带有运行成本函数的Hamilton-Jacobi可达性分析

水平集方法将可达集表示为Hamilton-Jacobi方程解的零水平集,保存多个不同时间范围的可达集则需要保存Hamilton-Jacobi方程在多个时刻的解,这不仅需要消耗大量的存储空间还为控制律的设计造成了困难.针对这些局限性,提出了一种改进的基于Hamilton-Jacobi方程的可达集表示方法.该方法在Hamilton-Jacobi方程中加入了一项运行成本函数,可以用同一个时刻的解的多个非零水平集表示多个不同时间范围的可达集,极大地节省了存储空间并为控制律的设计提供了便利.为了求解所构造的带有运行成本函数的Hamilton-Jacobi方程,采用了一种基于递归和插值的方法.最后,通过一些数值算例验证了所提出的方法的精确性、在存储空间方面的优越性以及设计的控制律的有效性.     

Topology optimization of steady Navier-Stokes flow via a piecewise constant level set method

This paper presents a piecewise constant level set method for the topology optimization of steady Navier-Stokes flow. Combining piecewise constant level set functions and artificial friction force, the optimization problem is formulated and analyzed based on a design variable. The topology sensitivities are computed by the adjoint method based on Lagrangian multipliers. In the optimization procedure, the piecewise constant level set function is updated by a new descent method, without the needing to solve the Hamilton-Jacobi equation. To achieve optimization, the piecewise constant level set method does not track the boundaries between the different materials but instead through the regional division, which can easily create small holes without topological derivatives. Furthermore, we make some attempts to avoid updating the Lagrangian multipliers and to deal with the constraints easily. The algorithm is very simple to implement, and it is possible to obtain the optimal solution by iterating a few steps. Several numerical examples for both two- and three-dimensional problems are provided, to demonstrate the validity and efficiency of the proposed method.     

Long range numerical simulation of short waves as nonlinear solitary waves

A new numerical method has been developed to propagate short wave equation pulses over indefinite distances and through regions of varying index of refraction, including multiple reflections. The method, “Wave Confinement”, utilizes a newly developed nonlinear partial differential equation that propagates basis functions according to the wave equation. These basis functions are generated as stable solitary waves where the discretized equation can be solved without any numerical dissipation. The method can also be used to solve for harmonic waves in the high frequency (Eikonal) limit, including multiple arrivals. The solution involves discretizing the wave equation on a uniform Eulerian grid and adding a simple nonlinear “Confinement” term. This term does not change the amplitude (integrated through each point on the pulse surface) or the propagation velocity, or arrival time, and yet results in capturing the waves as thin surfaces that propagate as thin nonlinear solitary waves and remain 2–3 grid cells in thickness indefinitely with no numerical spreading. With the method, only a simple discretized equation is solved each time step at each grid node. The method can be contrasted to Lagrangian Ray Tracing: it is an Eulerian based method that captures the waves directly on the computational grid, where the basic objects are codimension 1 surfaces (in the fine grid limit), defined on a regular grid, rather than collections of markers. In this way, the complex logic of current ray tracing methods, which involves allocation of markers to each surface and interpolation as the markers separate, is avoided.     

Simultaneous optimization of the material properties and the topology of functionally graded structures

A level set based method is proposed for the simultaneous optimization of the material properties and the topology of functionally graded structures. The objective of the present study is to determine the optimal material properties (via the material volume fractions) and the structural topology to maximize the performance of the structure in a given application. In the proposed method, the volume fraction and the structural boundary are considered as the design variables, with the former being discretized as a scalar field and the latter being implicitly represented by the level set method. To perform simultaneous optimization, the two design variables are integrated into a common objective functional. Sensitivity analysis is conducted to obtain the descent directions. The optimization process is then expressed as the solution to a coupled Hamilton-Jacobi equation and diffusion partial differential equation. Numerical results are provided for the problem of mean compliance optimization in two dimensions.     

A critical comparative assessment of differential equation-driven methods for structural topology optimization

In recent years, differential equation-driven methods have emerged as an alternate approach for structural topology optimization. In such methods, the design is evolved using special differential equations. Implicit level-set methods are one such set of approaches in which the design domain is represented in terms of implicit functions and generally (but not necessarily) use the Hamilton-Jacobi equation as the evolution equation. Another set of approaches are referred to as phase-field methods; which generally use a reaction-diffusion equation, such as the Allen-Cahn equation, for topology evolution. In this work, we exhaustively analyze four level-set methods and one phase-field method, which are representative of the literature. In order to evaluate performance, all the methods are implemented in MATLAB and studied using two-dimensional compliance minimization problems.     

