最速控制系統的分析与綜合

本文介绍一些关于最速系统的分析与综合理论的新结果。受控系统的运动设为变系数线性常微分方程组所描述,而系统的终点状态是相空间内的某一凸性区域。文内详细介绍了相空间内等时区的几何特性及与此有关的事实,它们对系统综合具有重要意义。此外,以定理形式所叙述的事实具有独立的意义。文末指出线性变系数和常系数系统的综合方法及应用这一方法的例题。     

Digital straight lines and convexity of digital regions

It is shown that a digital region is convex if and only if every pair of points in the region is connected by a digital straight line segment contained in the region. The midpoint property is shown to be a necessary but not a sufficient condition for the convexity of digital regions. However, it is shown that a digital region is convex if and only if it has the median-point property.     

On fundamental solutions for multidimensional Helmholtz equation with three singular coefficients

The main result of the present paper is the construction of fundamental solutions for a class of multidimensional elliptic equations with three singular coefficients, which could be expressed in terms of a confluent hypergeometric function of four variables. In addition, the order of the singularity is determined and the properties of the found fundamental solutions that are necessary for solving boundary value problems for degenerate elliptic equations of second order are found.     

Stress-constrained topology optimization with design-dependent loading

The purpose of this paper is to apply stress constraints to structural topology optimization problems with design-dependent loading. A comparison of mass-constrained compliance minimization solutions and stress-constrained mass minimization solutions is also provided. Although design-dependent loading has been the subject of previous research, only compliance minimization has been studied. Stress-constrained mass minimization problems are solved in this paper, and the results are compared with those of compliance minimization problems for the same geometries and loading. A stress-relaxation technique is used to avoid the singularity in the stress constraints, and these constraints are aggregated in blocks to reduce the total number of constraints in the optimization problem. The results show that these design-dependent loading problems may converge to a local minimum when the stress constraints are enforced. The use of a continuation method where the stress-constraint aggregation parameter is gradually increased typically leads to better convergence; however, this may not always be possible. The results also show that the topologies of compliance-minimization and stress-constrained solutions are usually vastly different, and the sizing optimization of a compliance solution may not lead to an optimum.     

A boundary oriented domain quadrature technique for arbitrary multiply-connected regions

An innovative two-dimensional domain quadrature technique inherently sensitive to functions which develop a singularity within the integration region is developed. The quadrature method utilizes a finite set of points located along the boundary and connected into a series of elements to represent the domain geometry, a feature which makes it extremely convenient for BEM work. The method combines the convenience of high order Gaussian quadrature with the practical advantages of implicit discretization of the domain. The application of the technique is to be illustrated with several useful examples of interest to engineers.     

Higher order numerical solution of the integral equation for the two-dimensional neumann problem

Interest in the problem of two-dimensional potential flow in arbitrary multiply-connected domains has been stimulated by the need to calculate flow about multiple airfoil configurations consisting of slats and flaps detached from the main airfoil. General methods of solution are based on the use of a singularity distribution over the boundary. The distribution is obtained as the solution of an integral equation over the boundary. In implementing this solution various investigators approximate the boundary by an inscribed polygon, whose faces are small flat surface elements. The singularity on each element is taken as constant by some investigators and linearly varying by others. This paper systematically investigates the effectiveness of higher order approximations of the integral equation, including use of curved surface elements and parabolically-varying singularity. It is found that the approach using flat elements with constant singularity is mathematically consistent as is the next higher-order approach with parabolic elements and linearly varying singularity. The popular approach based on flat elements with linearly varying singularity is shown to be mathematically inconsistent, and examples are presented for which the effect of element curvature is greater than that of the singularity derivative. A number of examples are presented to show that: (1) the higher order solutions give very little increase in accuracy for the important case of exterior flow about a convex body: (2) for bodies with substantial concave regions and for interior flows in ducts, the use of parabolic elements and linearly varying singularity can give a dramatic increase in accuracy; and (3) the use of still higher order solutions leads to a rather small additional gain in accuracy.     

