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This article proposes a space‐efficient approximation to empirical tail dependence coefficients of an indefinite bivariate stream of data. The approximation, which has stream‐length invariant error bounds, utilizes recent work on the development of a summary for bivariate empirical copula functions. The work in this paper accurately approximates a bivariate empirical copula in the tails of each marginal distribution, therefore modeling the tail dependence between the two variables observed in the data stream. Copulas evaluated at these marginal tails can be used to estimate the tail dependence coefficients. Modifications to the space‐efficient bivariate copula approximation, presented in this paper, allow the error of approximations to the tail dependence coefficients to remain stream‐length invariant. Theoretical and numerical evidence of this, including a case‐study using the Los Alamos National Laboratory netflow data‐set, is provided within this article. [ABSTRACT FROM AUTHOR]

This paper investigates an L1 adaptive fault-tolerant control scheme for nonlinear systems with input constraint, external disturbances, and multiple faults, which include actuator faults and sensor faults. Faults and input constraint are important factors that affect the stability and performance of a control system. Actuators and sensors are the most vulnerable components, with the former receiving more attention in comparison. In this paper, sensor faults are first transformed into pseudo-actuator faults through the augmented matrix approach, which facilitates their handling together with actuator faults. Saturation constraints on the control signal are not conducive to the design of the controller. The conversion of an input-saturated function to a time-varying linear system is completed based on function approximation and Lagrange's mean value theorem. Moreover, a nonlinear system with unknown input gain and uncertainties is constructed using these methods. Next, an L1 adaptive fault-tolerant controller is designed to cope with uncertainties, including system uncertainties, external disturbances, faults, and approximation errors. In the L1 adaptive controller, the online estimation of the time-varying parameters allows for updating of the system state, while the combination of the two is transmitted to the control law such that it can compensate for the effects of the uncertainties. The stability and performance boundaries are further derived using the Lyapunov theory and the L1 reference system. Finally, simulations are carried out to demonstrate the effectiveness of the proposed controller. [ABSTRACT FROM AUTHOR]

To address the problem of low accuracy and efficiency in trajectory planning algorithms for interceptors facing multiple constraints during the midcourse guidance phase, an improved trajectory convex programming method based on the lateral distance domain is proposed. This algorithm can achieve fast trajectory planning, reduce the approximation error of the planned trajectory, and improve the accuracy of trajectory guidance. First, the concept of lateral distance domain is proposed, and the motion model of the midcourse guidance segment in the interceptor is converted from the time domain to the lateral distance domain. Second, the motion model and multiple constraints are convexly and discretely transformed, and the discrete trajectory convex model is established in the lateral distance domain. Third, the deep reinforcement learning algorithm is used to learn and train the initial solution of trajectory convex programming, and a high-quality initial solution trajectory is obtained. Finally, a dynamic adjustment method based on the distribution of approximate solution errors is designed to achieve efficient dynamic adjustment of grid points in iterative solving. The simulation experiments show that the improved trajectory convex programming algorithm proposed in this paper not only improves the accuracy and efficiency of the algorithm but also has good optimization performance. [ABSTRACT FROM AUTHOR]

Structural symmetries of linear dynamical systems can be exploited for decoupling the dynamics and reducing the computational complexity of the controller implementation. However, in practical applications, inexact structural symmetries undermine the ability to decouple the system, resulting in the loss of any potential complexity reduction. To address this, we propose substituting an approximation with exact structural symmetries for the original system model, thereby introducing an approximation error. We focus on internal model controllers for cross-directional systems encountered in large-scale and high-speed control problems of synchrotrons or the process industry and characterise the stability, performance, and robustness properties of the resulting closed loop. While existing approaches replace the original system model with one that minimises the Frobenius norm of the approximation error, we show that this can lead to instability or poor performance. Instead, we propose approximations that are obtained from semidefinite programming problems. We show that our proposed approximations can yield stable systems even when the Frobenius norm approximation does not. The paper concludes with numerical examples and a case study of a synchrotron light source with inexact structural symmetries. Exploiting structural symmetries in large-scale and high-speed systems enables faster sampling times and the use of more advanced control techniques, even when the symmetries are approximate. [ABSTRACT FROM AUTHOR]

