Projection-Based Reduced Order Modeling for Spacecraft Thermal Analysis

This paper presents a mathematically rigorous, subspace projection-based reduced order modeling (ROM) methodology and an integrated framework to automatically generate reduced order models for spacecraft thermal analysis. Two key steps in the reduced order modeling procedure are described: (1) the acquisition of a full-scale spacecraft model in the ordinary differential equation (ODE) and differential algebraic equation (DAE) form to resolve its dynamic thermal behavior; and (2) the ROM to markedly reduce the dimension of the full-scale model. Specifically, proper orthogonal decomposition (POD) in conjunction with discrete empirical interpolation method (DEIM) and trajectory piece-wise linear (TPWL) methods are developed to address the strong nonlinear thermal effects due to coupled conductive and radiative heat transfer in the spacecraft environment. Case studies using NASA-relevant satellite models are undertaken to verify the capability and to assess the computational performance of the ROM technique in terms of speed-up and error relative to the full-scale model. ROM exhibits excellent agreement in spatiotemporal thermal profiles (<0.5% relative error in pertinent time scales) along with salient computational acceleration (up to two orders of magnitude speed-up) over the full-scale analysis. These findings establish the feasibility of ROM to perform rational and computationally affordable thermal analysis, develop reliable thermal control strategies for spacecraft, and greatly reduce the development cycle times and costs

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

Generalized nonlinear timing/phase macromodeling: Theory, numerical methods and applications

Abstract—We extend the concept of timing/phase macromodels, pre-viously established rigorously only for oscillators, to apply to gen-eral systems, both non-oscillatory and oscillatory. We do so by first establishing a solid foundation for the timing/phase response of any nonlinear dynamical system, then deriving a timing/phase macromodel via nonlinear perturbation analysis. The macromodel that emerges is a scalar, nonlinear time-varying equation that accurately characterizes the system’s phase/timing responses. We establish strong links of this technique with projection frameworks for model order reduction. We then present numerical methods to compute the phase model. The computation involves a full Floquet decomposition – we discuss numerical issues that arise if direct computation of the monodromy matrix is used for Floquet analysis, and propose an alternative method that are numerically superior. The new method has elegant connections to the Jacobian matrix in harmonic balance method (readily available in most RF simulators). We validate the technique on several highly nonlinear systems, in-cluding an inverter chain and a firing neuron. We demonstrate that the new scalar nonlinear phase model captures phase responses under various types of input perturbations, achieving accuracies consider-ably superior to those of reduced models obtained using LTI/LPTV MOR methods. Thus, we establish a powerful new way to extract timing models of combinatorial/sequential systems and memory (e.g., SRAMs/DRAMs), synchronization systems based on oscillator enslaving (e.g., PLLs, injection-locked oscillators, CDR systems, neural processing, energy grids), signal-processing blocks (e.g., ADCs/DACs, FIR/IIR filters), etc.. I

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

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

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

Snapshot-Based 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 second volume focuses on applications in engineering, biomedical engineering, computational physics and computer science

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 two-volume handbook covers methods as well as applications. This second volume focuses on applications in engineering, biomedical engineering, computational physics and computer science

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