Regulating the output of unstable process is a great challenge particularly in the presence of process parameters uncertainties, nonlinearity, transportation delays and load disturbances. These processes are even more sensitive to input changes of ramp types. In this communication, a fractional order Smith predictor scheme is analytically designed to handle the unstable process undergoing above mentioned unavoidable conditions. The suitable values of the design parameters β, α and λ are determined by exploring the stability region and investigating system robustness towards process model uncertainties. The suggested control is also well implemented on a nonlinear jacketed continuous stirred tank reactor and the performance enhancement of 71.5 % is achieved under the perfectly matched condition. When the process parameters are perturbed, 73.9 % percentage improvement is observed. Performance indices such as Integral of Squared Error (ISE), Integral of Time-weighted Absolute Error (ITAE), Integral of Absolute Error (IAE), and Total Variation (TV) are also calculated.

One major concern in control engineering is the problem of introducing an unstable system. Such systems are even more sensitive to ramp input changes, either set-point or disturbance. We have proposed an extended fractional-order IMC (FOIMC) control for an unstable system exhibiting a time delay. The complete design involves only three adjustable parameters, such as PID tuning. In the proposed structure, the inner loop control stabilises the system, whereas the fractional IMC filter improves the overall performance, together to tackle the ramp inputs. A systematic approach is developed to tune the required design parameters to obtain the desired peak of the sensitivity function and stability margins. The proposed control is simple and can easily calculate the FOIMC parameters from the explicit formulae. The method works under practical considerations, such as process parameter perturbations and load disturbances. The developed scheme is also tested on the nonlinear continuous stirred tank reactor system. The proposed control method results in a percentage enhancement of 61.9% under the perfectly ideal condition (when the process model is equal to the actual plant) whereas 81.3% enhancement is obtained when the process parameter variations are considered for the CSTR system.

A novel double-loop control architecture with a fractional-order IMC (internal model control) in the outer loop is suggested for integrating plants with dead time and inverse response behavior. The inner loop controller is tuned using the maximum sensitivity concept to stabilize the plant, and it also enhances the disturbance response. The IMC controller is analytically designed to achieve improved closed-loop performance and robustness. The proposed tuning rules involve three design parameters β ,α and λ , whose method of selection is explained through extensive simulations and stability analysis. The suitability of the proposed control is verified for a wide class of processes, including higher-order and double-integrating processes with non-minimum phase characteristics. The suggested control has the capability of producing a fast and smooth set-point tracking response and rejecting the load disturbance effectively even in the presence of measurement noise. Random perturbations are also introduced in the process parameters to further investigate the system’s robustness.

This paper proposes a digital morphological filtering method based on a novel structuring element (SE) formulated using fractional Fourier transform (FrFT) and cross-convolution of window functions. The highlighting feature of this newly formulated filter is its flexibility in adding an adaptive nature to classical morphological filtering. Until now, every method listed in the literature makes a convenient assumption that noises corrupting ECG signal are Gaussian and model the filter around this assumption. Addressing this shortcoming, the first-of-its-kind filter adopts the α -stable distribution model (of which Gaussian distribution is a special case) of noise to better replicate the real-time noises interfering with ECG signals. The designed filter can suppress the noise and adapt to the changes in ECG signal morphology for better reconstruction. The proposed filter is tested on MIT-BIH Arrhythmia Database. The simulation results show improved performance in several quantitative metrics, demonstrating the superiority of our suggested method over the currently used state-of-the-art techniques.