Approximating Bézier curve by polygon is a traditional technique in computer-aided geometric design, and there are several methods for constructing such polygons. In this paper, we propose to approximate Bézier curve by the least square polygon (LSP). LSP was derived by least square fitting which minimizes the integral of approximating errors. We provide a closed formula for the vertices of LSP. Because of its optimization method, LSP has lower approximation errors than existing approximating polygons. This observation is verified by many numerical examples we tested and several typical examples are included in this paper. [ABSTRACT FROM AUTHOR]

This paper proposes Brenke-type polynomials for uncertainty approximation in adaptive control of cooperative electrically driven manipulators. The contribution of this paper is reducing the number of signals needed in the construction of the regressor vector for uncertainty approximation due to the simplicity of Brenke-type polynomials in comparison with other alternatives. It is verified that uniformly bounded stability of the tracking/approximation errors can be assured if the Brenke-type polynomials are utilised as regressors. Moreover, it has been assumed that velocity signals are not available for feedback. Different trajectories, including simple and highly complicated and fast paths, have been tested in simulation results. The results are also compared with four state-of-the-art powerful approximation methods to show the accuracy and efficiency of the proposed controller structure. [ABSTRACT FROM AUTHOR]

The purpose of the present paper is to study a sequence of linear and positive operators which was introduced by A. Lupas. First, we obtain estimate of moments of the operators and then prove a basic convergence theorem for simultaneous approximation. Further, we find error in approximation in terms of modulus of continuity of function. Finally, we establish a Voronovskaja asymptotic formula for linear combination of the above operators. [ABSTRACT FROM AUTHOR]

This paper studies an adaptive fault tolerant control (AFTC) scheme for a continuum robot subjected to unknown actuator faults, dynamics uncertainties, unknown disturbances, and prescribed performance. Specifically, to deal with uncertainties, a function approximation technique (FAT) is employed to evaluate the unknown actuator faults and uncertain dynamics of the continuum robot. Then, a nonlinear disturbance observer (DO) is developed to estimate the unknown compounded disturbance, which contains the unknown disturbances and approximation errors of the FAT. Furthermore, the prescribed error bound is treated as a time-varying constraint, and the controller design method is based on an asymmetric barrier Lyapunov function (BLF), which is operated to strictly ensure the steady-state and transient performance of the continuum robot. Afterwards, the simulation results validate the effectiveness of the proposed AFTC in dealing with the unknown actuator faults, uncertainties, unknown disturbances, and prescribed performance. Finally, the effectiveness of the proposed AFTC scheme is verified through experiments. [ABSTRACT FROM AUTHOR]

In this paper, we propose a fast and accurate method for approximating the Minkowski sum of two rotational ellipsoids with a superellipsoid. The Minkowski sum is used in a variety of applications such as robot motion planning and particle flow simulation requiring collision detection. Many of them are computed based on Minkowski sum, whose accuracy and processing time depend on how the mesh is created. We approximate Minkowski sum with a superellipsoid function. The superellipsoid has various shapes with the value of the exponents, and the computation of the parameters including the exponents is an algebraic computation with the time complexity O (1) . As a result, approximation error is small and computation is fast in all combinations of rotational ellipsoids. In addition, collision is quickly detected by solving the inequality of superellipsoid. [ABSTRACT FROM AUTHOR]

This paper proposes a dead-reckoning (DR) method for vehicles using Lie theory. This approach treats the pose (position and attitude) and velocity of the vehicle as elements of the Lie group SE 2 (3) and follows the computations based on Lie theory. Previously employed DR methods, which have been widely used, suffer from cumulative errors over time due to inaccuracies in the calculated changes from velocity during the motion of the vehicle or small errors in modeling assumptions. Consequently, this results in significant discrepancies between the estimated and actual positions over time. However, by treating the pose and velocity of the vehicle as elements of the Lie group, the proposed method allows for accurate solutions without the errors introduced by linearization. The incremental updates for pose and velocity in the DR computation are represented in the Lie algebra. Experimental results confirm that the proposed method improves the accuracy of DR. In particular, as the motion prediction time interval of the vehicle increases, the proposed method demonstrates a more pronounced improvement in positional accuracy. [ABSTRACT FROM AUTHOR]

