Artificial neural networks: a practical review of applications involving fractional calculus.

In this work, a bibliographic analysis on artificial neural networks (ANNs) using fractional calculus (FC) theory has been developed to summarize the main features and applications of the ANNs. ANN is a mathematical modeling tool used in several sciences and engineering fields. FC has been mainly applied on ANNs with three different objectives, such as systems stabilization, systems synchronization, and parameters training, using optimization algorithms. FC and some control strategies have been satisfactorily employed to attain the synchronization and stabilization of ANNs. To show this fact, in this manuscript are summarized, the architecture of the systems, the control strategies, and the fractional derivatives used in each research work, also, the achieved goals are presented. Regarding the parameters training using optimization algorithms issue, in this manuscript, the systems types, the fractional derivatives involved, and the optimization algorithm employed to train the ANN parameters are also presented. In most of the works found in the literature where ANNs and FC are involved, the authors focused on controlling the systems using synchronization and stabilization. Furthermore, recent applications of ANNs with FC in several fields such as medicine, cryptographic, image processing, robotic are reviewed in detail in this manuscript. Works with applications, such as chaos analysis, functions approximation, heat transfer process, periodicity, and dissipativity, also were included. Almost to the end of the paper, several future research topics arising on ANNs involved with FC are recommended to the researchers community. From the bibliographic review, we concluded that the Caputo derivative is the most utilized derivative for solving problems with ANNs because its initial values take the same form as the differential equations of integer-order.

Online ANN-based fault diagnosis implementation using an FPGA: Application in the EFI system of a vehicle.

In this research, fault detection and diagnosis (FDD) scheme for isolating the damaged injector of an internal combustion engine is formulated and experimentally applied. The FDD scheme is based on a temporal analysis (statistical methods), as well as in a frequency analysis (fast Fourier transform) of the fuel rail pressure. The arrangement of the scheme consists of three coupled artificial neural networks (ANNs) to classify the faulty injector correctly. The ANNs were trained considering five different scenarios, one scenario without fault in the injection system, and the other four scenarios represent a fault per injector (1 to 4). The Levenberg-Marquardt (LM), BFGS quasi-Newton, gradient descent (GD), and extreme learning machine (ELM) algorithms were tested to select the best training algorithm to classify the faults. Experimental results obtained from the implementation in a VW four-cylinder CBU 2.5L vehicle in idle operating conditions (800 rpm) show the effectiveness of the proposed FDD scheme.

Fitting of experimental data using a fractional Kalman-like observer.

In this paper, a fractional order Kalman filter (FOKF) is presented, this is based on a system expressed by fractional differential equations according to the Riemann-Liouville definition. In order to get the best fitting of the FOKF, the cuckoo search optimization algorithm (CS) was used. The purpose of using the CS algorithm is to optimize the order of the observer, the fractional Riccati equation and the FOKF tuning parameters. The Grünwald-Letnikov approximation was used to compute the numerical solution of the FOKF. To show the effectiveness of the proposed FOKF, four examples are presented, the brain activity, the cutaneous potential recordings of a pregnant woman, the earthquake acceleration, and the Chua's circuit response.

Fault tolerant system based on non-integers order observers: Application in a heat exchanger.

This work presents a fault-tolerant (FT) scheme based on the application of non-integer order observers also called fractional observers, the case of study is a double pipe countercurrent heat exchanger (HE). The aim of the FT is to detect sensors faults as soon as possible, and to provide a healthy signal in order to replace the faulty sensor signal by the fractional observer estimation. To develop the FT scheme a bank of high gain fractional order observers (HGFOO) is proposed. The Riemann-Liouville (RL) fractional derivative definition is used to solve each fractional observer. Experimental measures from a HE were used to test the performance of the fractional observers and the control scheme. The results show the robustness of the proposed observers.