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  • 1.
    Ardeshiri, Tohid
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, Faculty of Science & Engineering.
    Chen, Tianshi
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, Faculty of Science & Engineering.
    MAXIMUM ENTROPY PROPERTY OF DISCRETE-TIME STABLE SPLINE KERNEL2015In: 2015 IEEE INTERNATIONAL CONFERENCE ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING (ICASSP), IEEE , 2015, p. 3676-3680Conference paper (Refereed)
    Abstract [en]

    In this paper, the maximum entropy property of the discrete-time first-order stable spline kernel is studied. The advantages of studying this property in discrete-time domain instead of continuous-time domain are outlined. One of such advantages is that the differential entropy rate is well-defined for discrete-time stochastic processes. By formulating the maximum entropy problem for discrete-time stochastic processes we provide a simple and self-contained proof to show what maximum entropy property the discrete-time first-order stable spline kernel has.

  • 2.
    Chen, Tianshi
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Comments on "State estimation for linear systems with state equality constraints" [Automatica 43 (2007) 1363-1368] in AUTOMATICA, vol 46, issue 11, pp 1929-19322010In: Automatica, ISSN 0005-1098, E-ISSN 1873-2836, Vol. 46, no 11, p. 1929-1932Article in journal (Refereed)
    Abstract [en]

    The state estimation problem for linear systems with linear state equality constraints was dealt with in Ko andamp; Bitmead [Ko, S., andamp; Bitmead, R. (2007). State estimation for linear systems with state equality constraints. Automatica, 43, 1363-1368]. In this correspondence, it is first shown that a necessary assumption on the covariance of the process noise is missing in the main result of the paper. It is then shown that the main result of the paper can be achieved in a convenient and more general way without any additional assumptions on the covariance of the process noise except positive definiteness.

  • 3.
    Chen, Tianshi
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Andersen, Martin S.
    Technical University of Denmark, Denmark.
    Ljung, Lennart
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Chiuso, Alessandro
    University of Padua, Italy.
    Pillonetto, Gianluigi
    University of Padua, Italy.
    System Identification Via Sparse Multiple Kernel-Based Regularization Using Sequential Convex Optimization Techniques2014In: IEEE Transactions on Automatic Control, ISSN 0018-9286, E-ISSN 1558-2523, Vol. 59, no 11, p. 2933-2945Article in journal (Refereed)
    Abstract [en]

    Model estimation and structure detection with short data records are two issues that receive increasing interests in System Identification. In this paper, a multiple kernel-based regularization method is proposed to handle those issues. Multiple kernels are conic combinations of fixed kernels suitable for impulse response estimation, and equip the kernel-based regularization method with three features. First, multiple kernels can better capture complicated dynamics than single kernels. Second, the estimation of their weights by maximizing the marginal likelihood favors sparse optimal weights, which enables this method to tackle various structure detection problems, e. g., the sparse dynamic network identification and the segmentation of linear systems. Third, the marginal likelihood maximization problem is a difference of convex programming problem. It is thus possible to find a locally optimal solution efficiently by using a majorization minimization algorithm and an interior point method where the cost of a single interior-point iteration grows linearly in the number of fixed kernels. Monte Carlo simulations show that the locally optimal solutions lead to good performance for randomly generated starting points.

  • 4.
    Chen, Tianshi
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, Faculty of Science & Engineering.
    Ardeshiri, Tohid
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, Faculty of Science & Engineering.
    Carli, Francesca P.
    Department of Engineering, University of Cambridge, Cambridge, United Kingdom.
    Chiuso, Alessandro
    Dept. of Information Engineering, University of Padova, Padova, Italy.
    Ljung, Lennart
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, Faculty of Science & Engineering.
    Pillonetto, Gianluigi
    Dept. of Information Engineering, University of Padova, Padova, Italy.
    Maximum entropy properties of discrete-time first-order stable spline kernel2016In: Automatica, ISSN 0005-1098, E-ISSN 1873-2836, Vol. 66, p. 34-38Article in journal (Refereed)
    Abstract [en]

    The first order stable spline (SS-1) kernel (also known as the tunedcorrelated kernel) is used extensively in regularized system identification, where the impulse response is modeled as a zero-mean Gaussian process whose covariance function is given by well designed and tuned kernels. In this paper, we discuss the maximum entropy properties of this kernel. In particular, we formulate the exact maximum entropy problem solved by the SS-1 kernel without Gaussian and uniform sampling assumptions. Under general sampling assumption, we also derive the special structure of the SS-1 kernel (e.g. its tridiagonal inverse and factorization have closed form expression), also giving to it a maximum entropy covariance completion interpretation.

