19 from January 23 to March 18, 2020, this paper proposes a fractional SEIHDR model based on the coupling effect of inter-city networks. At the same time, the proposed model considers the mortality rates (exposure, infection and hospitalization) and the infectivity of individuals during the incubation period. By applying the least squares method and prediction-correction method, the proposed system is fitted and predicted based on the real-data from January 23 to March 18 - m where m represents predict days. Compared with the integer system, the non-network fractional model has been verified and can better fit the data of Beijing, Shanghai, Wuhan and Huanggang. Compared with the no-network case, results show that the proposed system with inter-city network may not be able to better describe the spread of disease in China due to the lock and isolation measures, but this may have a significant impact on countries that has no closure measures. Meanwhile, the proposed model is more suitable for the data of Japan, the USA from January 22 and February 1 to April 16 and Italy from February 24 to March 31. Then, the proposed fractional model can also predict the peak of diagnosis. Furthermore, the existence, uniqueness and boundedness of a nonnegative solution are considered in the proposed system. Afterward, the disease-free equilibrium point is locally asymptotically stable when the basic reproduction numberR 0 ≤ 1 , which provide a theoretical basis for the future control of COVID-19.

Finite-time synchronization of fractional-order gene regulatory networks with time delay

As multi-gene networks transmit signals and products by synchronous cooperation, investigating the synchronization of gene regulatory networks may help us to explore the biological rhythm and internal mechanisms at molecular and cellular levels. We aim to induce a type of fractional-order gene regulatory networks to synchronize at finite-time point by designing feedback controls. Firstly, a unique equilibrium point of the network is proved by applying the principle of contraction mapping. Secondly, some sufficient conditions for finite-time synchronization of fractional-order gene regulatory networks with time delay are explored based on two kinds of different control techniques and fractional Lyapunov function approach, and the corresponding setting time is estimated. Finally, some numerical examples are given to demonstrate the effectiveness of the theoretical results.

Dynamic analysis of a fractional-order delayed model for hepatitis B virus with CTL immune response

In this paper, a fractional-order delayed model with Holling II functional response and CTL immune response is constructed to describe the transmission of hepatitis B virus. Firstly, the existence and uniqueness of positive solutions are proved. Secondly, the basic reproduction number and the sufficient conditions for the existence of three equilibriums are obtained. Thirdly, the stability of three equilibriums are investigated. In addition, some sufficient conditions for the occurrence of Hopf bifurcation near the endemic equilibrium are demonstrated by using time delay as the bifurcation parameter. After that, some numerical simulations are performed to verify the theoretical prediction. At last, a brief discussion is presented to end this paper.

Practical stabilization of time-delay fractional-order systems by parametric controllers

This paper stabilization of time-delayed fractional-order systems by unlimited controllers is considered. To achieve the best controller so that the system be stable, the parameters of the feedback matrices are determinate with the minimum norm. Various constraints applied by the designer to obtain the desired performance criteria. We use the partial eigenvalue assignment (PEVA) method to decrease the constraints and ranks of matrices. The presented method is implemented in two numerical examples.

Non-fragile state estimation for fractional-order delayed memristive BAM neural networks

This paper deals with the non-fragile state estimation problem for a class of fractional-order memristive BAM neural networks (FMBAMNNs) with and without time delays for the first time. By means of a novel transformation and interval matrix approach, non-fragile estimators are designed and parameter mismatch problem is averted. Sufficient criteria are established to ascertain the error system is asymptotically stable based on fractional-order Lyapunov functionals and linear matrix inequalities (LMIs). Two examples are put forward to show the effectiveness of the obtained results.

