Bessel polynomial solutions of high-order linear Volterra integro-differential equations

In this study, a practical matrix method, which is based on collocation points, is presented to find approximate solutions of high-order linear Volterra integro-differential equations (VIDEs) under the mixed conditions in terms of Bessel polynomials. Numerical examples are included to demonstrate the validity and applicability of the technique and comparisons are made with the existing results. The results show the efficiency and accuracy of the present work. All of the numerical computations have been performed on the computer using a program written in MATLAB v7.6.0 (R2008a).     

Numerical computation of sign-indefinite linear quadratic differential games for weakly coupled large-scale systems

In this paper, N-player linear quadratic differential games that are sign-indefinite for infinite horizon weakly coupled large-scale systems are discussed. After establishing the asymptotic structure and local uniqueness of the solution for cross-coupled sign-indefinite algebraic Riccati equations (CSARE), a new algorithm for solving CSARE is provided. It is shown that the proposed algorithm attains linear convergence. Moreover, in order to reduce the computational workspace, the recursive algorithm is combined. Finally, a high-order approximation strategy based on the proposed iterative solutions is described. As a result, it was recently proved that the numerical strategy achieves a high-order approximation of the equilibrium value. As another important feature, when the small parameters are unknown, a parameter-independent strategy is developed.     

Automated low-order to high-order mesh conversion

An algorithm is presented for the CAD-free conversion of linear unstructured meshes into curved high-order meshes, which are necessary for high-order flow simulations. The algorithm operates via three steps: (1) autonomous detection of feature curves along the mesh surface, (2) reconstruction of the surface curvature from the combination of surface node positions and feature curve positions, and (3) alignment of the mesh interior to the newly curved surface. The algorithm is implemented in our freely available cross-platform graphical software program meshCurve, which transforms existing linear meshes into high-order curved meshes

     

A highly accurate alternating 6-point group method for the dispersive equation

In this paper, we construct a group of Saul'yev type asymmetric difference formulas for the dispersive equation. Based on these formulas we derive a new alternating 6-point group algorithm to solve dispersive equations with periodic boundary conditions. The algorithm has a high-order accuracy in space and an unconditional stability. The theoretical results are conformed to the numerical simulation. A comparison of this algorithm with the previous Alternating Group Explicit method is presented.     

A numerical approach for solving the high-order linear singular differential-difference equations

In this paper, a numerical method which produces an approximate polynomial solution is presented for solving the high-order linear singular differential-difference equations. With the aid of Bessel polynomials and collocation points, this method converts the singular differential-difference equations into the matrix equation. The matrix equation corresponds to a system of linear equations with the unknown Bessel coefficients. This method gives the analytic solutions when the exact solutions are polynomials. Finally, some experiments and their numerical solutions are given; by comparing the numerical results obtained from the other methods, we show the high accuracy and efficiency of the proposed method. All of the numerical computations have been performed on a PC using some programs written in MATLAB v7.6.0 (R2008a).     

非线性系统高阶迭代学习算法

结合迭代学习控制算法中的开环和闭环方案，本文针对更一般的非线性系统，讨论高阶算法的广泛适用性。理论和仿真结果表明了高阶算法在输出跟踪和干扰抑制方面的有效性。     

Electroosmotic flow control in complex microgeometries

Numerical simulation results for pure electroosmotic and combined electroosmotic/pressure driven Stokes flows are presented in the cross-flow and Y-split junctions. The numerical algorithm is based on a mixed structured/unstructured spectral element formulation, which results in high-order accurate resolution of thin electric double layers with discretization flexibility for complex engineering geometries. The results for pure electroosmotic flows in cross-flow junctions under multiple electric fields show similarities between the electric and velocity fields. The combined electroosmotic/pressure driven flows are also simulated by regulating the flowrate in different branches of the cross-flow junctions. Flow control in the Stokes flow regime is shown to have linear dependence on the magnitude of the externally applied electric field, both for pure electroosmotic and combined flows. This linear behavior enables utilization of electroosmotic forces as nonmechanical means of flow control for microfluidic applications     

高阶参数优化迭代学习控制算法

针对线性时不变离散系统的跟踪问题提出一种高阶参数优化迭代学习控制算法.该算法通过建立考虑了多次迭代误差影响的参数优化目标函数,求解得出优化后的时变学习增益参数.从理论上证明了:对于线性离散时不变系统,该算法在被控对象不满足正定性的松弛条件下仍可保证跟踪误差单调收敛于零.同时,采用之前多次迭代信息的高阶算法具有更好的收敛性和鲁棒性.最后利用一个仿真实例验证了算法的有效性.     