  • 5.
    Chen, Tianshi
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Huang, Jie
    Chinese University of Hong Kong, China.
    A Small Gain Approach to Global Stabilization of Nonlinear Feedforward Systems with Input Unmodeled Dynamics2010In: Automatica, ISSN 0005-1098, E-ISSN 1873-2836, Vol. 46, no 6, p. 1028-1034Article in journal (Refereed)
    Abstract [en]

    In this paper, we study the global robust stabilization problem of strict feedforward systems subject to input unmodeled dynamics. We present a recursive design method for a nested saturation controller which globally stabilizes the closed-loop system in the presence of input unmodeled dynamics. One of the difficulties of the problem is that the Jacobian linearization of our system at the origin may not be stabilizable. We overcome this difficulty by employing a special version of the small gain theorem to address the local stability, and, respectively, the asymptotic small gain theorem to establish the global convergence property, of the closed-loop system An example is given to show that a redesign of the controller is required to guarantee the global robust asymptotic stability in the presence of the input unmodeled dynamics.

  • 6.
    Chen, Tianshi
    et al.
    Chinese Univ Hong Kong, Mech and Automat Engn Dept, Hong Kong, Hong Kong, Peoples R China.
    Huang, Jie
    Chinese Univ Hong Kong, Mech and Automat Engn Dept, Hong Kong, Hong Kong, Peoples R China.
    Global Robust Output Regulation by State Feedback for Strict Feedforward Systems2009In: IEEE Transactions on Automatic Control, ISSN 0018-9286, E-ISSN 1558-2523, Vol. 54, no 9, p. 2157-2163Article in journal (Refereed)
    Abstract [en]

    This note studies the global robust output regulation problem by state feedback for strict feedforward systems. By utilizing the general framework for tackling the output regulation problem [10], the output regulation problem is converted into a global robust stabilization problem for a class of feedforward systems that is subject to both time-varying static and dynamic uncertainties. Then the stabilization problem is solved by using a small gain based bottom-up recursive design procedure.

  • 7.
    Chen, Tianshi
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Ljung, Lennart
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Implementation of algorithms for tuning parameters in regularized least squares problems in system identification2013In: Automatica, ISSN 0005-1098, E-ISSN 1873-2836, Vol. 49, no 7, p. 2213-2220Article in journal (Refereed)
    Abstract [en]

    There has been recently a trend to study linear system identification with high order finite impulse response (FIR) models using the regularized least-squares approach. One key of this approach is to solve the hyper-parameter estimation problem that is usually nonconvex. Our goal here is to investigate implementation of algorithms for solving the hyper-parameter estimation problem that can deal with both large data sets and possibly ill-conditioned computations. In particular, a QR factorization based matrix-inversion-free algorithm is proposed to evaluate the cost function in an efficient and accurate way. It is also shown that the gradient and Hessian of the cost function can be computed based on the same QR factorization. Finally, the proposed algorithm and ideas are verified by Monte-Carlo simulations on a large data-bank of test systems and data sets.

  • 8.
    Chen, Tianshi
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Ljung, Lennart
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Andersen, Martin
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Chiuso, Alessandro
    University of Padova, Italy.
    Carli, Francesca
    University of Padova, Italy.
    Pillonetto, Gianluigi
    University of Padova, Italy.
    Sparse multiple kernels for impulse response estimation with majorization minimization algorithms2012In: Decision and Control (CDC), 2012, IEEE , 2012, p. 1500-1505Conference paper (Refereed)
    Abstract [en]

    This contribution aims to enrich the recently introduced kernel-based regularization method for linear system identification. Instead of a single kernel, we use multiple kernels, which can be instances of any existing kernels for the impulse response estimation of linear systems. We also introduce a new class of kernels constructed based on output error (OE) model estimates. In this way, a more flexible and richer representation of the kernel is obtained. Due to this representation the associated hyper-parameter estimation problem has two good features. First, it is a difference of convex functions programming (DCP) problem. While it is still nonconvex, it can be transformed into a sequence of convex optimization problems with majorization minimization (MM) algorithms and a local minima can thus be found iteratively. Second, it leads to sparse hyper-parameters and thus sparse multiple kernels. This feature shows the kernel-based regularization method with multiple kernels has the potential to tackle various problems of finding sparse solutions in linear system identification.

