A trajectory piecewise-linear approach to model order reduction of nonlinear dynamical systems

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2003.Includes bibliographical references (p. 117-126).(cont.) Finally, we present projection schemes which result in improved accuracy of the reduced order TPWL models, as well as discuss approaches leading to guaranteed stable and passive TPWL reduced-order models.In this study we discuss the problem of Model Order Reduction (MOR) for a class of nonlinear dynamical systems. In particular, we consider reduction schemes based on projection of the original state-space to a lower-dimensional space e.g. by using Krylov methods. In the nonlinear case, however, applying a projection-based MOR scheme does not immediately yield computationally efficient macromodels. In order to overcome this fundamental problem, we propose to first approximate the original nonlinear system with a weighted combination of a small set of linearized models of this system, and then reduce each of the models with an appropriate projection method. The linearized models are generated about a state trajectory of the nonlinear system corresponding to a certain 'training' input. As demonstrated by results of numerical tests, the obtained trajectory quasi-piecewise-linear reduced order models are very cost-efficient, while providing superior accuracy as compared to existing MOR schemes, based on single-state Taylor's expansions. In this dissertation, the proposed MOR approach is tested for a number of examples of nonlinear dynamical systems, including micromachined devices, analog circuits (discrete transmission line models, operational amplifiers), and fluid flow problems. The tests validate the extracted models and indicate that the proposed approach can be effectively used to obtain system-level models for strongly nonlinear devices. This dissertation also shows an inexpensive method of generating trajectory piecewise-linear (TPWL) models based on constructing the reduced models 'on-the-fly', which accelerates simulation of the system response. Moreover, we propose a procedure for estimating simulation errors, which can be used to determine accuracy of the extracted trajectory piecewise-linear reduced order models.by MichaÅ Jerzy RewieÅski.Ph.D

MODEL ORDER REDUCTION OF NONLINEAR DYNAMIC SYSTEMS USING MULTIPLE PROJECTION BASES AND OPTIMIZED STATE-SPACE SAMPLING

Model order reduction (MOR) is a very powerful technique that is used to deal with the increasing complexity of dynamic systems. It is a mature and well understood field of study that has been applied to large linear dynamic systems with great success. However, the continued scaling of integrated micro-systems, the use of new technologies, and aggressive mixed-signal design has forced designers to consider nonlinear effects for more accurate model representations. This has created the need for a methodology to generate compact models from nonlinear systems of high dimensionality, since only such a solution will give an accurate description for current and future complex systems.The goal of this research is to develop a methodology for the model order reduction of large multidimensional nonlinear systems. To address a broad range of nonlinear systems, which makes the task of generalizing a reduction technique difficult, we use the concept of transforming the nonlinear representation into a composite structure of well defined basic functions from multiple projection bases.We build upon the concept of a training phase from the trajectory piecewise-linear (TPWL) methodology as a practical strategy to reduce the state exploration required for a large nonlinear system. We improve upon this methodology in two important ways: First, with a new strategy for the use of multiple projection bases in the reduction process and their coalescence into a unified base that better captures the behavior of the overall system; and second, with a novel strategy for the optimization of the state locations chosen during training. This optimization technique is based on using the Hessian of the system as an error bound metric.Finally, in order to treat the overall linear/nonlinear reduction task, we introduce a hierarchical approach using a block projection base. These three strategies together offer us a new perspective to the problem of model order reduction of nonlinear systems and the tracking or preservation of physical parameters in the final compact model

Otimização e melhoria da modulação comportamental para os interfaces de E/S analógica e de sinal misto de alta velocidade

