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  • 1.
    Everitt, Niklas
    KTH, School of Electrical Engineering (EES), Automatic Control.
    Identification of Modules in Acyclic Dynamic Networks A Geometric Analysis of Stochastic Model Errors2015Licentiate thesis, monograph (Other academic)
  • 2.
    Everitt, Niklas
    KTH, School of Electrical Engineering (EES), Automatic Control. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    Module identification in dynamic networks: parametric and empirical Bayes methods2017Doctoral thesis, monograph (Other academic)
    Abstract [en]

    The purpose of system identification is to construct mathematical models of dynamical system from experimental data. With the current trend of dynamical systems encountered in engineering growing ever more complex, an important task is to efficiently build models of these systems. Modelling the complete dynamics of these systems is in general not possible or even desired. However, often, these systems can be modelled as simpler linear systems interconnected in a dynamic network. Then, the task of estimating the whole network or a subset of the network can be broken down into subproblems of estimating one simple system, called module, embedded within the dynamic network.

    The prediction error method (PEM) is a benchmark in parametric system identification. The main advantage with PEM is that for Gaussian noise, it corresponds to the so called maximum likelihood (ML) estimator and is asymptotically efficient. One drawback is that the cost function is in general nonconvex and a gradient based search over the parameters has to be carried out, rendering a good starting point crucial. Therefore, other methods such as subspace or instrumental variable methods are required to initialize the search. In this thesis, an alternative method, called model order reduction Steiglitz-McBride (MORSM) is proposed. As MORSM is also motivated by ML arguments, it may also be used on its own and will in some cases provide asymptotically efficient estimates. The method is computationally attractive since it is composed of a sequence of least squares steps. It also treats the part of the network of no direct interest nonparametrically, simplifying model order selection for the user.

    A different approach is taken in the second proposed method to identify a module embedded in a dynamic network. Here, the impulse response of the part of the network of no direct interest is modelled as a realization of a Gaussian process. The mean and covariance of the Gaussian process is parameterized by a set of parameters called hyperparameters that needs to be estimated together with the parameters of the module of interest. Using an empirical Bayes approach, all parameters are estimated by maximizing the marginal likelihood of the data. The maximization is carried out by using an iterative expectation/conditional-maximization scheme, which alternates so called expectation steps with a series of conditional-maximization steps. When only the module input and output sensors are used, the expectation step admits an analytical expression. The conditional-maximization steps reduces to solving smaller optimization problems, which either admit a closed form solution, or can be efficiently solved by using gradient descent strategies. Therefore, the overall optimization turns out computationally efficient. Using markov chain monte carlo techniques, the method is extended to incorporate additional sensors.

    Apart from the choice of identification method, the set of chosen signals to use in the identification will determine the covariance of the estimated modules. To chose these signals, well known expressions for the covariance matrix could, together with signal constraints, be formulated as an optimization problem and solved. However, this approach does neither tell us why a certain choice of signals is optimal nor what will happen if some properties change. The expressions developed in this part of the thesis have a different flavor in that they aim to reformulate the covariance expressions into a form amenable for interpretation. These expressions illustrate how different properties of the identification problem affects the achievable accuracy. In particular, how the power of the input and noise signals, as well as model structure, affect the covariance.

  • 3.
    Everitt, Niklas
    et al.
    KTH, School of Electrical Engineering (EES), Automatic Control. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    Bottegal, Giulio
    KTH, School of Electrical Engineering (EES), Automatic Control.
    Rojas, Cristian R.
    KTH, School of Electrical Engineering (EES), Automatic Control. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    Hjalmarsson, Håkan
    KTH, School of Electrical Engineering (EES), Automatic Control. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    Identification ofmodules in dynamic networks: An empirical Bayes approach2016In: 55th IEEE Conference on Decisision and Control (CDC), IEEE conference proceedings, 2016Conference paper (Refereed)
    Abstract [en]

    We address the problem of identifying a specific module in a dynamic network, assuming known topology. We express the dynamics by an acyclic network composed of two blocks where the first block accounts for the relation between the known reference signals and the input to the target module, while the second block contains the target module. Using an empirical Bayes approach, we model the first block as a Gaussian vector with covariance matrix (kernel) given by the recently introduced stable spline kernel. The parameters of the target module are estimated by solving a marginal likelihood problem with a novel iterative scheme based on the ExpectationMaximization algorithm. Numerical experiments illustrate the effectiveness of the proposed method.