  • 9.
    Chen, Tianshi
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Ljung, Lennart
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Zhao, Yanlong
    Academy of Sciences, China.
    Impulse Response Estimation with Binary Measurements: A Regularized FIR Model2012In: Proceedings of the 16th IFAC Symposium on System Identification, 2012, p. 113-118Conference paper (Refereed)
    Abstract [en]

    FIR (finite impulse response) model is widely used in tackling the problem of the impulse response estimation with quantized measurements. Its use is, however, limited, in the case when a high order FIR model is required to capture a slowly decaying impulse response. This is because the high variance for high order FIR models would override the low bias and thus lead to large MSE (mean square error). In this contribution, we apply the recently introduced regularized FIR model approach to the problem of the impulse response estimation with binary measurements. We show by Monte Carlo simulations that the proposed approach can yield both better accuracy and better robustness than a recently introduced FIR model based approach.

  • 10.
    Chen, Tianshi
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Ohlsson, Henrik
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Goodwin, Graham C.
    University of Newcastle, Australia.
    Ljung, Lennart
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Kernel Selection in Linear System Identification: Part II: A Classical Perspective2011In: Proceedings of the 50th IEEE Conference on Decision and Control, 2011, p. 4326-4331Conference paper (Refereed)
    Abstract [en]

    In this companion paper, the choice of kernels for estimating the impulse response of linear stable systems is considered from a classical, “frequentist”, point of view. The kernel determines the regularization matrix in a regularized least squares estimate of an FIR model. The quality is assessed from a mean square error (MSE) perspective, and measures and algorithms for optimizing the MSE are discussed. The ideas are tested on the same data bank as used in Part I of the companion papers. The resulting findings and conclusions in the two papers are very similar despite the different perspectives.

  • 11.
    Chen, Tianshi
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Ohlsson, Henrik
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Ljung, Lennart
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    On the Estimation of Transfer Functions, Regularizations and Gaussian Processes – Revisited2010In: Proceedings of the 18th IFAC World Congress, 2010, p. 2303-2308Conference paper (Refereed)
    Abstract [en]

    Intrigued by some recent results on impulse response estimation by kernel and nonparametric techniques, we revisit the old problem of transfer function estimation from input-output measurements.We formulate a classical regularization approach, focused on finite impulse response (FIR) models, and find that regularization is necessary to cope with the high variance problem. This basic, regularized least squares approach is then a focal point for interpreting other techniques, like Bayesian inference and Gaussian process regression.

  • 12.
    Chen, Tianshi
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Ohlsson, Henrik
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Ljung, Lennart
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    On the Estimation of Transfer Functions, Regularizations and Gaussian Processes - Revisited2012In: Automatica, ISSN 0005-1098, E-ISSN 1873-2836, Vol. 48, no 8, p. 1525-1535Article in journal (Refereed)
    Abstract [en]

    Intrigued by some recent results on impulse response estimation by kernel and nonparametric techniques, we revisit the old problem of transfer function estimation from input-output measurements. We formulate a classical regularization approach, focused on finite impulse response (FIR) models, and find that regularization is necessary to cope with the high variance problem. This basic, regularized least squares approach is then a focal point for interpreting other techniques, like Bayesian inference and Gaussian process regression. The main issue is how to determine a suitable regularization matrix (Bayesian prior or kernel). Several regularization matrices are provided and numerically evaluated on a data bank of test systems and data sets. Our findings based on the data bank are as follows. The classical regularization approach with carefully chosen regularization matrices shows slightly better accuracy and clearly better robustness in estimating the impulse response than the standard approach - the prediction error method/maximum likelihood (PEM/ML) approach. If the goal is to estimate a model of given order as well as possible, a low order model is often better estimated by the PEM/ML approach, and a higher order model is often better estimated by model reduction on a high order regularized FIR model estimated with careful regularization. Moreover, an optimal regularization matrix that minimizes the mean square error matrix is derived and studied. The importance of this result lies in that it gives the theoretical upper bound on the accuracy that can be achieved for this classical regularization approach.