Output-feedback stabilization of an unstable wave equation

We consider the problem of stabilization of a one-dimensional wave equation that contains instability at its free end and control on the opposite end. In contrast to classical collocated “boundary damper” feedbacks for the neutrally stable wave equations with one end satisfying a homogeneous boundary condition, the controllers and the associated observers designed in the paper are more complex due to the open-loop instability of the plant. The controller and observer gains are designed using the method of “backstepping,” which results in explicit formulae for the gain functions. We prove exponential stability and the existence and uniqueness of classical solutions for the closed-loop system. We also derive the explicit compensators in frequency domain. The results are illustrated with simulations.     

Application of the Boltzmann kinetic equation to the eddy problems

The main purpose of this work is to investigate the feasibility of applying a kinetic approach to the problem of modeling turbulent and unstable flows. First, initial value problems with the Taylor–Green (TG) type and isotropic velocity conditions for compressible flow in two-dimensional (2D) and three-dimensional (3D) periodic domains are considered. Further, 3D direct numerical simulation of decaying isotropic turbulence is performed. Macroscopic flow quantities of interest are examined. The simulation is based on the direct numerical solution of the Boltzmann kinetic equation using an explicit–implicit scheme for the relaxation stage. Comparison with the solution of the Bhatnagar–Gross–Krook (BGK) model equation obtained by using an implicit scheme is carried out for the decaying isotropic turbulence problem and demonstrates a small difference. For the TG initial condition results show a fragmentation of the large initial eddies and subsequently the full damping of the system. Numerical data are close to the analytic solution of TG problem. A dependence of the kinetic energy on the wave number is obtained by means of the Fourier expansion of velocity components. A power-law exponent for the kinetic energy spectrum tends to the theoretical value “−3” for 2D turbulence in 2D case and to the famous Kolmogorov value “−5/3” in 3D case.     

Mathews⁃Lakshmanan 振子方程的分析力学解法

本文以求解非线性非保守的Mathews-Lakshmanan振子(以下简写成M-L振子)为例,说明分析力学理论和方法在非线性系统研究中的应用.根据变分法逆问题理论,将M-L振子方程变换为自伴随形式方程;利用四种方法构造出振子的拉格朗日函数和哈密顿函数;分别基于诺特理论和哈密顿-雅可比方法得到M-L振子方程的解.     

Lyapunov function design for finite-time convergence analysis: “Twisting” controller for second-order sliding mode realization

A generalization of the Zubov method of a Lyapunov function design is presented. It is based on the characteristic method application and is related to resolving the first-order partial differential equation of a special type. A successful resolution of this equation guaranties a finite-time convergence for the corresponding dynamics given by an ordinary differential equation with a discontinuous right-hand side. The suggested method is illustrated by its application to the so-called “twisting” controller stability analysis. The constructed Lyapunov function as well as its level line sections is graphically illustrated.     

A new level set method for systematic design of hinge-free compliant mechanisms

This paper presents a new level set-based method to realize shape and topology optimization of hinge-free compliant mechanisms. A quadratic energy functional used in image processing applications is introduced in the level set method to control the geometric width of structural components in the created mechanism. A semi-implicit scheme with an additive operator splitting (AOS) algorithm is employed to solve the Hamilton-Jacobi partial differential equation (PDE) in the level set method. The design of compliant mechanisms is mathematically represented as a general non-linear programming with a new objective function augmented by the high-order energy term. The structural optimization is thus changed to a numerical process that describes the design as a sequence of motions by updating the implicit boundaries until the optimized structure is achieved under specified constraints. In doing so, it is expected that numerical difficulties such as the Courant-Friedrichs-Lewy (CFL) condition and periodically applied re-initialization procedures in most conventional level set methods can be eliminated. In addition, new holes can be created inside the design domain. The final mechanism configurations consist of strip-like members suitable for generating distributed compliance, and solving the de-facto hinge problem in the design of compliant mechanisms. Two widely studied numerical examples are studied to demonstrate the effectiveness of the proposed method in the context of designing distributed compliant mechanisms.     

A level set-based topology optimization incorporating arbitrary Lagrangian Eulerian method for wavelength filter using extraordinary optical transmission

This paper deals with structural optimization for designing periodic structures in a hole array wavelength filter. The hole array wavelength filter that consists of metallic thin film and dielectric enables to transmit narrow bandwidth light. It is known that transmission spectrum can be changed not only by the periodicity of hole array but also by the shape of holes. For optimizing the hole shape, the level set method is used in this study. In the ordinary level set method, the boundaries are implicitly expressed by the zero level set of a scalar function, called the level set function, within a fixed mesh. Therefore, the material interpolation becomes numerically awkward within the elements across the implicit zero level set because those elements inevitably take on intermediate material properties even if the boundary of zero level set are mathematically clear. As the result, the optimization is likely to yield wrong solution. Here, a new level set optimization method incorporating Arbitrary Lagrangian Eulerian method is proposed to eliminate intermediate values on the interfaces perfectly. As a result, the proposed method can successfully perform the structural optimization of hole shape without intermediate values.     

