In the last decade, kernel-based regularization methods (KRMs) have been widely used for stable impulse response estimation in system identification. Its favorable performance over classic maximum likelihood/prediction error methods (ML/PEM) has been verified by extensive simulations. Recently, we noticed a surprising observation: for some data sets and kernels, no matter how the hyper-parameters are tuned, the regularized least square estimate cannot have higher model fit than the least square (LS) estimate, which implies that for such cases, the regularization cannot improve the LS estimate. Therefore, this paper focuses on how to understand this observation. To this purpose, we first introduce the squared error (SE) criterion, and the corresponding oracle hyper-parameter estimator in the sense of minimizing the SE criterion. Then we find the necessary and sufficient conditions under which the regularization cannot improve the LS estimate, and we show that the probability that this happens is greater than zero. The theoretical findings are demonstrated through numerical simulations, and simultaneously the anomalous simulation outcome wherein the probability is nearly zero is elucidated, and due to the ill-conditioned nature of either the kernel matrix, the Gram matrix, or both. (c) 2023 Elsevier Ltd. All rights reserved.

System identification learns mathematical models of dynamic systems starting from input-output data. Despite its long history, such research area is still extremely active. New challenges are posed by identification of complex physical processes given by the interconnection of dynamic systems. Examples arise in biology and industry, e.g., in the study of brain dynamics or sensor networks. In the last years, regularized kernel-based identification, with inspiration from machine learning, has emerged as an interesting alternative to the classical approach commonly adopted in the literature. In the linear setting, it uses the class of stable kernels to include fundamental features of physical dynamical systems, e.g., smooth exponential decay of impulse responses. Such class includes also unknown continuous parameters, called hyperparameters, which play a similar role as the model discrete order in controlling complexity. In this paper, we develop a linear system identification procedure by casting stable kernels in a full Bayesian framework. Our models incorporate hyperparameters uncertainty and consist of a mixture of dynamic systems over a continuum spectrum of dimensions. They are obtained by overcoming drawbacks related to classical Markov chain Monte Carlo schemes that, when applied to stable kernels, are proved to become nearly reducible (i.e., unable to reconstruct posteriors of interest in reasonable time). Numerical experiments show that full Bayes frequently outperforms the state-of-the-art results on typical benchmark problems. Two real applications related to brain dynamics (neural activity) and sensor networks are also included.

Network reconstruction of dynamical continuous-time (CT) systems is motivated by applications in many fields. Due to experimental limitations, especially in biology, data can be sampled at low frequencies, leading to significant challenges in network inference. We introduce the concept of "system aliasing" and characterize the minimal sampling frequency that allows reconstruction of CT systems from low sampled data. A test criterion is also proposed to detect the presence of system aliasing. With no system aliasing, this article provides an algorithm to reconstruct dynamic networks from full-state measurements in the presence of noise. With system aliasing, we add additional prior information such as sparsity to overcome the lack of identifiability. This article opens new directions in modeling of network systems where samples have significant costs. Such tools are essential to process available data in applications subject to experimental limitations.

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 l1-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. (c) 2020 Elsevier Ltd. All rights reserved.

In the local approach to linear parameter varying (LPV) system identification, it is widely acknowledged that locally estimated linear state-space models should be made coherent before being interpolated, but the accurate meaning of the term "coherent" or "coherence" is rarely defined. The purpose of this article is to analyze the relevance of two existing definitions and to point out the consequence of this analysis on the practice of LPV system identification.

Nonlinear system identification is an extremely broad topic, since every system that is not linear is nonlinear. That makes it impossible to give a full overview of all aspects of the fi eld. For this reason, the selection of topics and the organization of the discussion are strongly colored by the personal journey of the authors in this nonlinear universe.

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 and estimates trapped in local minima. Recent interest in identification algorithms has focused on techniques that are centered around convex formulations. This is partly the result of developments in semidefinite programming, machine learning and statistical learning theory. The development concerns issues of regular-ization 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 special approach is to look for difference-of-convex programming (DCP) formulations, in case a pure convex criterion is not found. Other techniques are based on Lagrangian relaxation and contraction theory. A quite different route to convexity is to use algebraic techniques to manipulate the model parameterizations. This article will illustrate all this recent development.