Fault-Tolerant Wormhole Routing with 2 Virtual Channels in Meshes

In wormhole meshes, a reliable routing is supposed to be deadlock-free and fault-tolerant. Many routing algorithms are able to tolerate a large number of faults enclosed by rectangular blocks or special convex, none of them, however, is capable of handling two convex fault regions with distance two by using only two virtual networks. In this paper, a fault-tolerant wormhole routing algorithm is presented to tolerate the disjointed convex faulty regions with distance two or no less, which do not contain any nonfaulty nodes and do not prohibit any routing as long as nodes outside faulty regions are connected in the mesh network. The processors＇ overlapping along the boundaries of different fault regions is allowed. The proposed algorithm, which routes the messages by X-Y routing algorithm in fault-free region, can tolerate convex fault-connected regions with only two virtual channels per physical channel, and is deadlock- and livelock-free. The proposed algorithm can be easily extended to adaptive routing.     

Study and resolution of singularities for a 6-DOF PUMA manipulator

Upon solving the inverse kinematics problem of robot manipulators, the inherent singularity problem should always be considered. When a manipulator is approaching a singular configuration, a certain degree of freedom will be lost such that there are no feasible solutions of the manipulator to move into this singular direction. In this paper, the singularities of a 6-DOF PUMA manipulator are analyzed in detail and all the corresponding singular directions in task space are clearly identified. In order to resolve this singularity problem, an approach denoted Singularity Isolation Plus Compact QP (SICQP) method is proposed. The SICQP method decomposes the work space into achievable and unachievable (i.e., singular) directions. Then, the exactness in the singular directions are released such that extra redundancy is provided to the achievable directions. Finally, the Compact QP method is applied to maintain the exactness in the achievable directions, and to minimize the tracking errors in the singular directions under the condition that feasible joint solutions must be obtained. In the end, some simulation results for PUMA manipulator are given to demonstrate the effectiveness of the SICQP method.     

Piecewise quasilinearization techniques for singular boundary-value problems

Piecewise quasilinearization methods for singular boundary-value problems in second-order ordinary differential equations are presented. These methods result in linear constant-coefficients ordinary differential equations which can be integrated analytically, thus yielding piecewise analytical solutions. The accuracy of the globally smooth piecewise quasilinear method is assessed by comparisons with exact solutions of several Lane-Emden equations, a singular problem of non-Newtonian fluid dynamics and the Thomas-Fermi equation. It is shown that the smooth piecewise quasilinearization method provides accurate solutions even near the singularity and is more precise than (iterative) second-order accurate finite difference discretizations. It is also shown that the accuracy of the smooth piecewise quasilinear method depends on the kind of singularity, nonlinearity and inhomogeneities of singular ordinary differential equations. For the Thomas-Fermi equation, it is shown that the piecewise quasilinearization method that provides globally smooth solutions is more accurate than that which only insures global continuity, and more accurate than global quasilinearization techniques which do not employ local linearization.     

A New Approach to Fully Automatic Mesh Generation

Automatic mesh generation is one of the most important parts in CIMS (Computer Integrated Manufacturing System).A method based on mesh grading propagation which automatically produces a triangular mesh in a multiply connected planar region is presented in this paper.The method decomposes the planar region into convex subregions,using algorithms which run in linear time.For every subregion,an algorithm is used to generate shrinking polygons according to boundary gradings and form delaunay triangulation between two adjacent shrinking polygons,both in linear time.It automatically propagates boundary gradings into the interior of the region and produces satisfactory quasi-uniform mesh.     

Formal Solutions of Linear PDEs and Convex Polyhedra

The Newton polygon construction for ODEs, and Malgrange–Ramis polygon for partial differential equations in one variable are generalized in order to give an algorithm to find solutions of a linear partial differential equation at a singularity. The solutions found involve exponentials, logarithms and Laurent power series with exponents contained in a strongly convex cone.     

Analytic center cutting plane method for multiple kernel learning

Multiple Kernel Learning (MKL) is a popular generalization of kernel methods which allows the practitioner to optimize over convex combinations of kernels. We observe that many recent MKL solutions can be cast in the framework of oracle based optimization, and show that they vary in terms of query point generation. The popularity of such methods is because the oracle can fortuitously be implemented as a support vector machine. Motivated by the success of centering approaches in interior point methods, we propose a new approach to optimize the MKL objective based on the analytic center cutting plane method (accpm). Our experimental results show that accpm outperforms state of the art in terms of rate of convergence and robustness. Further analysis sheds some light as to why MKL may not always improve classification accuracy over naive solutions.     

