Model order reduction and sensitivity analysis

The electronics industry provides the core technology for numerous industrial innovations. Progress in the area of microelectronics is highlighted by several milestones in chip technology, for example microprocessors and memory chips. The ongoing increase in performance and memory density would not have been possible without the extensive use of computer simulation techniques, especially electronic circuit simulation. The basis of the latter is formed by a sound framework of methods from the area of numerical methods. In recent years, the demands on the capabilities of circuit simulation have become even more stringent. Circuit simulators have become the core of all simulations within the electronics industry. Crosstalk effects in interconnect structures are modeled by large extracted RLC networks. Also, substrate effects that start playing a crucial role in determining the performance are modeled by extracting, again, large resistive or RC networks. New algorithms are needed to cope with such situations that are extremely crucial for designers. The complexity caused by these parasitic extractions must be reduced to facilitate the simulation of the circuit while preserving accuracy. Fortunately, highly accurate parasitic extraction is not necessary for all parts of the design. Each layout contains critical blocks or paths whose timing and performance is crucial for the overall functionality of the chip. High precision interconnect modeling must be used for these circuit parts to verify the functionality of the design. On the other hand, there is interconnect outside of critical paths which adds to the complexity but whose exact model is not necessary and can be simplified. For the critical paths a so-called sensitivity analysis can bring a major achievement in speed-up, by automatically determining the critical parasitic elements that provide the most dominant influence. Another important aspect is the fact that there is an increasing deviation between design and manufacturing. Due to the ever decreasing feature sizes in modern chips, deviations from the intended dimensions are becoming more probable. Designers need to cope with this, and design the circuits in such a way that a deviation from intended dimensions does not alter the functionality of the circuit. In order to investigate this properly, one needs to assume that all components can possibly be slightly different after manufacturing.The effects this has on the performance of the circuit can be studied by introducing many thousands or even millions of parameters, describing the deviations, and performing a sensitivity analysis of the circuit w.r.t. parameter changes. The aforementioned problems form the inspiration for the study in this thesis. Sensitivity analysis is crucial for the correctness of virtual design environments based on electronic circuit simulators, and gives designers insight in how to alter the designs in order to guarantee more robustness with respect to variability in the design. The problem is that a thorough sensitivity analysis requires derivatives of the solution with respect to a large amount of parameters. This is not feasible using classical methods, being far too time-consuming for modern circuits. Recently proposed methods using the adjoint problem to calculate sensitivities are far more efficient, and these form the basis for our methodology. Our work has concentrated on making such methods even more efficient, by mixing them with concepts from the area of model order reduction. This leads to very efficient, robust and accurate methods for sensitivity analysis, even if the underlying circuit is large and the number of parameters is excessive

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

Theoretical and practical aspects of linear and nonlinear model order reduction techniques

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2008.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Includes bibliographical references (p. 133-142).Model order reduction methods have proved to be an important technique for accelerating time-domain simulation in a variety of computer-aided design tools. In this study we present several new techniques for model reduction of the large-scale linear and nonlinear systems. First, we present a method for nonlinear system reduction based on a combination of the trajectory piecewise-linear (TPWL) method with truncated-balanced realizations (TBR). We analyze the stability characteristics of this combined method using perturbation theory. Second, we describe a linear reduction method that approximates TBR model reduction and takes advantage of sparsity of the system matrices or available accelerated solvers. This method is based on AISIAD (approximate implicit subspace iteration with alternate directions) and uses low-rank approximations of a system's gramians. This method is shown to be advantageous over the common approach of independently approximating the controllability and observability gramians, as such independent approximation methods can be inefficient when the gramians do not share a common dominant eigenspace. Third, we present a graph-based method for reduction of parameterized RC circuits. We prove that this method preserves stability and passivity of the models for nominal reduction. We present computational results for large collections of nominal and parameter-dependent circuits. Finally, we present a case study of model reduction applied to electroosmotic flow of a marker concentration pulse in a U-shaped microfluidic channel, where the marker flow in the channel is described by a three-dimensional convection-diffusion equation. First, we demonstrate the effectiveness of the modified AISIAD method in generating a low order models that correctly describe the dispersion of the marker in the linear case; that is, for the case of concentration-independent mobility and diffusion constants.(cont) Next, we describe several methods for nonlinear model reduction when the diffusion and mobility constants become concentration-dependent.by Dmitry Missiuro Vasilyev.Ph.D

A least squares approach to reduce stable discrete linear systems preserving their stability

AbstractA new stability preserving model reduction algorithm for discrete linear SISO-systems based on a least squares approach is proposed. Similar to the Padé approximation, an equation system for the Markov parameters involving a high dimensional Hankel matrix is considered. It is proved that approximate solutions, computed via the Moore–Penrose pseudo-inverse, give rise to a stability preserving reduction scheme. Furthermore, the proposed algorithm is compared to the balanced truncation method, showing comparable performance of the reduced systems