In this paper, the optimal approximation algorithm is proposed to simplify non-linear functions and/or discrete data as piecewise polynomials by using the constrained least squares. In time-sensitive applications or in embedded systems with limited resources, the runtime of the approximate function is as crucial as its accuracy. The proposed algorithm searches for the optimal piecewise polynomial (OPP) with the minimum computational cost while ensuring that the error is below a specified threshold. This was accomplished by using smooth piecewise polynomials with optimal order and numbers of intervals. The computational cost only depended on polynomial complexity, i.e., the order and the number of intervals at runtime function call. In previous studies, the user had to decide one or all of the orders and the number of intervals. In contrast, the OPP approximation algorithm determines both of them. For the optimal approximation, computational costs for all the possible combinations of piecewise polynomials were calculated and tabulated in ascending order for the specific target CPU off-line. Each combination was optimized through constrained least squares and the random selection method for the given sample points. Afterward, whether the approximation error was below the predetermined value was examined. When the error was permissible, the combination was selected as the optimal approximation, or the next combination was examined. To verify the performance, several representative functions were examined and analyzed. [ABSTRACT FROM AUTHOR]

Cellular thin-shell structures are widely applied in ultralightweight designs due to their high bearing capacity and strength-to-weight ratio. In this paper, a full-scale isogeometric topology optimization (ITO) method based on Kirchhoff-Love shells for designing cellular tshin-shell structures with excellent damage tolerance ability is proposed. This method utilizes high-order continuous nonuniform rational B-splines (NURBS) as basis functions for Kirchhoff-Love shell elements. The geometric and analysis models of thin shells are unified by isogeometric analysis (IGA) to avoid geometric approximation error and improve computational accuracy. The topological configurations of thin-shell structures are described by constructing the effective density field on the controlmesh. Local volume constraints are imposed in the proximity of each control point to obtain bone-like cellular structures. To facilitate numerical implementation, the p-norm function is used to aggregate local volume constraints into an equivalent global constraint. Several numerical examples are provided to demonstrate the effectiveness of the proposed method. After simulation and comparative analysis, the results indicate that the cellular thin-shell structures optimized by the proposed method exhibit great load-carrying behavior and high damage robustness. [ABSTRACT FROM AUTHOR]

This paper proposes a Generalized Multilevel B-spline Approximation (GMBA) method, which addresses scattered data interpolation problems in image processing. Mathematically, the GMBA provides a better solution for the B-spline control lattice by superimposing identical level B-splines compared with traditional Multilevel B-spline Approximation (MBA). Specifically, the GMBA allows the spacing of next control lattice to be subdivided arbitrarily or remained unchanged, which is determined by a predefined spacing set or the current error level. These improvements bring higher approximation accuracy and more flexibility for algorithm design to avoid over-fitting. In this paper, basic GMBA algorithm and its refined algorithm are compiled for image processing. Finally, six relevant cases are involved to test the GMBA, including surface approximation, image enlargement, image completion, and Salt-and-Pepper (SAP) noise removal. The experimental results show that the GMBA has better performance than the MBA in surface approximation and image processing, performs comparatively fast with the best performance on more than half of the standard test images compared with traditional algorithms, and has partially better performance even than deep learning algorithms. The GMBA can effectively recover meaningful details in images contaminated with even extremely high SAP noise level (up to 99%). • We give a generalized multilevel B-spline approximation (GMBA) method. • GMBA allows the control spacing to be subdivided arbitrarily or remained unchanged. • GMBA offers higher approximation accuracy and greater flexibility than traditional MBA. • GMBA provides superior performance and less run time than many of the state-of-the-art methods for SAP noise removal. • GMBA can recovery meaningful detail at noise levels as high as 99% for SAP noise removal. [ABSTRACT FROM AUTHOR]