  • 13.
    Chen, Tianshi
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, Faculty of Science & Engineering.
    Pillonetto, Gianluigi
    University of Padua, Italy.
    Chiuso, Alessandro
    University of Padua, Italy.
    Ljung, Lennart
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, Faculty of Science & Engineering.
    Spectral analysis of the DC kernel for regularized system identification2015In: 2015 54TH IEEE CONFERENCE ON DECISION AND CONTROL (CDC), IEEE , 2015, p. 4017-4022Conference paper (Refereed)
    Abstract [en]

    System identification with regularization methods has attracted increasing attention recently and is a complement to the current standard maximum likelihood/ prediction error method. In this paper, we focus on the kernel-based regularization method and give a spectral analysis of the so-called diagonal correlated (DC) kernel, one family of kernel structures that has been proven useful for linear time-invariant system identification. In particular, using the theory of Bessel functions, we derive the eigenvalues and corresponding eigenfunctions of the DC kernel. Accordingly, we derive the Karhunen-Loeve expansion of the stochastic process whose covariance function is the DC kernel.

  • 14.
    Chen, Tianshi
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Schön, Thomas
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Ohlsson, Henrik
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Ljung, Lennart
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Decentralization of Particle Filters Using Arbitrary State Decomposition2010In: Proceedings of the 49th IEEE Conference on Decision and Control, 2010, p. 7383-7388Conference paper (Refereed)
    Abstract [en]

    In this paper, a new particle filter (PF) which we refer to as the decentralized PF (DPF) is proposed. By first decomposing the state into two parts, the DPF splits the filtering problem into two nested sub-problems and then handles the two nested sub-problems using PFs. The DPF has an advantage over the regular PF that the DPF can increase the level of parallelism of the PF. In particular, part of the resampling in the DPF bears a parallel structure and thus can be implemented in parallel. The parallel structure of the DPF is created by decomposing the state space, differing from the parallel structure of the distributed PFs which is created by dividing the sample space. This difference results in a couple of unique features of the DPF in contrast with the existing distributed PFs. Simulation results from a numerical example indicates that the DPF has a potential to achieve the same level of performance as the regular PF, in a shorter execution time.

  • 15.
    Chen, Tianshi
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Schön, Thomas
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Ohlsson, Henrik
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Ljung, Lennart
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Decentralized Particle Filter with Arbitrary State Decomposition2011In: IEEE Transactions on Signal Processing, ISSN 1053-587X, E-ISSN 1941-0476, Vol. 59, no 2, p. 465-478Article in journal (Refereed)
    Abstract [en]

    In this paper, a new particle filter (PF) which we refer to as the decentralized PF (DPF) is proposed. By first decomposing the state into two parts, the DPF splits the filtering problem into two nested subproblems and then handles the two nested subproblems using PFs. The DPF has the advantage over the regular PF that the DPF can increase the level of parallelism of the PF. In particular, part of the resampling in the DPF bears a parallel structure and can thus be implemented in parallel. The parallel structure of the DPF is created by decomposing the state space, differing from the parallel structure of the distributed PFs which is created by dividing the sample space. This difference results in a couple of unique features of the DPF in contrast with the existing distributed PFs. Simulation results of two examples indicate that the DPF has a potential to achieve in a shorter execution time the same level of performance as the regular PF.

  • 16.
    Ljung, Lennart
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Chen, Tianshi
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Convexity Issues in System Identification2013In: 10th IEEE International Conference on Control & Automation, IEEE , 2013, p. 1-9Conference paper (Refereed)
    Abstract [en]

    System Identification is about estimating models of dynamical systems from measured input-output data. Its traditional foundation is basic statistical techniques, such as maximum likelihood estimation and asymptotic analysis of bias and variance and the like. Maximum likelihood estimation relies on minimization of criterion functions that typically are non-convex, and may cause numerical search problems. Recent interest in identification algorithms has focused on techniques that are centered around convex formulations. This is partly the result of developments in machine learning and statistical learning theory. The development concerns issues of regularization for sparsity and for better tuned bias/variance trade-offs. It also involves the use of subspace methods as well as nuclear norms as proxies to rank constraints. A quite different route to convexity is to use algebraic techniques manipulate the model parameterizations. This article will illustrate all this recent development.                       

