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To construct a nonlinear fractional-order neural network reflecting the complex environment of the real world, this paper considers the common factors such as uncertainties, perturbations, and delays that affect the stability of the network system. In particular, not only does the activation function include multiple time delays, but the memristive connection weights also consider transmission delays. Stemming from the characteristics of neural networks, two different types of discontinuous controllers with state information and sign functions are devised to effectuate network synchronization objectives. Combining the finite-time convergence criterion and the theory of fractional-order calculus, Mittag-Leffler synchronization conditions for fractional-order multi-delayed memristive neural networks (FMMNNs) are derived, and the upper bound of the setting time can be confirmed. Unlike previous jobs, this article focuses on applying different inequality techniques in the synchronous analysis process, rather than comparison principles to manage the multi-delay effects. In addition, this study removes the restrictive requirement that the activation function has a zero value at the switching jumps, and the discontinuous control protocol in this paper makes the networks achieve synchronization over a finite time, with some advantages in terms of the convergence speed. [ABSTRACT FROM AUTHOR]

Synchronous motors have extended their presence in different applications, specifically in high-demand environments such as agronomy. These uses need advanced and better control strategies to improve energy efficiency. Within this context, sliding mode control has demonstrated effectiveness in electric machine control due to its advantages in robustness and quick adaptation to uncertain dynamic system disturbances. Nevertheless, this control technique presents the undesirable chattering phenomenon due to the discontinuous control action. This paper introduces a novel speed integral control scheme based on sliding modes for synchronous motors. This approach is designed to track smooth speed profiles and is evaluated through several numeric simulations to verify its robustness against variable torque loads. This approach addresses using electric motors for different applications such as irrigation systems, greenhouses, pumps, and others. Moreover, to address the chattering problem, different sign function approximations are evaluated in the control scheme. Then, the most effective functions for suppressing the chattering phenomenon through extensive comparative analysis are identified. Integral compensation in this technique demonstrates improvement in motor performance, while sign function approximations show a chattering reduction. Different study cases prove the robustness of this control scheme for large-scale synchronous motors. The simulation results validate the proposed control scheme based on sliding modes with integral compensation, by achieving chattering reduction and obtaining an efficient control scheme against uncertain disturbances in synchronous motors for agronomy applications. [ABSTRACT FROM AUTHOR]

To construct a nonlinear fractional-order neural network reflecting the complex environment of the real world, this paper considers the common factors such as uncertainties, perturbations, and delays that affect the stability of the network system. In particular, not only does the activation function include multiple time delays, but the memristive connection weights also consider transmission delays. Stemming from the characteristics of neural networks, two different types of discontinuous controllers with state information and sign functions are devised to effectuate network synchronization objectives. Combining the finite-time convergence criterion and the theory of fractional-order calculus, Mittag-Leffler synchronization conditions for fractional-order multi-delayed memristive neural networks (FMMNNs) are derived, and the upper bound of the setting time can be confirmed. Unlike previous jobs, this article focuses on applying different inequality techniques in the synchronous analysis process, rather than comparison principles to manage the multi-delay effects. In addition, this study removes the restrictive requirement that the activation function has a zero value at the switching jumps, and the discontinuous control protocol in this paper makes the networks achieve synchronization over a finite time, with some advantages in terms of the convergence speed.

Synchronous motors have extended their presence in different applications, specifically in high-demand environments such as agronomy. These uses need advanced and better control strategies to improve energy efficiency. Within this context, sliding mode control has demonstrated effectiveness in electric machine control due to its advantages in robustness and quick adaptation to uncertain dynamic system disturbances. Nevertheless, this control technique presents the undesirable chattering phenomenon due to the discontinuous control action. This paper introduces a novel speed integral control scheme based on sliding modes for synchronous motors. This approach is designed to track smooth speed profiles and is evaluated through several numeric simulations to verify its robustness against variable torque loads. This approach addresses using electric motors for different applications such as irrigation systems, greenhouses, pumps, and others. Moreover, to address the chattering problem, different sign function approximations are evaluated in the control scheme. Then, the most effective functions for suppressing the chattering phenomenon through extensive comparative analysis are identified. Integral compensation in this technique demonstrates improvement in motor performance, while sign function approximations show a chattering reduction. Different study cases prove the robustness of this control scheme for large-scale synchronous motors. The simulation results validate the proposed control scheme based on sliding modes with integral compensation, by achieving chattering reduction and obtaining an efficient control scheme against uncertain disturbances in synchronous motors for agronomy applications.

