SDP-based approximation of stabilising solutions for periodic matrix Riccati differential equations

Numerically finding stabilising feedback control laws for linear systems of periodic differential equations is a nontrivial task with no known reliable solutions. The most successful method requires solving matrix differential Riccati equations with periodic coefficients. All previously proposed techniques for solving such equations involve numerical integration of unstable differential equations and consequently fail whenever the period is too large or the coefficients vary too much. Here, a new method for numerical computation of stabilising solutions for matrix differential Riccati equations with periodic coefficients is proposed. Our approach does not involve numerical solution of any differential equations. The approximation for a stabilising solution is found in the form of a trigonometric polynomial, matrix coefficients of which are found solving a specially constructed finite-dimensional semidefinite programming (SDP) problem. This problem is obtained using maximality property of the stabilising solution of the Riccati equation for the associated Riccati inequality and sampling technique. Our previously published numerical comparisons with other methods shows that for a class of problems only this technique provides a working solution. Asymptotic convergence of the computed approximations to the stabilising solution is proved below under the assumption that certain combinations of the key parameters are sufficiently large. Although the rate of convergence is not analysed, it appeared to be exponential in our numerical studies.     

Some sets of orthogonal polynomial kernel functions

Kernel methods provide high performance in a variety of machine learning tasks. However, the success of kernel methods is heavily dependent on the selection of the right kernel function and proper setting of its parameters. Several sets of kernel functions based on orthogonal polynomials have been proposed recently. Besides their good performance in the error rate, these kernel functions have only one parameter chosen from a small set of integers, and it facilitates kernel selection greatly. Two sets of orthogonal polynomial kernel functions, namely the triangularly modified Chebyshev kernels and the triangularly modified Legendre kernels, are proposed in this study. Furthermore, we compare the construction methods of some orthogonal polynomial kernels and highlight the similarities and differences among them. Experiments on 32 data sets are performed for better illustration and comparison of these kernel functions in classification and regression scenarios. In general, there is difference among these orthogonal polynomial kernels in terms of accuracy, and most orthogonal polynomial kernels can match the commonly used kernels, such as the polynomial kernel, the Gaussian kernel and the wavelet kernel. Compared with these universal kernels, the orthogonal polynomial kernels each have a unique easily optimized parameter, and they store statistically significantly less support vectors in support vector classification. New presented kernels can obtain better generalization performance both for classification tasks and regression tasks.     

基于提升小波和自适应多项式拟合的传感器网络多模数据压缩算法

为提高无线传感器网络的感知精度,提出了一种基于提升小波变换和自适应多项式拟合的多模数据压缩算法（adaptive multiple-modalities data compression algorithm based on lifting wavelet and adaptive polynomial fitting,简称AMLP）。在给定相关度阈值的前提下,AMLP算法先对数据进行灰色关联聚类,再对类中的相关数据进行自适应的多项式拟合,然后把未拟合的特征数据抽象成一个矩阵,利用提升小波变换去除数据的时间和空间相关性。最后,通过游程编码对数据作进一步压缩。仿真结果表明,AMLP算法能够有效去除不同数据间的冗余信息以及同种数据间的时间和空间冗余信息,提高压缩比,降低网络能耗。与基于小波的自适应多模数据压缩算法（adaptive multiple-modalities data compression algorithm based on wavelet,简称AMMC）相比,AMLP算法的数据恢复精度大大优于AMMC算法,压缩比和能耗相近。因此,AMLP算法更适用于要求高精度数据的传感器网络应用,如地质灾害监测、医疗和军事领域。     

Cyclic codes over R = Fp + uFp +?+ uFp with length pn

In this paper, all cyclic codes with length p^sn, (n prime to p) over the ring R = F_p + uF_p +?+ u^k−1F_p are classified. It is first proved that Tor_j(C) is an ideal of , so that the structure of ideals over extension ring S_u^k(m,ω)=GR(u^k,m)[ω]/〈ω^ps-1〉 is determined. Then, an isomorphism between R[X]/〈X^N − 1〉 and a direct sum ⊕_h∈IS_u^k(m_h,ω) can be obtained using discrete Fourier transform. The generator polynomial representation of the corresponding ideals over F_p + uF_p +?+ u^k−1F_p is calculated via the inverse isomorphism. Moreover, torsion codes, MS polynomial and inversion formula are described.     