This paper presents a numerical comparison of finite-element methods resulting in local mass conservation at the element level for Poisson's problem, namely the primal hybrid and mixed methods. These formulations result in an indefinite system. Alternative formulations yielding a positive-definite system are obtained after hybridizing each method. The choice of approximation spaces yields methods with enhanced accuracy for the pressure variable, and results in systems with identical size and structure after static condensation. A regular pressure precision mixed formulation is also considered based on the classical RT k space. The simulations are accelerated using Open multi-processing (OMP) and Thread-Building Blocks (TBB) multithreading paradigms alongside either a coloring strategy or atomic operations ensuring a thread-safe execution. An additional parallel strategy is developed using C++ threads, which is based on the producer-consumer paradigm, and uses locks and semaphores as synchronization primitives. Numerical tests show the optimal parallel strategy for these finite-element formulations, and the computational performance of the methods are compared in terms of simulation time and approximation errors. Additional results are developed during the process. Numerical solvers often fail to find an accurate solution to the highly indefinite systems arising from finite-element formulations, and this paper documents a matrix ordering strategy to stabilize the resolution. A procedure to enable static condensation based on the introduction of piecewise constant functions that also fulfills Neumann's compatibility condition, and yet computes an average pressure per element is presented. • Numerical comparison of the mixed, primal hybrid, and other locally conservatives finite element methods. • Introduction of a twice hybridized primal hybrid method, yielding a positive definite matrix after static condensation. • Comparison of multi-threaded strategies and thread-safety mechanisms. • A numerical procedure to stabilize the linear system resolution for indefinite matrices is described. • A general procedure that enables static condensation and fulfills Neumann's compatibility condition. [ABSTRACT FROM AUTHOR]

This paper proposes an algorithm to obtain the sampling factors to model any frequency partitioning so that it is realized using low complexity rational decimated non-uniform filter banks (RDNUFBs). The proposed algorithm is employed to approximate the Bark scale to find a rational frequency partitioning that can be realized using RDNUFBs with less approximation error. The proposed Bark frequency partitioning is found to reduce the average deviation in centre frequency and bandwidth by 65.04% and 48.50%, respectively, and root-mean-square bandwidth deviation by 54.46% when compared to those of the existing perceptual wavelet approximation of Bark scale. In the first filter bank design approach discussed in the paper, a near-perfect reconstruction partially cosine modulated filter bank is employed to obtain the improved Bark frequency partitioning. In the second design approach, the hardware complexity of the PCM-based RDNUFB is reduced by deriving the channels with different sampling factors from the same prototype filter by the method of channel merging. There is a considerable reduction of 40.83% in the number of multipliers when merging of partially cosine modulated channels is employed. It is also found that both the proposed approaches can reduce the delay by 95.17% when compared to the existing rational tree approximation methods and hence, are suitable for real-time applications. [ABSTRACT FROM AUTHOR]

This paper presents general theoretical studies on asymptotic convergence rate (ACR) for finite dimensional optimization. Given the continuous problem function and discrete time stochastic optimization process, the ACR is the optimal constant for control of the asymptotic behaviour of the expected approximation errors. Under general assumptions, condition ACR < 1 implies the linear behaviour of the expected time of hitting the ε - optimal sublevel set with ε → 0 + and determines the upper bound for the convergence rate of the trajectories of the process. This paper provides general characterization of ACR and, in particular, shows that some algorithms cannot converge linearly fast for any nontrivial continuous optimization problem. The relation between asymptotic convergence rate in the objective space and asymptotic convergence rate in the search space is provided. Examples and numerical simulations with use of a (1+1) self-adaptive evolution strategy and other algorithms are presented. [ABSTRACT FROM AUTHOR]

