State space modeling of fractional differential equations and the initial condition problem

Design of fractional order modeling based extended non-minimal state space MPC for temperature in an industrial electric heating furnace

This paper is concerned with the discretization of the fractional-order differentiator and integrator, which is the foundation of the digital realization of fractional order controller. Firstly, the parameterized Al-Alaoui transform is presented as a general generating function with one variable parameter, which can be adjusted to obtain the commonly used generating functions (e.g. Euler operator, Tustin operator and Al-Alaoui operator). However, the following simulation results show that the optimal variable parameters are different for different fractional orders. Then the weighted square integral index about the magtitude and phase is defined as the objective functions to achieve the optimal variable parameter for different fractional orders. Finally, the simulation results demonstrate that there are great differences on the optimal variable parameter for differential and integral operators with different fractional orders, which should be attracting more attentions in the design of digital fractional order controller.

Pulse density modulation (PDM) can find applications in consumer electronics, power and switch control systems. PDM signals can be generated by asynchronous sigma delta modulator (ASDM) structures and improvement of ASDM performance provides significant contribution for practical applications. This study numerically investigates effects of fractional order integration (FOI) on the PDM signal, produced by ASDM. We implemented fractional order integratorin ASDM structure and present numerical analysis to discuss effects of fractional order integrator on PDM signals, obtained for bipolar digital signal transmission application. Theoretical analysis indicates that FOI provides finer pulse trains and thus improves representation of original signal. In simulations, we tested FOI for various fractional-order values. Signal to noise ratio and signal to noise and distortion ratio (SINAD) levels of demodulated signals are reported for sinusoidal and audio test signals. Simulation results demonstrate that FOI can improve SNR and SINAD performance of ASDMs.

This article investigates the optimal results of new improved fractional order integrators (FOIs) of different orders. Mathematical models of FOIs have been first developed by a single-step procedure of direct linear interpolation of fractional integrators based on Al-Alaoui operator in fractional domain itself, instead of using three steps of the well-known conventional method, namely, digital interpolation, series expansion and truncation. Later, these transfer functions (TFs) are optimised for their coefficient values for finding a minimum error function by particle swarm optimisation (PSO) algorithm. Simulation results of magnitude responses, phase responses and relative magnitude errors (dB) for all the proposed half integrators have validated the effectiveness of this new technique of interpolation of fractional order operators, mixed with PSO algorithm. A parallel comparison has been also drawn between the proposed optimised half integrators and those obtained by discretisation of PSO optimised integer order digital integrators (DIs) to properly support the proposed novel combination of interpolation and PSO, both applied together in fractional domain.

We propose in this article a procedure for testing unit and fractional orders of integration, with the roots simultaneously occurring in the trend, the seasonal and the cyclical component of the time series. The tests have standard null and local limit distributions. However, finite sample critical values are computed, and several Monte Carlo experiments conducted across the paper show that the rejection frequencies against unit (and fractional) orders of integration are relatively high in all cases. The tests are applied to the UK consumption and income series, the results showing the importance of the roots corresponding to the trend and the seasonal components and, though the unit roots are found to be fairly suitable models, we show that fractional processes (including one for the cyclical component) may also be plausible alternatives in some cases.

The paper is devoted to the application of fractional calculus concepts to modeling, identification and control of discrete-time systems. Fractional difference equations (FAE) models are presented and their use in identification, state estimation and control context is discussed. The fractional difference state-space model is proposed for that purpose. For such a model stability conditions are given. A fractional Kalman filter (FKF) for this model is recalled. The proposed state-space model and fractional order difference equation are used in an identification procedure which produces very accurate results. Finally, the state-space model is used in closed-loop state feedback control form together with FKF as a state estimator. The latter is also given in an adaptive form together with FKF and a modification of recursive least squares (RLS) algorithm as a parameters identification procedure. All the algorithms presented were tested in simulations and the example results are given in the paper

The initial conditions problem in longitudinal count process: A simulation study

Step forward on nonlinear differential equations with the Atangana–Baleanu derivative: Inequalities, existence, uniqueness and method

In this study, we present an effective R-wave detection method in the QRS complex of the electrocardiogram (ECG) based on digital differentiation and integration of fractional order. The detection algorithm is performed in two steps. The pre-processing step is based on a fractional order digital band-pass filter whose fractional order is obtained by maximising the signal to noise ratio of the ECG signal, followed by a five points differentiator of fractional order 1.5 then the squaring transformation and the smoothing are used to generate peaks corresponding to the ECG parts with high slopes. The detection step is a new and simple strategy which is also based on fractional order operators for the localisation of the R waves. The MIT/BIH arrhythmia database is used to test the effectiveness of the proposed method. The algorithm has provided very good performance and has achieved about 99.86% of the detection rate for the standard database. The results obtained are presented, discussed and compared to the most recent and efficient R-wave detection algorithms.

Parameter identification of fractional order linear system based on Haar wavelet operational matrix

In this paper, a fractional order LQG benchmark is proposed for the control performance assessment of fractional order control systems. Similar to the conventional LQG benchmark, the fractional order LQG performance benchmark curve is determined by the numerical calculation method, which avoids the calculation of the complex interaction matrix. The fractional order process model is discretized via fractional order calculus. Meanwhile, the fractional order integral is introduced into the conventional LQG cost function. Then solving the linear quadratic Gaussian problem under the fractional order model and fractional control, the optimal input and output variances are determined for different weighting factors and the performance curves can be achieved. The comparison between fractional order LQG and the conventional LQG shows the improvement of the proposed benchmark under the same condition. The proposed benchmark can provide a more direct and superior reference standard to evaluate the performance of fractional order control system. Finally, a case study of fractional order PID(FO-PID) controller in industrial heating furnace temperature control experiment with model matching and model mismatch conditions is used to verify the effectiveness of the proposed benchmark.