  • 17.
    Ljung, Lennart
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Chen, Tianshi
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    What can regularization offer for estimation of dynamical systems?2013In: 11th  IFAC International Workshop on Adaptation and Learning in Control and Signal Processing, ALCOSP13, IFAC , 2013, p. 1-8Conference paper (Refereed)
    Abstract [en]

    Estimation of unknown dynamics is what system identication is about and acore problem in adaptive control and adaptive signal processing. It has long been known thatregularization can be quite benecial for general inverse problems of which system identicationis an example. But only recently, partly under the inuence of machine learning, the use ofwell tuned regularization for estimating linear dynamical systems has been investigated moreseriously. In this presentation we review these new results and discuss what they may mean forthe theory and practice of dynamical model estimation in general.

  • 18.
    Mu, Biqiang
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, Faculty of Science & Engineering.
    Chen, Tianshi
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, Faculty of Science & Engineering. Chinese Univ Hong Kong, Peoples R China.
    Ljung, Lennart
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, Faculty of Science & Engineering.
    Tuning of Hyperparameters for FIR models - an Asymptotic Theory2017In: IFAC PAPERSONLINE, ELSEVIER SCIENCE BV , 2017, Vol. 50, no 1, p. 2818-2823Conference paper (Refereed)
    Abstract [en]

    Regularization of simple linear regression models for system identification is a recent much-studied problem. Several parameterizations ("kernels") of the regularization matrix have been suggested together with different ways of estimating ("tuning") its parameters. This contribution defines an asymptotic view on the problem of tuning and selection of kernels. It is shown that the SURE approach to parameter tuning provides an asymptotically consistent estimate of the optimal (in a MSE sense) hyperparameters. At the same time it is shown that the common marginal likelihood (empirical Bayes) approach does not enjoy that property. (C) 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.

  • 19.
    Ohlsson, Henrik
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Chen, Tianshi
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Khoshfetrat Pakazad, Sina
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Ljung, Lennart
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Sastry, Shankar
    University of California at Berkeley, USA.
    Distributed Change Detection2012In: Proceedings of the 16th IFAC Symposium on System Identification, 2012, p. 77-82Conference paper (Refereed)
    Abstract [en]

    Change detection has traditionally been seen as a centralized problem. Many change detection problems are however distributed in nature and the need for distributed change detection algorithms is therefore significant. In this paper a distributed change detection algorithm is proposed. The change detection problem is first formulated as a convex optimization problem and then solved distributively with the alternating direction method of multipliers (ADMM). To further reduce the computational burden on each sensor, a homotopy solution is also derived. The proposed method have interesting connections with Lasso and compressed sensing and the theory developed for these methods are therefore directly applicable.

  • 20.
    Ohlsson, Henrik
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Chen, Tianshi
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Khoshfetratpakazad, Sina
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Ljung, Lennart
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Sastry, S. Shankar
    University of Calif Berkeley, CA 94720 USA .
    Scalable anomaly detection in large homogeneous populations2014In: Automatica, ISSN 0005-1098, E-ISSN 1873-2836, Vol. 50, no 5, p. 1459-1465Article in journal (Refereed)
    Abstract [en]

    Anomaly detection in large populations is a challenging but highly relevant problem. It is essentially a multi-hypothesis problem, with a hypothesis for every division of the systems into normal and anomalous systems. The number of hypothesis grows rapidly with the number of systems and approximate solutions become a necessity for any problem of practical interest. In this paper we take an optimization approach to this multi-hypothesis problem. It is first shown to be equivalent to a non-convex combinatorial optimization problem and then is relaxed to a convex optimization problem that can be solved distributively on the systems and that stays computationally tractable as the number of systems increase. An interesting property of the proposed method is that it can under certain conditions be shown to give exactly the same result as the combinatorial multi-hypothesis problem and the relaxation is hence tight.