In this paper, we obtain an approximation for the distribution function of sum of two independent random variables using quantile based representation. The error of approximation is shown to be negligible under some mild conditions. We then use the approximation to obtain a non-parametric estimator for the distribution function of sum of two independent random variables. The exact distribution of the proposed estimator is derived. The estimator is shown to be strongly consistent and asymptotically normally distributed. Extensive Monte Carlo simulation studies are carried out to evaluate the bias and mean squared error of the estimator and also to assess the approximation error. We also compare the performance of the proposed estimator with other estimators available in the literature. Finally, we illustrate the use of the proposed estimator for estimating the reliability function of a standby redundant system. [ABSTRACT FROM AUTHOR]

In this paper, an adaptive bearing rigid formation control strategy for a class of nonlinear system with unknown dead-zone inputs and external disturbance is proposed. Firstly, the I-Type fuzzy system is used to approximate the unknown nonlinear dynamics of the formation model, and the approximation errors and unknown external disturbance are eliminated by the parameter adaptive estimation. Furthermore, the adaptive dynamic estimation algorithm is utilized to estimate and compensate the unknown dead-zone parameters, effectively suppressing the impact of dead-zone on formation system performance. Finally, the stability of the formation system is proved based on LaSalle's invariance principle, and the effectiveness of the algorithm is verified by simulation results. [ABSTRACT FROM AUTHOR]

Large-scale symmetric and asymmetric matrices have emerged in predicting the relationship between genes and diseases. The emergence of large-scale matrices increases the computational complexity of the problem. Therefore, using low-rank matrices instead of original symmetric and asymmetric matrices can greatly reduce computational complexity. In this paper, we propose an approximation conjugate gradient method for solving the low-rank matrix recovery problem, i.e., the low-rank matrix is obtained to replace the original symmetric and asymmetric matrices such that the approximation error is the smallest. The conjugate gradient search direction is given through matrix addition and matrix multiplication. The new conjugate gradient update parameter is given by the F-norm of matrix and the trace inner product of matrices. The conjugate gradient generated by the algorithm avoids SVD decomposition. The backtracking linear search is used so that the approximation conjugate gradient direction is computed only once, which ensures that the objective function decreases monotonically. The global convergence and local superlinear convergence of the algorithm are given. The numerical results are reported and show the effectiveness of the algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we develop a method for the estimation of airgap spatial variation from the harmonics that appear in the stator's current under static eccentricity conditions. We move beyond the detection of the eccentricity fault and endeavour to quantify the severity of this fault. To achieve this, we formulate the problem as a mathematical inverse problem and attempt to solve it in the Bayesian framework using the so-called Bayesian approximation error approach. The solution takes the form of an algorithm capable of estimating the parameter(s) that characterize the spatial variation of the airgap length from noisy and limited data. Furthermore, it can quantify the uncertainty associated with the results as a byproduct of the Bayesian analysis approach employed in this paper. The results demonstrate that with each iteration of the algorithm, estimation accuracy improves, and the uncertainty of the estimated results reduces. • Bayesian approximation error analysis shows promise in monitoring electrical machines with eccentric rotors. • The paper applies Bayesian inversion for online estimation of airgap length in induction machines with eccentric rotors. • The study estimates static eccentricity ratio from limited data on harmonics of the machine line currents. • This paper gives an analytical method for calculating all inductances of induction machines under static eccentricity. [ABSTRACT FROM AUTHOR]

In this paper, singularly perturbed pseudo-parabolic initial-boundary value problems with time-delay parameter are considered by numerically. Initially, the asymptotic properties of the analytical solution are investigated. Then, a discretization with exponential coefficient is suggested on a uniform mesh. The error approximations and uniform convergence of the presented method are estimated in the discrete energy norm. Finally, some numerical experiments are given to clarify the theory. [ABSTRACT FROM AUTHOR]

This paper presents application and tuning of the equilibrated residual method (ERM) of a posteriori error estimation for coupled electromechanical problems of direct, inverse and general piezoelectricity. In these three cases, either electric potential is induced by strains or strains appear due to the applied electric potential or both phenomena occur simultaneously. The mentioned ERM is assigned for the assessment of modeling and approximation errors of the numerical finite element solution. Such error values usually serve as indication for adaptive hierarchical modeling and adaptive mesh changes within thin and/or solid piezoelectric members so as to obtain the solution of assumed accuracy. [ABSTRACT FROM AUTHOR]