In this paper, we propose a distributed finite-time optimization protocol for single-integrator continuous-time multi-agent systems, which utilizes a modified Newton–Raphson method. The inverse of Hessian matrix, sign function and the gradient are adopted for the design of the algorithms. The proposed algorithms make agents converge to the network optimizer under any initial state and finite time, respectively. Lyapunov method and the properties of sign function are employed to verify the convergence of the proposed algorithms. Besides, the adaptive method is also considered to avoid complex parameter conditions and centralized parameters. Finally, numerical simulations are provided to testify the presented results. [ABSTRACT FROM AUTHOR]

In this article, we have proposed a novel matrix iteration for computing the matrix sign function numerically. An extensive convergence analysis is carried out in matrix case to achieve fourth‐order convergence. The basins of attraction are drawn to illustrate the global convergence behavior of the proposed method. Analytically, it is shown that the presented scheme is asymptotically stable. Further, the theoretical developments are verified numerically by discussing eigenvalues clustering around ±1$$ \pm 1 $$. Moreover, a class of numerical examples for matrices of various dimensions is assessed to exhibit the efficacy of proposed method. [ABSTRACT FROM AUTHOR]

We study the sequence f n of functions that is defined by recursive convolutions as f 1 (x) = Π (x) , f n + 1 (x) = (f n ∗ f 1) (x) , n ∈ N , where Π is the unit rectangle function. We find out the general closed-form of the sequence f n and apply it for the evaluation of the improper integral 2 π ∫ 0 ∞ sin (ξ) ξ n cos (2 x ξ) d ξ , x ∈ R , n ∈ N , n ≥ 2. We also study some interesting features of the numerical coefficients that appear in the closed-form expression of f n . In connection to the numerical coefficients that appear in closed-form expression of f n , we introduce a map F defined on N × N 0 , by the rule, F (n , s) = ∑ i = 0 n i s (- 1) n - i i ! (n - i) ! , and show that its range is in N 0 , where N 0 : = N ∪ { 0 } . [ABSTRACT FROM AUTHOR]

This article focuses on global finite-time output feedback stabilization for uncertain nonlinear systems with unknown measurement sensitivity. The existence of the continuous measurement error resulting from limited accuracy of sensors invalidates the existing design strategies depending on the use of the precise output in the construction of an observer, which highlights the contribution of this article. Essentially, different from related works, we propose a new finite-time convergent observer by avoiding the use of the information on nonlinearities. By combining the homogeneous domination with the addition of a power integrator method, an output feedback controller composed of multiple nested sign functions is successfully developed. Finally, the effectiveness of the presented scheme is exhibited by a numerical example. [ABSTRACT FROM AUTHOR]

In this article, the consensus, collision avoidance, and distributed convex optimization problems of a class of multi‐agent systems with a time‐varying objective function are studied. Moreover, the proposed method in this article can reduce the chattering problem of the controller input caused by the sign function. First, the sign function is used to design the controller to improve the convergence rate of the system, on this basis, the optimization item is designed for the more complex time‐varying objective function to ensure that the value of the objective function is the smallest when the system achieves the speed consensus. Second, considering the chattering problem caused by the sign function, the designed controller is improved to reduce the chattering phenomenon. Third, two novel Lyapunov functions are constructed to verify the stability of the system and the distributed optimization objective. Due to the controller is designed for the time‐varying objective function, moreover, the tasks of the formation and collision avoidance are needed to take into account, therefore, the designed Lyapunov functions in this article are more difficult. Finally, the formation function is designed to ensure that the multi‐agent system achieves the desired formation configuration. The simulation results show the correctness of the theoretical design of this article and the superiority in extending the running time of the system. [ABSTRACT FROM AUTHOR]

Homomorphic encryption with the ability to compute over encrypted data without access to the secret key provides benefits for the secure and powerful computation, storage, and communication of resources in the cloud. One of its important applications is fast-growing robot control systems for building lightweight, low-cost, smarter robots with intelligent brains consisting of data centers, knowledge bases, task planners, deep learning, information processing, environment models, communication support, synchronous map construction and positioning, etc. It enables robots to be endowed with secure, powerful capabilities while reducing sizes and costs. Processing encrypted information using homomorphic ciphers uses the sign function polynomial approximation, which is a widely studied research field with many practical results. State-of-the-art works are mainly focused on finding the polynomial of best approximation of the sign function (PBAS) with the improved errors on the union of the intervals [ − 1 , − ϵ ] ∪ [ ϵ , 1 ] . However, even though the existence of the single PBAS with the minimum deviation is well known, its construction method on the complete interval [ − 1 , 1 ] is still an open problem. In this paper, we provide the PBAS construction method on the interval [ − 1 , 1 ] , using as a norm the area between the sign function and the polynomial and showing that for a polynomial degree n ≥ 1 , there is (1) unique PBAS of the odd sign function, (2) no PBAS of the general form sign function if n is odd, and (3) an uncountable set of PBAS, if n is even. [ABSTRACT FROM AUTHOR]

