R-3 Repository :: Browsing by Author "Antoulas, Athanasios C."

Browsing by Author "Antoulas, Athanasios C."

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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.
A novel mathematical method for disclosing oscillations in gene transcription: A comparative study
(Public Library of Science, 2018) Antoulas, Athanasios C.; Zhu, Bokai; Zhang, Qiang; York, Brian; O'Malley, Bert W.; Dacso, Clifford C.
Circadian rhythmicity, the 24-hour cycle responsive to light and dark, is determined by periodic oscillations in gene transcription. This phenomenon has broad ramifications in physiologic function. Recent work has disclosed more cycles in gene transcription, and to the uncovering of these we apply a novel signal processing methodology known as the pencil method and compare it to conventional parametric, nonparametric, and statistical methods. Methods: In order to assess periodicity of gene expression over time, we analyzed a database derived from livers of mice entrained to a 12-hour light/12-hour dark cycle. We also analyzed artificially generated signals to identify differences between the pencil decomposition and other alternative methods. Results: The pencil decomposition revealed hitherto-unsuspected oscillations in gene transcription with 12-hour periodicity. The pencil method was robust in detecting the 24-hour circadian cycle that was known to exist, as well as confirming the existence of shorter-period oscillations. A key consequence of this approach is that orthogonality of the different oscillatory components can be demonstrated. thus indicating a biological independence of these oscillations, that has been subsequently confirmed empirically by knocking out the gene responsible for the 24-hour clock. Conclusion: System identification techniques can be applied to biological systems and can uncover important characteristics that may elude visual inspection of the data. Significance: The pencil method provides new insights on the essence of gene expression and discloses a wide variety of oscillations in addition to the well-studied circadian pattern. This insight opens the door to the study of novel mechanisms by which oscillatory gene expression signals exert their regulatory effect on cells to influence human diseases.
Adaptive Reduction of Large Spiking Neurons
(2013-11-21) Du, Bosen; Sorensen, Danny C.; Cox, Steven J.; Embree, Mark; Antoulas, Athanasios C.
This thesis develops adaptive reduction approaches for various models of large spiking neurons. Most neurons are like dendritic trees with many branches, and they communicate by nonlinear spiking behaviors. However, with the exception of Kellems' Strong-Weak model, most existing reduction approaches compromise the active ionic mechanisms that cause the spiking dynamics. The Strong-Weak model can predict the spikes caused by suprathreshold input traveling from the dendritic branches to the spike initiation zone (SIZ), but it is not able to reproduce the back propagation from SIZ to the dendritic branches after spikes. This thesis develops a new model called QAact, the mechanisms to incorporate QAact into the hybrid model to capture the back propagation behavior, and different model reduction techniques for each part of the new hybrid model where they are most advantageous. Computational tests of QAact and the new hybrid model as well as corresponding model reduction techniques on FitzHugh-Nagumo system, active nonuniform cable, and branched cell LGMD, demonstrate a significant reduction of dimension, computational complexity and running time.
Balanced truncation for linear switched systems
(Springer, 2018) Gosea, Ion Victor; Petreczky, Mihaly; Antoulas, Athanasios C.; Fiter, Christophe
We propose a model order reduction approach for balanced truncation of linear switched systems. Such systems switch among a finite number of linear subsystems or modes. We compute pairs of controllability and observability Gramians corresponding to each active discrete mode by solving systems of coupled Lyapunov equations. Depending on the type, each such Gramian corresponds to the energy associated to all possible switching scenarios that start or, respectively end, in a particular operational mode. In order to guarantee that hard to control and hard to observe states are simultaneously eliminated, we construct a transformed system, whose Gramians are equal and diagonal. Then, by truncation, directly construct reduced order models. One can show that these models preserve some properties of the original model, such as stability and that it is possible to obtain error bounds relating the observed output, the control input and the entries of the diagonal Gramians.
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.
Generalized realization
(2008) Mayo, Andrew; Antoulas, Athanasios C.
This thesis is broadly concerned with the following problem: given frequency response measurements construct a linear time-invariant system that is consistent with these measurements. This problem arises, for example, in V.L.S.I. design and testing. The major contribution of this thesis is to provide a new solution to this problem in which the celebrated Silverman-Ho algorithm is shown to be a special case. The major difference between this method and existing methods is the use of the shifted Loewner matrix in addition to the Loewner matrix. The approach advocated is then extended to deal with inexact data and rational approximation. The error introduced in these scenarios is analyzed. Finally, some progress is made in the "permissible order" problem of rational interpolation. The permissible orders of interpolants are shown to be related to the Kronecker structure of the matrix pencil associated to the Loewner and shifted Loewner matrices.
