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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.

A new nonparametric approach for system identification has been recently proposed 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 paper 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., Comment: Extends results of 2014 IEEE MSC Conference Proceedings (arXiv:1406.5706)

International audience; The problem of estimating continuous-time model parameters of linear dynamical systems using sampled time-domain input and output data has received considerable attention over the past decades and has been approached by various methods. The research topic also bears practical importance due to both its close relation to first principles modeling and equally to linear model-based control design techniques, most of them carried in continuous time. Nonetheless, as the performance of the existing algorithms for continuous-time model identification has seldom been assessed and, as thus far, it has not been considered in a comprehensive study, this practical potential of existing methods remains highly questionable. The goal of this brief paper is to bring forward a first study on this issue and to factually highlight the main aspects of interest. As such, an analysis is performed on a benchmark designed to be consistent both from a system identification viewpoint and from a control-theoretic one. It is concluded that robust initialization aspects require further research focus towards reliable algorithm development.; Ce papier traite de benchmarking de l'identification de modèles à temps continu qui sont très utilisés dans l'ingiénerie.

This paper studies the asymptotic properties of the hyperparameter estimators including the leave- $k$ -out cross validation (LKOCV) and r-fold cross validation (RFCV), and discloses their relation with the Stein's unbiased risk estimators (SURE) as well as the mean squared error (MSE). It is shown that as the number of data goes to infinity, the LKOCV shares the same asymptotic best hyperparameter minimizing the MSE estimator as the SURE does if the input is bounded and the ratio between the training data and the whole data tends to zero. We illustrate the efficacy of the theoretical result by Monte Carlo simulations.

Performing multiple experiments is common when learning internal mechanisms of complex systems. These experiments can include perturbations of parameters or external disturbances. A challenging problem is to efficiently incorporate all collected data simultaneously to infer the underlying dynamic network. This paper addresses the reconstruction of dynamic networks from heterogeneous datasets under the assumption that the underlying networks share the same Boolean structure across all experiments. Parametric models are derived for dynamical structure functions, which describe causal interactions between measured variables. Multiple datasets are integrated into one regression problem with additional demands on group sparsity to assure network sparsity and structure consistency. To acquire structured group sparsity, we propose a sampling-based method, together with extended versions of l 1 -methods and sparse Bayesian learning. The performance of the proposed methods is benchmarked in numerical simulation. In summary, this paper presents efficient methods on network reconstruction from multiple experiments, and reveals practical experience that could guide applications.

The first order stable spline (SS-1) kernel is used extensively in regularized system identification. In particular, the stable spline estimator models the impulse response as a zero-mean Gaussian process whose covariance is given by the SS-1 kernel. In this paper, we discuss the maximum entropy properties of this prior. In particular, we formulate the exact maximum entropy problem solved by the SS-1 kernel without Gaussian and uniform sampling assumptions. Under general sampling schemes, we also explicitly derive the special structure underlying the SS-1 kernel (e.g. characterizing the tridiagonal nature of its inverse), also giving to it a maximum entropy covariance completion interpretation. Along the way similar maximum entropy properties of the Wiener kernel are also given.

We analyze the statistical performance of identification of stochastic dynamical systems with non-linear measurement sensors. This includes stochastic Wiener systems, with linear dynamics, process noise and measured by a non-linear sensor with additive measurement noise. There are many possible system identification methods for such systems, including the Maximum Likelihood (ML) method and the Prediction Error Method (PEM). The focus has mostly been on algorithms and implementation, and less is known about the statistical performance and the corresponding Cram\'er-Rao Lower Bound (CRLB) for identification of such non-linear systems. We derive expressions for the CRLB and the asymptotic normalized covariance matrix for certain Gaussian approximations of Wiener systems to show how a non-linear sensor affects the accuracy compared to a corresponding linear sensor. The key idea is to take second order statistics into account by using a common parametrization of the mean and the variance of the output process. This analysis also leads to a ML motivated identification method based on the conditional mean predictor and a Gaussian distribution approximation. The analysis is supported by numerical simulations

Using Maximum Likelihood (or Prediction Error) methods to identify linear state space model is a prime technique. The likelihood function is a nonconvex function and care must be exercised in the numerical maximization. Here the focus will be on affine parameterizations which allow some special techniques and algorithms. Three approaches to formulate and perform the maximization are described in this contribution: (1) The standard and well known Gauss-Newton iterative search, (2) a scheme based on the EM (expectation-maximization) technique, which becomes especially simple in the affine parameterization case, and (3) a new approach based on lifting the problem to a higher dimension in the parameter space and introducing rank constraints.

