Design of orbitally stable zero dynamics for a class of nonlinear systems

A method for the generation of globally attractive limit cycles for nonlinear systems is presented. It consists in designing an output that, when regulated through a suitable feedback, forces a limit cycle in the zero dynamics. Conditions are then given to ensure that a globally attractive limit cycle in the zero dynamics results in a globally attractive limit cycle in the whole system. The method is illustrated on the torque control of an induction motor.     

浅谈《微积分》中极限的地位

从《微积分》教学结构入手，对极限的各种地位详细说明，再以实例进行深入说明，阐述极限思想在微积分中的地位和重要性，将极限思想贯穿到微积分教学当中．从而达到轻松学习《微积分》的目的。     

《微积分》教学中函数极限的计算方法

函数的极限问题是微积分的基础,极限知识掌握的好坏会直接影响到微积分后续课程的学习,但是学生常常感觉到极限知识较难掌握。结合多年的教学经验,针对函数极限部分知识的学习提出一些相应的教学方案。     

一类切换系统极限环的反馈镇定研究

利用多Lyapunov函数法讨论一类切换系统极限环的反馈镇定问题．首先给出切换系统极限环的有界状态反馈、有界输出反馈以及动态反馈镇定定理；然后利用Jurdjevic—Quinn方法给出反馈控制器的具体形式，并证明了在给定反馈控制器的作用下，切换系统的极限环渐近稳定．仿真例子验证了结论的正确性．     

Parameter estimation from relay autotuning with asymmetric limit cycle data

Obtaining the parameters for PID controllers based on limit cycle information for the process in a relay controlled feedback loop has become an accepted practical procedure. Often the only information used is the frequency and amplitude of the limit cycle, which then using the describing function analysis provides estimates for the ultimate gain and frequency of the process. Alternatively, limit cycle measurements may be used to estimate the parameters of an assumed transfer function, usually again using describing function analysis. Both these methods can work well but may lead to problems in cases where the describing function analysis is inaccurate. In this paper, it is shown how the parameters of first and second order plant transfer functions, stable or unstable, with time delay can be found exactly, assuming no measurement errors, from measurements of the parameters of a single asymmetrical limit cycle in a relay controlled feedback loop. Examples are given to compare the approach with results using the describing function method and also to show how good controller design based on a first or second order time delay model can be achieved for plants of higher order.     

Computable majorants of the limit load in Hencky’s plasticity problems

We propose a new method for analyzing the limit (safe) load of elastoplastic media governed by the Hencky plasticity law and deduce fully computable bounds of this load. The main idea of the method is based on a combination of kinematic approach and new estimates of the distance to the set of divergence free fields. We show that two sided bounds of the limit load are sharp and the computational efficiency of the method is confirmed by numerical experiments.     

决策树分类准确率极限的研究

采用最大分类树作为分析经验风险与结构风险的工具，对决策树分类准确率极限进行了研究。针对决策树模型的分类效果难以客观评价的问题，讨论了决策树分类准确率极限的存在条件，给出了求出该极限的方法。以最大分类树作为分析工具，提出了在经验风险和结构风险4种分布条件下分类准确率极限是否存在的4个定理，并从机器学习理论和工程建模实践2个角度进行了讨论。实验验证了该理论的正确性。     

潮流场中质点运动的可视化*

提出了一种适用于非结构化网格系统中质点运动轨迹的跟踪算法。利用数值计算方法的稳定性限制条件对质点位置的搜索算法进行了优化,极大地降低了算法的时间复杂度,提高了计算效率。用Visual Fortran语言实现了质点运动的动态显示。将其用于渤海潮流场中质点运动规律的研究,初步得出了渤海中质点运动和水交换的一些规律。     

关于汽轮机DEH控制系统超速限制功能的讨论

本交分析了传统DEH控制系统超速限制功能,指出了在甩负荷工况下,超速103％OPC超速限制功能可能引起系统振荡,提出了一种全新的超速限制方法——加速度大超速限制功能。此功能不但可以有效地限制系统甩负荷后机组转速的超调量,而且可以避免系统振荡。     

The use of walsh functions in multivariable limit cycle prediction

The describing function method has played a leading role in the prediction of limit cycles in scalar systems; in recent years this approach has been extended to the multivariable case. Derived techniques assume the nonlinearity inputs to be sinusoidal. However, this condition is not always satisfied, and as an alternative, the present paper considers the use of a method which is based on a set of non-sinusoidal functions. Novel concepts related to the characteristic loci are defined and employed for the derivation of new graphical tests.     

基于PLC的环境模拟系统温度控制算法的研究与实现

针对环境模拟试验温度控制系统中被控对象存在的非线性、时滞等特点,本文采用区间限幅PID控制算法和模糊PID控制算法对传统控制方法进行了改进。首先为了解决模拟量三通粗调阀调节缓慢的缺点,建立了区间限幅PID控制算法的控制规则表,并将其在PLC中实现。其次提出用模糊PID控制算法来解决电加热器的非线性、大时滞性问题,并结合实际控制经验建立了模糊控制规则表,然后将模糊PID控制算法在PLC中进行实现。最后将限幅PID和模糊PID控制算法应用于某大型环境模拟试验控制系统,实验结果表明利用改进算法对温度控制具有良好的稳定性及精确度。     

基于深度小波去噪自动编码器的轴承智能故障诊断方法

针对原始振动数据无监督特征学习问题,提出一种深度小波去噪自动编码器与鲁棒极限学习机相结合的滚动轴承的智能故障诊断方法。利用小波函数作为非线性激活函数设计小波去噪自动编码器,从而有效地捕获信号特征;利用多个小波去噪自动编码器构造一个深度小波去噪自动编码器来增强无监督特征学习能力;采用鲁棒极限学习机作为分类器,对不同的轴承故障进行分类识别。对实验所得的轴承振动信号进行对比分析,结果验证了该方法在原始振动数据无监督特征学习的条件下优于传统方法和标准深度学习方法。     

Prediction error methods for limit cycle data

Prediction error methods are considered for identification of the forward linear dynamics of nonlinear feedback closed-loop systems which operate in a perturbed stable limit cycle. A model of the signals measured in a neighborhood of the limit cycle is presented and shown to satisfy a quasistationarity property. Quasistationarity is then used to prove that prediction error methods are both convergent and consistent for our data model.     

