Stability and robustness analysis of nonlinear systems via contraction metrics and SOS programming

A wide variety of stability and performance questions about linear dynamical systems can be reformulated as convex optimization problems involving linear matrix inequalities (LMIs). These techniques have been recently extended to nonlinear systems with polynomial or rational dynamics through the use of sum of squares (SOS) programming.In this paper we further extend the class of systems that can be analyzed with convexity-based methods. We show how to analyze the robust stability properties of uncertain nonlinear systems with polynomial or rational dynamics, via contraction analysis and SOS programming. Since the existence of a global contraction metric is a sufficient condition for global stability of an autonomous system, we develop an algorithm for finding such contraction metrics using SOS programming. The search process is made computationally tractable by relaxing matrix definiteness constraints, the feasibility of which indicates the existence of a contraction metric, to SOS constraints on polynomial matrices. We illustrate our results through examples from the literature and show how our contraction-based approach offers advantages when compared with traditional Lyapunov analysis.     

Computing sum of squares decompositions with rational coefficients

Sum of squares (SOS) decompositions for nonnegative polynomials are usually computed numerically, using convex optimization solvers. Although the underlying floating point methods in principle allow for numerical approximations of arbitrary precision, the computed solutions will never be exact. In many applications such as geometric theorem proving, it is of interest to obtain solutions that can be exactly verified. In this paper, we present a numeric–symbolic method that exploits the efficiency of numerical techniques to obtain an approximate solution, which is then used as a starting point for the computation of an exact rational result. We show that under a strict feasibility assumption, an approximate solution of the semidefinite program is sufficient to obtain a rational decomposition, and quantify the relation between the numerical error versus the rounding tolerance needed. Furthermore, we present an implementation of our method for the computer algebra system Macaulay 2.     

快速AVS帧内预测算法

针对AVS(Audio Video code Standard)帧内预测中选择最优编码模式流程复杂的问题,提出一种快速帧内预测编码的算法。通过分析AVS帧内预测编码的原理及其在参考软件GDM2.1中的代码实现,推导出判定零系数块的阈值。该快速算法根据零系数块的判定阈值条件判定当前块,对于零系数块,直接编码cbp元素,提前中止帧内预测编码,避免了后续的复杂流程;对于非零系数块,采用时域SAD(Sum of Absolute Difference)与频域SATD(Sum of Absolute Transformed Difference)相联合的快速选择算法,排除了40%~80%的不可能预测模式,对余下的模式采用率失真代价函数的方法选取最佳预测模式。经实验验证,在PSNR(Peak Signal to Noise Ratio)降低0.25 db和码率增加2%之内,AVS编码时间减少了40%以上,大幅度提高了编码效率。     

Optimized algorithms for predictive range and KNN queries on moving objects

There have been many studies on management of moving objects recently. Most of them try to optimize the performance of predictive window queries. However, not much attention is paid to two other important query types: the predictive range query and the predictive k nearest neighbor query. In this article, we focus on these two types of queries. The novelty of our work mainly lies in the introduction of the Transformed Minkowski Sum, which can be used to determine whether a moving bounding rectangle intersects a moving circular query region. This enables us to use the traditional tree traversal algorithms to perform range and kNN searches. We theoretically show that our algorithms based on the Transformed Minkowski Sum are optimal in terms of the number of tree node accesses. We also experimentally verify the effectiveness of our technique and show that our algorithms outperform alternative approaches.     

State estimation of nonlinear dynamical systems using nonlinear update based Unscented Gaussian Sum Filter

Two attractive features of Unscented Kalman Filter (UKF) are: (1) use of deterministically chosen points (called sigma points), and (2) only a linear dependence of the number of sigma points on the number of states. However, an implicit assumption in UKF is that the prior conditional state probability density and the state and measurement noise densities are Gaussian. To avoid the restrictive Gaussianity assumption, Gaussian Sum-UKF (GS-UKF) has been proposed in literature that approximates all the underlying densities using a sum of Gaussians. However, the number of sigma points required in this approach is significantly higher than in UKF, thereby making GS-UKF computationally intensive. In this work, we propose an alternate approach, labeled Unscented Gaussian Sum Filter (UGSF), for state estimation of nonlinear dynamical systems, corrupted by Gaussian state and measurement noises. Our approach uses a Sum of Gaussians to approximate the non-Gaussian prior density. A key feature of this approximation is that it is based on the same number of sigma points as used in UKF, thereby resulting in similar computational complexity as UKF. We implement the proposed approach on two nonlinear state estimation case studies and demonstrate its utility by comparing its performance with UKF and GS-UKF.     