New coprimeness over multivariable polynomial matrices and its application to control delay systems

This paper presents a new notion of coprimeness over multivariable polynomial matrices, where a single variable is given priority over the remaining variables. From a characterization through a set of common zeros of the minors, it is clarified that the presented coprimeness is equivalent to weakly zero coprimeness in the particular variable. An application of the presented coprimeness to control systems with non-commensurate delays and finite spectrum assignment is also presented. Because the presented coprimeness is stronger than minor coprimeness, non-commensurate delays are difficult to deal with in algebraic control theory. The “rational ratio condition” between delays, which can reduce non-commensurate delays to commensurate delays, proves to be both powerful and practical concept in algebraic control theory for delay systems.     

基于DPT的非线性调频信号的DOA估计

提出了一种基于DPT的宽带非线性调频信号的DOA估计算法.首先将非线性调频(NLFM)信号建模为高阶多项式相位信号(PPS)模型,然后通过高阶瞬时矩进行多项式相位变换.接收信号将变换为单个正弦信号和新的噪声.再利用ROOT-MUSIC或者ESPRIT算法对变换得到的正弦信号的波达方向进行估计.理论分析和仿真结果表明,该...     

Computing monodromy via continuation methods on random Riemann surfaces

We consider a Riemann surface X defined by a polynomial f(x,y) of degree d, whose coefficients are chosen randomly. Hence, we can suppose that X is smooth, that the discriminant δ(x) of f has d(d−1) simple roots, Δ, and that δ(0)≠0, i.e. the corresponding fiber has d distinct points {y₁,…,y_d}. When we lift a loop 0∈γ⊂C−Δ by a continuation method, we get d paths in X connecting {y₁,…,y_d}, hence defining a permutation of that set. This is called monodromy.Here we present experimentations in Maple to get statistics on the distribution of transpositions corresponding to loops around each point of Δ. Multiplying families of “neighbor” transpositions, we construct permutations and the subgroups of the symmetric group they generate. This allows us to establish and study experimentally two conjectures on the distribution of these transpositions and on transitivity of the generated subgroups.Assuming that these two conjectures are true, we develop tools allowing fast probabilistic algorithms for absolute multivariate polynomial factorization, under the hypothesis that the factors behave like random polynomials whose coefficients follow uniform distributions.     

Dynamic normal forms and dynamic characteristic polynomial

We present the first fully dynamic algorithm for computing the characteristic polynomial of a matrix. In the generic symmetric case, our algorithm supports rank-one updates in O(n²logn) randomized time and queries in constant time, whereas in the general case the algorithm works in O(n²klogn) randomized time, where k is the number of invariant factors of the matrix. The algorithm is based on the first dynamic algorithm for computing normal forms of a matrix such as the Frobenius normal form or the tridiagonal symmetric form. The algorithm can be extended to solve the matrix eigenproblem with relative error 2^−b in additional O(nlog²nlogb) time. Furthermore, it can be used to dynamically maintain the singular value decomposition (SVD) of a generic matrix. Together with the algorithm, the hardness of the problem is studied. For the symmetric case, we present an Ω(n²) lower bound for rank-one updates and an Ω(n) lower bound for element updates.     

Sparse interpolation of multivariate rational functions

Consider the black box interpolation of a τ-sparse, n-variate rational function f, where τ is the maximum number of terms in either numerator or denominator. When numerator and denominator are at most of degree d, then the number of possible terms in f is O(dⁿ) and explodes exponentially as the number of variables increases. The complexity of our sparse rational interpolation algorithm does not depend exponentially on n anymore. It still depends on d because we densely interpolate univariate auxiliary rational functions of the same degree. We remove the exponent n and introduce the sparsity τ in the complexity by reconstructing the auxiliary function’s coefficients via sparse multivariate interpolation.The approach is new and builds on the normalization of the rational function’s representation. Our method can be combined with probabilistic and deterministic components from sparse polynomial black box interpolation to suit either an exact or a finite precision computational environment. The latter is illustrated with several examples, running from exact finite field arithmetic to noisy floating point evaluations. In general, the performance of our sparse rational black box interpolation depends on the choice of the employed sparse polynomial black box interpolation. If the early termination Ben-Or/Tiwari algorithm is used, our method achieves rational interpolation in O(τd) black box evaluations and thus is sensitive to the sparsity of the multivariate f.