Auditory attention identification methods attempt to identify the sound source of a listeners interest by analyzing measurements of electrophysiological data. We present a tutorial on the numerous techniques that have been developed in recent decades, and we present an overview of current trends in multivariate correlation-based and model-based learning frameworks. The focus is on the use of linear relations between electrophysiological and audio data. The way in which these relations are computed differs. For example, canonical correlation analysis (CCA) finds a linear subset of electrophysiological data that best correlates to audio data and a similar subset of audio data that best correlates to electrophysiological data. Model-based (encoding and decoding) approaches focus on either of these two sets. We investigate the similarities and differences between these linear model philosophies. We focus on (1) correlation-based approaches (CCA), (2) encoding/decoding models based on dense estimation, and (3) (adaptive) encoding/decoding models based on sparse estimation. The specific focus is on sparsity-driven adaptive encoding models and comparing the methodology in state-of-the-art models found in the auditory literature. Furthermore, we outline the main signal processing pipeline for how to identify the attended sound source in a cocktail party environment from the raw electrophysiological data with all the necessary steps, complemented with the necessary MATLAB code and the relevant references for each step. Our main aim is to compare the methodology of the available methods, and provide numerical illustrations to some of them to get a feeling for their potential. A thorough performance comparison is outside the scope of this tutorial.

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.

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. (C) 2017 Elsevier Ltd. All rights reserved.

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.

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.

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.

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.

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.

This paper develops and illustrates a new maximum-likelihood based method for the identification of Hammerstein-Wiener model structures. A central aspect is that a very general situation is considered wherein multivariable data, non-invertible Hammerstein and Wiener nonlinearities, and colored stochastic disturbances both before and after the Wiener nonlinearity are all catered for. The method developed here addresses the blind Wiener estimation problem as a special case.

The presence of abrupt changes, such as impulsive and load disturbances, commonly occur in applications, but make the state estimation problem considerably more difficult than in the standard setting with Gaussian process disturbance. Abrupt changes often introduce a jump in the state, and the problem is therefore readily and often treated by change detection techniques. In this paper, we take a different approach. The state smoothing problem for linear state space models is here formulated as a constrained least-squares problem with sum-of-norms regularization, a generalization of l1-regularization. This novel formulation can be seen as a convex relaxation of the well known generalized likelihood ratio method by Willsky and Jones. Another 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 jumps. Good practical choices of this parameter along with an extension to nonlinear state space models are given. 

The framework of differential algebra, especially Ritts algorithm, has turned out to be a useful tool when analyzing the identifiability of certain nonlinear continuous-time model structures. This framework provides conceptually interesting means to analyze complex nonlinear model structures via the much simpler linear regression models. One difficulty when working with continuous-time signals is dealing with white noise in nonlinear systems. In this paper, difference algebraic techniques, which mimic the differential-algebraic techniques, are presented. Besides making it possible to analyze discrete-time model structures, this opens up the possibility of dealing with noise. Unfortunately, the corresponding discrete-time identifiability results are not as conclusive as in continuous time. In addition, an alternative elimination scheme to Ritts algorithm will be formalized and the resulting algorithm is analyzed when applied to a special form of the NFIR model structure.

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.

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 andamp; 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.

The Two-Stage Algorithm (TSA) has been extensively used and adapted for the identification of Hammerstein systems. It is essentially based on a particular formulation of Hammerstein systems in the form of bilinearly parameterized linear regressions. This paper has been motivated by a somewhat contradictory fact: though the optimality of the TSA has been established by Bai in 1998 only in the case of some special weighting matrices, the unweighted TSA is usually used in practice. It is shown in this paper that the unweighted TSA indeed gives the optimal solution of the weighted nonlinear least squares problem formulated with a particular weighting matrix. This provides a theoretical justification of the unweighted TSA, and also leads to a generalization of the obtained result to the case of colored noise with noise whitening. Numerical examples of identification of Hammerstein systems are presented to validate the theoretical analysis.

The subject of this paper is the direct identification of continuous-time autoregressive moving average (CARMA) models. The topic is viewed from the frequency domain perspective which then turns the reconstruction of the continuous-time power spectral density (CT-PSD) into a key issue. The first part of the paper therefore concerns the approximate estimation of the CT-PSD from uniformly sampled data under the assumption that the model has a certain relative degree. The approach has its point of origin in the frequency domain Whittle likelihood estimator. The discrete- or continuous-time spectral densities are estimated from equidistant samples of the output. For low sampling rates the discrete-time spectral density is modeled directly by its continuous-time spectral density using the Poisson summation formula. In the case of rapid sampling the continuous-time spectral density is estimated directly by modifying its discrete-time counterpart.

