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

A Two-Step Global Maximum Error Controller-Based TPWL MOR with POD Basis Vectors and Its Applications to MEMS

In our previous study, we have proposed a linearization point (LP) selection method based on a global maximum error controller for the trajectory piecewise-linear (TPWL) method. It has been demonstrated that this method has many advantages over other existing methods. In this paper, a more efficient version of this method is presented, which introduces a preliminary LP selection procedure and constructs projection matrix by the proper orthogonal decomposition (POD) method. Compared with the original method, the improved method takes much less time for extracting a reduced-order model (ROM) of similar quality and gets some other benefits (such as being easier to implement, having lower memory requirement, and enhanced flexibility). The effectiveness of the new method is fully demonstrated by a diode transmission line RLC circuit. And then, the method is applied to three more complicated microelectromechanical systems (MEMS) devices, which are a micromachined switch, an electrostatic micropump diaphragm, and a thermomechanical in-plane microactuator

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

Stabilizing schemes for piecewise-linear reduced order models via projection and weighting functions

Abstract—In this paper we present several results concerning the stabilization of piecewise-linear reduced order models. We include proofs of internal and external stability for models whose system matrices possess special structures. We then introduce a new projection scheme, and a new set of weighting functions which allow us to extend some of these results to piecewise-linear systems comprised of arbitrary matrices, at least one of which is Hurwitz. Included are an algorithm for creating switching piecewise-linear reduced models comprised of globally exponentially stable systems, and stable simulation results for a system which produces unstable results when using the standard TPWL method. I

Stability-preserving model reduction for linear and nonlinear systems arising in analog circuit applications

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2010.Cataloged from PDF version of thesis.Includes bibliographical references (p. 221-229).Despite the increasing presence of RF and analog components in personal wireless electronics, such as mobile communication devices, the automated design and optimization of such systems is still an extremely challenging task. This is primarily due to the presence of both parasitic elements and highly nonlinear elements, which makes simulation computationally expensive and slow. The ability to generate parameterized reduced order models of analog systems could serve as a first step toward the automatic and accurate characterization of geometrically complex components and subcircuits, eventually enabling their synthesis and optimization. This thesis presents techniques for reduced order modeling of linear and nonlinear systems arising in analog applications. Emphasis is placed on developing techniques capable of preserving important system properties, such as stability, and parameter dependence in the reduced models. The first technique is a projection-based model reduction approach for linear systems aimed at generating stable and passive models from large linear systems described by indefinite, and possibly even mildly unstable, matrices. For such systems, existing techniques are either prohibitively computationally expensive or incapable of guaranteeing stability and passivity. By forcing the reduced model to be described by definite matrices, we are able to derive a pair of stability constraints that are linear in terms of projection matrices.(cont.) These constraints can be used to formulate a semidefinite optimization problem whose solution is an optimal stabilizing projection framework. The second technique is a projection-based model reduction approach for highly nonlinear systems that is based on the trajectory piecewise linear (TPWL) method. Enforcing stability in nonlinear reduced models is an extremely difficult task that is typically ignored in most existing techniques. Our approach utilizes a new nonlinear projection in order to ensure stability in each of the local models used to describe the nonlinear reduced model. The TPWL approach is also extended to handle parameterized models, and a sensitivity-based training system is presented that allows us to efficiently select inputs and parameter values for training. Lastly, we present a system identification approach to model reduction for both linear and nonlinear systems. This approach utilizes given time-domain data, such as input/output samples generated from transient simulation, in order to identify a compact stable model that best fits the given data. Our procedure is based on minimization of a quantity referred to as the 'robust equation error', which, provided the model is incrementally stable, serves as up upper bound for a measure of the accuracy of the identified model termed 'linearized output error'. Minimization of this bound, subject to an incremental stability constraint, can be cast as a semidefinite optimization problem.by Bradley Neil Bond.Ph.D

Parameterized model order reduction for nonlinear dynamical systems

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2006.Includes bibliographical references (p. 67-70).The presence of several nonlinear analog circuits and Micro-Electro-Mechanical (MEM) components in modern mixed signal System-on-Chips (SoC) makes the fully automatic synthesis and optimization of such systems an extremely challenging task. The research presented in this thesis concerns the development of techniques for generating Parameterized Reduced Order Models (PROMs) of nonlinear dynamical systems. Such reduced order models could serve as a first step towards the automatic and accurate characterization of geometrically complex components and subcircuits, eventually enabling their synthesis and optimization. This work combines elements from a non-parameterized trajectory piecewise linear method for nonlinear systems with a moment matching paramneterized technique for linear systems. Exploiting these two methods one can create four different algorithms or generating PROMs of nonlinear systems. The algorithms were tested on three different systems: a MEM switch and two nonlinear analog circuits. All three examples contain distributed strong nonlinearities and possess dependence on several geometric parameters.(cont.) Using the proposed algorithms, the local and global parameter-space accuracy of the reduced order models can be adjusted as desired. Models call be created which are extremely accurate over a narrow range of parameter values. as well as models which are less accurate locally but still provide adequate accuracy over a much wider range of parameter values.by Bradley N. Bond.S.M

Scalable trajectory methods for on-demand analog macromodel extraction.

ABSTRACT Trajectory methods sample the state trajectory of a circuit as it simulates in the time domain, and build macromodels by reducing and interpolating among the linearizations created at a suitably spaced subset of the time points visited during training simulations. Unfortunately, moving from simple to industrial circuits requires more extensive training, which creates models too large to interpolate efficiently. To make trajectory methods practical, we describe a scalable interpolation architecture, and the first implementation of a complete trajectory "infrastructure" inside a full SPICE engine. The approach supports arbitrarily large training runs, automatically prunes redundant trajectory samples, supports limited hierarchy, enables incremental macromodel updates, and gives 3-10X speedups for larger circuits

A NOVEL AUTOMATED MODEL GENERATION ALGORITHM FOR HIGH LEVEL FAULT MODELING OF ANALOG CIRCUITS

gh level modelling techniques have been used by researchers from few decades to increase fault simulation speed of analog circuits. However, due to manual model generation, the techniques are tedious and time consuming and unable to reduce analog testing time. To overcome manual modelling limitation, researchers adopt algorithmic support and start using automated model generation (AMG) methods to generate models for high level modelling of analog circuits. AMG models successfully perform HLFM but unfortunately fail to increase high level fault simulation (HLFS) speed compared to full SPICE-circuit simulations. The failure is mainly occurred due to the consumption of multiple models and computational overhead of model switching required capturing nonlinear effects

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