Generalized Hilbert transform (GHT) has been widely and successfully used in signal processing and interpretation for many years. However, the order parameter in GHT can only be chosen in a certain range, and the GHT result is not stable with respect to the chosen order parameter, which usually leads to poor-stability and low-resolution outputs. To improve the stability of GHT and broaden its order parameter chosen range, and finally achieve high-efficiency and high-resolution results, we extend GHT by introducing the sign function, the two frequency integration parameters, and propose the Lp-order time–frequency spectra (LTFS). In LTFS, the sign function aims to broaden the chosen range of the order parameter into any positive numbers, and then leads to high-stability and high-resolution LTFS results; the two frequency integration parameters aim to optimize the computation efficiency and output resolution of LTFS. Synthetic one-dimensional (1D) channel data testing, synthetic two-dimensional (2D) seismic data, and actual three-dimensional (3D) seismic data examples demonstrate favorable capability of LTFS for edge detection. [ABSTRACT FROM AUTHOR]

This paper deals with estimation of interval controls for interval Linear Time Invariant (LTI) plants using pole placement technique. Generally, the parameters associated with LTI plant system are assumed to be crisp values, but due to the occurrence of error in experimentation, measurement or due to maintenance induced errors, the parameters are imprecise. As such, the LTI problem reduces to interval LTI. The pole placement problem is then reduced to interval algebraic equation viz. interval Diophantine equation using proper transfer function. Further, the interval Diophantine equation is transformed to Interval System of Linear Equations (ISLE) viz. interval Sylvester matrix equation. Initially, using interval arithmetic an interval algebraic approach has been applied to solve the ISLE for computing non-negative or non-positive controls. Then, another approach using Artificial Neural Network (ANN) is proposed for computing the interval controls. Further, algorithms based on sign function and ANN procedures have been discussed. Finally, the efficiency of the proposed algorithms has been verified using different order interval plants. • Pole placement technique gets converted to Diophantine equation. • Interval Diophantine equation have been associated to interval linear systems. • Sign function and ANN procedures are studied for solving interval linear equations. • Interval controls have been obtained using Sign function and ANN procedures. [ABSTRACT FROM AUTHOR]

Abstract The Hilbert transform (HT) has been widely used in seismic data processing and interpretation. However, HT results exhibit lower noise robustness and resolution. This paper extends HT by introducing three orders greater than or equal to 0 and the sign function in frequency domain and propose an extended Hilbert transform (EHT) which has many advantages over the HT—particularly noise robustness, resolution, and a variety of applications. The output of EHT depends on the choice of the three orders and the sign function. Analysis of synthetic cosine signal and hypothetical channel model shows that EHT helps to retain the desired characteristics of HT while overcoming its shortcomings and exhibits better noise robustness and higher resolution than HT. Analysis of real 3‐D seismic data application demonstrates that EHT shows a very desirable ability to capture subtle geologic features and can be well used for seismic attributes and other fields involving digital data processing.

This work develops novel rational Krylov methods for updating a large-scale matrix function f(A) when A is subject to low-rank modifications. It extends our previous work in this context on polynomial Krylov methods, for which we present a simplified convergence analysis. For the rational case, our convergence analysis is based on an exactness result that is connected to work by Bernstein and Van Loan on rank-one updates of rational matrix functions. We demonstrate the usefulness of the derived error bounds for guiding the choice of poles in the rational Krylov method for the exponential function and Markov functions. Low-rank updates of the matrix sign function require additional attention; we develop and analyze a combination of our methods with a squaring trick for this purpose. A curious connection between such updates and existing rational Krylov subspace methods for Sylvester matrix equations is pointed out. [ABSTRACT FROM AUTHOR]