基于高阶结构约束的稀疏人脸识别算法

为了挖掘人脸稀疏表示的高阶结构信息,从而获得更精确的人脸分类判别信息,提出一种新的基于高阶结构约束的稀疏表示的人脸识别算法。在提取人脸图像的Gabor特征后,选取适当的训练样本构建稀疏线性编码模型,利用样本特征间的高阶结构信息约束和最优化求解测试样本的稀疏系数,将稀疏系数作为最终特征对人脸进行识别,实验结果表明,与结构约束稀疏表示和非结构约束稀疏表示的方法比较,人脸的高阶结构稀疏表示能显著提高人脸的识别准确率。     

Numerical solutions of systems of linear Fredholm integro-differential equations with Bessel polynomial bases

In this paper, a numerical matrix method based on collocation points is presented for the approximate solution of the systems of high-order linear Fredholm integro-differential equations with variable coefficients under the mixed conditions in terms of the Bessel polynomials. Numerical examples are included to demonstrate the validity and the applicability of the technique and also the results are compared with the different methods. The results show the efficiently and the accuracy of the present work. All of the numerical computations have been performed on a PC using some programs written in MATLAB v7.6.0 (R2008a).     

Stability criteria of high-order delay differential systems

ABSTRACT

In this paper, the delay-dependent stability of linear, high-order delay differential systems is investigated. First, two bounds of the unstable eigenvalues of the systems are derived. The two bounds are based on the spectral radius and the norms of the parameter matrices of the systems, respectively. We emphasise that the bounds of the unstable eigenvalues involve only the spectral radius and norms of the matrices of lower size. They can be obtained with much less computational effort and work well in practice for large problems. Then, using the argument principle, a computable stability criterion is presented which is a necessary and sufficient condition for the delay-dependent stability of the systems. Furthermore, a numerical algorithm is provided for checking the delay-dependent stability of the systems. Numerical examples are given to illustrate the main results.     

线性连续系统的仿真与解析算法

本文提出了在一类典型函数输入激励下,线性定常系统的快速仿真算法以及一种求取系 统解析解的算法.快速仿真算法具有高精度、高数值稳定性的特点,可在其他算法无法收敛的 步长下进行仿真;并对线性系统"自治化"的状态增广方法及对高阶微分方程描述系统的初值 变换算法.系统解析解算法可处理含有复特征根的情形.本文提出的各种算法均在CAAD 成果中实现[3],使用极为方便.     

Stability of BDF methods for nonlinear volterra integral equations with delay

This paper deals with the stability analysis of the backward differential formulas (or BDF methods) for nonlinear Volterra integral equations with delay (VIDEs). The presented approach is based on a nonclassical Lipschitz condition. In particular, the criteria on the global and the asymptotic stability of the methods are given.     

含仿射不确定参数的多项式非线性系统控制器设计

针对含仿射时变不确定参数的多项式非线性系统，提出了基于多项式分解的控制方法. 多项式分解方法主要思想是将多项式系统转化成带自由变量的系数矩阵，从而偶次多项式的非负性验证问题可转化成线性矩阵不等式或双线性矩阵不等式求解问题. 文中多项式系统控制器综合基于 Lyapunov 稳定定理. 构造 Lyapunov 函数以及寻找反馈控制器可由所给的算法通过计算机程序自动完成. 对于多维系统相对高阶的控制器，由多项式全基构造的控制器将有很多项单项式. 为克服这一问题，文中算法给出含最少单项式的简约型控制器设计方法，并提出针对最小代价性能目标优化的增益受约次优控制. 数值仿真例子表明，文中所给的控制方法取得良好性能.     

Distributed Containment Control of Multi-agent Systems with General Linear Dynamics and Time-delays

Containment control problems for high-order linear time-invariant multi-agent systems with fixed communication time-delays are investigated. Based on Lyapunov-Krasovskii functional method and the linear matrix inequality (LMI) method, sufcient conditions on the communication digraph, the feedback gains, and the allowed upper bound of the delays to ensure containment control of the multi-agent systems under the different containment control algorithms are given. Finally, numerical simulations are presented to demonstrate theoretical results.     

