Induced rational approximations of transcendental curves

A novel technique for the construction of positive weight high-precision rational approximations to a class of transcendental curves is presented. The approximations are induced from the rational parametrisations of the circle. The previously published rational parametrisations of the circle are not suited to the induction process and the new parametrisations are constructed in the paper for the purpose. Explicit rational approximations of a number of transcendental curves are then given. The work is a development of the authors’ previous work on induced rational parametrisations of special algebraic curves.     

Hermitian solutions of the discrete algebraic Riccati equation

Hermitian solutions of the discrete algebraic Riccati equation play an important role in the least-squares optimal control problem for discrete linear systems. In this paper we describe the set of hermitian solutions in various ways: in terms of factorizations of rational matrix functions which take hermitian values on the unit circle; in terms of certain invariant subspaces of a matrix which is unitary in an indefinite scalar product; and in terms of all invariant subspaces of a certain matrix. These results are inspired by known results for the algebraic Riccati equation arising in the least-squares optimal control problem for continuous linear systems.     

Frequency domain evaluation of circle criterion,Popov criterion and off-axis circle criterion in the MIMO case

Many control applications can be described as non-linear feedback systems whose stability properties can, for example, be analysed with the circle criterion, the Popov criterion or the off-axis circle criterion. In the SISO case these sector criteria became very popular because of their simple graphic interpretations; the criteria's significance for multivariable feedback systems remained rather poor since there are almost no practicable ways to evaluate them. In this paper the MIMO versions of the three stability criteria are derived using Popov's theory of hyperstability. The main contribution is the presentation of new methods for a frequency domain evaluation of the sector criteria in the multivariable case. The new methods are based on graphic elements: the 'positivity curves'.     

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.     

On Stochastic Simulation of Forward-Looking Models

The solution of a nonlinear macroeconometric model with expectations of future-dated variables generally has to be approximated by numerical simulation. A brief review of deterministic, and stochastic dynamic simulations of a backward-looking model is followed by a conceptual presentation of methods for dynamic simulation of a forward-looking (rational expectations) model. I distinguish between uncertainty faced by rational agents and by the modeller, and suggest different ways of simulating random variables in the model. Simulations of simple linear and nonlinear univariate time-series models illustrate the methods.     

Least eccentric ellipses for geometric Hermite interpolation

We present a rational Bézier solution to the geometric Hermite interpolation problem. Given two points and respective unit tangent vectors, we provide an interpolant that can reproduce a circle if possible. When the tangents permit an ellipse, we produce one that deviates least from a circle. We cast the problem as a theorem and provide its proof, and a method for determining the weights of the control points of a rational curve. Our approach targets ellipses, but we also present a cubic interpolant that can find curves with inflection points and space curves when an ellipse cannot satisfy the tangent constraints.     

闭合圆锥曲线的2次周期性NURBS表示

本文探讨了用2次周期性NURBS表示闭合圆锥曲线的方法,导出了精确表示椭圆、抛物线和双曲线的条件,并给出了精确表示圆弧的充要条件;提出了用2次NURBS精确表示圆的新方法.和其它基于矩形控制多边形的同类方法相比,其控制点数最少.用2次非周期性NURBS(2次有理Bezier样条)和2次均匀有理B样条描述圆锥曲线的方法均是本文的一个特例.     

Algebraic stability criteria and symbolic derivation of stability conditions for feedback control systems

This article provides algebraic settings of the stability criteria of Nyquist and Popov and the circle criterion for closed-loop linear control systems with linear or nonlinear feedback whose transfer functions are rational ones with integer coefficients. The proposed settings make use of algebraic methods of parametric curve implicitisation, real root isolation, symbolic integration and quantifier elimination and allow one to derive exact stability conditions for feedback control systems with symbolic computation. An example is presented to illustrate the algebraic approach and its effectiveness. Some numerical stability results obtained previously are confirmed.     

Comparison of different methods for computing the two-dimensional envelope for surface finish measurements

In the envelope system of surface finish measurement, an envelope traced by a circle of definite radius gives the reference line for a two-dimensional surface profile. Geometrical construction of this envelope, on actual surface profiles, is fairly easy. For evaluating the benefits of the E-system it is advantageous to use digitized surface profiles, to compute the envelope from the profile ordinates. There are several ways of computing the envelope from digitized surface profiles and this paper explains the various methods and gives the times taken for computation.     

Rational quadratic circles are parametrized by chord length

We show that the chord length parameter assignment is exact for circle segments in standard rational quadratic form.     