Geometric nonlinear filtering theory with application to the maneuvering aircraft tracking problem
(1990) Bishop, Robert H.; Antoulas, Athanasios C.
A geometric nonlinear filter (GNF) is designed for application to the problem of tracking a maneuvering aircraft. The aircraft tracking problem is a state estimation problem and a state prediction problem. A nonlinear aircraft maneuver model is proposed for use in the state estimation as well as the state prediction. This nonlinear model is based on the so-called coordinated turn and describes planar trajectories. The GNF design approach involves state transformations with output injection to transform the nonlinear system model to a linear form, known as the observer canonical form. For many nonlinear systems, such as the proposed aircraft maneuver model, this linearizing transformation does not exist. Therefore, for the maneuvering aircraft model, a transformation to an approximate observer canonical form is given. Utilizing a Lyapunov stability approach, sufficient conditions for stability of the GNF estimation error are derived. No such conditions exist for the extended Kalman filter (EKF). The GNF was found to be stable in cases where the EKF was not stable. The tracking performance of the GNF compares favorably with the EKF for various levels of measurement noise. However, the GNF offers a substantial savings in computational time making it more attractive than the EKF for use in a fire control computer.
Introducing the Loewner Method as a Data-Driven and Regularization-Free Approach for the Distribution of Relaxation Times Analysis of Lithium-Ion Batteries
(MDPI, 2023) Rüther, Tom; Gosea, Ion Victor; Jahn, Leonard; Antoulas, Athanasios C.; Danzer, Michael A.
For the identification of processes in lithium-ion batteries (LIB) by electrochemical impedance spectroscopy, frequency data is often transferred into the time domain using the method of distribution of relaxation times (DRT). As this requires regularization due to the ill-conditioned optimization problem, the investigation of data-driven methods becomes of interest. One promising approach is the Loewner method (LM), which has already had a number of applications in different fields of science but has not been applied to batteries yet. In this work, it is first deployed on synthetic data with predefined time constants and gains. The results are analyzed concerning the choice of model order, the type of processes , i.e., distributed and discrete, and the signal-to-noise ratio. Afterwards, the LM is used to identify and analyze the processes of a cylindrical LIB. To verify the results of this assessment a comparison is made with the generalized DRT at two different states of health of the LIB. It is shown that both methods lead to the same qualitative results. For the assignment of processes as well as for the interpretation of minor gains, the LM shows advantageous behavior, whereas the generalized DRT shows better results for the determination of lumped elements and resistive–inductive processes.
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.
Numerically Stable and Statistically Efficient Algorithms for Large Scale Exponential Fitting
(2013-12-06) Hokanson, Jeffrey; Embree, Mark; Cox, Steven J.; Antoulas, Athanasios C.; Heinkenschloss, Matthias
The exponential fitting problem appears in diverse applications such as magnetic resonance spectroscopy, mechanical resonance, chemical reactions, system identification, and radioactive decay. In each application, the exponential fitting problem decomposes measurements into a sum of exponentials with complex coefficients plus noise. Although exponential fitting algorithms have existed since the invention of Prony's Method in 1795, the modern challenge is to build algorithms that stably recover statistically optimal estimates of these complex coefficients while using millions of measurements in the presence of noise. Existing variants of Prony's Method prove either too expensive, most scaling cubically in the number of measurements, or too unstable. Nonlinear least squares methods scale linearly in the number of measurements, but require well-chosen initial estimates lest these methods converge slowly or find a spurious local minimum. We provide an analysis connecting the many variants of Prony's Method that have been developed in different fields over the past 200 years. This provides a unified framework that extends our understanding of the numerical and statistical properties of these algorithms. We also provide two new algorithms for exponential fitting that overcome several practical obstacles. The first algorithm is a modification of Prony's Method that can recover a few exponential coefficients from measurements containing thousands of exponentials, scaling linearly in the number of measurements. The second algorithm compresses measurements onto a subspace that minimizes the covariance of the resulting estimates and then recovers the exponential coefficients using an existing nonlinear least squares algorithm restricted to this subspace. Numerical experiments suggest that small compression spaces can be effective; typically we need fewer than 20 compressed measurements per exponential to recover the parameters with 90% efficiency. We demonstrate the efficacy of this approach by applying these algorithms to examples from magnetic resonance spectroscopy and mechanical vibration. Finally, we use these new algorithms to help answer outstanding questions about damping in mechanical systems. We place a steel string inside vacuum chamber and record the free response at multiple pressures. Analyzing these measurements with our new algorithms, we recover eigenvalue estimates as a function of pressure that illuminate the mechanism behind damping.
Passivity preserving model reduction in the context of spectral zero interpolation
(2008) Ionutiu, Roxana; Antoulas, Athanasios C.