The malware spreading in Wireless Sensor Network (WSN) has lately attracted the attention of many researchers as a hot problem in nonlinear systems. WSN is a collection of sensor nodes that communicate with each other wirelessly. These nodes are linked in a decentralised and distributed structure, allowing for efficient data collection and communication. Due to their decentralised architecture and limited resources, WSN is vulnerable to security risks, including malware attacks. Malware can attack sensor nodes, causing them to malfunction and consume more energy. These attacks can spread from one infected node to others in the network, making it essential to protect WSN against malware attacks. In this paper, we focus on the analysis of a novel fractional epidemiology model, specifically the fractional order SEIVR epidemic model in the sense of Caputo’s fractional derivative of order <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$0{ &lt; {\alpha }{\leq }}1$ </tex-math></inline-formula> with the goal of examining the efficacy of vaccination strategies and the heterogeneity of a scale-free network on epidemic spreading. First, using the next-generation technique and obtain the basic reproduction number of the proposed epidemic model, which is essential for determining both the locally asymptotically stable equilibrium point of the worm-free system and the unique existence of the endemic equilibrium point. To numerically solve the model, the Adam-Bashforth-Moulton predictor-corrector (ABM) method is applied. The fractional calculus enables us to deal directly with the “memory effect” of numerous phenomena, taking into account the system’s dependence on previous stages. This method provides the results of a complex system. Additionally, research demonstrates that vaccine treatments are quite effective at preventing the spread of malware. The outcome of the study reveals that the applied ABM predictor-corrector method is computationally strong and effective to analyse fractional order dynamical systems in the SEIVR epidemic model for malware propagation in WSN. The results show that the order of the fractional derivative has a significant effect on the dynamic process. Also, from the result, it is obvious that the memory effect is zero for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\alpha }$ </tex-math></inline-formula> = 1. When the fractional order <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\alpha }$ </tex-math></inline-formula> is decreased from 1, the memory effect appears, and its dynamics vary according to the time. This memory effect points out the difference between derivatives of fractional and integer orders. The theorems and their proofs are presented to validate the validity of the proposed model. To validate the proposed model, extensive theoretical study and computational analysis have also been applied.

The integer order differentiation by integration method based on the Jacobi orthogonal polynomials for noisy signals was originally introduced by Mboup, Join and Fliess. We propose to extend this method from the integer order to the fractional order to estimate the fractional order derivatives of noisy signals. Firstly, two fractional order differentiators are deduced from the Jacobi orthogonal polynomial filter, using the Riemann-Liouville and the Caputo fractional order derivative definitions respectively. Exact and simple formulae for these differentiators are given by integral expressions. Hence, they can be used for both continuous-time and discrete-time models in on-line or off-line applications. Secondly, some error bounds are provided for the corresponding estimation errors. These bounds allow to study the design parameters' influence. The noise error contribution due to a large class of stochastic processes is studied in discrete case. The latter shows that the differentiator based on the Caputo fractional order derivative can cope with a class of noises, whose mean value and variance functions are polynomial time-varying. Thanks to the design parameters analysis, the proposed fractional order differentiators are significantly improved by admitting a time-delay. Thirdly, in order to reduce the calculation time for on-line applications, a recursive algorithm is proposed. Finally, the proposed differentiator based on the Riemann-Liouville fractional order derivative is used to estimate the state of a fractional order system and numerical simulations illustrate the accuracy and the robustness with respect to corrupting noises.

The self-excited induction generator (SEIG) is widely applied in the wind energy conversion system (WECS) to enhance power generation. The power generation from WECS is varied in terms of varying wind speed. Hence, to improve the working of SEIG-based WECS, the multi-stage coati optimized proportional integral (CPI) fractional order proportional integral derivative (FOPID) controller is proposed in this work. The proposed coati optimized proportional integral - fractional order proportional integral derivative controls the SEIG’s grid side converter and generator side converter. The coati optimization algorithm optimizes the controller parameters of a proposed multi-state controller. The proposed CPI-FOPID controls both the voltage at the grid side converter and the current at a generator side converter. Moreover, the pitch angle of the wind turbine (WT) is controlled by the fuzzy-based tilt integral derivative (F-TID) controller. The proposed work will be implemented on the Matlab/Simulink platform, and the THD of 0.63% has proven the efficacy of the proposed methodology.

The presence of fixed and stochastic parameters in a mixed linear regression model has been dealt with for the case where the stochastic parameters follow a stationary process. The solution is given either by Generalized Least Squares (e.g. Rao, 1965, p. 192) or by a recursive state space estimation procedure such as the Kalman filter and smoother (e.g. Sallas and Harville 1981). The estimation of non-stationary stochastic parameters has been mainly approached in the state space framework, either as an initial condition problem (see among others, Ansley and Kohn, 1985, 1989; Kohn and Ansley, 1986; Bell and Hillmer, 1991 and De Jong, 1989, 1991); or as a hierarchical model with “fixed” effects in the hierarchy given a flat prior distribution (see Sallas and Harville, 1981, 1988; Tsimikas and Ledolter, 1994).KeywordsKalman FilterTime Series AnalysisAmerican Statistical AssociationARIMA ModelGeneralize Little SquareThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.