Class of linear time-varying discrete systems

A class of linear time-varying discrete systems is considered, and closed-form solutions are obtained in different cases. Some comments on stability are also included.

UNIFORM DISSIPATIVENESS, ROBUST SYNCHRONIZATION AND LOCATION OF THE ATTRACTOR OF PARAMETRIZED NONAUTONOMOUS DISCRETE SYSTEMS

In this series of papers, we study issues related to the synchronization of two coupled chaotic discrete systems arising from secured communication. The first part deals with uniform dissipativeness with respect to parameter variation via the Liapunov direct method. We obtain uniform estimates of the global attractor for a general discrete nonautonomous system, that yields a uniform invariance principle in the autonomous case. The Liapunov function is allowed to have positive derivative along solutions of the system inside a bounded set, and this reduces substantially the difficulty of constructing a Liapunov function for a given system. In particular, we develop an approach that incorporates the classical Lagrange multiplier into the Liapunov function method to naturally extend those Liapunov functions from continuous dynamical system to their discretizations, so that the corresponding uniform dispativeness results are valid when the step size of the discretization is small. Applications to the discretized Lorenz system and the discretization of a time-periodic chaotic system are given to illustrate the general results. We also show how to obtain uniform estimation of attractors for parametrized linear stable systems with nonlinear perturbation.

Classification of discrete systems on a square lattice

We consider the classification up to a Möbius transformation of real linearizable and integrable partial difference equations with dispersion defined on a square lattice by the multiscale reduction around their harmonic solution. We show that the A1, A2, and A3 linearizability and integrability conditions constrain the number of parameters in the equation, but these conditions are insufficient for a complete characterization of the subclass of multilinear equations on a square lattice.

Steady state undamped vibrations of a class of nonlinear discrete systems

"Materials Laboratory, Contract no. AF 33(616)-5449. Project no. 7360."

The steady-state response of multidegree-of-freedom systems with a spatially localized nonlinearity

This thesis is concerned with the dynamic response of a General multidegree-of-freedom linear system with a one dimensional nonlinear constraint attached between two points. The nonlinear constraint is assumed to consist of rate-independent conservative and hysteretic nonlinearities and may contain a viscous dissipation element. The dynamic equations for general spatial and temporal load distributions are derived for both continuous and discrete systems. The method of equivalent linearization is used to develop equations which govern the approximate steady-state response to generally distributed loads with harmonic time dependence.

The qualitative response behavior of a class of undamped chainlike structures with a nonlinear terminal constraint is investigated. It is shown that the hardening or softening behavior of every resonance curve is similar and is determined by the properties of the constraint. Also examined are the number and location of resonance curves, the boundedness of the forced response, the loci of response extrema, and other characteristics of the response. Particular consideration is given to the dependence of the response characteristics on the properties of the linear system, the nonlinear constraint, and the load distribution.

Numerical examples of the approximate steady-state response of three structural systems are presented. These examples illustrate the application of the formulation and qualitative theory. It is shown that disconnected response curves and response curves which cross are obtained for base excitation of a uniform shear beam with a cubic spring foundation. Disconnected response curves are also obtained for the steady-state response to a concentrated load of a chainlike structure with a hardening hysteretic constraint. The accuracy of the approximate response curves is investigated.

Synchronization in climate dynamics and other extended systems

Synchronization is now well established as representing coherent behaviour between two or more otherwise autonomous nonlinear systems subject to some degree of coupling. Such behaviour has mainly been studied to date, however, in relatively low-dimensional discrete systems or networks. But the possibility of similar kinds of behaviour in continuous or extended spatiotemporal systems has many potential practical implications, especially in various areas of geophysics. We review here a range of cyclically varying phenomena within the Earth's climate system for which there may be some evidence or indication of the possibility of synchronized behaviour, albeit perhaps imperfect or highly intermittent. The exploitation of this approach is still at a relatively early stage within climate science and dynamics, in which the climate system is regarded as a hierarchy of many coupled sub-systems with complex nonlinear feedbacks and forcings. The possibility of synchronization between climate oscillations (global or local) and a predictable external forcing raises important questions of how models of such phenomena can be validated and verified, since the resulting response may be relatively insensitive to the details of the model being synchronized. The use of laboratory analogues may therefore have an important role to play in the study of natural systems that can only be observed and for which controlled experiments are impossible. We go on to demonstrate that synchronization can be observed in the laboratory, even in weakly coupled fluid dynamical systems that may serve as direct analogues of the behaviour of major components of the Earth's climate system. The potential implications and observability of these effects in the long-term climate variability of the Earth is further discussed. © 2010 Springer-Verlag Berlin Heidelberg.