This paper considers the input design of kernelbased regularization methods for LTI system identification. We first derive the Bayesian mean squared error matrix under the Bayesian perspective, and then use some typical scalar measures (e.g., the A-optimality, D-optimality, and E-optimality) as optimization criteria for the input design problem. Instead of directly solving the nonconvex optimization problem, we propose a two-step procedure. The first step is to solve a convex optimization and the second one is to determine the inverse image of a quadratic map. Both of these two steps can be solved efficiently by the proposed method and hence all the globally optimal inputs are found. In particular, we show that for some kernels, the optimal input under the D-optimality has an explicit expression.

An established method for grey-box identification is to use maximum-likelihood estimation for the nonlinear case implemented via extended Kalman filtering. In applications of (nonlinear) model pred ...

For the identification of a linear parameter varying (LPV) system steered by a scheduling variable evolving within a finite set, the local approach consists in separately estimating local linear time invariant (LTI) models corresponding to fixed values of the scheduling variable. It is shown in this paper that, without any global structural assumption of the considered LPV system, the local state-space LTI models do not contain the necessary information about the similarity transformations making them coherent. Nevertheless, it is possible to estimate these similarity transformations from input-output data under appropriate input excitation conditions. These estimations result in a common state basis of the transformed local LTI models, so that they form a coherent global LPV model, suitable for numerical simulations in the case of fast scheduling variable evolutions.

A widely used approach for identification of linear, time-invariant, MIMO (multi-input/multi output) systems from continuous-time frequency response data is to solve it in discrete-time domain using subspace based identification algorithm incorporated with a bilinear transformation. However, the bilinear transformation maps the distribution of the frequency lines from continuous-time domain to discrete-time domain in a non-linear fashion which may make identification algorithm to be ill-conditioned. In this paper we propose a solution to get around this problem by designing a dedicated frequency sampling strategy. Promising results are obtained when the algorithm is applied to synthetic data from a 6DOF mass-spring model.

To estimate a model of useful complexity for control design, at the same time as having a good insight into its reliability is a central issue in system identification, in particular for identification for control. Basically one can think of a (simpler) design model and a (more complex and exible) error model. These concepts are discussed in terms of model error modeling and stochastic embedding in a framework that allows the error model to vary over time. It is then of interest to estimate a probabilistic description of this error model. This can be accomplished by estimating parameters that describe the pdf of the errors. In this contribution explicit expressions for the likelihood function for these parameters are derived by marginalization over the error model, in case this is a linear regression.

A center issue for system identification is how to get a model estimate that achieves a good balance between the data fit and the model complexity. For the recently introduced kernel-based regularization method for linear system identification, the problem becomes first how to design a suitable kernel structure and second how to determine a right kernel among the kernel structure. In this paper and its companion one, we will focus on the issue of kernel structure design. Depending on the type of the prior knowledge, we provide two different ways: from a machine learning perspective or from a system theory perspective. We will focus on the first perspective here. In particular, we show that both the stable spline kernel and the diagonal correlated kernel belong to the class of the so-called exponentially convex locally stationary (ECLS) kernels. This finding motivates to construct ECLS or LS kernels for this regularization method in different ways, e.g., based on carefully designed state space models.

Regularization is a well known technique in estimation methodology. Its usefulness to assure well conditioned calculations and to handle reliable prior information about partly known parameters is a classical theme in statistics. Recently some deeper understanding about the advantages for general estimation and identification methods has been found and discussed. This is often done using the term "kernel methods". It has some links to machine learning and statistical function learning as well as to reproducing kernel Hilbert spaces. But algorithmically, it is all a question of regularization with an appropriate (quadratic norm) regularization matrix. Regularization was introduced into the MATLAB System Identification Toolbox in the 2013a version. It is a general option for all linear and nonlinear model estimation. Several specialized commands for estimation impulse responses (impulseest), tuning of kernels for ARX models (arxRegul), and general linear state space models with regularization (ssregest) have also been implemented. The paper describes and illustrates the used of these new features in the toolbox.

In this companion paper, we will continue our discussions on the issue of kernel structure design and we will focus on the system theory perspective. In particular, we will study the stability, maximum entropy property, Markov property and model complexity of the recently introduced state space model induced kernel. These findings motivate a system theory perspective to understand the behavior of kernels and to construct more general kernels.