Fractional lower-order time–frequency distributions (FLO-TFDs) exhibit all the advantages of established time–frequency (TF) tools and offer robustness for even non-Gaussian noise environments. Therefore, this paper presents a novel extension to the existing FLO-TFDs known as fractional lower-order fractional Stockwell transform (FLO-FrST), with the aim of enhancing resolution , reconstruction and robustness . The proposed tool is analytically designed by amalgamating the advantages of fractional Fourier transform (FrFT), Stockwell transform (ST) and fractional lower-order statistics (FLOS). To demonstrate the efficacy of the suggested FLO-FrST tool, an experimental study is demonstrated, which includes a comparison with established methodologies in terms of qualitative and quantitative analysis using performance metric parameters; Jones–Park (JP) measure and root-mean-square error (RMSE) . The high value of JP measure and the low value of RMSE obtained establish the superiority of the proposed tool. Finally, an application of using this tool as a 2-dimensional (2-D) mapping tool is illustrated in electroencephalogram (EEG) epileptic classification using a deep learning approach. The proposed classification methodology is validated and compared with established TF and FLO-TF methods in terms of sensitivity , positive predictivity , accuracy , error rate , F1-score and Matthew’s correlation coefficient . The overall performance of the proposed tool presented in current study showcases its precedence over state-of-the-art methods, indicating its potential as a tool for achieving high-resolution and improved reconstruction in both non-Gaussian α -stable and Gaussian environments.

The diagnostic accuracy and reliability of an unsupervised electrocardiogram (ECG) analysis system entirely depend on the response of ECG preprocessing stage. Unfortunately, ECG signal analysis faces the challenge of getting distorted by various noises and artifacts (physiological and non-physiological origin). Thus, designing a denoising technique capable of dealing with different noises in real time is challenging, so selecting a noise analysis model is particularly important. Based on the extensive survey of state-of-the-art techniques, it is noticed that all the denoising techniques are designed with an implicit assumption that noises distorting ECG signals are of Gaussian nature and therefore, are based on the Gaussian distribution noise analysis model. However, in practical scenarios, noises may not always have a Gaussian nature. Therefore, this paper puts forward a non-Gaussian α -stable distribution model for noise analysis from the perspective of ECG signal analysis. This distribution model encompasses Gaussian distribution as a special case. From rigorous simulations and analytical studies, this research offers statistical proof that the α -stable distribution noise model may effectively capture background noises corrupting ECG signals. Ultimately, the effectiveness of R-peak detection techniques and deep learning models is evaluated in the presence of two types of noise: Gaussian distribution and α -stable distribution. Finally, through intensive simulation studies, it is discovered that relying on the assumption of Gaussian background noise can be misleading when the actual noise follows a non-Gaussian α -stable nature.

Abstract In this work, a proportional–integral (PI) controller with a set point filter is designed using the direct synthesis method for unstable plus time delay process. The Suggested method involves design parameters whose suitable values are recommended based on robust stability and robust performance constraints. The absence of derivative term makes PI controllers less sensitive to noise and, therefore, PI controllers are more preferable than PID in industrial applications. Despite a simple control architecture, the proposed method gives improved or comparable performance to previously presented approaches, which are comparatively complex. Four case studies are considered to evaluate the suitability and superiority of the suggested control technique. Proposed controller may be applied to the integrating plus time delay plants after some elementary transformations in the process model.

Cardiac ailments are increasing at an alarming rate globally due to the sedentary lifestyle and increased desk-bound activities. For decades, the ECG signal is used in aiding the analysis of human heart. Arrhythmia is a disorder that alters the normal cardiac cycle of ECG signal. The automatic classification of arrhythmia is a highly desirable and tough task. Mainly, the traditional methods of arrhythmia classification, are evaluated on the intra-patient criterion that may not befit the inter-patient criterion. A complete classification method has been proposed in the paper which performs well in intra-patient, inter-patient criterion and it is effective for minority class of MIT-BIH arrhythmia database (MIT-BIH-AD). In the proposed method, after preliminary processing based on Riesz fractional-order digital differentiator and R-peak detection, various features are extracted from ECG beat. The feature set proposed is a fusion of time-domain, time–frequency domain features and the new and novel features based on the Fibonacci series and coefficients of fractional-order Riesz based derivative signals which are proposed in this work. The results are evaluated on the MIT-BIH-AD and the results for intra-patients' criterion have achieved an overall accuracy, sensitivity and positive predictivity values of 99.85%, 99.16%, 99.93% respectively, and for the inter-patient scheme 92.5%, 89.89% and 95.54% an average value of accuracy, sensitivity and positive predictivity of six ECG classes. The obtained results in both criterions have outperformed other methods in the literature and the proposed work has also attained better results for minority class of MIT-BIH-AD in both the scheme.

