Optimization-based computation of analytic interpolants of bounded complexity

This paper provides a unifying algorithm for computing any analytic interpolant of bounded complexity. Such computation can be performed by solving an optimization problem, due to a theorem by Georgiou and Lindquist. This optimization problem is numerically solvable by a continuation method. The proposed numerical algorithm is useful, among other cases, for designing a low-degree controller for a benchmark problem in robust control. The algorithm unifies previously developed algorithms for the Carathéodory extension and the Nevanlinna–Pick interpolation to one for more general interpolation problems.     

On the solutions of the rational covariance extension problem corresponding to pseudopolynomials having boundary zeros

In this note, we study the rational covariance extension problem with degree bound when the chosen pseudopolynomial of degree at most n has zeros on the boundary of the unit circle and derive some new theoretical results for this special case. In particular, a necessary and sufficient condition for a solution to be bounded (i.e., has no poles on the unit circle) is established. Our approach is based on convex optimization, similar in spirit to the recent development of a theory of generalized interpolation with a complexity constraint. However, the two treatments do not proceed in the same way and there are important differences between them which we discuss herein. An implication of our results is that bounded solutions can be computed via methods that have been developed for pseudopolynomials which are free of zeros on the boundary, extending the utility of those methods. Numerical examples are provided for illustration.     

Spectral estimation by least-squares optimization based on rational covariance extension

This paper proposes a new spectral estimation technique based on rational covariance extension with degree constraint. The technique finds a rational spectral density function that approximates given spectral density data under constraint on a covariance sequence. Spectral density approximation problems are formulated as nonconvex optimization problems with respect to a Schur polynomial. To formulate the approximation problems, the least-squares sum is considered as a distance. Properties of optimization problems and numerical algorithms to solve them are explained. Numerical examples illustrate how the methods discussed in this paper are useful in stochastic model reduction and stochastic process modeling.     

On the partial stochastic realization problem

We describe a complete parameterization of the solutions to the partial stochastic realization problem in terms of a nonstandard matrix Riccati equation. Our analysis of this covariance extension equation (CEE) is based on a complete parameterization of all strictly positive real solutions to the rational covariance extension problem, answering a conjecture due to Georgiou (1987) in the affirmative. We also compute the dimension of partial stochastic realizations in terms of the rank of the unique positive semidefinite solution to the CEE, yielding some insights into the structure of solutions to the minimal partial stochastic realization problem. By combining this parameterization with some of the classical approaches in partial realization theory, we are able to derive new existence and robustness results concerning the degrees of minimal stochastic partial realizations. As a corollary to these results, we note that, in sharp contrast with the deterministic case, there is no generic value of the degree of a minimal stochastic realization of partial covariance sequences of fixed length     

Directional interpolation in the state space

This paper presents a new state-space characterization of the directional interpolation based on the observation that the generalized Pick matrix is regarded as a solution of a Lyapunov-type equation. A simple representation of the set of all solutions is derived in terms of linear fractional transformations. The computation of the associated J-unitary matrix is simpler than the well-known procedure based on the Glover's scheme or the J-spectral factorization of Ball and Ran.     

一种改进的边缘方向插值算法

已有的边缘方向插值算法利用高低分辨率图像局部方差之间存在的对偶性实现自适应非线性插值。其性能明显优于传统线性插值算法，但图像边缘噪声可能仍然比较明显。为了获得更好的视觉效果。提出了一种改进算法。充分利用局部降采样像素之间的相关信息，用和被插点相邻的6个降采样像素估计高分辨率图像的局部协方差。该算法应用于灰度和彩色图像的分辨率增强，不仅降低了运算复杂度，而且有效地抑制了边缘噪声，进一步提高了插值图像的视觉质量。     

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.     

