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On Pade-type model order reduction of J-Hermitian linear dynamical systems

linear ALGEBRA AND ITS APPLICATIONS 2008年第10期429卷 2451-2464页

作者： Freund, Roland W. Univ Calif Davis Dept Math Davis CA 95616 USA

A simple, yet powerful approach to model order reduction of large-scale linear dynamical systems is to employ projection onto block Krylov subspaces. The transfer functions of the resulting reduced-order models of such projection methods can be characterized as Pade-type approximants of the transfer function of the original large-scale system. If the original system exhibits certain symmetries, then the reduced-order models are considerably more accurate than the theory for general systems predicts. In this paper, the framework of J-Hermitian linear dynamical systems is used to establish a general result about this higher accuracy. In particular, it is shown that in the case of J-Hermitian linear dynamical systems, the reduced-order transfer functions match twice as many Taylor coefficients of the original transfer function as in the general case. An application to the SPRIM algorithm for order reduction of general RCL electrical networks is discussed. (C) 2008 Elsevier Inc. All rights reserved.

关键词： linear dynamical system Descriptor system Model order reduction Block Krylov subspace Projection Pade-type approximation J-Hermitian Sesquilinear form SPRIM algorithm RCL network

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Variational Inference and Learning of Piecewise linear dynamical systems

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IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING systemS 2022年第8期33卷 3753-3764页

作者： Alameda-Pineda, Xavier Drouard, Vincent Horaud, Radu INRIA Grenoble Rhone Alpes F-38330 Montbonnot St Martin France Image Metr Manchester M1 3HZ Lancs England

Modeling the temporal behavior of data is of primordial importance in many scientific and engineering fields. Baseline methods assume that both the dynamic and observation equations follow linear-Gaussian models. However, there are many real-world processes that cannot be characterized by a single linear behavior. Alternatively, it is possible to consider a piecewise linear model which, combined with a switching mechanism, is well suited when several modes of behavior are needed. Nevertheless, switching dynamical systems are intractable because their computational complexity increases exponentially with time. In this article, we propose a variational approximation of piecewise linear dynamical systems. We provide full details of the derivation of two variational expectation-maximization algorithms: a filter and a smoother. We show that the model parameters can be split into two sets: static and dynamic parameters, and that the former parameters can be estimated offline together with the number of linear modes, or the number of states of the switching variable. We apply the proposed method to the head-pose tracking, and we thoroughly compare our algorithms with several state of the art trackers.

关键词： Mathematical model Hidden Markov models Switches Computational modeling Nonlinear dynamical systems Heuristic algorithms Predictive models Bayesian inference expectation-maximization inverse regression Kalman filter (KF) linear dynamical system switching state space model variational approximation

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Gradient Descent Learns linear dynamical systems

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JOURNAL OF MACHINE LEARNING RESEARCH 2018年第1期19卷 1025-1068页

作者： Hardt, Moritz Ma, Tengyu Recht, Benjamin Univ Calif Berkeley Dept Elect Engn & Comp Sci Berkeley CA 94720 USA Facebook AI Res Menlo Pk CA USA

We prove that stochastic gradient descent efficiently converges to the global optimizer of the maximum likelihood objective of an unknown linear time-invariant dynamical system from a sequence of noisy observations generated by the system. Even though the objective function is non-convex, we provide polynomial running time and sample complexity bounds under strong but natural assumptions. linear systems identification has been studied for many decades, yet, to the best of our knowledge, these are the first polynomial guarantees for the problem we consider.

关键词： non-convex optimization linear dynamical system stochastic gradient descent generalization bounds time series over-parameterization

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Solution of the matrix eigenvalue problem VA plus A*V=μV with applications to the study of free linear dynamical systems

