Explicit stability condition for delta fractional order systems with α∈(0,+∞).

This paper delves into the stability of time-advance delta fractional order systems, with a specific emphasis on the order range (0,+∞) rather than the conventional range (0,1). The delta Laplace transform is used to investigate the stability of the suggested system, and a mapping relation ρ=ss+1 is introduced. The explicit stability condition is provided, articulated in relation to a specific distribution of eigenvalues of the system matrix. To validate this condition, the paper establishes equivalence between the delta difference and the nabla difference. Furthermore, both quantitative and qualitative analyses are conducted on the range of the unstable region. Finally, the correctness of the developed results is validated by three examples.

Projective Synchronization for Uncertain Fractional Reaction-Diffusion Systems via Adaptive Sliding Mode Control Based on Finite-Time Scheme.

In this article, the projective synchronization of uncertain fractional-order (FO) reaction-diffusion systems is studied for the first time via the fractional adaptive sliding mode control (SMC) method. A FO integral type switching function is designed, and corresponding adaptive SMC laws are derived which ensure the FO sliding mode surface (SMS) is reachable after a finite time interval. The improved version of these control laws which have smaller oscillation and better control performance are also derived. A new lemma for proving the finite-time reachability of the FO SMS is developed. At last, numerical examples are provided to verify the effectiveness of our theories.

Multiple models for outbreak decision support in the face of uncertainty.

Policymakers must make management decisions despite incomplete knowledge and conflicting model projections. Little guidance exists for the rapid, representative, and unbiased collection of policy-relevant scientific input from independent modeling teams. Integrating approaches from decision analysis, expert judgment, and model aggregation, we convened multiple modeling teams to evaluate COVID-19 reopening strategies for a mid-sized United States county early in the pandemic. Projections from seventeen distinct models were inconsistent in magnitude but highly consistent in ranking interventions. The 6-mo-ahead aggregate projections were well in line with observed outbreaks in mid-sized US counties. The aggregate results showed that up to half the population could be infected with full workplace reopening, while workplace restrictions reduced median cumulative infections by 82%. Rankings of interventions were consistent across public health objectives, but there was a strong trade-off between public health outcomes and duration of workplace closures, and no win-win intermediate reopening strategies were identified. Between-model variation was high; the aggregate results thus provide valuable risk quantification for decision making. This approach can be applied to the evaluation of management interventions in any setting where models are used to inform decision making. This case study demonstrated the utility of our approach and was one of several multimodel efforts that laid the groundwork for the COVID-19 Scenario Modeling Hub, which has provided multiple rounds of real-time scenario projections for situational awareness and decision making to the Centers for Disease Control and Prevention since December 2020.

The effect mitigation measures for COVID-19 by a fractional-order SEIHRDP model with individuals migration.

In this paper, the generalized SEIHRDP (susceptible-exposed-infective-hospitalized-recovered-death-insusceptible) fractional-order epidemic model is established with individual migration. Firstly, the global properties of the proposed system are studied. Particularly, the sensitivity of parameters to the basic reproduction number are analyzed both theoretically and numerically. Secondly, according to the real data in India and Brazil, it can all be concluded that the bilinear incidence rate has a better description of COVID-19 transmission. Meanwhile, multi-peak situation is considered in China, and it is shown that the proposed system can better predict the next peak. Finally, taking individual migration between Los Angeles and New York as an example, the spread of COVID-19 between cities can be effectively controlled by limiting individual movement, enhancing nucleic acid testing and reducing individual contact.

Enhanced fractional order sliding mode control for a class of fractional order uncertain systems with multiple mismatched disturbances.

To improve the control performance of the fractional order uncertain systems with multiple mismatched disturbances, an enhanced fractional order sliding mode control (FOSMC) method is developed in this paper. The multiple disturbances and uncertainties are estimated by the finite-time disturbance observers (FTDO) and a fractional order extended state observer (FOESO), respectively. A fractional order switching law is designed to provide a fast convergence mode for the system states. Then a novel FOSMC law is developed by incorporating the feedforward compensation, the fractional order switching law, and the auxiliary state for input saturation. The proposed method is applied to numerical examples and to a motor speed control problem. The effectiveness of the proposed method is demonstrated by the performance comparisons with some existing control methods.

A Controller Synthesis Method to Achieve Independent Reference Tracking Performance and Disturbance Rejection Performance.