Homomorphic encryption with the ability to compute over encrypted data without access to the secret key provides benefits for the secure and powerful computation, storage, and communication of resources in the cloud. One of its important applications is fast-growing robot control systems for building lightweight, low-cost, smarter robots with intelligent brains consisting of data centers, knowledge bases, task planners, deep learning, information processing, environment models, communication support, synchronous map construction and positioning, etc. It enables robots to be endowed with secure, powerful capabilities while reducing sizes and costs. Processing encrypted information using homomorphic ciphers uses the sign function polynomial approximation, which is a widely studied research field with many practical results. State-of-the-art works are mainly focused on finding the polynomial of best approximation of the sign function (PBAS) with the improved errors on the union of the intervals [−1,−ϵ]∪[ϵ,1]. However, even though the existence of the single PBAS with the minimum deviation is well known, its construction method on the complete interval [−1,1] is still an open problem. In this paper, we provide the PBAS construction method on the interval [−1,1], using as a norm the area between the sign function and the polynomial and showing that for a polynomial degree n≥1, there is (1) unique PBAS of the odd sign function, (2) no PBAS of the general form sign function if n is odd, and (3) an uncountable set of PBAS, if n is even.

A brief review of fractional differentiation, fractional integration and differo-integral operators is given based on the properties of the Fourier transform. This is undertaken to provide the reader with a quick-guide and a short background to the fractional calculus and includes a brief discussion on some of the principal characteristics of fractional differointegral operators. The paper then presents a new definition for a fractional differo-integral based on the properties of the sign function and explores some related results. Using the properties of the Dirac delta function, a generalisation is developed in order to quantify the issue as to whether there is an upper bound to the number of definitions for a fractional differo-integral operator that can be developed. Finally, an approximation of a fractional differential is considered and used in the construction of a self-affine stochastic time series model based on the Kolmogorov-Feller equation for the memory function. [ABSTRACT FROM AUTHOR]

Generalized Hilbert transform (GHT) has been widely and successfully used in signal processing and interpretation for many years. However, the order parameter in GHT can only be chosen in a certain range, and the GHT result is not stable with respect to the chosen order parameter, which usually leads to poor-stability and low-resolution outputs. To improve the stability of GHT and broaden its order parameter chosen range, and finally achieve high-efficiency and high-resolution results, we extend GHT by introducing the sign function, the two frequency integration parameters, and propose the Lp-order time–frequency spectra (LTFS). In LTFS, the sign function aims to broaden the chosen range of the order parameter into any positive numbers, and then leads to high-stability and high-resolution LTFS results; the two frequency integration parameters aim to optimize the computation efficiency and output resolution of LTFS. Synthetic one-dimensional (1D) channel data testing, synthetic two-dimensional (2D) seismic data, and actual three-dimensional (3D) seismic data examples demonstrate favorable capability of LTFS for edge detection.

A brief review of fractional differentiation, fractional integration and differo-integral operators is given based on the properties of the Fourier transform. This is undertaken to provide the reader with a quick-guide and a short background to the fractional calculus and includes a brief discussion on some of the principal characteristics of fractional differointegral operators. The paper then presents a new definition for a fractional differo-integral based on the properties of the sign function and explores some related results. Using the properties of the Dirac delta function, a generalisation is developed in order to quantify the issue as to whether there is an upper bound to the number of definitions for a fractional differo-integral operator that can be developed. Finally, an approximation of a fractional differential is considered and used in the construction of a self-affine stochastic time series model based on the Kolmogorov-Feller equation for the memory function t −α /Γ(1 − α), α ∈ (0, 1). [ABSTRACT FROM AUTHOR]

A new computational approach that combines the extended finite element method associated with variable-node elements and cohesive zone model is developed. By using a new enriched technique based on sign function, the proposed model using 4-node quadrilateral elements can eliminate the blending element problem. It also allows modeling the equal stresses at both sides of the crack in the crack-tip as assumed in the cohesive model, and is able to simulate the arbitrary crack-tip location. The multiscale mesh technique associated with variable-node elements and the arc-length method further improve the efficiency of the developed approach. The performance and accuracy of the present approach are illustrated through numerical experiments considering both mode-I and mixed-mode fracture in concrete. [ABSTRACT FROM AUTHOR]