This paper proposes a generalized fractional differential (GFD) mask which incorporates various fractional-order kernels such as power-law kernel, exponentional kernel, and Mittag-Leffler kernel, corresponding to Riemann-Liouville (RL)/Caputo, Caputo-Fabrizio (CF), and Atangana-Baleanu (AB) fractional-order differentials, respectively. Furthermore, a generalized fractional integral (GFI) mask incorporating RL and AB fractional integrals is put forward. The proposed generalized fractional masks provide a more flexible and appropriate tool for image enhancement and image denoising applications, and hence, avoids the construction of different masks for each kernel separately. Additionally, a generalized fractional integral and fractional differential based adaptive algorithm for image denoising (GFIFD-AA) is proposed. The proposed GFIFD-AA incorporates a novel noise detection method (NDM) for the detection of salt and pepper (SP) noise in images, which exhibits a profound advantage in avoiding misclassification of original pixels as noisy pixels when original image itself has some pixels with intensity values 0 or 255. The detected noisy pixels are processed by utilizing the proposed GFI-based adaptive mask (GFI-AM). The noise-free pixels are further updated by utilizing the proposed GFD-based adaptive mask (GFD-AM) so as to enhance the details of the image. Several standard and medical images of different characteristics are examined to evaluate the performance of the proposed approach on images affected by SP noise at various noise densities (i.e., 10%-95%). Furthermore, the performance of the proposed GFI mask is also tested for images affected by Gaussian noise and is compared against conventional fractional-order masks. The simulation results based upon several quantitative parameters validate the effectiveness of the proposed method against conventional methods.

In this paper, a novel closed-form analytical expression for the design of recently introduced non-singular and non-local Mittag–Leffler kernel-based Atangana–Baleanu–Caputo (ABC) fractional-order digital filter (FODF) has been proposed. The design has been obtained by first numerically approximating the ABC fractional differential operator using backward finite difference approach. Then, MacLaurin series expansion-based fractional delay interpolation formula is applied to obtain closed-form FIR filter approximation of ABC-FODF (herein, specified as ML-ABC-FODF). Various design examples are presented to show the performance of the proposed ML-ABC-FODF. From simulation and analytical study conducted, it has been seen that ML-ABC-FODF yields better performance over the entire Nyquist band of frequencies. Furthermore, an exact approximation to the first-order digital differentiator has been obtained when fractional-order α→ 1 , which highlights the efficacy of the proposed method. Finally, 1-D and 2-D applications of the proposed ML-ABC-FODF are established and comparison is made with conventional approaches for accurate delineation of R-peaks in ECG signals as well as sharpening of medical images.

This paper intends to apply a new mathematical approach based on Riemann–Liouville fractional–differential operator to the design of fractional-order digital differentiators (FODDs). Under the research area of fractional-order calculus, Grünwald–Letnikov (GL) and Riemann–Liouville (RL) are most widely used fractional–differential operators. The GL-based methods have been extensively investigated by research community to design FODD, but there seems to be an improvement window for designing FIR filters based upon RL fractional–differential operator. Therefore, this paper establishes a generalized framework for the design of RL-based FODD (RL-FODD) and compares its performance with well-established GL-FODD designs, based upon their ability to yield ideal frequency response of FODD. Initially, closed-form analytical expression is formulated for computing FIR filter coefficients of RL-FODD. Then, the design accuracy of proposed filter in the high-frequency region is improved by incorporating non-integer sample delay into the design process. Several design examples are presented to illustrate the comparative analysis between conventional GL-FODD and proposed RL-FODD for varying fractional orders. The proposed method is evaluated, taking into account several amplitude-modulated and harmonic signals corrupted by AWGN and high-frequency chirp noise. Furthermore, the application in parameter estimation of fractional noise process is investigated. Compared with conventional GL-FODD, results of the proposed study validate its superiority and robustness based upon various experimental simulations.