一种用于圆检测的快速HOUGH算法

分析了现有的几种于圆检测的ＨＯＵＧＨ算法,针对它们的变换速度较慢等不足之处,提出了一种新的快速算法,这种算法应用于圆检测时能提高变换速度接近一个数量级,并且非常适应于多个圆嵌套的图象检测,给出了与传统ＨＯＵＧＨ变换相对照的结果。     

Identification of rational approximate models inH∞ using generalized orthonormal basis

Robust identification with FIR models fails to be successful when the number of coefficients to be estimated becomes large, caused by lightly damped modes of the plant or poles very close to the unit circle. The paper presents a two-stage algorithm to obtain a low-order approximate model in frequency domain in a generalized orthonormal basis with guaranteed H^∞ error bound for deterministic linear time-invariant stable systems, the first stage being an L^∞ rational approximation and the second nonlinear step being an H ^∞ rational approximation     

Multivariate rational interpolation

Many papers have already been published on the subject of multivariate polynomial interpolation and also on the subject of multivariate Padé approximation. But the problem of multivariate rational interpolation has only very recently been considered; we refer among others to [8] and [3]. The computation of a univariate rational interpolant can be done in various equivalent ways: one can calculate the explicit solution of the system of interpolatory conditions, or start a recursive algorithm, or calculate the convergent of a continued fraction. In this paper we will generalize each of those methods from the univariate to the multivariate case. Although the generalization is simple, the equivalence of the computational methods is completely lost in the multivariate case. This was to be expected since various authors have already remarked [2,7] that there is no link between multivariate Padé approximants calculated by matching the Taylor series and those obtained as convergents of a continued fraction.     

有理高斯函数曲线模拟技术

讨论了有理高斯函数曲线模拟技术。通过控制点位置和权值节点分布以及高斯标准误差来控制曲线形状。对于给定的插值点,根据曲线封闭和不封闭两种情况,分别反向求出控制点,并给出相应高斯标准误差的估计值。在此基础上,运用高斯函数曲线模拟技术对直线、圆进行模拟,并给出最大误差。     

Squaring the Circle: Marriage of Heaven and Earth

It is impossible to construct circles and squares of equal areas or perimeters precisely, for circles are measured by the incommensurable value pi (π) and squares by rational whole numbers. But from early times, geometers have attempted to reconcile these two orders of geometry. “Squaring the circle” can represent the union of opposing eternal and finite qualities, symbolizing the fusion of matter and spirit and the marriage of heaven and earth. In this column, we consider various methods for squaring the circle and related geometric constructions.     

带法向约束的圆平均非线性细分曲线设计

目的 对采样设备获取的测量数据进行拟合,可实现原模型的重建及功能恢复。但有些情况下,获取的数据点不仅包含位置信息,还包含法向量信息。针对这一问题,本文提出了基于圆平均的双参数4点binary非线性细分法与单参数3点ternary插值非线性细分法。方法 首先将线性细分法改写为点的重复binary线性平均,然后用圆平均代替相应的线性平均,最后用加权测地线平均计算的法向量作为新插入顶点的法向量。基于圆平均的双参数4点binary细分法的每一次细分过程可分为偏移步与张力步。基于圆平均的单参数3点ternary细分法的每一次细分过程可分为左插步、插值步与右插步。结果 对于本文方法的收敛性与C¹连续性条件给出了理论证明;数值实验表明,与相应的线性细分相比,本文方法生成的曲线更光滑且具有圆的再生力,可以较好地实现3个封闭曲线重建。结论 本文方法可以在带法向量的初始控制顶点较少的情况下,较好地实现带法向约束的离散点集的曲线重建问题。     

圆形目标实时检测的改进算法研究

文中针对如何快速检测圆形目标展开研究,在随机霍夫变换的基础上提出了两步筛选法和区域约束法,对可能存在的错误采样进行了分析,利用图像的梯度信息和圆的几何特性筛选采样点,避免了大量的无效累积;同时通过设计检验窗口,减少了真实性检验的计算量,有效提高了检测速度和精度。该算法应用于圆形目标实时检测系统,在实时性与准确性上取得了良好的效果。     

结合多阶段优化的圆检测算法

在常规圆检测算法中,Hough变换、随机Hough变换以及随机圆检测算法的检测效率低,导致难以适用于复杂场景或者对检测速度有较高要求的情况。为了提高圆检测的效率,本文从采样点的选取、候选圆的确定以及真圆的确认3个阶段进行分析,结合这3个阶段的优化方法,提出一种结合多阶段优化的圆检测算法。人工图像和实际图像的实验结果表明：该算法较其他算法有效地提高了圆检测的速度,并且具有较好的检测鲁棒性和检测精度。     

圆形截面的四边形元素网格剖分

叙述了一种对圆形截面进行网格剖分的方法。圆形截面剖分后由不同类型的元素表示,节点的创建和元素的生成依据节点和元素在圆内的位置而采用不同的方式分别生成。     

基于特征点匹配的全局运动估计

传统块匹配法只考虑平移运动,对大角度旋转无法进行有效估计。为此,根据圆形的旋转不变性,提出一种基于圆形块的匹配方法,提取圆投影矢量作为匹配特征,采用特征匹配的策略,得到当前块的运动矢量。为提高匹配效率,利用Harris算子检测特征点,获取特征点的运动矢量。通过局部运动矢量和刚体变换模型得到线性参数方程,并采用RANSAC算法进行求解,以获得全局运动参数。实验结果证明,该方法对较大的平移及旋转角度具有较强的适用性。