Transient simulation of complex electronic circuits and systems operating at ultra high frequencies

The electronics industry worldwide faces increasingly difficult challenges in a bid to produce ultra-fast, reliable and inexpensive electronic devices. Electronic manufacturers rely on the Electronic Design Automation (EDA) industry to produce consistent Computer A id e d Design (CAD) simulation tools that w ill enable the design of new high-performance integrated circuits (IC), the key component of a modem electronic device. However, the continuing trend towards increasing operational frequencies and shrinking device sizes raises the question of the capability of existing circuit simulators to accurately and efficiently estimate circuit behaviour. The principle objective of this thesis is to advance the state-of-art in the transient simulation of complex electronic circuits and systems operating at ultra high frequencies. Given a set of excitations and initial conditions, the research problem involves the determination of the transient response o f a high-frequency complex electronic system consisting of linear (interconnects) and non-linear (discrete elements) parts with greatly improved efficien cy compared to existing methods and with the potential for very high accuracy in a way that permits an effective trade-off between accuracy and computational complexity. High-frequency interconnect effects are a major cause of the signal degradation encountered b y a signal propagating through linear interconnect networks in the modem IC. Therefore, the development of an interconnect model that can accurately and efficiently take into account frequency-dependent parameters of modem non-uniform interconnect is of paramount importance for state-of-art circuit simulators. Analytical models and models based on a set of tabulated data are investigated in this thesis. Two novel, h igh ly accurate and efficient interconnect simulation techniques are developed. These techniques combine model order reduction methods with either an analytical resonant model or an interconnect model generated from frequency-dependent sparameters derived from measurements or rigorous full-wave simulation. The latter part o f the thesis is concerned with envelope simulation. The complex mixture of profoundly different analog/digital parts in a modern IC gives rise to multitime signals, where a fast changing signal arising from the digital section is modulated by a slower-changing envelope signal related to the analog part. A transient analysis of such a circuit is in general very time-consuming. Therefore, specialised methods that take into account the multi-time nature o f the signal are required. To address this issue, a novel envelope simulation technique is developed. This technique combines a wavelet-based collocation method with a multi-time approach to result in a novel simulation technique that enables the desired trade-off between the required accuracy and computational efficiency in a simple and intuitive way. Furthermore, this new technique has the potential to greatly reduce the overall design cycle

SCEE 2008 book of abstracts : the 7th International Conference on Scientific Computing in Electrical Engineering (SCEE 2008), September 28 – October 3, 2008, Helsinki University of Technology, Espoo, Finland

This report contains abstracts of presentations given at the SCEE 2008 conference.reviewe

System- and Data-Driven Methods and Algorithms

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 two-volume handbook covers methods as well as applications. This first volume focuses on real-time control theory, data assimilation, real-time visualization, high-dimensional state spaces and interaction of different reduction techniques

Instrumented Footwear and Machine Learning for Gait Analysis and Training

Gait analysis allows clinicians and researchers to quantitatively characterize the kinematics and kinetics of human movement. Devices that quantify gait can be either portable, such as instrumented shoes, or non-portable, such as motion capture systems and instrumented walkways. There is a tradeoff between these two classes of systems in terms of portability and accuracy. However, recent computer advances allow for the collection of meaningful data outside of the clinical setting. In this work, we present the DeepSole system combined with the different neural network models. This system is a fully capable to characterize the gait of the individuals and provide vibratory feedback to the wearer. Thanks to the flexible construction and its wireless capabilities, it can be comfortably worn by wide arrange of people, both able-bodied and people with pathologies that affect their gait. It can be used for characterization, training, and as an abstract sensor to measure human gait in real-time. Three neural network models were designed and implemented to map the sensors embedded in the DeepSole system to gait characteristics and events. The first one is a recurrent neural network that classifies the gait into the correct gait phase of the wearer. This model was validated with data from healthy young adults and children with Cerebral Palsy. Furthermore, this model was implemented in real-time to provide vibratory feedback to healthy young adults to create temporal asymmetry on the dominant side during regular walking. During the experiment, the subjects who walked had an increased stance time on both sides, but the dominant side was affected more. The second model is encoder-decoder recurrent neural network that maps the sensors into current gait cycle percentage. This model is useful to provide continuous feedback that is synchronized to the gait. This model was implemented in real-time to provide vibratory feedback to six muscle groups used during regular walking. The effects of the vibration were analyzed. It was found that depending on the feedback, the subjects changed their spatial and temporal gait parameters. The third model uses all the sensors in the instrumented footwear to identify a motor phenomenon called freezing of gait in patients with Parkinson’s Disease. This phenomenon is characterized by transient periods, usually lasting for several seconds, in which attempted ambulation is halted. The model has better performance than the state-of-the-art and does not require any pre-processing. The DeepSole system when used in conjunction with the presented models is able to characterize and provide feedback in a wide range of scenarios. The system is portable, comfortable, and can accommodate a wide range of populations who can benefit from this wearable technology