Multiplicative calculus (MUC) measures the rate of change of function in terms of ratios, which makes the exponential functions significantly linear in the framework of MUC. Therefore, a generally non-linear optimization problem containing exponential functions becomes a linear problem in MUC. Taking this as motivation, this paper lays mathematical foundation of well-known classical Gauss-Newton minimization (CGNM) algorithm in the framework of MUC. This paper formulates the mathematical derivation of proposed method named as multiplicative Gauss-Newton minimization (MGNM) method along with its convergence properties. The proposed method is generalized for n number of variables, and all its theoretical concepts are authenticated by simulation results. Two case studies have been conducted incorporating multiplicatively-linear and non-linear exponential functions. From simulation results, it has been observed that proposed MGNM method converges for 12972 points, out of 19600 points considered while optimizing multiplicatively-linear exponential function, whereas CGNM and multiplicative Newton minimization methods converge for only 2111 and 9922 points, respectively. Furthermore, for a given set of initial value, the proposed MGNM converges only after 2 iterations as compared to 5 iterations taken by other methods. A similar pattern is observed for multiplicatively-non-linear exponential function. Therefore, it can be said that proposed method converges faster and for large range of initial values as compared to conventional methods.

Atangana–Baleanu–Caputo (ABC) fractional differential operator based upon Mittag-Leffler kernel exhibits all the advantages of conventional Riemann–Liouville and Caputo fractional differential operators; in addition, the kernel associated is non-singular. Therefore, this paper puts forward a closed-form analytical formulation for the design of an ABC-based fractional-order FIR filter for various signal processing and filtering applications. The closed-form expression is derived by utilizing backward finite difference method and fractional sample delay interpolation techniques. Furthermore, several design examples are considered to illustrate the effectiveness of the proposed method. From the analytical and simulation studies done, it is observed that the proposed design efficiently approximates the ideal frequency response of ABC-fractional differential operator. Finally, one-dimensional and two-dimensional applications of the proposed method are validated and compared against state-of-the-art methods for electrocardiogram (ECG) R-peak detection as well as for digital image sharpening.

Electrocardiogram (ECG) is a non-invasive technique, used by physicians for prognosis of underlying heart diseases. ECG being the time-varying electric signal evolving from heart, is susceptible to various kinds of low and high-frequency noises. Therefore, this paper presents a novel method for denoising ECG signals, in order to assist physicians to monitor cardiovascular disorders with good accuracy. The proposed denoising method is based on the time-frequency analysis tool known as fractional Stockwell transform (FrST). Method exploits the characteristics of FrST to perform operations in a transformed domain where required signal information is highly concentrated. This aids in suppressing noise from ECG and hence making its analysis effective under noisy environment. Performance of the proposed denoising method is tested on two databases, namely; MIT-BIH Arrhythmia Database and European ST-T Database. The proposed method is used to remove background noise corrupting ECG signal from the time of its acquisition in the form of additive noise. Performance of the proposed method is evaluated by artificially corrupting these ECG signals with additive white Gaussian noise at 5 dB, 10 dB, and 15 dB levels and with real-time noises like baseline wander and motion artifact. Simulation results prove superiority of the proposed method over existing denoising methods in terms of Root-Mean-Square Error, Percent Root Mean Square Difference, and improved Signal-to-Noise Ratio values. (C) 2020 Elsevier Ltd. All rights reserved.

Clinically,electrocardiogram (ECG) is a powerful tool for determining the health and functioning of the human heart. Faster detection and diagnosis of heart functioning would aid cardiologists to provide appropriate treatment to the patients (subjects). In this paper, the concept of fractional-order calculus is employed for noise cancellation and artifacts removal in ECG signal as fractional-order differentiator proved to provide more peculiar details about signals than an integer-order differentiator. In the proposed method, R-peaks are detected using Riesz fractional-order digital differentiator (RFODD) based on the differencing method. The differentiation operation enhances the high-frequency components of the signal. So, QRS complex which is a high-frequency component in ECG is accentuated and the R-peaks are detected using appropriate threshold technique. The proposed method is tested on Massachusetts Institute of Technology-Beth Israel Hospital (MIT-BIH) arrhythmia database, and the experimental results of the proposed method have achieved a sensitivity of 99.95%, positive predictivity of 99.949% and an error rate of 0.095%. ECG waveforms are analyzed on various fractional orders of RFODD, and their performance parameters, i.e., sensitivity, positive predictivity and error rate, are also calculated.