基于Hamilton-Jacobi方程的飞行器机动动作可达集分析

为了给驾驶员完成标准机动动作提供决策支持, 提出一种使用哈密尔顿-雅克比(Hamilton-Jacobi)方程求解机动动作可行状态空间的研究方法.使用关键点将机动动作划分为不同阶段, 将各关键点的标准状态约束作为目标集, 逆时间求解目标集对应的可达集得到各阶段的边界状态范围, 目标集和可达集均由零水平集表示.使用该方法得到斤斗动作三维度运动模型下各阶段的可达集及斤斗动作的可行状态空间, 为了使运动模型的控制量与驾驶员实际操纵更为接近, 构建了以迎角变化率为控制量的四维度运动模型, 在此基础上对斤斗动作各阶段的可达集进行了分析.     

Robotic vocabulary building using extension inference and implicit contrast

TWIG (“Transportable Word Intension Generator”) is a system that allows a robot to learn compositional meanings for new words that are grounded in its sensory capabilities. The system is novel in its use of logical semantics to infer which entities in the environment are the referents (extensions) of unfamiliar words; its ability to learn the meanings of deictic (“I,” “this”) pronouns in a real sensory environment; its use of decision trees to implicitly contrast new word definitions with existing ones, thereby creating more complex definitions than if each word were treated as a separate learning problem; and its ability to use words learned in an unsupervised manner in complete grammatical sentences for production, comprehension, or referent inference. In an experiment with a physically embodied robot, TWIG learns grounded meanings for the words “I” and “you,” learns that “this” and “that” refer to objects of varying proximity, that “he” is someone talked about in the third person, and that “above” and “below” refer to height differences between objects. Follow-up experiments demonstrate the system's ability to learn different conjugations of “to be”; show that removing either the extension inference or implicit contrast components of the system results in worse definitions; and demonstrate how decision trees can be used to model shifts in meaning based on context in the case of color words.     

A Level Set Framework for Capturing Multi-Valued Solutions of Nonlinear First-Order Equations

We introduce a level set method for the computation of multi-valued solutions of a general class of nonlinear first-order equations in arbitrary space dimensions. The idea is to realize the solution as well as its gradient as the common zero level set of several level set functions in the jet space. A very generic level set equation for the underlying PDEs is thus derived. Specific forms of the level set equation for both first-order transport equations and first-order Hamilton-Jacobi equations are presented. Using a local level set approach, the multi-valued solutions can be realized numerically as the projection of single-valued solutions of a linear equation in the augmented phase space. The level set approach we use automatically handles these solutions as they appear     

Application of AVIRIS data in detection of oil-induced vegetation stress and cover change at Jornada, New Mexico

On June 1, 2000, an oil spill accident occurred along transportation pipeline located in the Jornada Experimental Range (USDA), Jornada, New Mexico, a long-term ecological research (LTER). In order to detect potential vegetation stress caused by the accident, two AVIRIS data sets of the oil spill area, before and after the oil release, are analyzed and the reliability of several techniques in the detection of vegetation stress is examined.The polynomial fitting and Lagrangian interpolation, and spectral mixture analysis (SMA) are applied to the AVIRIS data sets. The first two methods are applied for the detection of the “red-edge” shift in vegetation reflectance spectra, and the last for the detection of change in vegetation fraction. The results indicate that the polynomial fitting and Lagrangian interpolation both are able to detect a red-shift of the vegetation “red-edge”, but the latter's performance depends on the band combination used and is sensitive to data noise. The polynomial fitting results are inconsistent in detection of “the red-edge” shift, while Lagrangian interpolation is not. Within the oil spill area, the fraction estimates of vegetation resulting from SMA demonstrate a decrease (10-30%) of the vegetation fraction after the accident, indicating stressed vegetation and cover change. The result also indicates that areas of extremely large decrease (>40%) in plant cover outside of the oil spill area is due to the response of grasses due to the water stress in 2000, and that the integration of some auxiliary data on ecological and climatological data with the analysis of remotely sensed data is thus very important to the interpretation of the detection results. A sensitivity analysis indicates that the detected vegetation cover change is insensitive to the noise introduced by the radiometric normalization.     