We study identification of stochastic Wiener dynamic systems using so-called indirect inference. The main idea is to first fit an auxiliary model to the observed data and then in a second step, oft ...

The local approach to linear parameter varying (LPV) system identification consists in interpolating individually estimated local linear time invariant (LTI) models corresponding to fixed values of the scheduling variable. It is shown in this paper that, without any global structural assumption of the considered LPV system, individually estimated local state-space LTI models do not contain sufficient information for determining similarity transformations making them coherent. It is possible to estimate these similarity transformations from input-output data under appropriate excitation conditions. (C) 2017 Published by Elsevier Ltd.

The kernel-based regularization method has two core issues: kernel design and hyperparameter estimation. In this paper, we focus on the second issue and study the properties of several hyperparameter estimators including the empirical Bayes (EB) estimator, two Steins unbiased risk estimators (SURE) (one related to impulse response reconstruction and the other related to output prediction) and their corresponding Oracle counterparts, with an emphasis on the asymptotic properties of these hyperparameter estimators. To this goal, we first derive and then rewrite the first order optimality conditions of these hyperparameter estimators, leading to several insights on these hyperparameter estimators. Then we show that as the number of data goes to infinity, the two SUREs converge to the best hyperparameter minimizing the corresponding mean square error, respectively, while the more widely used EB estimator converges to another best hyperparameter minimizing the expectation of the EB estimation criterion. This indicates that the two SUREs are asymptotically optimal in the corresponding MSE senses but the EB estimator is not. Surprisingly, the convergence rate of two SUREs is slower than that of the EB estimator, and moreover, unlike the two SUREs, the EB estimator is independent of the convergence rate of Phi(T)Phi/N to its limit, where Phi is the regression matrix and N is the number of data. A Monte Carlo simulation is provided to demonstrate the theoretical results. (C) 2018 Elsevier Ltd. All rights reserved. Funding Agencies|National Natural Science Foundation of China [61773329, 61603379]; central government of China; Shenzhen Science and Technology Innovation Council [Ji-20170189, Ji-20160207]; Chinese University of Hong Kong, Shenzhen [PF. 01.000249, 2014.0003.23]; Swedish Research Council [2014-5894]; National Key Basic Research Program of China (973 Program) [2014CB845301]; Presidential Fund of the Academy of Mathematics and Systems Science, CAS [2015-hwyxqnrc-mbq]

There are two key issues for the kernel-based regularization method: the kernel structure design and the hyper-parameter estimation. In this contribution, we introduce a new family of kernel structures based on state space models. It has more flexible and more general structure, and includes some of stable spline kernels and diagonal correlated kernels as special cases. We also tested a different method for the hyper-parameter estimation by maximizing a profile marginal likelihood and examined three methods dealing with the initialization. Monte Carlo simulations show that the tested kernels are on the average a bit better than the tuned correlated kernel and the profile marginal likelihood maximization and the pre-windowing method work well for hyper-parameter estimation and initialization.

In the last years, nonparametric linear dynamical systems modeling has regained attention in the system identification world. In particular, the application of regularization techniques that were already widely used in statistics and machine learning, has proven beneficial for the estimation of the impulse response of linear systems. The low-rank approximation of the impulse response obtained by the truncated singular value decomposition (SVD) also leads to reduced complexity estimates. In this paper, the link between regularization and SVD truncation for finite impulse response (FIR) model estimation is made explicit. The SVD truncation is reformulated as a regularization problem with a specific choice of the regularization matrix. Both approaches (regularization and SVD truncation) are applied on a FIR modeling example and compared with the classic prediction error method/maximum likelihood approach. The results show the advantage of these techniques for impulse response estimation.

International audience; The identification of continuous-time models of dynamical systems based on sampled measurements of input and output signals is a research topic that has received much attention during the past decade. However, a framework for the correct assessment of the performance of various estimation methods, as well as their numerical reliability, is still missing due to a number of benchmarking difficulties, equally applicable to both discrete- and continuous-time identification problems. This paper revisits this topic, reports new numerical results, highlights several fundamental aspects regarding the definition of an appropriate benchmark for the evaluation of continuous-time linear model identification algorithms and discusses several means of addressing the related existing problems.