  • 4.
    Everitt, Niklas
    et al.
    KTH, School of Electrical Engineering (EES), Automatic Control. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    Bottegal, Giulio
    KTH, School of Electrical Engineering (EES), Automatic Control. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    Rojas, Cristian R.
    KTH, School of Electrical Engineering (EES), Automatic Control. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    Hjalmarsson, Håkan
    KTH, School of Electrical Engineering (EES), Automatic Control. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    On the Effect of Noise Correlation in Parameter Identification of SIMO Systems2015In: IFAC-PapersOnLine, ISSN 2405-8963, Vol. 48, no 28, 326-331 p.Article in journal (Refereed)
    Abstract [en]

    The accuracy of identified linear time-invariant single-input multi-output (SIMO) models can be improved when the disturbances affecting the output measurements are spatially correlated. Given a linear parametrization of the modules composing the SIMO structure, we show that the correlation structure of the noise sources and the model structure of the othe modules determine the variance of a parameter estimate. In particular we show that increasing the model order only increases the variance of other modules up to a point. We precisely characterize the variance error of the parameter estimates for finite model orders. We quantify the effect of noise correlation structure, model structure and signal spectra.

  • 5.
    Everitt, Niklas
    et al.
    KTH, School of Electrical Engineering (EES), Automatic Control. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    Bottegal, Giulio
    KTH, School of Electrical Engineering (EES), Automatic Control. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    Rojas, Cristian R.
    KTH, School of Electrical Engineering (EES), Automatic Control. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    Hjalmarsson, Håkan
    KTH, School of Electrical Engineering (EES), Automatic Control. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    On the Variance Analysis of identified Linear MIMO Models2015In: IEEE Explore, Institute of Electrical and Electronics Engineers (IEEE), 2015Conference paper (Refereed)
    Abstract [en]

    We study the accuracy of identified linear time-invariant multi-input multi-output (MIMO) systems. Under a stochastic framework, we quantify the effect of the spatial correlation and choice of model structure on the covariance matrix of the transfer function estimates. In particular, it is shown how the variance of a transfer function estimate depends on signal properties and model orders of other modules composing the MIMO system.

  • 6.
    Everitt, Niklas
    et al.
    KTH, School of Electrical Engineering (EES), Automatic Control. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    Bottegal, Giulio
    KTH, School of Electrical Engineering (EES), Automatic Control. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    Rojas, Cristian R.
    KTH, School of Electrical Engineering (EES), Automatic Control. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    Hjalmarsson, Håkan
    KTH, School of Electrical Engineering (EES), Automatic Control. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    Variance analysis of linear SIMO models with spatially correlated noise2017In: Automatica, ISSN 0005-1098, Vol. 77, 68-81 p.Article in journal (Refereed)
    Abstract [en]

    In this paper we address the identification of linear time-invariant single-input multi-output (SIMO) systems. In particular, we assess the performance of the prediction error method by quantifying the variance of the parameter estimates. Using an orthonormal representation for the modules composing the SIMO structure, we show that the parameter estimate of a module depends on the model structure of the other modules, and on the correlation structure of the output disturbances. We provide novel results which quantify the variance-error of the parameter estimates for finite model orders, where the effects of noise correlation structure, model structure and input spectrum are visible. In particular, we show that a sensor does not increase the accuracy of a module if common dynamics have to be estimated. When a module is identified using less parameters than the other modules, we derive the noise correlation structure that gives the minimum total variance. The implications of our results are illustrated through numerical examples and simulations.

  • 7.
    Everitt, Niklas
    et al.
    KTH, School of Electrical Engineering (EES), Automatic Control. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    Bottegal, Giulio
    KTH, School of Electrical Engineering (EES), Automatic Control.
    Rojas, Cristian R.
    KTH, School of Electrical Engineering (EES), Automatic Control. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    Hjalmarsson, Håkan
    KTH, School of Electrical Engineering (EES), Automatic Control. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    Variance Analysis of Linear SIMO Models with Spatially Correlated NoiseManuscript (preprint) (Other academic)
    Abstract [en]

    Substantial improvement in accuracy of identied linear time-invariant single-input multi-output (SIMO) dynamical models ispossible when the disturbances aecting the output measurements are spatially correlated. Using an orthogonal representation for the modules composing the SIMO structure, in this paper we show that the variance of a parameter estimate of a module is dependent on the model structure of the other modules, and the correlation structure of the disturbances. In addition, we quantify the variance-error for the parameter estimates for finite model orders, where the effect of noise correlation structure, model structure and signal spectra are visible. From these results, we derive the noise correlation structure under which the mentioned model parameterization gives the lowest variance, when one module is identied using less parameters than the other modules.

  • 8.
    Everitt, Niklas
    et al.
    KTH, School of Electrical Engineering (EES), Automatic Control. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    Rojas, Cristian R.
    KTH, School of Electrical Engineering (EES), Automatic Control. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    Hjalmarsson, Håkan
    KTH, School of Electrical Engineering (EES), Automatic Control. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    A Geometric Approach to Variance Analysis of Cascaded Systems2013In: Proceedings of the 52nd Conference On Decision And Control, IEEE conference proceedings, 2013, 6496-6501 p.Conference paper (Refereed)
    Abstract [en]

    Modeling complex and interconnected systems is a key issue in system identification. When estimating individual subsystems of a network of interconnected system, it is of interest to know the improvement of model-accuracy in using different sensors and actuators. In this paper, using a geometric approach, we quantify the accuracy improvement from additional sensors when estimating the first of a set of subsystems connected in a cascade structure. We present results on how the zeros of the first subsystem affect the accuracy of the corresponding model. Additionally we shed some light on how structural properties and experimental conditions determine the accuracy. The results are particularized to FIR systems, for which the results are illustrated by numerical simulations. A surprising special case occurs when the first subsystem contains a zero on the unit circle; as the model orders grows large, thevariance of the frequency function estimate, evaluated at thecorresponding frequency of the unit-circle zero, is shown to be the same as if the other subsystems were completely known.