A scale-adjusted measure of “Urban sprawl” using nighttime satellite imagery

“Urban Sprawl” is a growing concern of citizens, environmental organizations, and governments. Negative impacts often attributed to urban sprawl are traffic congestion, loss of open space, and increased pollutant runoff into natural waterways. Definitions of “Urban Sprawl” range from local patterns of land use and development to aggregate measures of per capita land consumption for given contiguous urban areas (UA). This research creates a measure of per capita land use consumption as an aggregate index for the spatially contiguous urban areas of the conterminous United States with population of 50,000 or greater. Nighttime satellite imagery obtained by the Defense Meteorological Satellite Program's Operational Linescan System (DMSP OLS) is used as a proxy measure of urban extent. The corresponding population of these urban areas is derived from a grid of the block group level data from the 1990 U.S. Census. These numbers are used to develop a regression equation between Ln(Urban Area) and Ln(Urban Population). The ‘scale-adjustment’ mentioned in the title characterizes the “Urban Sprawl” of each of the urban areas by how far above or below they are on the “Sprawl Line” determined by this regression. This “Sprawl Line” allows for a more fair comparison of “Urban Sprawl” between larger and smaller metropolitan areas because a simple measure of per capita land consumption or population density does not account for the natural increase in aggregate population density that occurs as cities grow in population. Cities that have more “Urban Sprawl” by this measure tended to be inland and Midwestern cities such as Minneapolis-St. Paul, Atlanta, Dallas-Ft. Worth, St. Louis, and Kansas City. Surprisingly, west coast cities including Los Angeles had some of the lowest levels of “Urban Sprawl” by this measure. There were many low light levels seen in the nighttime imagery around these major urban areas that were not included in either of the two definitions of urban extent used in this study. These areas may represent a growing commuter-shed of urban workers who do not live in the urban core but nonetheless contribute to many of the impacts typically attributed to “Urban Sprawl”. “Urban Sprawl” is difficult to define precisely partly because public perception of sprawl is likely derived from local land use planning decisions, spatio-demographic change in growing urban areas, and changing values and social mores resulting from differential rates of international migration to the urban areas of the United States. Nonetheless, the aggregate measures derived here are somewhat different than similar previously used measures in that they are ‘scale-adjusted’; also, the spatial patterns of “Urban Sprawl” shown here shed some insight and raise interesting questions about how the dynamics of “Urban Sprawl” are changing.     

Dominant pole placement for multi-loop control systems

This paper proposes a method for multi-loop PI controller design which can achieve dominant pole placement for two input two output processes. It is an extension of the original dominant pole design (PID Controllers: Theory, Design, and Tuning, Instrument Society of America, Research Triangle park, NC, 1995.) for SISO systems. Unlike its SISO counterpart, where the controller parameters can be obtained analytically, the multi-loop version amounts to solving some coupled nonlinear equation with complex coefficients, for which closed-form formulae are not possible. A novel approach is developed to solve the equation using a “root trajectory” method, in which the solution to our pole placement problem is found from intersection points between the “root trajectories” and the positive real axis. The design procedure is given and simulation examples are provided to show the effectiveness of the proposed method and comparisons are made with the BLT method.     

An object class-uncertainty induced adaptive force and its application to a new hybrid snake

Object segmentation is of paramount interest in many imaging applications, especially, those involving numeric, symbolic, syntactic, or even high level cognitive knowledge perception. Among others, “snake”—an “active contour” model—is a popular boundary-based segmentation approach where a smooth curve is continuously deformed to lock onto an object boundary. The dynamics of a snake is governed by different internal and external forces. A major limitation of the present framework has been the difficulty of incorporating object-intensity driven features into snake dynamics so as to prevent uncontrolled expansion/contraction once the snake leaks through a weak boundary region. In this paper, a local-intensity-driven “adaptive force” is introduced into the model using object class-uncertainty theory. Given a priori knowledge of object/background intensity distributions, class-uncertainty theory yields object/background classification of every location and establishes its confidence level. It has been demonstrated earlier that confidence level is high inside homogeneous regions and low near boundaries. In the current paper, object class-uncertainty theory has been applied to control snake deformation leading to a new adaptive force acting outward (expanding) inside intensity-defined object regions and inward (squeezing) inside background regions. It has been demonstrated that the method possesses potential to resist uncontrolled expansion of a snake contour (for an expanding type) inside background after leaking through a weak boundary. Further, it has been shown that the adaptive force operates in a complementary fashion with the image intensity gradient by reducing its strength near boundaries using the confidence level of classification. Another major contribution of this paper is the formulation of a “hybrid snake” (HS)—a new model, where an initial contour is gradually deformed over a hybrid energy surface composed of some direct energies (e.g., internal energies) and other indirect energies contributed by local contour displacements over a force-field (e.g., image or user-constrained force-field). Applications of the proposed adaptive force-enabled HS on different phantom and real images have been presented and comparisons have been made with a conventional snake (CS). Finally, a quantitative comparison based on computer-generated phantoms at various levels of blur and noise has been provided.