Real time manufacturing systems simulator

A área da simulação computacional teve um rápido crescimento desde o seu apareciment, sendo actualmente uma das ciências de gestão e de investigação operacional mais utilizadas. O seu princípio baseia-se na replicação da operação de processos ou sistemas ao longo de períodos de tempo, tornando-se assim uma metodologia indispensável para a resolução de variados problemas do mundo real, independentemente da sua complexidade. Das inúmeras áreas de aplicação, nos mais diversos campos, a que mais se destaca é a utilização em sistemas de produção, onde o leque de aplicações disponível é muito vasto. A sua aplicação tem vindo a ser utilizada para solucionar problemas em sistemas de produção, uma vez que permite às empresas ajustar e planear de uma maneira rápida, eficaz e ponderada as suas operações e os seus sistemas, permitindo assim uma rápida adaptação das mesmas às constantes mudanças das necessidades da economia global. As aplicações e packages de simulação têm seguindo as tendências tecnológicas pelo que é notório o recurso a tecnologias orientadas a objectos para o desenvolvimento das mesmas. Este estudo baseou-se, numa primeira fase, na recolha de informação de suporte aos conceitos de modelação e simulação, bem como a respectiva aplicação a sistemas de produção em tempo real. Posteriormente centralizou-se no desenvolvimento de um protótipo de uma aplicação de simulação de ambientes de fabrico em tempo real. O desenvolvimento desta ferramenta teve em vista eventuais fins pedagógicos e uma utilização a nível académico, sendo esta capaz de simular um modelo de um sistema de produção, estando também dotada de animação. Sem deixar de parte a possibilidade de integração de outros módulos ou, até mesmo, em outras plataformas, houve ainda a preocupação acrescida de que a sua implementação recorresse a metodologias de desenvolvimento orientadas a objectos.

Energy-Entropy-Momentum integration of discrete thermo-visco-elastic dynamics.

A novel time integration scheme is presented for the numerical solution of the dynamics of discrete systems consisting of point masses and thermo-visco-elastic springs. Even considering fully coupled constitutive laws for the elements, the obtained solutions strictly preserve the two laws of thermo dynamics and the symmetries of the continuum evolution equations. Moreover, the unconditional control over the energy and the entropy growth have the effect of stabilizing the numerical solution, allowing the use of larger time steps than those suitable for comparable implicit algorithms. Proofs for these claims are provided in the article as well as numerical examples that illustrate the performance of the method.

Additions to the z transform table

The ztransform method is a widely used tool for the analysis and synthesis of discrete systems. In this note a table of ztransform pairs when F(z) is an irrational function of z is given. The table is also useful for obtaining closed-form sums for some infinite series.

一类线性MIMO离散系统的切换控制方法

提出一种针对线性MIMO离散系统的切换控制方法.该切换控制方法不仅利用了系统输出误差,而且利用了控制器的状态变量信息,使系统具有理想的动态响应特性.算法需要较少的被控对象信息,实时计算量小.仿真结果表明了该算法的有效性.

Convergence of attractors

The system of coupled oscillators and its time-discretization (with constant stepsize h) are considered in this paper. Under some conditions, it is showed that the discrete systems have one-dimensional global attractors l(h) converging to l which is the global attractor of continuous system.

一般广义离散系统动态输出反馈H_∞控制器设计

提出一种用于一般广义离散系统的严格真动态输出反馈H_∞控制器设计方法。首先，构造辅助广义离散系统，给出该系统的状态反馈H_∞控制器设计方法，在此基础上，用两组矩阵不等式给出使一般广义离散系统是允许的且满足H_∞范数限制的控制器存在的充分条件，并给出了控制器的解析表达式。通过解这两组矩阵不等式，即可获得所需的控制器。控制器的可解性条件由系统的系数矩阵表达，因此不需要矩阵分解，可避免由矩阵分解产生的数值问题。仿真结果证实算法的有效性。

About the Stabilization of a Nonlinear Perturbed Difference Equation

This paper investigates the local asymptotic stabilization of a very general class of instable autonomous nonlinear difference equations which are subject to perturbed dynamics which can have a different order than that of the nominal difference equation. In the general case, the controller consists of two combined parts, namely, the feedback nominal controller which stabilizes the nominal (i.e., perturbation-free) difference equation plus an incremental controller which completes the stabilization in the presence of perturbed or unmodeled dynamics in the uncontrolled difference equation. A stabilization variant consists of using a single controller to stabilize both the nominal difference equation and also the perturbed one under a small-type characterization of the perturbed dynamics. The study is based on Banach fixed point principle, and it is also valid with slight modification for the stabilization of unstable oscillatory solutions.

Analysis and optimization of stress wave propagation in two-dimensional granular crystals with defects

Granular crystals are compact periodic assemblies of elastic particles in Hertzian contact whose dynamic response can be tuned from strongly nonlinear to linear by the addition of a static precompression force. This unique feature allows for a wide range of studies that include the investigation of new fundamental nonlinear phenomena in discrete systems such as solitary waves, shock waves, discrete breathers and other defect modes. In the absence of precompression, a particularly interesting property of these systems is their ability to support the formation and propagation of spatially localized soliton-like waves with highly tunable properties. The wealth of parameters one can modify (particle size, geometry and material properties, periodicity of the crystal, presence of a static force, type of excitation, etc.) makes them ideal candidates for the design of new materials for practical applications. This thesis describes several ways to optimally control and tailor the propagation of stress waves in granular crystals through the use of heterogeneities (interstitial defect particles and material heterogeneities) in otherwise perfectly ordered systems. We focus on uncompressed two-dimensional granular crystals with interstitial spherical intruders and composite hexagonal packings and study their dynamic response using a combination of experimental, numerical and analytical techniques. We first investigate the interaction of defect particles with a solitary wave and utilize this fundamental knowledge in the optimal design of novel composite wave guides, shock or vibration absorbers obtained using gradient-based optimization methods.