This thesis presents a new passivity preserving model reduction method for circuit simulation, based on interpolation of dominant spectral zeros. Implemented as an eigenvalue approximation problem, the dominant spectral zero method (dominant SZM) is based on the subspace accelerated dominant pole algorithm (SADPA), which computes dominant spectral zeros automatically. The application of dominant SZM is extended beyond its interpolatory nature, proposing solutions for several problems in passive reduction. In particular, better approximation is achieved when combined with partial realization for descriptor systems, a framework for SISO reduction of the voltage transfer function in transmission lines is presented, and the implementation of MIMO dominant SZM is developed. Dominant SZM reduces automatically passive circuits irrespective of how the system equations are formulated, transmission line models with controlled sources, or circuits containing susceptance elements. Results show that dominant SZM gives comparable and often more accurate reduced models than state of the art techniques.
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.
Recursive identification and model reduction from time domain data
(2009) Ionita, Antonio Cosmin; Antoulas, Athanasios C.
In a system theoretic setting, identification from time domain data can be viewed as interpolating derivatives of a rational function. Typically, rational interpolation of derivatives requires computing the singular value decomposition (SVD) of large Loewner matrices constructed directly from the data. As a result, significant computational overhead is introduced through the SVD. The main result of the present thesis is simple—we construct interpolants efficiently without forming large Loewner matrices. A previously known recursive procedure is revisited with new insights, then further developed in a state-space setting. The key is to construct an interpolant recursively from ground up, by using the minimum amount of data. The resulting recursive interpolant is minimal and given in a state space form with rich structure. An important special case is the interpolation of impulse response measurements. This case is addressed separately and an efficient implementation requiring only matrix-vector multiplications is put forward. Furthermore, we extend the method to data corrupted by noise, where an additional model order reduction step is used to identify a low order model from the data. The newly developed recursive procedure is then tested on two examples involving actual noisy time-domain responses of a beaded elastic string and a cantilever beam.
Some a posteriori error bounds for reduced-order modelling of (non-)parametrized linear systems
(EDP Sciences, 2017) Feng, Lihong; Antoulas, Athanasios C.; Benner, Peter
We propose a posteriori error bounds for reduced-order models of non-parametrized linear time invariant (LTI) systems and parametrized LTI systems. The error bounds estimate the errors of the transfer functions of the reduced-order models, and are independent of the model reduction methods used. It is shown that for some special non-parametrized LTI systems, particularly efficiently computable error bounds can be derived. According to the error bounds, reduced-order models of both non-parametrized and parametrized systems, computed by Krylov subspace based model reduction methods, can be obtained automatically and reliably. Simulations for several examples from engineering applications have demonstrated the robustness of the error bounds.
System identification for robust control
(1998) Zhang, Huipin; Antoulas, Athanasios C.
In the design of a robust control system, one needs a nominal model together with a quantitative bound on the uncertainty that results from under-modeling and disturbances. In this thesis we do not intentionally seek a nominal model and a quantitative bound, instead, the uncertainty is directly parameterized so that the resulting uncertain model family can be characterized by means of a real parameter vector with at most unit length. This is an innovative approach to the control-oriented system identification, since it is not in accordance with the general philosophy of robust identification. However, it is applicable to the robust synthesis problem by taking advantage of a convex parameterization of robust controllers that simultaneously stabilize the uncertain models in the family. The robust performance problem becomes tractable since it can be converted into a quasi-convex optimization problem with Linear Matrix Inequality (LMI) constraints. The relation between the optimal robust performance and the uncertainty is studied by analyzing the explicit bounds of the maximal robust margin. Model (in)validation is a complement to system identification. In our approach it is an integral ingredient of the process of obtaining robust control-oriented system models. A single model is not invalidated if it is inside the ellipsoid, and thus the intersection of the ellipsoids is not invalidated. In order to make the unfalsified model set (the intersection) fit in our framework, we can compute an optimal ellipsoid bounding the intersection of the ellipsoids. (Abstract shortened by UMI.)
Towards a behavioral approach to linear approximate modeling
(1993) Gatt, George John; Antoulas, Athanasios C.
In this thesis, the foundations for the development of a behavioral approach to linear approximate modeling, are established. A particular data set, consisting of stable, discrete-time, purely exponential time series and a specific class of dynamical models are considered. A misfit function, between the data measurements and a system, belonging to this model class, is defined and the problem of characterizing all members of our model class, for which the value of the misfit function remains below a prespecified error level, is addressed. The concept of the block Hankel matrix, constructed from the data measurements, is then introduced, and it is shown that the optimal Hankel-norm approximation theory provides the main tool for a partial solution of the above problem.

Browsing by Author "Antoulas, Athanasios C."

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