ECE Theses and Dissertations

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https://hdl.handle.net/1911/8316

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A behavioral approach to positive interpolation
(2005) Mayo, Andrew; Antoulas, Athanasios C.
We study interpolation by positive functions from a behavioural point of view. In particular, by considering the notion of mirror image data, the interpolation problem with passivity constraint is transformed into an unconstrained behavioural modeling one. It will be shown that the generating system for this problem has to be unitary with respect to an indefinite matrix. Using this approach, several results in the theory of interpolation by positive functions are derived in a very natural manner. The use of generating systems leads in a natural way to the recent results obtained by Byrnes et al concerning parametrizing the set of interpolants by the spectral zeros. We then apply the same approach to interpolation on the boundary.
Congestion control and complexity reduction of large-scale networks
(2002) Sanayei, Shahab; Antoulas, Athanasios C.
The current version of Transport Control Protocol (TCP) does not meet the high demands of the exponentially growing Internet. The packet loss is one of the major limiting factors on the performance and the quality of services over Internet (especially for multimedia applications). Also, other improved versions of TCP (Vegas) cannot be deployed in the heterogeneous environment of current Internet. In this paper we propose a new protocol which not only enhances the network performance, but also is deployable in the large scale networks. We study the stability and the fairness of the proposed protocol in the framework of dynamical systems and finally verify the results by simulation.
Discrete-time linear periodically time-varying systems: Analysis, realization and model reduction
(2003) Hu, Jing; Antoulas, Athanasios C.
Discrete-time linear periodically time-varying (LPTV) systems, considered as a bridge between the well-studied linear time-invariant (LTI) model and the nonlinear time-varying problems in real world, have been receiving increasing attention in recent a few decades. In this research project, we try to understand discrete-time LPTV systems both internally and externally and derive basic theories for analysis, realization and model reduction of LPTV systems. Firstly we review the system model for LPTV systems, define its transfer function matrix, Markov parameters, stability, reachability and observability. Then we emphasize on the numerically efficient and stable methods to compute LPTV system grammians and to approximate the eigenvalue decay rate. Another main result of this thesis is Krylov-based moment matching algorithm for model reduction of LPTV systems, which is derived afterwards, and is also compared to the other approach: balancing and balanced truncation of LPTV systems. Almost any application of discrete-time LPTV systems, including periodic digital filters and periodic control theories, demands a periodic state-space model from input-output maps. This periodic realization problem is treated at the end of the thesis with demonstration of applicable non-minimal and quasi-minimal realization methods.
Lagrange rational interpolation and its applications to approximation of large-scale dynamical systems
(2013-11-06) Ionita, Antonio; Antoulas, Athanasios C.; Zhong, Lin; Embree, Mark
We present several new, efficient algorithms that extract low complexity models from frequency response measurements of large-scale dynamical systems. Our work is motivated by the fact that, in many applications, analytical models of a dynamical system are seldom available. Instead, we may only have access to its frequency response measurements. For example, for a system with multiple inputs and outputs, we may only have access to data sets of S-parameters. In this setting, our new approach extracts models that interpolate the given measurements. The extracted models have low complexity (or reduced order) and, thus, lead to short simulation times and low data storage requirements. The main tool used by our approach is Lagrange rational interpolation -- a generalization of the classic result of Lagrange polynomial interpolation. We present an in-depth look at Lagrange rational interpolation and provide several new insights and simplified proofs. This analysis leads to new algorithms that rely on the singular value decomposition (SVD) of the Loewner matrix pencil formed directly from the measurements. We show several new results on rational interpolation for measurements of linear, bi-linear and quadratic-linear systems. Furthermore, we generalize these results to parametrized measurements, that is, we show how to interpolate frequency response measurements that depend on parameters. We showcase this new approach through a series of relevant numerical examples such as n-port systems and parametrized partial differential equations.
Model validation and consistency
(2000) Gugercin, Serkan; Antoulas, Athanasios C.
This thesis addresses model validation, important in robust control system modeling, for the identification method developed by Antoulas. Given a system model, the problem is to assess whether the model is consistent with the data. This work formulates the validation problem in the form of a quadratic optimization problem subject to a spherical constraint. This new, computationally tractable method allows us to find a necessary and sufficient condition on the energy of the input sequence required to invalidate a given model. Therefore, for a given energy level, not all the models can be invalidated. For fixed noise level, the set of invalidatable models decreases as the energy of the input sequence decreases. Moreover, even if infinite length measurements are taken, the set of plants which cannot be invalidated does not shrink to the true model. The true model, in addition, can never be invalidated using an input of finite energy.
Modeling Systems from Measurements of their Frequency Response