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2008年第1期213卷 142-165页

作者： Kohaupt, L. TFH Berlin FB 2 D-13353 Berlin Germany

The new idea is to study the stability behavior of the solution x = x(1) of the initial value problem x = Ax, t >= t(0), x(t(0)) = x(0), with A epsilon C-nxn, in a weighted (semi-) norm parallel to center dot parallel to(R) where R is taken as an appropriate solution of the matrix eigenvalue problem RA + A* R = rho R, rather than as the solution of the algebraic Lyapunov matrix equation RA + A* R = -S with given positive (semi-) definite matrix S. Substantially better results are obtained by the new method. For example, if A is diagonalizable and all eigenvalues lambda(i)(A) have negative real pans, i.e., Re lambda(i)(A) < 0, i = 1,..., n, then rho=rho(i) = 2Re lambda(i)(A) < 0,the associated eigenmatrices R= R-i are positive semi-definite, and parallel to x(t)parallel to R-i = parallel to x(0)parallel to(Ri) e(Re lambda i(A)(t-t0)) -> 0(t -> infinity), which is much more than the old result, which only states that lim(t -> infinity) x(t) = 0. Especially, the semi-norms parallel to center dot parallel to(Ri) have a decoupling and filter effect on x(t). Further, new two-sided bounds (depending on x(0)) for the asymptotic behavior can be derived. The best constants in the bounds are obtained by the differential calculus of norms. Applications are made to free linear dynamical systems, and computations underpin the theoretical findings. (c) 2007 Elsevier B.V. All rights reserved.

关键词： matrix eigenvalue problem VA plus A* V = mu V algebraic Lyapunov equation RA plus A*R = -S linear dynamical system decoupling and filter effect of weighted semi-norms stability two-sided bounds depending on the initial conditions differential calculus of norms

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A linear dynamical systems Approach to Streamflow Reconstruction Reveals History of Regime Shifts in Northern Thailand

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WATER RESOURCES RESEARCH 2018年第3期54卷 2057-2077页

作者： Nguyen, Hung T. T. Galelli, Stefano Singapore Univ Technol & Design Pillar Engn Syst & Design Singapore Singapore

Catchment dynamics is not often modeled in streamflow reconstruction studies;yet, the streamflow generation process depends on both catchment state and climatic inputs. To explicitly account for this interaction, we contribute a linear dynamic model, in which streamflow is a function of both catchment state (i.e., wet/dry) and paleoclimatic proxies. The model is learned using a novel variant of the Expectation-Maximization algorithm, and it is used with a paleo drought recordthe Monsoon Asia Drought Atlasto reconstruct 406 years of streamflow for the Ping River (northern Thailand). Results for the instrumental period show that the dynamic model has higher accuracy than conventional linear regression;all performance scores improve by 45-497%. Furthermore, the reconstructed trajectory of the state variable provides valuable insights about the catchment historye.g., regime-like behaviorthereby complementing the information contained in the reconstructed streamflow time series. The proposed technique can replace linear regression, since it only requires information on streamflow and climatic proxies (e.g., tree-rings, drought indices);furthermore, it is capable of readily generating stochastic streamflow replicates. With a marginal increase in computational requirements, the dynamic model brings more desirable features and value to streamflow reconstructions.

关键词： linear dynamical system streamflow reconstruction stochastic streamflow generator regime shifts Chao Phraya River Basin Thailand

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Switching linear dynamical systems for noise robust speech recognition

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IEEE TRANSACTIONS ON AUDIO SPEECH AND LANGUAGE PROCESSING 2007年第6期15卷 1850-1858页

作者： Mesot, Bertrand Barber, David IDIAP Res Inst CH-1920 Martigny Switzerland

Real world applications such as hands-free dialling in cars may have to deal with potentially very noisy environments. Existing state-of-the-art solutions to this problem use feature-based HMMs, with a preprocessing stage to clean the noisy signal. However, the effect that raw signal noise has on the induced HMM features is poorly understood, and limits the performance of the HMM system. An alternative to feature-based HMMs is to model the raw signal, which has the potential advantage that including an explicit noise model is straightforward. Here we jointly model the dynamics of both the raw speech signal and the noise, using a Switching linear dynamical system (SLDS). The new model was tested on isolated digit utterances corrupted by Gaussian noise. Contrary to the Autoregressive HMM and its derivatives, which provides a model of uncorrupted raw speech, the SLDS is comparatively noise robust and also significantly outperforms a state-of-the-art feature-based HMM. The computational complexity of the SLDS scales exponentially with the length of the time series. To counter this we use Expectation Correction which provides a stable and accurate linear-time approximation for this important class of models, aiding their further application in acoustic modeling.

