Fully Petrov–Galerkin spectral method for the distributed-order time-fractional fourth-order partial differential equation

Distributed fractional derivative operators can be used for modeling of complex multiscaling anomalous transport, where derivative orders are distributed over a range of values rather than being just a fixed integer number. In this paper, we consider the space-time Petrov–Galerkin spectral method for a two-dimensional distributed-order time-fractional fourth-order partial differential equation. By applying a proper Gauss-quadrature rule to discretize the distributed integral operator, the problem is converted to a multi-term time-fractional equation. Then, the proposed method for solving the obtained equation is based on using Jacobi polyfractonomial, which are eigenfunctions of the first kind fractional Sturm–Liouville problem (FSLP), as temporal basis and Legendre polynomials for the spatial discretization. The eigenfunctions of the second kind FSLP are used as temporal basis in test space. This approach leads to finding the numerical solution of the problem through solving a system of linear algebraic equations. Finally, we provide some examples with smooth solutions and finite regular solutions to numerically demonstrate the efficiency, accuracy, and exponential convergence of the proposed method.

     

Introduction

The Journal of Supercomputing -     

Robust non-aggressive three-axis attitude control of spacecraft: dynamic sliding mode approach

Conventional sliding mode control (SMC) has been extensively applied in controlling spacecrafts because of its appealing characteristics such as robustness and a simple design procedure. Several methods such as second-order sliding modes and discontinuous controllers are applied for the SMC implementation. However, the main problems of these methods are convergence and error tracking in a finite amount of time. This paper combines an improved dynamic sliding mode controller and model predictive controller for spacecrafts to solve the chattering phenomenon in traditional sliding mode control. To this aim, this paper develops dynamic sliding mode control for spacecraft’s applications to omit the chattering issue. The proposed approach shows robust attitude tracking by a set of reaction wheels and stabilizes the spacecraft subject to disturbances and uncertainties. The proposed method improves the performance of the SMC for spacecraft by avoiding chattering. A set of simulation results are provided that show the advantages and improvements of this approach (in some sense) compared to SMC approaches.     

Using hash tables to manage the time-storage complexity in a point location problem: Application to explicit model predictive control

The online computational burden of linear model predictive control (MPC) can be moved offline by using multi-parametric programming, so-called explicit MPC. The solution to the explicit MPC problem is a piecewise affine (PWA) state feedback function defined over a polyhedral subdivision of the set of feasible states. The online evaluation of such a control law needs to determine the polyhedral region in which the current state lies. This procedure is called point location; its computational complexity is challenging, and determines the minimum possible sampling time of the system. A new flexible algorithm is proposed which enables the designer to trade off between time and storage complexities. Utilizing the concept of hash tables and the associated hash functions, the proposed method solves an aggregated point location problem that overcomes prohibitive complexity growth with the number of polyhedral regions, while the storage–processing trade-off can be optimized via scaling parameters. The flexibility and power of this approach is supported by several numerical examples.     

PBSVM: Partitioning and biased support vector machine for vocal fold pathology assessment using labeled and unlabeled data sets

Most of the existing classification methods, used for voice pathology assessment, are built based on labeled pathological and normal voice signals. This paper studies the problem of building a classifier using labeled and unlabeled data. We propose a novel learning technique, called Partitioning and Biased Support Vector Machine Classification (PBSVM), which tries to utilize all the available data in two steps: (1) a new heuristically partition-based algorithm, which extracts high quality pathological and normal samples from an unlabeled set, and (2) a more principle approach based on biased formulation of support vector machine, which is fairly robust to mislabeling and unbalance data problem. Experiments with wavelet-based energy features extracted from sustained vowels show that the new recognition scheme is highly feasible and significantly outperform the baseline classical SVM classifier, especially in the situation where the labeled training data is small.     

The voting analytic hierarchy process method for discriminating among efficient decision making units in data envelopment analysis

Making optimal use of available resources has always been of interest to humankind, and different approaches have been used in an attempt to make maximum use of existing resources. Limitations of capital, manpower, energy, etc., have led managers to seek ways for optimally using such resources. In fact, being informed of the performance of the units under the supervision of a manager is the most important task with regard to making sensible decisions for managing them. Data envelopment analysis (DEA) suggests an appropriate method for evaluating the efficiency of homogeneous units with multiple inputs and multiple outputs. DEA models classify decision making units (DMUs) into efficient and inefficient ones. However, in most cases, managers and researchers are interested in ranking the units and selecting the best DMU. Various scientific models have been proposed by researchers for ranking DMUs. Each of these models has some weakness(es), which makes it difficult to select the appropriate ranking model. This paper presents a method for ranking efficient DMUs by the voting analytic hierarchy process (VAHP). The paper reviews some ranking models in DEA and discusses their strengths and weaknesses. Then, we provide the method for ranking efficient DMUs by VAHP. Finally we give an example to illustrate our approach and then the new method is employed to rank efficient units in a real world problem.     

Comparison of two tools for the measurement of interfragmentary movement in femoral neck fractures stabilised by cannulated screws

Achieving stability at the site of femoral neck fracture is an important factor for callus formation in the post-operative period. However, measuring interfragmentary movement in vivo is not currently possible as telemetric screws have not been manufactured for surgical use. Understanding how the implantation of the screws can affect the stability of the fracture allows the surgeon to tailor the procedure to the patient and produce the best possible outcome. Two techniques have been developed that measure interfragmentary movement between fractured surfaces. The first was a FEA model of the proximal femur with screws represented by nodal links. Movement was quantified by the amount of relative motion occurring between paired nodes either side of the fracture. The second was a mechanical compression test of a composite femur that allowed the motion analysis of paired markers on the external surface of the femur. Movement was digitised with markers selected and displacements calculated by transforming the global coordinate system to a local system relative to the fracture plane.     

Multiple robust H∞ controller design using the nonsmooth optimization method

This paper proposes a systematic technique to design multiple robust H_∞ controllers. The proposed technique achieves a desired robust performance objective, which is impossible to achieve with a single robust controller, by dividing the uncertainty set into several subsets and by designing a robust controller to each subset. To achieve this goal with a small number of divisions of the uncertainty set, an optimization problem is formulated. Since the cost function of this optimization problem is not a smooth function, a numerical nonsmooth optimization algorithm is proposed to solve this problem. This method avoids the use of Lyapunov variables, and therefore it leads to a moderate size optimization problem. A numerical example shows that the proposed multiple robust control method can improve the closed‐loop performance when a single robust controller cannot achieve satisfactory performance. Copyright © 2009 John Wiley & Sons, Ltd.