A new result on passivity preserving model reduction

The problem of model reduction with preservation of passivity is investigated. The approach is based on positive real interpolation, and is inspired by the similarity between Löwner and Pick matrices. The former are important in problems of general rational interpolation while the latter in problems of interpolation by positive real functions. It follows that interpolation of the original set of data together with an appropriately defined mirror-image set of data yields automatically positive real interpolants. Subsequently, we show how this result can be implemented using a Krylov projection procedure. The ensuing model reduction method preserves stability and passivity and can be implemented efficiently for the large-scale systems.     

渐进式约束扩展的机械臂运动规划算法

为了保证冗余机械臂带约束运动规划时的路径连续性,针对目前的直接插值和连续插值方法潜在的约束失效和路径失效问题,提出渐进式约束扩展的快速搜索随机树算法（PCE-RRT）。该算法在进行搜索树扩展时引入了一种渐进式约束扩展方法,该方法使用带有最近点信息的逆运动学计算策略,可以在保证任务约束的情况下迭代缩短最近点和扩展点的距离,让每一次扩展都能够保证搜索树的约束性和连续性。PCE-RRT可以使最终生成的搜索树很好地贴合于约束流形子空间,且树上任意两个相连节点都能满足连续条件。通过仿真实验对算法进行验证,实验结果表明该算法能够为机械臂带约束运动规划问题提供同时满足约束性和连续性条件的路径解。     

一类加权有理插值样条曲面及局部约束控制

目的 构造一类新的基于函数值与偏导数值的加权有理插值样条曲面,讨论该样条曲面的相关性质并分析曲面的局部约束控制。方法 一方面,先从x方向构造有理三次插值样条,再从y方向构造二元有理插值样条曲面;另一方面,按相反次序构造另一个二元有理插值样条曲面;最后将两种插值曲面加权得到一类新的有理插值样条曲面。结果 讨论插值曲面的性质,包括基函数、边界性质、积分加权系数的性质以及误差估计。通过选择合适的参数和加权系数,在不改变插值数据的前提下实现对插值区域内的局部约束控制。结论 实验结果表明,新的加权有理插值样条曲面具有良好的约束控制性质。     

The Inverse Problem of Analytic Interpolation With Degree Constraint and Weight Selection for Control Synthesis

The minimizers of certain weighted entropy functionals are the solutions to an analytic interpolation problem with a degree constraint, and all solutions to this interpolation problem arise in this way by a suitable choice of weights. Selecting appropriate weights is pertinent to feedback control synthesis, where interpolants represent closed-loop transfer functions. In this paper we consider the correspondence between weights and interpolants in order to systematize feedback control synthesis with a constraint on the degree. There are two basic issues that we address: we first characterize admissible shapes of minimizers by studying the corresponding inverse problem, and then we develop effective ways of shaping minimizers via suitable choices of weights. This leads to a new procedure for feedback control synthesis.      

约束双圆弧插值

提出一种构造C-型双圆弧和S-型双圆弧的算法,该双圆弧的2个端点为给定的点,在端点处的2个切向量为给定的切向量,且完全位于一条给定直线的一侧.对于C-型双圆弧,分12种情况给出不等式直接判断双圆弧是否满足位于给定直线一侧的约束条件.如果存在多个同时满足插值条件和直线约束条件的双圆弧,则通过求解一个最小值问题选出最优的双圆弧;否则,通过添加一个额外的点构造2段双圆弧.对于S-型双圆弧也给出一些带直线约束的插值结果.     

一种改进的同伦算法与H∞鲁棒控制器设计

提出了一种具有阶次限制的鲁棒控制器设计方法, 该算法将控制系统的性能指标转化为灵敏度函数问题, 并利用Nevanlinna-Pick插值算法进行求解. 提出了一种改进的同伦算法, 将其用于求解由灵敏度函数产生的非线性方程. 基于改进同伦算法设计的鲁棒控制器 不仅避免了传统H∞控制中加权函数的选择问题, 而且克服了鲁棒控制器阶次较高的缺陷. 最后,文章以4阶系统为例, 设计了具有阶次限制的H∞鲁棒控制器, 通过与传统鲁棒控制器的比较可以看出, 基于本文方法设计的控制器不仅具有较低的阶次, 而且其控制性能也具有明显的优越性.     