This paper deals with the conflict between the input-output response and the disturbance-output response, which cannot be completely eliminated by traditional and advanced control strategies without using the accurate process model. The inherently close association of these two responses and the unavailability of the accurate process model pose a great challenge to field test engineers of a coal-fired power plant, that is, the design requirements of reference tracking and disturbance rejection are compromised. In this paper, a novel two-degree-of-freedom controller-feedforward compensated (FC) desired dynamic equational (DDE) proportional-integral-derivative (PID) (FC-DDE PID)-is proposed as a viable alternative. In addition to achieving independent reference tracking performance and disturbance rejection performance, its simple structure and tuning procedure are specifically appealing to practitioners. Simulations, experiments, and field tests demonstrate the advantages of the proposed controller in both reference tracking and disturbance rejection, thus making FC-DDE PID a convenient and effective controller for the control of the coal-fired power plants, readily implementable on the distributed control system (DCS).

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.

Lyapunov stability criteria in terms of class K functions for Riemann-Liouville nabla fractional order systems.

This paper focuses on the problem of stability analysis for Riemann-Liouville nabla fractional order systems. On one hand, a useful comparison principle is built and then a rigorous proof is constructed for the well-known Lyapunov stability criterion in terms of class K functions. On the other hand, the constraint of the Lyapunov function is refined using a positive constant γ4 or a sequence h(k), resulting two practical theorems. Finally, three illustrative examples are given to show the applicability of the proposed method.

A controller design method for high-order unstable linear time-invariant systems.

This paper deals with high-order unstable systems, which are dangerous and more difficult to control. Their presence is increasingly prevalent, posing a great challenge to both traditional PID-based industrial designs and various advanced control strategies which are difficult to implement on common industrial control platforms. In this paper, the generalized desired dynamic equational (G-DDE) PID controller, developed by authors earlier, is proposed as a viable alternative. In addition to guarantee the closed-loop stability, its simple structure and tuning procedure are specifically appealing to practitioners. Simulations and experimental results show advantages of G-DDE PID in reference tracking, disturbance rejection and robustness, thus making G-DDE PID a convenient and effective control strategy for high-order unstable systems, readily implementable on common industrial platforms.

Sensor Fault Diagnostics Using Physics-Informed Transfer Learning Framework.

The field of smart health monitoring, intelligent fault detection and diagnosis is expanding dramatically in order to maintain successful operation in many engineering applications. Considering possible fault scenarios that can occur in a system, indicating the type of fault in a sensor is one of the most important and challenging problems in the area of intelligent sensor fault diagnostics. Within this frame of reference, we extended the physics-informed transfer learning framework, first presented previously for a fault cause assignment, to the level of sensor fault diagnostics for a range of different fault scenarios. Hence, the framework is utilized to perform intelligent sensor fault diagnostics for the first time. The underlying dynamics of the reference system are extracted using a completely data-driven methodology and dynamic mode decomposition with control (DMDc) in order to generate time-frequency illustrations of each sample with continuous wavelet transform (CWT). Then, sensor fault diagnostics for bias, drift over time, sine disturbance and increased noise sensor fault scenarios are achieved using the idea of transfer learning with a pre-trained image classification algorithm. The classification results yields a good performance on sensor fault diagnostics with 91.5% training and 84.7% test accuracy along with a fair robustness level with a set of reference benchmark system parameters.

Stability analysis of a nonlocal SIHRDP epidemic model with memory effects.

The prediction and control of COVID-19 is critical for ending this pandemic. In this paper, a nonlocal SIHRDP (S-susceptible class, I-infective class (infected but not hospitalized), H-hospitalized class, R-recovered class, D-death class and P-isolated class) epidemic model with long memory is proposed to describe the multi-wave peaks for the spread of COVID-19. Based on the basic reproduction number R 0 , which is completely controlled by fractional order, the stability of the proposed system is studied. Furthermore, the numerical simulation is conducted to gauge the performance of the proposed model. The results on Hunan, China, reveal that R 0 < 1 suggests that the disease-free equilibrium point is globally asymptotically stable. Likewise, the situation of the multi-peak case in China is presented, and it is clear that the nonlocal epidemic system has a superior fitting effect than the classical model. Finally an adaptive impulsive vaccination is introduced based on the proposed system. Then employing the real data of France, India, the USA and Argentina, parameters identification and short-term forecasts are carried out to verify the effectiveness of the proposed model in describing the case of multiple peaks. Moreover, the implementation of vaccine control is expected once the hospitalized population exceeds 20 % of the total population. Numerical results of France, Indian, the USA and Argentina shed light on the varied effect of vaccine control in different countries. According to the vaccine control imposed on France, no obvious effect is observed even consider reducing human contact. As for India, although there will be a temporary increase in hospitalized admissions after execution of vaccination control, COVID-19 will eventually disappear. Results on the USA have seen most significant effect of vaccine control, the number of hospitalized individuals drops off and the disease is eventually eradicated. In contrast to the USA, vaccine control in Argentina has also been very effective, but COVID-19 cannot be completely eradicated.