The purpose of Imaging Inverse problems are to recover original true signals/images from observations by plethora physical methods. The recovered signals/images are measured by appropriate quantity and quality measures. Given a specific physical process, imaging is classically a mathematical algorithm design to estimate the original signals/images. In this paper starting from inverse imaging problem of Electrical Impedance Tomography (EIT), we will address several aspects of loss landscape and its regularization properties of a neuron network solution for imaging inverse problems. Loss landscape and regularization strategy for imaging inverse problems is first introduced in I), and is followed by II) Sampling from unbounded landscape to bounded discrete. In III), an asymptotic approximation property when neuron network width goes to infinity is analyzed. Complexity of training and approximation error is discussed in IV). In V), case study for indicator function approximation and application in imaging EIT by neuron network is highlighted. Finally in VI), discussions will be devoted to the limitations of regularization capability of neuron network as a new tool, especially in medical imaging applications. [ABSTRACT FROM AUTHOR]

Uncompensated motion errors can seriously affect the imaging quality of synthetic aperture sonars (SASs). In the existing line-by-line motion compensation (MOCO) algorithms for wide-beam multiple-receiver SAS systems, the approximate form of the range history error usually introduces a significant approximation error, and the residual two-dimensional (2D) range cell migration (RCM) caused by aperture-dependent motion errors is not corrected, resulting in the severe defocus of the image. In this paper, in the presence of translational and rotational errors in a multiple-receiver SAS system, the exact range history error concerning the five-degree-of-freedom (DOF) motion errors of the sway, heave, yaw, pitch, and roll under the non-stop-hop-stop case is derived. Based on this, a two-stage subaperture MOCO algorithm for wide-beam multiple-receiver SAS systems is proposed. We decompose the range history error into the beam-center term (BCT) and the residual spatial-variant term (RSVT) to compensate successively. In the first stage, the time delay and phase error caused by the BCT are compensated receiver-by-receiver through interpolation and phase multiplication in the azimuth-time domain. In the second stage, the data of a single pulse are regarded as a subaperture, and the RSVT is compensated in the subaperture range-Doppler (RD) domain. We divide the range into several blocks to correct RCM caused by the RSVT in the subaperture RD domain, and the phase error caused by the RSVT is compensated by phase multiplication. After compensation, the wide-beam RD algorithm is used for imaging. Simulated and real-data experiments verify the superiority and robustness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

The paper presents a posteriori error approximation concept based on residuals in the two-dimensional discontinuous Galerkin (DG) method. The approach is relatively simple and effective in application, and it takes advantage of some unique properties of the DG method. The error function is constructed in an enriched approximation space, utilizing the hierarchical nature of the basis functions. Among many versions of the DG method, the most popular one is based on the interior penalty approach. However, in this paper a DG method with finite difference (DGFD) is utilized, where the continuity of the approximate solution is enforced by finite difference conditions applied on the mesh skeleton. In the DG methods arbitrarily shaped finite elements can be used, so in this paper the meshes with polygonal finite elements are considered, including quadrilateral and triangular elements. Some benchmark examples are presented, in which Poisson's and linear elasticity problems are considered. The examples use various mesh densities and approximation orders to evaluate the errors. The error estimation maps, generated for the discussed tests, indicate a good correlation with the exact errors. In the last example, the error approximation concept is applied for an adaptive hp mesh refinement. [ABSTRACT FROM AUTHOR]