Robust stability of time-delay systems

There are two fundamental results available when we study stability of a polynomial family that is described by convex polytope in the coefficient space: the edge theorem and the theory based on the concept of convex directions. Many known results can be explained with these two results. This paper deals with a generalization of this line of research to the case of quasipolynomials that are entire functions which include both degree of the independent variable and exponential functions. The main objects of the paper are the developing of the concept of convex directions for quasipolynomials and exploiting this concept for construction of testing sets for quasipolynomial families. One of the primary sources of motivation for the class of problems considered in this paper is derived from process control. A typical problem formulation almost always includes a delay element in each subsystem process block. When we interconnect a number of such blocks in a feedback system, the study of robust stability involves quasipolynomials of the sort considered in this paper     

Application of the finite method in micromechanical analyses of creep fracture problems

The finite element method is applied to determine the generalized stress intensity factors for two problems fundamental in the micromechanical study of intergranular creep fracture. Special techniques to simulate multipoint constraints arising from symmetry and periodicity, and a singularity element have been developed and proved effective. The good agreements of finite element results with local field solutions are discussed.     

Non-equivalence of the conventional boundary integral formulation and its elimination for two-dimensional mixed potential problems

For a given mixed type potential problem, the corresponding conventional boundary integral equation is shown to yield non-equivalent solutions. Numerical results show that the conventional boundary integral formulation yields incorrect potential and flux results when the distance scale in the fundamental solution approaches its degenerate value. Such a kind of non-equivalence of the conventional boundary integral equation can be eliminated by the use of the necessary and sufficient boundary integral formulation which always ensures the equivalence of solutions.     

Singularity and positioning constants in Poisson's equation with a point source of unit intensity

The singularity constants and positioning constants are described in the potential field governed by Poisson's equation with a point source of unit intensity. It is then demonstrated that the singularity constants play an important role in the integral equation formulations utilizing rather complicated fundamental solutions.     

On the Stability of the Quadruple Solutions of the Forward Kinematic Problem in Analytic Parallel Robots

Many parallel robots can change between different assembly modes (solutions of the forward kinematic problem) without crossing singularities, either by enclosing cusps or alpha-curves of the planar sections of their singularity loci. Both the cusps and the alpha-curves are stable singularities, which do not disappear under small perturbations of the geometry of the robot. Recently, it has been shown that some analytic parallel robots can also perform these nonsingular changes of assembly mode by encircling isolated points of their singularity loci at which the forward kinematic problem admits solutions with multiplicity four. In this paper, we study the stability of these quadruple solutions when the design of the robot deviates from the analytic geometry, and we show that such quadruple solutions are not stable since the isolated singular points at which they occur degenerate into closed deltoid curves. However, we also demonstrate that, although the quadruple solutions are unstable, the behavior of the robot when moving around them is practically unaffected by the perturbations from the analytic geometry. This means that the robot preserves its ability to perform nonsingular transitions by enclosing the quadruple solutions, even when its geometry is not exactly analytic due to small manufacturing tolerances.     

On the determination of the solutions and maximum admissible power of the load flow equation via LMIS

It is well‐known that the load flow equation plays a key role in power systems. This paper investigates the load flow equation addressing two fundamental problems, namely the determination of the solutions of this equation in a given region of interest, and the computation of the maximum admissible power for which this equation has a solution in such a region. For the first problem, it is shown that the sought solutions can be computed by solving an eigenvalue problem (EVP), i.e. a convex optimization problem with linear matrix inequality (LMI) constraints, and by looking for vectors with special structure into a linear subspace. For the second problem, it is shown that an upper bound of the sought maximum admissible power can be found by solving another EVP, moreover a sufficient and necessary condition for establishing the tightness of the found upper bound is provided. The proposed methodology is illustrated by some numerical examples. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Taxonomies of logically defined qualitative spatial relations

This paper develops a taxonomy of qualitative spatial relations for pairs of regions, which are all logically defined from two primitive (but axiomatized) notions. The first primitive is the notion of two regions being connected, which allows eight jointly exhaustive and pairwise disjoint relations to be defined. The second primitive is the convex hull of a region which allows many more relations to be defined. We also consider the development of the useful notions of composition tables for the defined relations and networks specifying continuous transitions between pairs of regions. We conclude by discussing what kind of criteria to apply when deciding how fine grained a taxonomy to create.