Probabilistic Analysis of MULTIPLE QUICK SELECT

We investigate the distribution of the number of comparisons made by MULTIPLE QUICK SELECT (a variant of QUICK SORT for finding order statistics). By convergence in the Wasserstein metric space, we show that a limit distribution exists for a suitably normalized version of the number of comparisons. We characterize the limiting distribution by an inductive convolution and find its variance. We show that the limiting distribution is smooth and prove that it has a continuous density with unbounded support. Received November 9, 1997; revised March 10, 1998.     

Study of limit cycle in antagonistically coupled magneto-rheological actuators

In this paper, the presence of limit cycles in the behaviour of antagonistically coupled magneto-rheological (MR) actuators is investigated. The actuator considered in this paper offers high torque-to-mass and torque-to-inertia ratios, for inherent safe actuation. While the antagonistic arrangement is beneficial in improving the actuator performance and eliminating backlash, it may result in limit cycles when the actuator operates in a position control loop. The occurrence of a limit cycle depends on the parameters of the actuator as well as the controller. An in-depth analysis is carried out in this paper to establish a connection between the system parameters and the limit cycle occurrence. Moreover, sufficient conditions for avoiding limit cycles are derived specifically for a proportional-derivative (PD) controller. Simulations and experimental results validate the analysis and provide insights into conditions under which limit cycles are observed in the operation of antagonistic MR actuators.     

Heavy traffic analysis of integrated services systems

The performance analysis of integrated voice and data systems with movable boundary, ATM multiplexers, etc. is often based on Neuts' classical theory of Markov chains with matrix-geometric invariant measure or on more general theory of structured stochastic matrices of M/G/1 type. But under heavy traffic algorithmic approaches based on these theories cannot be used for computational purposes. To analyze the systems behavior under heavy traffic we suggest the use of limit theorems for scaled performance characteristics and develop a simple method to find asymptotic behavior of the mean characteristics.     

Characterization of limit cycles for self-regulating and integral processes with PI control and send-on-delta sampling

This work is focused on the study of limit cycles that appear in a control scheme which is based on the use of a PI controller with an event-based send-on-delta sampling (SOD). The processes investigated are integrator processes plus time delay (IPTD) and first and second order processes plus time delay, which are of interest because they are frequently used to model many industrial processes. The SOD sampling is characterized as a non-linearity of n levels with hysteresis. An algorithm to calculate the limit cycles properties is proposed, and then the results obtained in simulations are compared with experiments performed on a real plant, a distributed solar collector field at the Solar Platform of Almería (PSA, Spain).     

Computation of feedback control for time optimal state transfer using Groebner basis

Computation of time optimal feedback control law for a controllable linear time invariant system with bounded inputs is considered. Unlike a recent paper by the authors, the target final state is not limited to the origin of state-space, but is allowed to be in the set of constrained controllable states. Switching surfaces are formulated as semi-algebraic sets using Groebner basis based elimination theory. Using these semi-algebraic sets, a nested switching logic is synthesized to generate the time optimal feedback control. However, the optimal control law enforces an unavoidable limit cycle in the time-optimal trajectory for most non-origin target points. The time-period of this limit-cycle is dependent on the target position. This dependence is algebraically characterized and a method to compute the time-period of the limit-cycle is provided. As a natural extension, the set of constrained controllable states is also computed.     

Linear-quadratic tracking of coupled slow and fast targets

The paper examines a singular perturbation model of tracking two targets, one slow and a fast one which is assumed periodic. The plant is linear and the cost is quadratic. A limit problem is displayed. It corresponds to the limit behavior of the perturbed system when the small parameter tends to zero. This limit is not of reduced order, as the fast target is periodic. Rather, an infinite-horizon optimal averaging is performed on the fast scale, and serves as an input to slow-scale optimization. This limit design is used to approximate the optimal solution of the singularly perturbed system. Bounds for the rates of the approximation are given. The optimal limit design is computed in a feedback form in the general case, and for illustrative concrete examples.Zvi Artstein is an incumbent of the Hettie H. Heineman Professorial Chair in Mathematics. The work by Vladimir Gaitsgory was done while visiting the Department of Theoretical Mathematics, The Weizmann Institute of Science, Rehovot, Israel.     

Closed curve solutions and limit cycles in a class of second-order switched nonlinear systems

This paper studies the problem of finding the initial states for which the solution of a class of switched systems consisting of unstable second-order nonlinear subsystems is convergent. A method is described and applied to establish the regions in the plane where it is possible to define a switching law such that the solution of a class of switched nonlinear systems converges to the origin. We prove that, under certain conditions, these regions are delimited by closed curve solutions of the switched system. Furthermore, a sufficient condition for the closed curve solution to be a limit cycle is presented. Finally, a numerical example is included in order to illustrate the results.