Optimal adaptive interval type-2 fuzzy fractional-order backstepping sliding mode control method for some classes of nonlinear systems

In this research, a novel adaptive interval type-2 fuzzy fractional-order backstepping sliding mode control (AIT2FFOBSMC) method is presented for some classes of nonlinear fully-actuated and under-actuated mechanical systems with uncertainty. The AIT2FFOBSMC method exploits the advantages of backstepping and sliding mode methods to improve the performance of closed-loop control systems by lowering the tracking error and increasing robustness. To mitigate chattering and the tracking error, a fractional sliding surface is designed. In addition to the fractional sliding surface, an adaptive interval type-2 fuzzy compensator is used to estimate the uncertainty and perturbation of the nonlinear system in order to further reduce chattering caused by switching term as well as to enhance the perturbation rejection. In order to achieve an optimal performance, the multi-tracker optimization algorithm (MTOA) is used. Finally, a number of simulations and experimental tests are carried out to examine the performance of the AIT2FFOBSMC method.

Distributed formation control of fractional-order multi-agent systems with relative damping and nonuniform time-delays

A novel formation control law in the case of relative damping and nonuniform time-delays is proposed for fractional-order multi-agent systems(FOMASs) in this paper. The nonuniform time-delays can be generally divided into symmetric and asymmetric time-delays. Hence, the formation control algorithm for FOMASs in the case of symmetric time-delays and relative damping is first studied under an undirected network topology. Then, the formation control algorithm for FOMASs in the case of asymmetric time-delays and relative damping is studied under a directed network topology. By the means of frequency-domain theory, algebra graph theory and matrix theory, sufficient conditions are derived to ensure the formation control of FOMASs in the case of nonuniform time-delays and relative damping. Finally, several numerical examples are given and the corresponding simulations are provided to demonstrate the correctness of obtained results.

Intelligent fractional-order backstepping control for an ironless linear synchronous motor with uncertain nonlinear dynamics

This study aims to develop an intelligent fractional-order backstepping controller to control the mover position of an ironless permanent magnet linear synchronous motor. First, we investigated the operating principle and dynamic modeling of the linear synchronous motor based on the field-oriented control method. Next, to improve the convergence speed and control accuracy of the conventional backstepping controller, we designed a fractional-order backstepping controller that has more degrees of freedom in the control parameters. However, designing the switching control gain is difficult owing to the unknown degree of uncertainty. To address this problem, we proposed an intelligent fractional-order backstepping controller to further enhance the adaptiveness and robustness of the fractional-order backstepping controller. In this intelligent controller, we proposed a Hermite-polynomial-based functional-link fuzzy neural network as an uncertainty estimator that can directly estimate system uncertainty, thereby improving the disturbance rejection ability and requiring no uncertainty bound information. Additionally, to compensate for the estimation error introduced by the estimator, we designed an exponential compensator that employs a smooth exponential self-regulation mechanism. We utilized the Lyapunov theorem to derive estimation laws for the online tuning of the control parameters. Experimental results demonstrate the effectiveness and high positioning performance of the proposed intelligent fractional-order backstepping controller in comparison with the backstepping controller and fractional-order backstepping controller in the linear synchronous motor control system.

Single image super-resolution using self-optimizing mask via fractional-order gradient interpolation and reconstruction

Image super-resolution using self-optimizing mask via fractional-order gradient interpolation and reconstruction aims to recover detailed information from low-resolution images and reconstruct them into high-resolution images. Due to the limited amount of data and information retrieved from low-resolution images, it is difficult to restore clear, artifact-free images, while still preserving enough structure of the image such as the texture. This paper presents a new single image super-resolution method which is based on adaptive fractional-order gradient interpolation and reconstruction. The interpolated image gradient via optimal fractional-order gradient is first constructed according to the image similarity and afterwards the minimum energy function is employed to reconstruct the final high-resolution image. Fractional-order gradient based interpolation methods provide an additional degree of freedom which helps optimize the implementation quality due to the fact that an extra free parameter α-order is being used. The proposed method is able to produce a rich texture detail while still being able to maintain structural similarity even under large zoom conditions. Experimental results show that the proposed method performs better than current single image super-resolution techniques.