Qualitative and quantitative analysis of the COVID-19 pandemic by a two-side fractional-order compartmental model.

Global efforts are focused on discussing effective measures for minimizing the impact of COVID-19 on global community. It is clear that the ongoing pandemic of this virus caused an immense threat to public health and economic development. Mathematical models with real data simulations are powerful tools that can identify key factors of pandemic and improve control or mitigation strategies. Compared with integer-order and left-hand side fractional models, two-side fractional models can better capture the state of pandemic spreading. In this paper, two-side fractional models are first proposed to qualitative and quantitative analysis of the COVID-19 pandemic. A basic framework are given for the prediction and analysis of infectious diseases by these types of models. By means of asymptotic stability analysis of disease-free and endemic equilibrium points, basic reproduction number R0 can be obtained, which is helpful for estimating the severity of an outbreak qualitatively. Sensitivity analysis of R0 is performed to identify and rank key epidemiological parameters. Based on the real data of the United States, numerical tests reveal that the model with both left-hand side fractional derivative and right-hand side fractional integral terms has a better forecast ability for the epidemic trend in the next ten days. Our extensive computational results also quantitatively reveal that non-pharmaceutical interventions, such as isolation, stay at home, strict control of social distancing, and rapid testing can play an important role in preventing the pandemic of the disease. Thus, the two-side fractional models are proposed in this paper can successfully capture the change rule of COVID-19, which provide a strong tool for understanding and analyzing the trend of the outbreak.

Adaptive current harmonics suppression strategy for grid-tie inverters.

Since the observed angle of a phase-locked loop (PLL) is relatively accurate even under distorted grid conditions, the mathematical model of the voltage errors caused by the switching modulation and the dead-time effect are derived as a function of the grid voltage angle in this paper. Based on the model analysis, an adaptive compensation algorithm is proposed to suppress the grid-side current harmonics in three-phase converters. The proposed algorithm fits the unmeasurable voltage errors by a truncated Fourier series expansion, and then takes it as an equivalent disturbance in the current control loop to achieve harmonic compensation. By the feed-forward compensation, the design and tuning of the controller parameters are simplified and separated from the dynamic performance. In addition, the controller can adapt to the grid frequency variation by updating the grid voltage angle with the PLL block. To reduce the computational burden, a simplified version of the proposed method is also presented. Simulation and experiment results show that the proposed methods can suppress the current harmonics and achieve better performance in terms of total harmonic distortion, strong robustness and insensitivity to the grid frequency variations, compared with the traditional PI control or repetitive control strategy.

Active disturbance rejection control design for high-order integral systems.

Active disturbance rejection control (ADRC) is designed for regular processes frequently, due to its strong ability to reject disturbances and handle system uncertainties. However, ADRC design for high-order integral systems existing in many natural systems is always ignored. The controller design for high-order integral systems is different from the regular systems due to no pole in the left and right half s-planes. The ADRC design for high-order integral systems is studied to explain this question theoretically in this paper. Based on the equivalent form of ADRC, a theorem about the necessary condition of ADRC is proven which can guarantee the closed-loop system's stability. Additionally, the advantages of ADRC over proportional-integral-derivative (PID) controller in sensor noise rejection and control signal variation are analyzed theoretically. In order to achieve expected control performance for high-order integral systems, a practical design procedure of ADRC is summarized by the single variable method. Several comparative simulations and an experiment based on a ball and beam system are carried out. Running data verify that ADRC can obtain better control performance with strong robustness than PID controller. Eventually, a 100th order ADRC is designed for a 100th order integral system, and simulation results show that ADRC can be designed for high-order integral systems conveniently. Based on theoretical analyses and experimental verifications, ADRC shows some advantages for high-order integral systems.

Uniform Stability of Complex-Valued Neural Networks of Fractional Order With Linear Impulses and Fixed Time Delays.

