A quintic polynomial differential system with eleven limit cycles at the infinity

In this article, a recursion formula for computing the singular point quantities of the infinity in a class of quintic polynomial systems is given. The first eleven singular point quantities are computed with the computer algebra system Mathematica. The conditions for the infinity to be a center are derived as well. Finally, a system that allows the appearance of eleven limit cycles in a small enough neighborhood of the infinity is constructed.     

Pseudo-isochronicity in a class of septic differential systems

This paper deals with the pseudo-isochronicity for a class of septic differential systems. In this paper, we transform infinity into the origin so that the properties of infinity can be investigated with the methods developed for finite critical points. By calculating the singular point quantities and period constants of the origin, the problem of infinity being a pseudo-isochronous center has been solved in this case.     

GENERALIZED QUADRATIC STABILIZATION FOR DISCRETE‐TIME SINGULAR SYSTEMS WITH TIME‐DELAY AND NONLINEAR PERTURBATION

This paper discusses a generalized quadratic stabilization problem for a class of discrete‐time singular systems with time‐delay and nonlinear perturbation (DSSDP), which the satisfies Lipschitz condition. By means of the S‐procedure approach, necessary and sufficient conditions are presented via a matrix inequality such that the control system is generalized quadratically stabilizable. An explicit expression of the static state feedback controllers is obtained via some free choices of parameters. It is shown in this paper that generalized quadratic stability also implies exponential stability for linear discrete‐time singular systems or more generally, DSSDP. In addition, this new approach for discrete singular systems (DSS) is developed in order to cast the problem as a convex optimization involving linear matrix inequalities (LMIs), such that the controller can stabilize the overall system. This approach provides generalized quadratic stabilization for uncertain DSS and also extends the existing robust stabilization results for non‐singular discrete systems with perturbation. The approach is illustrated here by means of numerical examples.     

A high-order integral algorithm for highly singular PDE solutions in Lipschitz domains

We present a new algorithm, based on integral equation formulations, for the solution of constant-coefficient elliptic partial differential equations (PDE) in closed two-dimensional domains with non-smooth boundaries; we focus on cases in which the integral-equation solutions as well as physically meaningful quantities (such as, stresses, electric/magnetic fields, etc.) tend to infinity at singular boundary points (corners). While, for simplicity, we restrict our discussion to integral equations associated with the Neumann problem for the Laplace equation, the proposed methodology applies to integral equations arising from other types of PDEs, including the Helmholtz, Maxwell, and linear elasticity equations. Our numerical results demonstrate excellent convergence as discretizations are refined, even around singular points at which solutions tend to infinity. We demonstrate the efficacy of this algorithm through applications to solution of Neumann problems for the Laplace operator over a variety of domains—including domains containing extremely sharp concave and convex corners, with angles as small as π/100 and as large as 199π/100.     

An approach to a singular optimal control problem for non-linear systems using differential form theory

By applying differential form theory, we consider the singular control problem for non-linear systems with control variables appearing linearly in both the system dynamics and the performance index. First, we derive necessary conditions of singular optimality for a single-input system, including the relation to the Euler-Poisson equation and to the generalized Legendre-Clebsch condition. Defining the degree of singularity, we develop necessary conditions satisfied by the singular trajectory embedded in a reduced space. For a time-invariant system, we clarify the relation between the dynamic and the related static optimality. Second, we derive necessary conditions for singular optimality for a multi-input system where the dimension of the control vector is equal to that of the state space. We show that the Shima-Sawaragi condition for the optimality of boundary controls and the generalized Legendre-Clebsch condition are obtained from these conditions. The results are also applied to the analysis of a time-invariant system.     

Robust stability and stabilization of discrete singular systems: an equivalent characterization

This note deals with the problems of robust stability and stabilization for uncertain discrete-time singular systems. The parameter uncertainties are assumed to be time-invariant and norm-bounded appearing in both the state and input matrices. A new necessary and sufficient condition for a discrete-time singular system to be regular, causal and stable is proposed in terms of a strict linear matrix inequality (LMI). Based on this, the concepts of generalized quadratic stability and generalized quadratic stabilization for uncertain discrete-time singular systems are introduced. Necessary and sufficient conditions for generalized quadratic stability and generalized quadratic stabilization are obtained in terms of a strict LMI and a set of matrix inequalities, respectively. With these conditions, the problems of robust stability and robust stabilization are solved. An explicit expression of a desired state feedback controller is also given, which involves no matrix decomposition. Finally, an illustrative example is provided to demonstrate the applicability of the proposed approach.     

