Inverted pendulum stabilization: Characterization of codimension-three triple zero bifurcation via multiple delayed proportional gains

The paper considers the problem of stabilization of systems possessing a multiple zero eigenvalue at the origin. The controller that we propose, uses multiple delayed measurements instead of derivative terms. Doing so, we increase the performances of the closed loop in presence of system uncertainties and/or noisy measurements. The problem formulation and the analysis is presented through a classical engineering problem which is the stabilization of an inverted pendulum on a cart moving horizontally. On one hand, we perform a nonlinear analysis of the center dynamics described by a three dimensional system of ordinary differential equations with a codimension-three triple zero bifurcation. On the other hand, we present the complementary stability analysis of the corresponding linear time invariant system with two delays describing the behavior around the equilibrium. The aim of this analysis is to characterize the possible local bifurcations. Finally, the proposed control scheme is numerically illustrated and discussed.     

Intriguing nonlinear dynamics of a controller with a sluggish actuator

We apply a bifurcation-theoretic approach to the standard Moore–Greitzer compressor model in effort to understand a curious phenomenon. When one forces the actuator to respond sluggishly to commanded inputs, performance of the controller improves dramatically. We show that the behavior can be explained via Hopf–Hopf interactions of surge and rotating stall. Also, we argue that there is a broad class of controllers which possess the general dynamic features responsible for the surprising effectiveness of sluggish actuators.     

Bifurcation points and bifurcated branches by an asymptotic numerical method and Padé approximants

The aim of this work is to develop a reliable and fast algorithm to compute bifurcation points and bifurcated branches. It is based upon the asymptotic numerical method (ANM) and Padé approximants. The bifurcation point is detected by analysing the poles of Padé approximants or by evaluating, along the computed solution branch, a bifurcation indicator well adapted to ANM. Several examples are presented to assess the effectiveness of the proposed method, that emanate from buckling problems of thin elastic shells. Especially problems involving large rotations are discussed. Copyright © 2004 John Wiley & Sons, Ltd.     

壳体的有限元线法分析(Ⅰ)——基本理论

本文从退化壳理论[6]出发构造了任意曲面壳体的四边形有限元线法[1][2]单元。该单元满足 连续，为协调单元。对于所构造的单元，本文从最小势能原理出发推导出用该单元作壳体静力计算的控制微分方程和边界条件，得到一致的线法方程体系。全文共分两篇，此为上篇，主要介绍基本理论，数值算例将在下篇中给出。     

Application of Galerkin Method to Dynamical Behavior of Viscoelastic Timoshenko Beam with Finite Deformation

The motion equations governing the dynamical behavior of a viscoelasticTimoshenko beam with finite deformation are derived and simplified byGalerkin method. The viscoelastic material is assumed to obey thethree-dimensional fractional derivative constitutive relation. Thedynamical behaviors of the simplified systems with order 1 and order 2are numerically computed and compared by using the computational methodpresented by the authors. The dynamical behaviors of the systems areuniform qualitatively, but there is a little deviation quantitatively.And the truncated system with order 1 is safer than the one of order 2.It is also shown that the lower order system is reasonable. Theinfluences of the load parameter and the fractional derivative parameter(material parameter) on the deflection of the beam are consideredrespectively. The numerical methods in nonlinear dynamics, such as phasediagram, and Poincaré section, are applied to reveal dynamical behaviorsof the nonlinear viscoelastic Timoshenko beam. There are plenty ofdynamical behaviors, such as periodicity, bifurcation, quasi-periodicityand chaos in the dynamical system.     

含损伤软化材料层合板承压时的分叉分析

本文讨论了由纤维和树脂-损伤应变软化材料组成的层合板在面内受压时的压缩分叉。用有限变开理论和半解析有限元法建立了相应的分叉方程,并进行数值模拟。计算结果表明,分叉将在损伤材料的应变软化阶段发生,为剪切带形分叉,损伤材料骨有剪应变集中,而且在纤维和树脂的界面附近有较大的剪应变突变等现象。     

Fast‐scale bifurcation in power‐factor‐correction buck‐boost converters and effects of incompatible periodicities

In this paper, we derive the discrete‐time model for the power‐factor‐correction (PFC) buck‐boost converter in terms of a stroboscopic switching map. Fast‐scale instability is analysed through a fold diagram, which exposes the periodicity of the operation as well as the locations of the critical phase angles of the line voltage at which instability begins to occur along a half‐line cycle. The asymmetrical locations of the critical phase angles along a half‐line cycle is explained in terms of ‘under‐developed’ bifurcation. Border collision bifurcations are observed and analysed in detail. Copyright © 2006 John Wiley & Sons, Ltd.     

固定化酵母粒子中生产乙醇的定态分岔行为研究

运用变分原理和打靶法将分布参数系统的非线性边值问题转化为集总参数系统的初值问题，基于延拓算法，研究了固定床中球形固定化酵母粒子表面乙醇和葡萄糖浓度随参数的变化关系，得出了单颗粒固定化酵母生产乙醇的多定态特性。结果表明：随系统尺度三的变化，方程出现分岔点；方程的解与液相主体浓度有关。在某些条件下酵母粒子表面浓度有三个定态：在多定态区域，随三的增加，方程一个解支的乙醇浓度迅速增加，从而可以在固定床中通过控制粒径而使反应器达到较高的产率。研究结果为固定床反应器的开工及控制提供了重要的理论依据。