高维三次样条插值法在气动参数计算中的应用

根据一维三次样条插值算法能够达到二阶连续的光滑程度,将其扩展到高维插值,应用到由多自变量决定的气动参数计算。相对于高维线性插值算法,提高了插值精度,同时又避免了高次拉格朗日插值中的失真现象。并针对于高维三次样条插值算法,提出一种改进的边界条件,能够提高插值精度,简便工程应用。通过标准函数的数值试验,证明该高维插值方法及其改进边界条件对气动参数计算精度的提高是明显的。     

改进型快速NEDI图像插值实现算法

针对经典新边缘指导插值(NEDI)算法存在的计算复杂度高、硬件实现困难、插值系数误差累计导致放大图像边缘噪声大的缺陷,提出一种改进的快速NEDI算法。算法采用圆形窗口计算插值系数,且该插值系数在高倍放大中可重复使用,避免了迭代计算插值系数引入的误差,并节省了迭代计算的时间。同时,对边缘区域非中心像素插值时,采用和被插点相邻的6个原像素点估计高分辨率图像的局部协方差。最后给出实验,并与双立方插值及经典NEDI算法进行比较。实验结果表明,使用改进算法插值后的图像边缘更加清晰,消除了大比例缩放时锯齿现象,提高了图像的视觉质量,计算复杂度也较经典NEDI算法大大降低。     

基于再生核W空间的图像插值算法

提出了一种基于再生核的图像插值方法.再生核源于不同学科分支,已成为函数逼近的重要工具.该方法将再生核的再生公式离散,并按照证明的再生核数值积分方法导出了新型的图像插值算法.新插值算法的优点是利用再生核的数学模型特点保持图像的边界信息和光滑性,促进了插值的整体性能.实验结果表明:该算法能够克服其它算法的缺点,获取的高分辨率图像既能够保持图像的边界信息,又能保证图像的光滑性.     

二次样条插值研究

样条技术在计算机辅助设计,计算机辅助制造,和计算机图形系统得到了广泛应用。分析了二次样条函数插值的条件,分5种边值条件给出了二次样条插值的求解方法,最后给出实例验证求解方法。     

A State‐Space Based New Approach To The Directional Interpolation Problem

Directional interpolation plays an important role in robust control, system realization and model reduction. Several solutions to various directional interpolation problems have been proposed. In this paper, we consider the directional interpolation problem in a general setting and present a statespace based new approach to solving the problem. The solution is simple, and its exposition is as self‐contained as possible. We describe all the (strictly) bounded real rational matrix functions that satisfy the directional interpolation requirements by means of linear fractional transformation. Moreover, we give a necessary and sufficient condition for the interpolating function to be unique and show that the unique interpolating function is an inner (a co‐inner). The main procedures used to generate the interpolating function consist of standard matrix operations consisting of easy numerical computations, so the present solution is significant from the numerical viewpoint as well as the analytical viewpoint.     

Modified N–P Interpolation Theory for Closed Loop Pole Placement

Khargonekar and Tannenbaum have shown how to modify the Nevanlinna‐Pick (N–P) interpolation theory to design an internally stable and robust controller. In this paper, we present a new modification of the N‐P theory so that closed‐loop poles can be placed inside a circular region in the left half of the complex plane, in addition to the control system design being robust and internally stable. This ensures a minimum damping of the closed‐loop system. Numerical examples illustrate the theory.     

Weight Selection in Feedback Design With Degree Constraints

We present an approach for feedback design which is based on recent developments in analytic interpolation with a degree constraint. Performance is cast as an interpolation problem with bounded analytic functions. Minimizers of a certain weighted-entropy functional provide interpolants having degree less than the number of constraints. The choice of weight parameterizes all such bounded degree solutions. However, the relationship between the weights and the shape of corresponding transfer functions is not direct. Thus, in this paper we develop a formalism that guides weight selection.