A counterexample to an algorithm for computing monotone hulls of simple polygons

A two-stage algorithm was recently proposed by Sklansky (1982) for computing the convex hull of a simple polygon P. The first step is intended to compute a simple polygon $\text{P}{}^1$ which is monotonic in both the x and y directions and which contains the convex hull vertices of P. The second step applies a very simple convex hull algorithm on $\text{P}{}^1$. In this note we show that the first step does not always work correctly and can even yield non-simple polygons, invalidating the use of the second step. It is also shown that the first step can discard convex hull vertices thus invalidating the use of any convex hull algorithm in the second step.     

The Krakovskii-LaSalle Invariance Principle for a Class of Unilateral Dynamical Systems

This paper is devoted to the study of the extension of the invariance lemma to a class of hybrid dynamical systems, namely evolution variational inequalities. Applications can be found in models of electrical circuits with ideal diodes or oligopolistic market equilibrium.Acknowledgements. This work was partially supported by the European project SICONOS IST2001-37172.     

A method for solving the n-dimensional convex hull problem

A method is presented for finding all vertices and all hyperplanes containing the faces of a convex polyhedron spanned by a given finite set X in Euclidean space Eⁿ. The present paper indicates how this method can be applied to the investigation of linear separability of two given finite sets X₁ and X₂ in Eⁿ. In the case of linear separability of these sets the proposed method makes it possible to find the separating hyperplane.     

平面二级倒立摆的保性能控制

摆系统是一个强耦合、非线性、高阶次的不稳定系统,常规控制方法如PID算法、最优控制算法等。由于摆系统的数学模型是在忽略了次要因素的基础上得出来的,当系统受到外部的干扰时,这些次要因素的影响比较突出,控制系统采用基于线性矩阵不等式的方法设计一系统性能上界最小的最优保性能控制律,以改善系统的瞬态性与鲁棒性。然后采用Matlab的工具箱设计该控制律,并对系统进行仿真。最后采用VC++6.0编写控制程序,实现对平面二级摆系统的实时控制。     

A new design method for PI–PD control of unstable processes with dead time

PID controllers are still widely practiced in the industrial systems. In the literature, many publications can be found considering PID controller design for unstable processes. However, owing to the structural limitations of PID controllers, generally, good closed loop performance cannot be achieved with a PID for controlling unstable processes and usually a step response with a high overshoot and oscillation is obtained. On the other hand, PI–PD controllers are proved to give very satisfactory closed loop performances for unstable processes. The paper presents a simple design method to tune parameters of a PI–PD controller for the control of the unstable processes with time delay. The proposed method is based on plotting the stability boundary locus, which is a locus dependent on the parameters of the controller and frequency, in the parameter plane. The method uses a new concept named centroid of the convex stability region. Simulation examples and an experimental application are given to illustrate the superiority of the proposed method over some existing ones.     

Prediction of cutting forces and machining error in ball end milling of curved surfaces -II experimental verification

This paper presents the results of a series of experiments performed to examine the validity of a theoretical model for evaluation of cutting forces and machining error in ball end milling of curved surfaces. The experiments are carried out at various cutting conditions, for both contouring and ramping of convex and concave surfaces. A high precision machining center is used in the cutting tests. In contouring, the machining error is measured with an electric micrometer, while in ramping it is measured on a 3-coordinate measuring machine. The results show that in contouring, the cutting force component that influences the machining error decreases with an increase in milling position angle, while in ramping, the two force components that influence the machining error are hardly affected by the milling position angle. Moreover, in contouring, high machining accuracy is achieved in “Up cross-feed, Up cut” and “Down cross-feed, Down cut” modes, while in ramping, high machining accuracy is achieved in “Left cross-feed, Downward cut” and “Right cross-feed, Upward cut” modes. The theoretical and experimental results show reasonably good agreement.     

Generating parametric G2 non-four-sided blending surfaces

A method of generating n-sided (n=3, 5, 6) G² blending surfaces (except that the 3-sided blending surface is C⁰ continuous at the three vertexes and the 5-sided blending surface is C⁰ continuous at a vertex) is presented in this paper. When this method is used to generate n-sided (n=3, 5, 6) G² blending surfaces between the base surfaces, the base surfaces are expressed as n-sided (n=3, 5, 6) by reparameterising the base surfaces; then the n-sided (n=3, 5, 6) G² blending surfaces are convex combinations of the base surfaces expressed as n-sided (n=3, 5, 6) surfaces. When the contact curves are arbitrary curves in the base surfaces, the regions near the contact curves in the base surfaces are reparameterised and expressed as n-sided (n=3, 5, 6) surfaces; then the n-sided (n=3, 5, 6) G² blending surfaces are convex combinations of the reparameterised local base surfaces expressed as n-sided (n=3, 5, 6) surface styles. The shape of the n-sided (n=3, 5, 6) blending surfaces can be adjusted by changing the size of the reparameterised local base surfaces and the combination weights. An example of a 5-sided surface generated between an aerofoil, the body and the fringe surface of a missile is given.     

低秩矩阵近似与优化问题研究进展

首先以高维数据压缩与恢复为背景,详细阐述由香农采样理论到稀疏表示和压缩感知理论再到低秩矩阵问题的发展历程,引出低秩矩阵近似与优化问题的重要性.然后,从低秩矩阵最小化问题、低秩矩阵分解问题、低秩矩阵的优化与应用三方面对现有方法进行详细的综述.最后对当前研究的不足之处与未来的研究方向提出合理的建议.