含仿射不确定参数的多项式非线性系统控制器设计

针对含仿射时变不确定参数的多项式非线性系统，提出了基于多项式分解的控制方法. 多项式分解方法主要思想是将多项式系统转化成带自由变量的系数矩阵，从而偶次多项式的非负性验证问题可转化成线性矩阵不等式或双线性矩阵不等式求解问题. 文中多项式系统控制器综合基于 Lyapunov 稳定定理. 构造 Lyapunov 函数以及寻找反馈控制器可由所给的算法通过计算机程序自动完成. 对于多维系统相对高阶的控制器，由多项式全基构造的控制器将有很多项单项式. 为克服这一问题，文中算法给出含最少单项式的简约型控制器设计方法，并提出针对最小代价性能目标优化的增益受约次优控制. 数值仿真例子表明，文中所给的控制方法取得良好性能.     

离散时间高阶不确定线性多个体系统保性能一致性分析

本文研究了存在参数不确定性的离散时间高阶多个体系统保性能一致性问题,给出了一种设计其线性一致性协议的方法.首先,通过模型转换的方法将该问题转换为一组离散时间不确定系统的稳定性问题;然后,构造合适的Lyapunov函数并利用离散时间系统稳定性理论,推导出一个使离散时间高阶不确定多个体系统获得保性能一致的LMI充分条件;接着,以一致性序列的形式给出参数不确定条件下的离散时间高阶多个体系统的一致性收敛结果.最后,参数不确定的对比数值仿真验证了本文理论的正确性和有效性.     

An algorithm for state-space frequency domain identificationwithout windowing distortions

An algorithm is presented for identifying state-space models from frequency-domain data. The main advantage of this approach is that it avoids windowing distortions associated with other frequency-domain methods. Other advantages are that an arbitrary frequency weighting can be introduced to shape the estimation error, and the system order can be overspecified. The numerical properties are demonstrated on real physical data taken on a complex flexible structure, leading to the successful identification of a multivariable (four-input/three-output) 100-state model over a bandwidth of 100 Hertz. The results indicate that the algorithm would be useful in applications requiring the accurate identification of high-order systems over wide bandwidths     

Output containment control for swarm systems with general linear dynamics: A dynamic output feedback approach

Output containment control problems for high-order linear time-invariant swarm systems under directed interaction topologies are investigated using a dynamic output feedback approach. Firstly, to propel the outputs of followers to converge to the convex hull formed by the outputs of leaders, a dynamic output containment protocol is presented. Then necessary and sufficient conditions for swarm systems to achieve output containment are proposed. To ensure the scalability of the criteria, a sufficient condition which only includes two linear matrix inequality constraints independent of the number of agents is further presented. Moreover, an approach independent of the number of agents is proposed to determine the gain matrices in the dynamic output containment protocols. Finally, numerical simulations are presented to demonstrate theoretical results.     

A Spectral Finite Element Approach to Modeling Soft Solids Excited with High-Frequency Harmonic Loads

An approach for efficient and accurate finite element analysis of harmonically excited soft solids using high-order spectral finite elements is presented and evaluated. The Helmholtz-type equations used to model such systems suffer from additional numerical error known as pollution when excitation frequency becomes high relative to stiffness (i.e. high wave number), which is the case, for example, for soft tissues subject to ultrasound excitations. The use of high-order polynomial elements allows for a reduction in this pollution error, but requires additional consideration to counteract Runge's phenomenon and/or poor linear system conditioning, which has led to the use of spectral element approaches. This work examines in detail the computational benefits and practical applicability of high-order spectral elements for such problems. The spectral elements examined are tensor product elements (i.e. quad or brick elements) of high-order Lagrangian polynomials with non-uniformly distributed Gauss-Lobatto-Legendre nodal points. A shear plane wave example is presented to show the dependence of the accuracy and computational expense of high-order elements on wave number. Then, a convergence study for a viscoelastic acoustic-structure interaction finite element model of an actual ultrasound driven vibroacoustic experiment is shown. The number of degrees of freedom required for a given accuracy level was found to consistently decrease with increasing element order. However, the computationally optimal element order was found to strongly depend on the wave number.