  • 21.
    Paola Carli, Francesca
    et al.
    University of Liege, Belgium; University of Cambridge, England.
    Chen, Tianshi
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, Faculty of Science & Engineering. Chinese University of Hong Kong, Peoples R China.
    Ljung, Lennart
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, Faculty of Science & Engineering.
    Maximum Entropy Kernels for System Identification2017In: IEEE Transactions on Automatic Control, ISSN 0018-9286, E-ISSN 1558-2523, Vol. 62, no 3, p. 1471-1477Article in journal (Refereed)
    Abstract [en]

    Bayesian nonparametric approaches have been recently introduced in system identification scenario where the impulse response is modeled as the realization of a zero-mean Gaussian process whose covariance (kernel) has to be estimated from data. In this scheme, quality of the estimates crucially depends on the parametrization of the covariance of the Gaussian process. A family of kernels that have been shown to be particularly effective in the system identification framework is the family of Diagonal/Correlated (DC) kernels. Maximum entropy properties of a related family of kernels, the Tuned/Correlated (TC) kernels, have been recently pointed out in the literature. In this technical note, we show that maximum entropy properties indeed extend to the whole family of DC kernels. The maximum entropy interpretation can be exploited in conjunction with results on matrix completion problems in the graphical models literature to shed light on the structure of the DC kernel. In particular, we prove that the DC kernel admits a closed-form factorization, inverse, and determinant. These results can be exploited both to improve the numerical stability and to reduce the computational complexity associated with the computation of the DC estimator.

  • 22.
    Pillonetto, Gianluigi
    et al.
    University of Padua, Italy.
    Chen, Tianshi
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, Faculty of Science & Engineering.
    Chiuso, Alessandro
    University of Padua, Italy.
    De Nicolao, Giuseppe
    University of Pavia, Italy.
    Ljung, Lennart
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, Faculty of Science & Engineering.
    Regularized linear system identification using atomic, nuclear and kernel-based norms: The role of the stability constraint2016In: Automatica, ISSN 0005-1098, E-ISSN 1873-2836, Vol. 69, p. 137-149Article in journal (Refereed)
    Abstract [en]

    Inspired by ideas taken from the machine learning literature, new regularization techniques have been recently introduced in linear system identification. In particular, all the adopted estimators solve a regularized least squares problem, differing in the nature of the penalty term assigned to the impulse response. Popular choices include atomic and nuclear norms (applied to Hankel matrices) as well as norms induced by the so called stable spline kernels. In this paper, a comparative study of estimators based on these different types of regularizers is reported. Our findings reveal that stable spline kernels outperform approaches based on atomic and nuclear norms since they suitably embed information on impulse response stability and smoothness. This point is illustrated using the Bayesian interpretation of regularization. We also design a new class of regularizers defined by "integral" versions of stable spline/TC kernels. Under quite realistic experimental conditions, the new estimators outperform classical prediction error methods also when the latter are equipped with an oracle for model order selection. (C) 2016 Elsevier Ltd. All rights reserved.

  • 23.
    Pillonetto, Gianluigi
    et al.
    University of Padua, Italy .
    Dinuzzo, Francesco
    Max Planck Institute Intelligent Syst, Germany .
    Chen, Tianshi
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    De Nicolao, Giuseppe
    University of Pavia, Italy .
    Ljung, Lennart
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Kernel methods in system identification, machine learning and function estimation: A survey2014In: Automatica, ISSN 0005-1098, E-ISSN 1873-2836, Vol. 50, no 3, p. 657-682Article in journal (Refereed)
    Abstract [en]

    Most of the currently used techniques for linear system identification are based on classical estimation paradigms coming from mathematical statistics. In particular, maximum likelihood and prediction error methods represent the mainstream approaches to identification of linear dynamic systems, with a long history of theoretical and algorithmic contributions. Parallel to this, in the machine learning community alternative techniques have been developed. Until recently, there has been little contact between these two worlds. The first aim of this survey is to make accessible to the control community the key mathematical tools and concepts as well as the computational aspects underpinning these learning techniques. In particular, we focus on kernel-based regularization and its connections with reproducing kernel Hilbert spaces and Bayesian estimation of Gaussian processes. The second aim is to demonstrate that learning techniques tailored to the specific features of dynamic systems may outperform conventional parametric approaches for identification of stable linear systems.

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