It is well known that the Kolmogorov complexity function (the minimal length of a program producing a given string, when an optimal programming language is used) is not computable and, moreover, does not have computable lower bounds. In this paper we investigate a more general question: can this function be approximated? By approximation we mean two things: firstly, some (small) difference between the values of the complexity function and its approximation is allowed; secondly, at some (rare) points the values of the approximating function may be arbitrary. For some values of the parameters such approximation is trivial (e.g., the length function is an approximation with error d except for a O (2 − d) fraction of inputs). However, if we require a significantly better approximation, the approximation problem becomes hard, and we prove it in several settings. Firstly, we show that a finite table that provides good approximations for Kolmogorov complexities of n-bit strings, necessarily has high complexity. Secondly, we show that there is no good computable approximation for Kolmogorov complexity of all strings. In particular, Kolmogorov complexity function is neither generically nor coarsely computable, as well as its approximations, and the time-bounded Kolmogorov complexity (for any computable time bound) deviates significantly from the unbounded complexity function. We also prove hardness of Kolmogorov complexity approximation in another setting: the mass problem whose solutions are good approximations for Kolmogorov complexity function is above the halting problem in the Medvedev lattice. Finally, we mention some proof-theoretic counterparts of these results. A preliminary version of this paper was presented at CiE 2019 conference (In Computing with Foresight and Industry – 15th Conference on Computability in Europe, CiE 2019, Durham, UK, July 15–19, 2019, Proceedings (2019) 230–239 Springer). [ABSTRACT FROM AUTHOR]

In this paper, the composite control of position and orientation of the quadrotor for tracking purposes are investigated. The quadrotor is an underactuated nonlinear system. So, based on the introduced virtual inputs the quadrotor is transformed into an actuated system. External disturbances and different sources of uncertainty such as input saturation are considered in the dynamic model of the quadrotor. Also, after transform implementation, some complicated terms appear which are considered as other sources of uncertainty. To guarantee the stabilization and tracking goals of the quadrotor in the perturbed model, the composite adaptive fuzzy PID-Like controller is proposed. This controller consists of three parts where the innovation of the paper appears in all terms. The main term is an adaptive fuzzy PID-like controller which able to approximate the coefficients of PID based on the decentralized fuzzy logic. Then, a H∞ compensator based on a modified Riccati-like equation is presented; which can attenuate the fuzzy approximation errors. Also, as the last term of the composite controller, the modified disturbance observer is designed that estimates the uncertainty and disturbance. The experimental result shows that the proposed method leads to resilient stabilization and tracking in comparison with conventional methods. [ABSTRACT FROM AUTHOR]

The paper refers to interdisciplinary research in the areas of hierarchical cluster analysis of big data and ordering of primary data to detect objects in a color or in a grayscale image. To perform this on a limited domain of multidimensional data, an NP-hard problem of calculation of close to optimal piecewise constant data approximations with the smallest possible standard deviations or total squared errors (approximation errors) is solved. The solution is achieved by revisiting, modernizing, and combining classical Ward's clustering, split/merge, and K-means methods. The concepts of objects, images, and their elements (superpixels) are formalized as structures that are distinguishable from each other. The results of structuring and ordering the image data are presented to the user in two ways, as tabulated approximations of the image showing the available object hierarchies. For not only theoretical reasoning, but also for practical implementation, reversible calculations with pixel sets are performed easily, as with individual pixels in terms of Sleator–Tarjan Dynamic trees and cyclic graphs forming an Algebraic Multi-Layer Network (AMN). The detailing of the latter significantly distinguishes this paper from our prior works. The establishment of the invariance of detected objects with respect to changing the context of the image and its transformation into grayscale is also new. [ABSTRACT FROM AUTHOR]

In this paper, a new criterion for the evaluation of the theoretical efficiency of a greedy algorithm is suggested. Using this criterion, we prove some results on the rate of convergence of greedy algorithms, which provide expansions. We consider both the case of Hilbert spaces and the more general case of Banach spaces. The new component of this paper is that we bound the error of approximation by the product of two norms—the norm of f and the A 1 -norm of f. Typically, only the A 1 -norm of f is used. In particular, we establish that some greedy algorithms (Pure Greedy Algorithm (PGA) and its modifications) are as good as the Orthogonal Greedy Algorithm (OGA) in this new sense of the rate of convergence, while it is known that the PGA is much worse than the OGA in the standard sense. Our new results provide better bounds for the accuracy than known results in the case of small ∥ f ∥ . [ABSTRACT FROM AUTHOR]