(2012) Lefteriu, Sanda; Antoulas, Athanasios C.
The problem of modeling systems from frequency response measurements is of interest to many engineers. In electronics, we wish to construct a macromodel from tabulated impedance, admittance or scattering parameters to incorporate it into a circuit simulator for performing circuit analyses. Structural engineers employ frequency response functions to determine the natural frequencies and damping coefficients of the underlying structure. Subspace identification, popular among control engineers, and vector fitting, used by electronics engineers, are examples of algorithms developed for this problem. This thesis has three goals. 1. For multi-port devices, currently available algorithms arc expensive. This thesis therefore proposes an approach based on the Loewner matrix pencil constructed in the context of tangential interpolation with several possible implementations. They are fast, accurate, build low dimensional models, and are especially designed for a large number of terminals. For noise-free data, they identify the underlying system, rather than merely fitting the measurements. For noisy data, their performance is analyzed for different noise levels introduced in the measurements and an improved version, which identifies an approximation of the original system even for large noise values, is proposed. 2. This thesis addresses the problem of generating parametric models from measurements performed with respect to the frequency, but also with respect to one or more design parameters, which could relate to geometry or material properties. These models are suited for performing optimization over the design variables. The proposed approach generalizes the Loewner matrix to data depending on two variables. 3. This thesis analyzes the convergence properties of vector fitting, an iterative algorithm that relocates the poles of the model, given some "starting poles" chosen heuristically. It was recognized as a reformulation of the Sanathanan-Koerner iteration and several authors attempted to improve its convergence properties, but a thorough convergence analysis has been missing. Numerical examples show that for high signal to noise ratios, the iteration is convergent, while for low ones, it may diverge. Hence, incorporating a Newton step aims at making the iteration always convergent for "starting poles" chosen close to the solution. A connection between vector fitting and the Loewner framework is exhibited, which resolves the issue of choosing the starting poles.
New approaches to modeling multi-port scattering parameters
(2009) Lefteriu, Sanda; Antoulas, Athanasios C.
This work addresses the problem of building a macromodel from frequency response measurements by means of a stable and passive linear dynamical system in state-space representation. The proposed algorithms are based on a system-theoretic tool, the matrix pencil of the shifted Loewner and Loewner matrices. Their performance is compared with that of the widely-used vector fitting in terms of the computational time required to build such a model and the accuracy of the interpolating system, when the same order model is constructed, and it is shown that our algorithms render better models in less time. Even though the main application we have in mind is modeling the scattering parameters of an electromagnetic device, no modifications are needed when the admittance parameters are provided instead. Last, our algorithms are especially suited for devices with a large number of ports because the data matrices are collapsed into vectors.
Projection methods for model reduction of large-scale dynamical systems
(2003) Gugercin, Serkan; Antoulas, Athanasios C.
Numerical simulation of dynamical systems have been a successful means for studying complex physical phenomena. However, in large-scale settings, the system dimension makes the computations infeasible due to memory and time limitations, and ill-conditioning. The remedy is model reduction. This dissertation focuses on projection methods to efficiently construct reduced order models for large linear dynamical systems. A modified cyclic low-rank Smith method is introduced to compute low-rank approximations to solutions of large-scale Lyapunov equations. Unlike the original cyclic low-rank Smith method of Penzl, the number of columns in the modified approximant does not necessarily increase at each step and is much lower. Fundamental convergence results are established for the errors in the approximate solutions and also in the approximate Hankel singular values. For positive real balancing, this work derives a multiplicative error bound and develops a modified scheme with an absolute error bound for a certain subclass of positive real systems. Moreover, a frequency weighted balancing method with guaranteed stability and a simple Hinfinity error bound is introduced. Unlike the existing approaches, the method avoids the explicit computation of the input and output weightings. This dissertation derives an exact expression for the H2 norm of the error system of the Lanczos procedure, the first such result for Krylov based methods. The resulting expression shows that the H2 error is due to the mismatch at the mirror images of the poles of the original and reduced systems, and hence suggests choosing the mirror images as the interpolation points for the rational Krylov method. In addition two algorithms are proposed to overcome the rank deficiencies occurring in the MIMO version of the rational Krylov method. Finally, a novel model reduction algorithm by least-squares is developed, one of the cornerstones of this dissertation. The method is a projection and combines Krylov and singular value decomposition methods. The reduced model is asymptotically stable, matches a certain number of moments; and minimizes a weighted H2 error in the discrete time case. The effectiveness of the proposed approaches is tested by means of various numerical experiments.

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