QRS complex present in Electrocardiogram (ECG) is the most vital component which is used as a basis for determining the condition of a human heart. However, due to the non-stationary nature of ECG, QRS detectors are unable to accurately delineate the R-peaks which may result in significant false negatives and false positives. So, in order to improve the detection rate of ECG monitoring system, this paper introduces a novel technique by amalgamating fractional Fourier transform and Stockwell transform i.e., fractional Stockwell transform (FrST) for improving the accuracy and simultaneously suppressing artifacts affecting the ECG. The proposed technique employed in this paper not only assures good detection rate but also provides an effective basis to various front end ECG signal processing measures. It also focuses on accurately identifying the QRS complex of unclassifiable beats which are among the five beat classes of Arrhythmia recommended by the Association for Advancement of Medical Instrumentation (AAMI). The proposed approach follows the five-stage methodology for correctly identifying the occurrence of R-peaks in the presence of noise. Performance is validated against ECG records taken from the MIT-BIH Arrhythmia database. The results prove the superiority of the proposed technique by achieving a sensitivity of 99.99%, positive predictivity of 99.97%, detection accuracy of 99.97%, and error rate of 0.03%. (C) 2019 Elsevier Ltd. All rights reserved.

R-peak detection is the most important step in Electrocardiography (ECG) signal processing, automatic identification of anomalies in it and further classification of it into normal and abnormal subjects (patients). There are several thresholding techniques used in the literature for R-peak detection of ECG after pre-processing. In this paper, wavelet transform is employed for pre-processing and the thresholding techniques is proposed which is used for R-peak detection. The proposed thresholding method is compared with one of the previously established threshold methods and results show that the proposed method outperforms in performance evaluation parameters: sensitivity, positive predictivity and error rate.

Electrocardiogram (ECG) signal processing and analysis is becoming more and more popular as it is useful in diagnosis and prognosis of human heart and clinically automatic machine estimation is based upon it. R-peak is the most important component in ECG beat and is widely used to investigate normal and abnormal subjects (patients). From the last few decades, R-peak detection in ECG has been the most challenging topic in the biomedical research. As QRS complex has high frequency in ECG as compared to other waves (P, T, U-wave), so majority of algorithms estimate QRS complex by either filtering or suppressing the lower frequency waves, including various artifacts like baseline wander, power line interference, and electromyograph noises. This paper demonstrates a new kind of ECG denoising algorithm based on self-convolution window (SCW) concept. The SCW based on Hamming window, herein referred to as Hamming self-convolution window, is used to design a new kind of filter which possesses negligible ripples in the stop band, as compared to the conventional window-based filters. This algorithm is validated on MIT-BIH arrhythmia database and the results outperform in terms of sensitivity, positive predictivity, and error rate obtained as 99.93%, 99.95%, and 0.117%, respectively, as compared to the other well-established works. The approach has also outperformed the results of well-established window-based filters (Hamming and Kaiser) in terms of reduced false negative, false positive, and error rate.

In this paper, the generalized framework for higher-order fractional derivatives along with its numerical approximation is proposed. Due to the various shortcomings of the traditional fractional-order derivatives incorporating power-law kernel, many other derivatives with different kernels have been proposed, such as Caputo-Fabrizio, Atangana-Baleanu, and so on. A unified framework has been presented in the literature to put together these various definitions of fractional-order derivatives, but only up to the order one. Therefore, this paper provides a more flexible tool to describe the dynamics of fractional calculus by extending this unified framework to higher-order fractional derivatives. Furthermore, the numerical approximation of the proposed generalized higher-order fractional derivative is formulated and authenticated by simulation results. As an illustration, the proposed mathematical formulation is utilized to compute the higher-order fractional derivatives of Gaussian function, which is considered as a powerful tool for signal processing applications, such as, sampling, smoothing, change and blob detection, etc.