As a generation of the real-valued neural network (RVNN), complex-valued neural network (CVNN) is based on the complex-valued (CV) parameters and variables. The fractional-order (FO) CVNN with linear impulses and fixed time delays is discussed. By using the sign function, the Banach fixed point theorem, and two classes of activation functions, some criteria of uniform stability for the solution and existence and uniqueness for equilibrium solution are derived. Finally, three experimental simulations are presented to illustrate the correctness and effectiveness of the obtained results.

Observer-Based Boundary Stabilization of Coupled Semilinear Reaction-Diffusion Neural Networks With Spatially Varying Coefficients via Event-Triggered Controller.

The aim of this article is to propose an observer-based event-triggered Robin boundary control strategy for the exponential stabilization of the coupled semilinear reaction-diffusion neural networks with spatially varying coefficients. Toward this aim, we design an observer to estimate the value of system states by using some of these system values as the available measurement. An observer-based event-triggered boundary stabilizer is then presented to exponentially stabilize the considered systems with the Zeno behavior being excluded. Throughout this article, the main used method is backstepping, which yields an explicit expression of the control formulae. Moreover, we see that the proposed event-triggered boundary control scheme can ensure the desired level of control performance with fewer control law updates. A numerical example is finally given to illustrate the effectiveness of our proposed method.

On a Method of Solution of Systems of Fractional Pseudo-Differential Equations.

This paper is devoted to the general theory of linear systems of fractional order pseudo-differential equations. Single fractional order differential and pseudo-differential equations are studied by many authors and several monographs and handbooks have been published devoted to its theory and applications. However, the state of systems of fractional order ordinary and partial or pseudo-differential equations is still far from completeness, even in the linear case. In this paper we develop a new method of solution of general systems of fractional order linear pseudo-differential equations and prove existence and uniqueness theorems in the special classes of distributions, as well as in the Sobolev spaces.

Integrated Time-Fractional Diffusion Processes for Fractional-Order Chaos-Based Image Encryption.

The purpose of this paper is to explore a novel image encryption algorithm that is developed by combining the fractional-order Chua's system and the 1D time-fractional diffusion system of order α∈(0,1]. To this end, we first discuss basic properties of the fractional-order Chua's system and the 1D time-fractional diffusion system. After these, a new spatiotemporal chaos-based cryptosystem is proposed by designing the chaotic sequence of the fractional-order Chua's system as the initial condition and the boundary conditions of the studied time-fractional diffusion system. It is shown that the proposed image encryption algorithm can gain excellent encryption performance with the properties of larger secret key space, higher sensitivity to initial-boundary conditions, better random-like sequence and faster encryption speed. Efficiency and reliability of the given encryption algorithm are finally illustrated by a computer experiment with detailed security analysis.

RLIM: a recursive and latent infection model for the prediction of US COVID-19 infections and turning points.

Initially found in Hubei, Wuhan, and identified as a novel virus of the coronavirus family by the WHO, COVID-19 has spread worldwide at exponential speed, causing millions of deaths and public fear. Currently, the USA, India, Brazil, and other parts of the world are experiencing a secondary wave of COVID-19. However, the medical, mathematical, and pharmaceutical aspects of its transmission, incubation, and recovery processes are still unclear. The classical susceptible-infected-recovered model has limitations in describing the dynamic behavior of COVID-19. Hence, it is necessary to introduce a recursive, latent model to predict the number of future COVID-19 infection cases in the USA. In this article, a dynamic recursive and latent infection model (RLIM) based on the classical SEIR model is proposed to predict the number of COVID-19 infections. Given COVID-19 infection and recovery data for a certain period, the RLIM is able to fit current values and produce an optimal set of parameters with a minimum error rate according to actual reported numbers. With these optimal parameters assigned, the RLIM model then becomes able to produce predictions of infection numbers within a certain period. To locate the turning point of COVID-19 transmission, an initial value for the secondary infection rate is given to the RLIM algorithm for calculation. RLIM will then calculate the secondary infection rates of a continuous time series with an iterative search strategy to speed up the convergence of the prediction outcomes and minimize the maximum square errors. Compared with other forecast algorithms, RLIM is able to adapt the COVID-19 infection curve faster and more accurately and, more importantly, provides a way to identify the turning point in virus transmission by searching for the equilibrium between recoveries and new infections. Simulations of four US states show that with the secondary infection rate ω initially set to 0.5 within the selected latent period of 14 days, RLIM is able to minimize this value at 0.07 and reach an equilibrium condition. A successful forecast is generated using New York state's COVID-19 transmission, in which a turning point is predicted to emerge on January 31, 2021. Supplementary Information: The online version contains supplementary material available at 10.1007/s11071-021-06520-1.