Summary: This paper focuses on the problem of disturbance attenuation with fast global finite‐time convergence (FTC) for a class of generalized high‐order uncertain nonlinear systems. Combining the fast finite‐time stabilization technique with a delicate manipulation of sign functions, a new control approach is proposed to attenuate the serious uncertainties substantially, including time‐varying control coefficients, nonlinear parameters, and external disturbances, while achieving the performance evaluated in terms of L2‐L2p gain. A notable feature of the control strategy is the fast FTC, which greatly shortens the convergent time when the initial state is far away from the origin. A numerical example is provided to demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, a new iteration scheme for computing the sign of a matrix which has no pure imaginary eigenvalues is presented. Then, by applying a well-known identity in matrix functions theory, an algorithm for computing the geometric mean of two Hermitian positive deffnite matrices is constructed. Moreover, another effcient algorithm for this purpose is derived free from the computation of principal matrix square root. Finally, some tests are given to show their applicabilities. [ABSTRACT FROM AUTHOR]

Based on the existing threshold denoising algorithm,a threshold denoising algorithm based on the new symbolic function was proposed.The new threshold function has advantages of continuous guidance,small deviation of wavelet coefficient,strong threshold adaptability and so on.It not only preserved the low-frequency wavelet coefficients,but also filtered the noise coefficients in the high-frequency coefficients effectively,so that the reconstructed image was closer to the original image.The simulation results of Bridge image,Lena image and B-mode Fetus image with Gaussian white noise show that the visual effect of both the new threshold function and the quantitative indicators PSNR and MSE are better than the existing threshold image denoising algorithm.The edge and detail information can be better protected,have no obvious oscillation,the image is smoother and even,and the method has good stubbornness under the background of complex noise.

Summary: This paper investigates the problem of finite‐time output‐feedback stabilization of a class of high‐order nonholonomic systems under weaker conditions on system powers and nonlinearities. By constructing the appropriate Lyapunov function and observer, skillfully combining generalized adding a power integrator technique, sign function, and homogeneous domination method, and successfully introducing a new mathematical method, an output‐feedback controller is constructed to guarantee that all the states of the closed‐loop system converge to origin in a finite time. [ABSTRACT FROM AUTHOR]

The Hilbert transform (HT) has been widely used in seismic data processing and interpretation. However, HT results exhibit lower noise robustness and resolution. This paper extends HT by introducing three orders greater than or equal to 0 and the sign function in frequency domain and propose an extended Hilbert transform (EHT) which has many advantages over the HT—particularly noise robustness, resolution, and a variety of applications. The output of EHT depends on the choice of the three orders and the sign function. Analysis of synthetic cosine signal and hypothetical channel model shows that EHT helps to retain the desired characteristics of HT while overcoming its shortcomings and exhibits better noise robustness and higher resolution than HT. Analysis of real 3‐D seismic data application demonstrates that EHT shows a very desirable ability to capture subtle geologic features and can be well used for seismic attributes and other fields involving digital data processing. Key Points: EHT extends HT by introducing three orders greater than or equal to 0 and the sign function in frequency domainEHT exhibits better noise robustness and higher resolution than HTEHT can be well used for seismic attributes and other fields [ABSTRACT FROM AUTHOR]

Summary: This paper considers the problem of state feedback finite‐time stabilization for a class of high‐order nonlinear systems with an output constraint. By proposing a novel tan‐type barrier Lyapunov function combined with manipulating sign functions, the technique of adding a power integrator is skillfully revamped to develop a systematic approach that guides us to construct a state feedback finite‐time stabilizer for high‐order nonlinear systems while preventing the violation of a prespecified output constraint during operation. The proposed approach is a unified tool in the sense that it can provide a finite‐time stabilizer design even when the constraint is infinite, or equivalently, there is no need for a constraint. A simple example is presented to demonstrate the effectiveness of the proposed strategy. [ABSTRACT FROM AUTHOR]

This article proposes an application of a discrete-time adaptive backstepping control strategy for a hydraulic process pumping station. The proposed solution leads to improved control system performances in terms of pressure and flow tracking in transient and standstill operation and improvement of pressure response time. The proposed design methodology is based on accurate model for pumping station, which is developed in previous works using fuzzy-C means algorithm. The control law design is based on discrete-time adaptive backstepping control, which is developed in the sense of Lyapunov stability theory using sign function, in order to satisfy various objectives of a stable pumping station like the asymptotic stability of the tracking error. To validate the proposed solution, simulation and experimental tests are made and analyzed. Compared to the conventional proportional-integral approach, the results show that the discrete-time adaptive backstepping control allows exhibiting excellent transient response over a wide range of operating conditions and especially is easier to be implemented in practice. [ABSTRACT FROM AUTHOR]