Limits of generalized state space systems under proportional and derivative feedback

In this paper we study the high-gain feedback classification problem for generalized state space systems. We solve this problem for proportional and derivative feedback transformations of regularizable systems, i.e., we give necessary and sufficient conditions for a regularizable system to be a limit of a given system under high-gain proportional and derivative feedback. We also derive a new complete set of invariants for proportional feedback equivalence and specify a set of necessary conditions for a system to be the limit of another system under these feedback transformations. The necessary conditions are sufficient for arbitrary state space systems and for controllable singular systems.     

An improved algorithm for the computation of structural invariants of a system pencil and related geometric aspects

In this paper we propose a new recursive algorithm for computing the staircase form of a matrix pencil, and implicitly its Kronecker structure. The algorithm compares favorably to existing ones in terms of elegance, versatility, and complexity. In particular, the algorithm without any modification yields the structural invariants associated with a generalized state-space system and its system pencil. Two related geometric aspects are also discussed: we show that an appropriate choice of a set of nested spaces related to the pencil leads directly to the staircase form; we extend the notion of deflating subspace to the singular pencil case.     

A Lyapunov approach to analysis of discrete singular systems

In this paper, a new type generalized Lyapunov equation for discrete singular systems is proposed. Then it is applied to study problems such as pole clustering, controllability and observability for discrete singular systems. First, some necessary and sufficient conditions for pole clustering are derived via the solution of this new type Lyapunov equation. Further, the relationship between the solution of the Lyapunov equation and structure properties of discrete singular systems will be investigated based on these results. Finally, a type of generalized Riccati equation is proposed and its solution is used to design state feedback law for discrete singular systems such that all the finite poles of the closed-loop systems are clustered into a specified disk.     

用动态补偿法讨论奇异系统的强模型匹配问题*

讨论了一类非线性奇异系统的强模型匹配问题,在扩展结构算法的基础上,导出了该问题中解的充要条件,并给出适当的动态补偿控制器。     

Extension of the divide-and-conquer algorithm for the efficient inverse dynamics analysis of multibody systems

This paper presents a new mathematical framework to extend the Generalized Divide-and-Conquer Algorithm (GDCA) for the inverse dynamics analysis of fully actuated constrained multibody systems. Inverse-GDCA (iGDCA) is a highly parallelizable method which does not create the mass and Jacobian matrices of the entire system. In this technique, generalized driving forces and constraint loads due to kinematic pairs are clearly and separately differentiated from each other in the equations of motion. As such, it can be easily used for control scheme purposes. iGDCA works based on a series of recursive assembly and disassembly passes to form and solve the equations governing the inverse dynamics of the system. Herein, the mathematical formulations to efficiently combine the dynamics of consecutive bodies in the assembly pass for the purpose of inverse dynamics analysis are presented. This is followed by generating the disassembly pass algorithm to efficiently compute generalized actuating forces. Furthermore, this paper presents necessary mathematical formulations to efficiently treat the inverse dynamics of multibody systems involving kinematic loops with various active and passive boundary conditions. This is followed by the design of a new strategy to efficiently perform the assembly–disassembly pass in these complex systems while avoiding unnecessary computations. Finally, the presented method is applied to selected open-chain and closed-chain multibody systems.     

广义系统的无穷远特征结构

研究了广义系统的无穷远征结构问题,给出了一种计算了广义系统的无穷远特征向量和广义无穷远特征向量的方法,在此基础上,讨了消除广义系统脉冲运动的问题,得到了保证系统无脉冲运动的充要条件。     

First-order and second-order necessary conditions for optimality of singular controls

The derivation of necessary conditions for the optimality of singular controls (in the sense of Pontryagin's maximum principle) in optimal control problems is inves- tigated. A new class of the necessary conditions is developed. This class of necessary conditions is composed of two parts, the first-order and second-order necessary conditions. As an application of these necessary conditions, an illustrative example is given, which shows that the necessary conditions developed are stronger than the generalized Legendre-Clebsch condition. That is, the singular control in the example satisfies the generalized Legendre-Clebsch condition, but does not satisfy the necessary conditions developed in this paper.     