The stable spline kernel and the diagonal correlated kernel are two kernels that have been tested extensively in kernel-based regularization methods for LTI system identification. As shown in our recent works, although these two kernels are introduced in different ways, they share some common features, e.g., they all belong to the class of exponentially convex locally stationary kernels, and state-space model induced kernels. In this work, we further show that similar to the derivation of the stable spline kernel, the continuous-time diagonal correlated kernel can be derived by applying the same “stable” coordinate change to a “generalized” first order spline kernel, and thus can be interpreted as a stable generalized first order spline kernel. This interpretation provides new facets to understand the properties of the diagonal correlated kernel. Due to this interpretation, new eigendecompositions, explicit expression of the norm, and new maximum entropy interpretation of the diagonal correlated kernel are derived accordingly.

The purpose of this paper is to advance project theory on how distinctive significant characteristics in different project types can be identified and utilized for the effective management of projects and project portfolios. The since long dominating view that all projects can be managed using a standardized set of methods and techniques is insufficient, since the most apparent feature of projects is that each project is unique. Several attempts have been made to develop typologies for diagnosing projects. The process perspective, based on a rationalistic standpoint, has long been the prevailing perspective in both project practice and project research. This perspective has since the mid ninetieth been challenged by an organizational perspective. We argue that the “result perspective”, a trivialized perspective within the project research field, can be utilized to increase the understanding of the project phenomenon, and to identify distinctive characteristics in projects. Partly conceptual, and partly based on findings from case studies in three multi project environments carried out during a doctoral study 2004-2009, we describe a framework for a Structured Project Analysis (SPA) based on variations in project deliverables, goals and effects. Three analysis models are outlined: 1) Operational logics - based on variations in the character of the deliverables. 2) Strategic archetypes - based on variations in the project goal and effects. 3) Complexity - based on the nature of complexity in the project deliverables, goal and effects.

The ability to use the project work form tends to be increasingly important for long-term profitability in most organizations, thus, strategic management nowadays normally comprises management of projects and project portfolios. Many significant decisions concerning the organization's vision, goals and operations origins from a few basic questions in a strategic perspective. Some of these are related to developing a strategic position: Which products should we offer? How can we retain old customers and attract new ones? How can we increase our internal efficiency? The answers to these questions often results in decisions to initiate projects for product development, marketing campaigns or internal improvements. Other projects are related to operating from an established strategic position: delivering customer orders and producing events. The purpose of this paper is to describe and discuss a project typology, derived from a strategic management perspective. The typology consists of five project archetypes: Product development projects, Marketing projects, Internal improvement projects, Customer order projects, and Event projects. The typology highlights distinctive characteristics in the result perspective, i.e. variations in project deliverables, goals and intended effects, which have significant consequences for business oriented project steering in practice. Variations between project archetypes are described focusing on business decision processes, the purpose and content of project phases, progress control/follow-up, and organizational principles. The typology represents one of three analysis models in a framework for Structured Project Analysis (the SPA framework) developed during a doctoral study 2004-2009. (C) 2013 The Authors. Published by Elsevier Ltd.

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. Funding Agencies|MIUR FIRB project [RBFR12M3AC]; Progetto di Ateneo [CPDA147754/14]; Linnaeus Center CADICS; Swedish Research Council; ERC advanced grant LEARN [267381]; European Research Council; Swedish Research Council (VR) [2014-5894]

State-space smoothing has found many applications in science and engineering. Under linear and Gaussian assumptions, smoothed estimates can be obtained using efficient recursions, for example Rauch-Tung-Striebel and Mayne-Fraser algorithms. Such schemes are equivalent to linear algebraic techniques that minimize a convex quadratic objective function with structure induced by the dynamic model. These classical formulations fall short in many important circumstances. For instance, smoothers obtained using quadratic penalties can fail when outliers are present in the data, and cannot track impulsive inputs and abrupt state changes. Motivated by these shortcomings, generalized Kalman smoothing formulations have been proposed in the last few years, replacing quadratic models with more suitable, often nonsmooth, convex functions. In contrast to classical models, these general estimators require use of iterated algorithms, and these have received increased attention from control, signal processing, machine learning, and optimization communities. In this survey we show that the optimization viewpoint provides the control and signal processing community great freedom in the development of novel modeling and inference frameworks for dynamical systems. We discuss general statistical models for dynamic systems, making full use of nonsmooth convex penalties and constraints, and providing links to important models in signal processing and machine learning. We also survey optimization techniques for these formulations, paying close attention to dynamic problem structure. Modeling concepts and algorithms are illustrated with numerical examples., Comment: 29 pages, 11 figures

Version 8.0 of MATLAB's System Identification toolbox is released with version R2012a of MATLAB in the spring of 2012. This release presents a re-engineered implementation of the code using the new ...