O(t-α)-synchronization and Mittag-Leffler synchronization for the fractional-order memristive neural networks with delays and discontinuous neuron activations

This paper investigates O(t^-α)-synchronization and adaptive Mittag-Leffler synchronization for the fractional-order memristive neural networks with delays and discontinuous neuron activations. Firstly, based on the framework of Filippov solution and differential inclusion theory, using a Razumikhin-type method, some sufficient conditions ensuring the global O(t^-α)-synchronization of considered networks are established via a linear-type discontinuous control. Next, a new fractional differential inequality is established and two new discontinuous adaptive controller is designed to achieve Mittag-Leffler synchronization between the drive system and the response systems using this inequality. Finally, two numerical simulations are given to show the effectiveness of the theoretical results. Our approach and theoretical results have a leading significance in the design of synchronized fractional-order memristive neural networks circuits involving discontinuous activations and time-varying delays.

Lagrange α-exponential stability and α-exponential convergence for fractional-order complex-valued neural networks

This paper deals with the problem on Lagrange α-exponential stability and α-exponential convergence for a class of fractional-order complex-valued neural networks. To this end, some new fractional-order differential inequalities are established, which improve and generalize previously known criteria. By using the new inequalities and coupling with the Lyapunov method, some effective criteria are derived to guarantee Lagrange α-exponential stability and α-exponential convergence of the addressed network. Moreover, the framework of the α-exponential convergence ball is also given, where the convergence rate is related to the parameters and the order of differential of the system. These results here, which the existence and uniqueness of the equilibrium points need not to be considered, generalize and improve the earlier publications and can be applied to monostable and multistable fractional-order complex-valued neural networks. Finally, one example with numerical simulations is given to show the effectiveness of the obtained results.

Dissipativity and stability analysis of fractional-order complex-valued neural networks with time delay

As we know, the notion of dissipativity is an important dynamical property of neural networks. Thus, the analysis of dissipativity of neural networks with time delay is becoming more and more important in the research field. In this paper, the authors establish a class of fractional-order complex-valued neural networks (FCVNNs) with time delay, and intensively study the problem of dissipativity, as well as global asymptotic stability of the considered FCVNNs with time delay. Based on the fractional Halanay inequality and suitable Lyapunov functions, some new sufficient conditions are obtained that guarantee the dissipativity of FCVNNs with time delay. Moreover, some sufficient conditions are derived in order to ensure the global asymptotic stability of the addressed FCVNNs with time delay. Finally, two numerical simulations are posed to ensure that the attention of our main results are valuable.

Stability analysis of memristor-based fractional-order neural networks with different memductance functions

In this paper, the problem of the existence, uniqueness and uniform stability of memristor-based fractional-order neural networks (MFNNs) with two different types of memductance functions is extensively investigated. Moreover, we formulate the complex-valued memristor-based fractional-order neural networks (CVMFNNs) with two different types of memductance functions and analyze the existence, uniqueness and uniform stability of such networks. By using Banach contraction principle and analysis technique, some sufficient conditions are obtained to ensure the existence, uniqueness and uniform stability of the considered MFNNs and CVMFNNs with two different types of memductance functions. The analysis results establish from the theory of fractional-order differential equations with discontinuous right-hand sides. Finally, four numerical examples are presented to show the effectiveness of our theoretical results.

Dynamic behavior analysis of fractional-order Hindmarsh-Rose neuronal model

Previous experimental work has shown that the firing rate of multiple time-scales of adaptation for single rat neocortical pyramidal neurons is consistent with fractional-order differentiation, and the fractional-order neuronal models depict the firing rate of neurons more verifiably than other models do. For this reason, the dynamic characteristics of the fractional-order Hindmarsh-Rose (HR) neuronal model were here investigated. The results showed several obvious differences in dynamic characteristic between the fractional-order HR neuronal model and an integer-ordered model. First, the fractional-order HR neuronal model displayed different firing modes (chaotic firing and periodic firing) as the fractional order changed when other parameters remained the same as in the integer-order model. However, only one firing mode is displayed in integer-order models with the same parameters. The fractional order is the key to determining the firing mode. Second, the Hopf bifurcation point of this fractional-order model, from the resting state to periodic firing, was found to be larger than that of the integer-order model. Third, for the state of periodically firing of fractional-order and integer-order HR neuron model, the firing frequency of the fractional-order neuronal model was greater than that of the integer-order model, and when the fractional order of the model decreased, the firing frequency increased.