关键词： approximate inference expectation correction isolated digit recognition linear dynamical system noise robustness switching autoregressive process

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Belitskii's canonical forms of linear dynamical systems

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linear ALGEBRA AND ITS APPLICATIONS 2017年 531卷 533-536页

作者： Chen, Yuan Nie, Liujie Xu, Yunge Hubei Univ Fac Math & Stat Hubei Key Lab Appl Math Wuhan 430062 Hubei Peoples R China

关键词： Belitskii's canonical form linear dynamical system Indecomposability

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Compressive Acquisition of linear dynamical systems

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SIAM JOURNAL ON IMAGING SCIENCES 2013年第4期6卷 2109-2133页

作者： Sankaranarayanan, Aswin C. Turaga, Pavan K. Chellappa, Rama Baraniuk, Richard G. Carnegie Mellon Univ Dept Elect & Comp Engn Pittsburgh PA 15213 USA Arizona State Univ Arts Media & Engn Dept Tempe AZ 85287 USA Univ Maryland Dept Elect & Comp Engn College Pk MD 20740 USA Rice Univ Dept Elect & Comp Engn Houston TX 77005 USA

Compressive sensing (CS) enables the acquisition and recovery of sparse signals and images at sampling rates significantly below the classical Nyquist rate. Despite significant progress in the theory and methods of CS, little headway has been made in compressive video acquisition and recovery. Video CS is complicated by the ephemeral nature of dynamic events, which makes direct extensions of standard CS imaging architectures and signal models difficult. In this paper, we develop a new framework for video CS for dynamic textured scenes that models the evolution of the scene as a linear dynamical system (LDS). This reduces the video recovery problem to first estimating the model parameters of the LDS from compressive measurements and then reconstructing the image frames. We exploit the low-dimensional dynamic parameters (the state sequence) and high-dimensional static parameters (the observation matrix) of the LDS to devise a novel compressive measurement strategy that measures only the time-varying parameters at each instant and accumulates measurements over time to estimate the time-invariant parameters. This enables us to lower the compressive measurement rate considerably. We validate our approach and demonstrate its effectiveness with a range of experiments involving video recovery and scene classification.

关键词： compressive sensing linear dynamical system video compressive sensing

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Affine-periodic orbits for linear dynamical systems

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MATHEMATICAL METHODS IN THE APPLIED SCIENCES 2023年第6期46卷 7230-7238页

作者： Yang, Xue Cheng, Ming Li, Yong Jilin Univ Coll Math Changchun 130012 Peoples R China Northeast Normal Univ Ctr Math & Interdisciplinary Sci Changchun 130024 Peoples R China

We observe the relationship between forward bounded orbits and affine periodic orbits for infinite-dimensional linear dynamical systems and prove a Massera type criterion, which asserts that a forward bounded orbit implies the existence of affine periodic orbits. Such an affine-periodic orbit might be periodic, quasi(almost)-periodic, or spiral one. We also give an analog for hybrid (switching) discrete dynamical systems.

关键词： fixed-point theorems affine-periodic orbit massera-type criterion linear dynamical system

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A Hankel norm for quadrature rules solving random linear dynamical systems

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2017年 316卷 307-318页

作者： Pulch, Roland Ernst Moritz Arndt Univ Greifswald Dept Math & Comp Sci Walther Rathenau Str 47 D-17489 Greifswald Germany

In linear time-invariant dynamical systems, random variables are included to quantify uncertainties. The solution can be expanded into a series with predetermined orthogonal basis functions, which depend on the random variables. We define a norm of Hankel-type associated to a truncated series. A quadrature rule or a sampling method yields approximations of the unknown time-dependent coefficient functions in the truncated series. We arrange a Hankel norm for the quadrature technique in this context. Assuming a convergent sequence of quadrature rules, we show that the Hankel norms of the quadrature methods converge to the Hankel norm of the truncated series. Hence a numerical method is obtained to calculate the Hankel norm of the truncated series approximately. Results of numerical computations confirm the convergence property in a test example. (C) 2016 Elsevier By. All rights reserved.

关键词： linear dynamical system Random variables Quadrature rule Polynomial chaos expansion Hankel norm Uncertainty quantification

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