Sum‐of‐Squares‐Based Finite‐Time Adaptive Sliding Mode Control of Uncertain Polynomial Systems With Input Nonlinearities

This paper proposes a novel adaptive sliding mode control (ASMC) for a class of polynomial systems comprising uncertain terms and input nonlinearities. In this approach, a new polynomial sliding surface is proposed and designed based on the sum‐of‐squares (SOS) decomposition. In the proposed method, an adaptive control law is derived such that the finite‐time reachability of the state trajectories in the presence of input nonlinearity and uncertainties is guaranteed. To do this, it is assumed that the uncertain terms are bounded and the input nonlinearities belong to sectors with positive slope parameters. However, the bound of the uncertain terms is unknown and adaptation law is proposed to effectively estimate the uncertainty bounds. Furthermore, based on a novel polynomial Lyapunov function, the finite‐time convergence of the sliding surface to a pre‐chosen small neighborhood of the origin is guaranteed. To eliminate the time derivatives of the polynomial terms in the stability analysis conditions, the SOS variables of the Lyapunov matrix are optimally selected. In order to show the merits and the robust performance of the proposed controller, chaotic Chen system is provided. Numerical simulation results demonstrate chattering reduction in the proposed approach and the high accuracy in estimating the unknown parameters.     

覆盖Value集

覆盖粗糙集和Vague集都是处理不确定性问题的数学工具,它们分别是粗糙集和模糊集的扩展。已有的覆盖粗糙集模型在求上、下近似时,可能将一些实际上并非肯定属于给定集合的元素纳入到下近似中,而一些可能属于给定集合的元素却没有纳入到上近似中,这就会改变一些元素与给定集合的关系。通过深入分析论域中的元素与其相关覆盖元之间的关系,建立了覆盖Vague集。该覆盖Vague集能够从一种新的角度反映出论域中各元素与给定集合之间的从属程度。进一步研究了覆盖Vague集与覆盖粗糙集中一些重要概念之间的关系。最后讨论了当覆盖退化为划分时覆盖Vague集的特性。     

用弹性力学方法求解圆形平面体的应力问题

求解有关圆形平面单联体在边界上受各种外力作用时的应力解的问题 ,采用了弹性力学中关于半平面体在边界上受集中力作用时的应力解 ,以及应力叠加原理 ,将这两者结合起来 ,该方法特别适用于圆形平面体在边界上受平衡外力作用时的情况 ,并通过具体实例加以说明。此外 ,该方法了可以推广到其它形状的平面单联体上。     

矿业投资风险分析

本文提出了一种新的风险分析方法——方差和法。它使用三角形分布来代替各种分布。因此可简单地按有关专家的经验用最低极限值,最可能值与最高极限值三点来确定概率密度的分布特性。它可以在矿业工程经济评价初期数据资料不充分的情况下应用。     

用于线纹显微图像的边缘检测算法

为了在简化计算的同时达到较高的定位精度,提出一种轴向邻域和差边缘检测算法,用于低信噪比、缓慢过渡的微结构显微图像的边缘检测。首先,结合显微图像采集系统的配置,分析了线纹显微图像边缘灰度轮廓特征和基于导数的边缘检测算法的不足。然后,基于方向信息测度,定义了轴向邻域和差运算,依据矩不变理论推导出轴向邻域和差边缘检测算法。实验结果表明:轴向邻域和差边缘检测算法能够适应不同分辨率的显微图像,具有较强的抗噪能力和较高的定位精度,边缘检测效果优于基于导数的算法。该算法用于显微图像时,其边缘坐标定位方差为0.57pixel,微米级线条宽度的测量结果与扫描电子显微镜的测量结果(1.35μm)相差0.17μm,基本满足了测量精度的要求。