Doutoramento em Engenharia ElectrotécnicaA integridade do sinal em sistemas digitais interligados de alta velocidade, e avaliada através da simulação de modelos físicos (de nível de transístor) é custosa de ponto vista computacional (por exemplo, em tempo de execução de CPU e armazenamento de memória), e exige a disponibilização de detalhes físicos da estrutura interna do dispositivo. Esse cenário aumenta o interesse pela alternativa de modelação comportamental que descreve as características de operação do equipamento a partir da observação dos sinais eléctrico de entrada/saída (E/S). Os interfaces de E/S em chips de memória, que mais contribuem em carga computacional, desempenham funções complexas e incluem, por isso, um elevado número de pinos. Particularmente, os buffers de saída são obrigados a distorcer os sinais devido à sua dinâmica e não linearidade. Portanto, constituem o ponto crítico nos de circuitos integrados (CI) para a garantia da transmissão confiável em comunicações digitais de alta velocidade. Neste trabalho de doutoramento, os efeitos dinâmicos não-lineares anteriormente negligenciados do buffer de saída são estudados e modulados de forma eficiente para reduzir a complexidade da modelação do tipo caixa-negra paramétrica, melhorando assim o modelo standard IBIS. Isto é conseguido seguindo a abordagem semi-física que combina as características de formulação do modelo caixa-negra, a análise dos sinais eléctricos observados na E/S e propriedades na estrutura física do buffer em condições de operação práticas. Esta abordagem leva a um processo de construção do modelo comportamental fisicamente inspirado que supera os problemas das abordagens anteriores, optimizando os recursos utilizados em diferentes etapas de geração do modelo (ou seja, caracterização, formulação, extracção e implementação) para simular o comportamento dinâmico não-linear do buffer. Em consequência, contributo mais significativo desta tese é o desenvolvimento de um novo modelo comportamental analógico de duas portas adequado à simulação em overclocking que reveste de um particular interesse nas mais recentes usos de interfaces de E/S para memória de elevadas taxas de transmissão. A eficácia e a precisão dos modelos comportamentais desenvolvidos e implementados são qualitativa e quantitativamente avaliados comparando os resultados numéricos de extracção das suas funções e de simulação transitória com o correspondente modelo de referência do estado-da-arte, IBIS.Signal integrity (SI) simulation of high-speed digital interconnected system via transistor level models is computational expensive (e.g. CPU time and memory storage), and requires the availability of physical details information of device’s internal structure. This scenario raises the interest for a behavioral modeling alternative which describes the device’s operation characteristics based on the observed input/output (I/O) electrical signal. I/O buffers that interface memory’s interconnects have major share in the computational load containing a very active complex functional part and high numbers of pins. Particularly, output buffers/drivers are forced to distort the I/O signals due to their nonlinear dynamics. In this concern, they constitute the integrated circuit (IC) bottleneck of ensuring reliable data transmission in the high-speed digital communication link. In this PhD work, the previously neglected driver’s nonlinear dynamic effects are efficiently captured to significantly reduce the state of the art black-box parametric modeling complexities and enhance the input/output buffers information specifications (IBIS). This is achieved by following the gray-box approach that merges the features of the black-box model’s formulation, the analysis of the observed I/O electrical signals and the buffer’s physical structure properties under practical operation conditions. This approach leads to physically inspired behavioral model’s construction procedure that overcomes the issues of the previous modeling approaches by optimizing the resources used at different model’s generation steps (i.e. characterization, formulation, extraction, and implementation) to mimic the driver’s nonlinear dynamic behavior. Moreover, the most important achievement is the development of a new two-port analog behavioral model for overclocking simulation that copes with the recent trends in I/O memory interfaces characterized by higher data rate transmission. The effectiveness and the accuracy of the developed and implemented behavioral models are qualitatively and quantitatively assessed by comparing the numerical results of their functions extraction and transient simulation to the ones simulated and extracted with transistor level models and the state of the art IBIS in order to validate their predictive and the generalization capabilities

Model Order Reduction

An increasing complexity of models used to predict real-world systems leads to the need for algorithms to replace complex models with far simpler ones, while preserving the accuracy of the predictions. This three-volume handbook covers methods as well as applications. This third volume focuses on applications in engineering, biomedical engineering, computational physics and computer science

Technical accomplishments of the NASA Lewis Research Center, 1989

Topics addressed include: high-temperature composite materials; structural mechanics; fatigue life prediction for composite materials; internal computational fluid mechanics; instrumentation and controls; electronics; stirling engines; aeropropulsion and space propulsion programs, including a study of slush hydrogen; space power for use in the space station, in the Mars rover, and other applications; thermal management; plasma and radiation; cryogenic fluid management in space; microgravity physics; combustion in reduced gravity; test facilities and resources

90th Annual Meeting of the Virginia Academy of Science: Proceedings

Full proceedings of the 90th Annual Meeting of the Virginia Academy of Science, Norfolk State University, Norfolk Virginia, May 23-25, 201

Aeronautical engineering. A continuing bibliography with indexes, supplement 114

This bibliography lists 394 reports, articles, and other documents introduced into the NASA scientific and technical information system in September 1979