输入约束不确定系统的点对点迭代学习控制与优化

为解决工业过程中机械臂等特殊重复运行系统的输出在有限时间内无需实现全轨迹跟踪,仅需跟踪期望轨迹上某些特殊关键点的控制问题,针对线性时不变离散系统提出一种基于范数最优的点对点迭代学习控制算法.通过输入输出时间序列矩阵模型变换构建综合性多目标点性能指标函数,求解二次型最优解得到优化迭代学习控制律,同时给出模型标称和不确定情形下最大奇异值形式鲁棒控制算法收敛的充分条件,并进一步推广得到输入约束系统优化控制算法的收敛性结果,最后在三轴龙门机器人模型上验证算法的有效性.     

基于奇异值分解的非均匀采样非线性系统的模糊模型辨识

针对复杂、不确定、非均匀采样数据的非线性系统,提出一种基于矩阵奇异值分解(SVD)的模型结构辨识和参数估计的建模方法.首先,利用矩阵奇异值(SVD)分解算法分析各局部模型与奇异值、积累贡献率的关系,确定模糊模型的规则数,从而实现模型的结构优化;然后,为了克服递推最小二乘出现的误差积累、传递现象,采用奇异值分解的递推最小二乘估计模型的结论参数;最后,通过仿真实例验证所提出算法的有效性.     

Towards an efficient evaluation of recursive aggregates in deductive databases

This paper is devoted to the evaluation of aggregates (avg, sum,…) in deductive databases. Aggregates have proved to be a necessary modeling tool for a wide range of applications in non-deductive relational databases. They also appear to be important in connection with recursive rules, as shown by thebill of materials example. Several recent papers have studied the problem of semantics for aggregate programs. As in these papers, we distinguish between the classes of stratified (non-recursive) and recursive aggregate programs. For each of these two classes, the declarative semantics is recalled and an efficient evaluation algorithm is presented. The semantics and computation of aggregate programs in the recursive case are more complex: we rely on the notion of graph traversal to motivate the semantics and the evaluation method proposed. The algorithms presented here are integrated in the QSQ framework. Our work extends the recent work on aggregates by proposing an efficient algorithm in the recursive case. Recursive aggregates have been implemented in the EKS-V1 system.     

二阶自治广义Birkhoff系统的奇点分析

建立二阶自治广义Birkhoff系统的微分方程.给出该系统的线性化方程,得到该线性方程转化为梯度系统的条件,利用梯度系统的性质对线性系统的奇点进行了分析,然后再利用Perron定理探讨了相应的非线性系统的奇点类型.结果表明,如果线性系统能成为梯度系统,那么相应的非线性系统的奇点可能是结点或者鞍点.     

A prediction-error-method for recursive identification of nonlinear systems

Nonlinear systems can be identified with nonlinear filters by combined estimation of parameters and state. Often these methods are very complex or have convergence problems, as for example the Extended Kalman Filter. In this paper a new algorithm for recursive nonlinear system identification is presented. Convergence problems are eliminated by an improved calculation of the gradient as the total derivative of the prediction error. By separately estimating parameters and states, computation time is reduced. The efficacy of the new identification algorithm is illustrated by studying its performance in a gravimetric filling system.     

基于图像传感器的圆心定位技术实用化研究

基于图像传感器的圆心定位技术适用于多种非接触式测量系统。这类技术对测量环境的要求比较苛刻。以提高圆心定位技术的适应能力为目的,提出了针对抗干扰、边界识别和圆心拟合等测量过程关键技术环节的实用化改进方案,综合运用了时间递推滤波、深度优先搜索算法和迭代最小二乘法等方法。改进后的测量系统具有良好的稳定性和适应性,在地铁隧道形变监测项目中成功投入使用。     

Generalized quadratic stability for continuous-time singular systems with nonlinear perturbation

This note considers the generalized quadratic stability problem for continuous-time singular system with nonlinear perturbation. The perturbation is a function of time and system state and satisfies a Lipschitz constraint. In this work, a sufficient condition for the existence and uniqueness of solution to the singular system is firstly presented. Then by using S-procedure and matrix inequality approach, a necessary and sufficient condition is presented in terms of linear matrix inequality, under which the maximal perturbation bound is obtained to guarantee the generalized quadratic stability of the system. That is, the system remains exponential stable and the nominal system is regular and impulse free. Furthermore, robust stability for nonsingular systems with perturbation can be obtained as a special case. Finally, the effectiveness of the developed approach for both singular and nonsingular systems is illustrated by numerical examples.