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.

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.

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.

In this paper, we propose a new method to identify biochemical reaction networks (i.e. both reactions and kinetic parameters) from heterogeneous datasets. Such datasets can contain (a) data from several replicates of an experiment performed on a biological system; (b) data measured from a biochemical network subjected to different experimental conditions, for example, changes/perturbations in biological inductions, temperature, gene knock-out, gene over-expression, etc. Simultaneous integration of various datasets to perform system identification has the potential to avoid non-identifiability issues typically arising when only single datasets are used.

Segmentation of time series data is of interest in many applications, as for example in change detection and fault detection. In the area of convex optimization, the sum-of-norms regularization has ...

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. CADICS

This paper develops and illustrates methods for the identification of Wiener model structures. These techniques are capable of accommodating the “blind” situation where the input excitation to the linear block is not observed. Furthermore, the algorithm developed here can accommodate a nonlinearity which need not be invertible, and may also be multivariable. Central to these developments is the employment of the Expectation Maximisation (EM) method for computing maximum likelihood estimates, and the use of a new approach to particle smoothing to efficiently compute stochastic expectations in the presence of nonlinearities.

Segmentation of time-varying systems and signals into models whose parameters are piecewise constant in time is an important and well studied problem. Here it is formulated as a least-squares problem with sum-of-norms regularization over the state parameter jumps. a generalization of L1-regularization. A nice property of the suggested formulation is that it only has one tuning parameter, the regularization constant which is used to trade-off fit and the number of segments. CADICS

In applications of (nonlinear) model predictive control a more and more common approach for the state estimation is to use moving horizon estimation, which employs (nonlinear) optimization directly ...

The standard continuous time state space model with stochastic disturbances contains the mathematical abstraction of continuous time white noise. To work with well defined, discrete time observations, it is necessary to sample the model with care. The basic issues are well known, and have been discussed in the literature. However, the consequences have not quite penetrated the practice of estimation and identification. One example is that the standard model of an observation, being a snapshot of the current state plus noise independent of the state, cannot be reconciled with this picture. Another is that estimation and identification of time continuous models require a more careful treatment of the sampling formulas. We discuss and illustrate these issues in the current contribution. An application of particular practical importance is the estimation of models based on irregularly sampled observations.

System identification is the art and science of building mathematical models of dynamic systems from observed input-output data. It can be seen as the interface between the real world of applications and the mathematical world of control theory and model abstractions. As such, it is an ubiquitous necessity for successful applications. System identification is a very large topic, with different techniques that depend on the character of the models to be estimated: linear, nonlinear, hybrid, nonparametric etc. At the same time, the area can be characterized by a small number of leading principles, e.g. to look for sustainable descriptions by proper decisions in the triangle of model complexity, information contents in the data, and effective validation. The area has many facets and there are many approaches and methods. A tutorial or a survey in a few pages is not quite possible. Instead, this presentation aims at giving an overview of the “science” side, i.e. basic principles and results and at pointing to open problem areas in the practical, “art”, side of how to approach and solve a real problem.

This paper treats several aspects relevant to the identification of continuous-time output error (OE) models based on non-uniformly sampled output data. The exact method for doing this is well known in the time domain, where the continuous-time system is discretized, simulated and the result is fitted in a mean square sense to measured data. The material presented here is based on a method proposed in a companion paper (Gillberg & Ljung, 2010) which deals with the same topic but for the case of uniformly sampled data. In this text it will be shown how that method suggests that the output should be reconstructed using a B-spline with uniformly distributed knots. This representation can then be used to directly identify the continuous-time system without proceeding via discretization. Only the relative degree of the model is used to choose the order of the spline.

This paper treats several aspects relevant to identification of continuous-time output error (OE) models based on sampled data. The exact method for doing this is well known both for data given in the time and frequency domains. This approach becomes somewhat complex, especially for non-uniformly sampled data. We study various ways to approximate the exact method for reasonably fast sampling. While an objective is to gain insights into the non-uniform sampling case, this paper only gives explicit results for uniform sampling.