  • 9.
    Everitt, Niklas
    et al.
    KTH, School of Electrical Engineering (EES), Automatic Control. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    Rojas, Cristian R.
    KTH, School of Electrical Engineering (EES), Automatic Control. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    Hjalmarsson, Håkan
    KTH, School of Electrical Engineering (EES), Automatic Control. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    Variance Results for Parallel Cascade Serial Systems2014In: Proceedings of 19th IFAC World Congress, 2014Conference paper (Refereed)
    Abstract [en]

    Modelling dynamic networks is important in different fields of science. At present, little is known about how different inputs and sensors contribute to the statistical properties concerning an estimate of a specific dynamic system in a network. We consider two forms of parallel serial structures, one multiple-input-multiple-output structure and one single-input multiple-output structure. The quality of the estimated models is analysed by means of the asymptotic covariance matrix, with respect to input signal characteristics, noise characteristics, sensor locations and previous knowledge about the remaining systems in the network. It is shown that an additive property applies to the information matrix for the considered structures. The impact of input signal selection, sensor locations and incorporation of previous knowledge isillustrated by simple examples.

  • 10.
    Galrinho, Miguel
    et al.
    KTH, School of Electrical Engineering (EES), Automatic Control. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    Everitt, Niklas
    KTH, School of Electrical Engineering (EES), Automatic Control. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    Hjalmarsson, Håkan
    KTH, School of Electrical Engineering (EES), Automatic Control. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    ARX modeling of unstable linear systems2017In: Automatica, ISSN 0005-1098, Vol. 75, 167-171 p.Article in journal (Refereed)
    Abstract [en]

    High-order ARX models can be used to approximate a quite general class of linear systems in a parametric model structure, and well-established methods can then be used to retrieve the true plant and noise models from the ARX polynomials. However, this commonly used approach is only valid when the plant is stable or if the unstable poles are shared with the true noise model. In this contribution, we generalize this approach to allow the unstable poles not to be shared, by introducing modifications to correctly retrieve the noise model and noise variance.

  • 11.
    Mårtensson, Jonas
    et al.
    KTH, School of Electrical Engineering (EES), Automatic Control. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    Everitt, Niklas
    KTH, School of Electrical Engineering (EES), Automatic Control. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    Hjalmarsson, Håkan
    KTH, School of Electrical Engineering (EES), Automatic Control. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    Covariance analysis in SISO linear systems identification2017In: Automatica, ISSN 0005-1098, E-ISSN 1873-2836, Vol. 77, 82-92 p.Article in journal (Refereed)
    Abstract [en]

    In this paper, we analyze the asymptotic covariance of models of causal single-input single-output linear time invariant systems. Expressions for the asymptotic (co)variance of system properties estimated using the prediction error method are derived. These expressions delineate the impacts of model structure, model order, true system dynamics, and experimental conditions. A connection to results on frequency function estimation is established. Also, simple model structure independent upper bounds are derived. Explicit variance expressions and bounds are provided for common system properties such as impulse response coefficients and non-minimum phase zeros. As an illustration of the insights the expressions provide, they are used to derive conditions on the input spectrum which make the asymptotic variance of non-minimum phase zero estimates independent of the model order and model structure.

  • 12.
    Mårtensson, Jonas
    et al.
    KTH, School of Electrical Engineering (EES), Automatic Control.
    Everitt, Niklas
    Hjalmarsson, Håkan
    KTH, School of Electrical Engineering (EES), Automatic Control. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
    Variance Analysis in SISO Linear Systems Identification2015In: Automatica, ISSN 0005-1098, E-ISSN 1873-2836Article in journal (Refereed)
    Abstract [en]

    Causal single input single output linear time invariant systems are considered. Expressions for the asymptotic (co)variance of system properties estimated using the prediction error method are derived. These expressions delineate the impacts of model structure, model order, true system dynamics, and experimental conditions. A connection to results on frequency function estimation is established. Also, simple model structure independent upper bounds are derived. Explicit variance expressions and bounds are provided for common system properties such as impulse response coefficients and non-minimum phase zeros. As an illustration of the insights the expressions provide, they are used to derive conditions on the input spectrum which makethe asymptotic variance of non-minimum phase zero estimates independent of the model order and model structure.

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