The Wiener model is a block oriented model, having a linear dynamic system followed by a static nonlinearity. The dominating approach to estimate the components of this model has been to minimize the error between the simulated and the measured outputs. We show that this will, in general, lead to biased estimates if there are other disturbances present than measurement noise. The implications of Bussgang's theorem in this context are also discussed. For the case with general disturbances, we derive the Maximum Likelihood method and show how it can be efficiently implemented. Comparisons between this new algorithm and the traditional approach, confirm that the new method is unbiased and also has superior accuracy.

The current demand for more complex models has initiated a shift away from state-space models towards models described by differential-algebraic equations (DAEs). These models arise as the natural product of object-oriented modeling languages, such as Modelica. However, the mathematics of DAEs is somewhat more involved than the standard state-space theory. The aim of this work is to present a well-posed description of a linear stochastic differential-algebraic equation and more importantly explain how well-posed estimation problems can be formed. We will consider both the system identification problem and the state estimation problem. Besides providing the necessary theory we will also explain how the procedures can be implemented by means of efficient numerical methods.

This article presents a parameter estimation of continuous-time polytopic models for a linear parameter varying (LPV) system. The prediction error method of linear time invariant (LTI) models is modified for polytopic models. The modified prediction error method is applied to an LPV aircraft system whose varying parameter is the flight velocity and model parameters are the stability and control derivatives (SCDs). In an identification simulation, the polytopic model is more suitable for expressing the behaviours of the LPV aircraft system than the LTI model from the viewpoints of the time and the frequency responses. The SCDs of the initial polytopic model are adjusted so as to fit the model output to the output data obtained from the LPV aircraft system.

Subspace identification methods (SIMs) for estimating state-space models have been proven to be very useful and numerically efficient. They exist in several variants, but all have one feature in common: as a first step, a collection of high-order ARX models are estimated from vectorized input-output data. In order not to obtain biased estimates, this step must include future outputs. However, all but one of the submodels include non-causal input terms. The coefficients of them will be correctly estimated to zero as more data become available. They still include extra model parameters which give unnecessarily high variance, and also cause bias for closed-loop data. In this paper, a new model formulation is suggested that circumvents the problem. Within the framework, the system matrices (A,B,C,D) and Markov parameters can be estimated separately. It is demonstrated through analysis that the new methods generally give smaller variance in the estimate of the observability matrix and it is supported by simulation studies that this gives lower variance also of the system invariants such as the poles.

Identification of non-linear dynamical models of a black box nature involves both structure decisions, i.e., which regressors to use, the selection of a regressor function, and the estimation of the parameters involved. The typical approach in system identification seems to be to mix all these steps, which for example means that the selection of regressors is based on the fits that is achieved for different choices. Alternatively one could then interpret the regressor selection as based on hypothesis tests (F-tests) at a certain confidence level that depends on the data. It would in many cases be desirable to decide which regressors to use independently of the other steps. In this paper we investigate what the well-known method of analysis of variance (ANOVA) can offer for this problem. System identification applications violate many of the ideal conditions for which ANOVA was designed and we study how the method performs under such non-ideal conditions. ANOVA is much faster than a typical parametric estimation method, using e.g. neural networks. It is actually also more reliable, in our tests, in picking the correct structure even under non-ideal conditions. One reason for this may be that ANOVA requires the data set to be balanced, that is, all parts of the regressor space are weighted equally. Just applying tests of fit for the recorded data may give, for structure identification, improper weight to areas with many, or few, samples.

Much attention in robust identification and control has been focused on linear low order models approximating high order linear systems. We consider the more realistic situation with a linear model approximating a nonlinear system. We describe how a nonlinear model error model can be developed, that allows a complete linear design process that results in a closed loop system with performance robustness guarantees (in terms of gain from disturbance to output) against the nonlinear error. Clearly the design can be successful only if the linear model is a reasonably good approximation of the system. A particular aspect of the design process is to define a workable definition of 'practical stability' for robust control design, with possible non-linear model errors. We use affine norms for that purpose.

An expression for the variance of the estimated spectrum based on auto-regressions is developed. This expression is asymptotic in the number of data, but exact in the model order. As the order tends to infinity it converges to the well known result that the variance is proportional to the model order times the square of the spectrum itself. The exact expression gives insight into the character of this convergence, its speed and its dependence on the poles of the underlying AR-process.

A framework for identification oriented robust controller design is developed. The model is identified from open-loop i/o-data and contains parametric uncertainties as well as an additive and norm-bounded error. The set of all robustly stabilising controllers, that additionally guarantee robust performance is characterised by a system of second order cones, which can be efficiently solved by interior point algorithms.

Mechanisms for adapting models, filters, decisions, regulators, and so on to changing properties of a system or a signal are of fundamental importance in many modern signal processing and control algorithms. This contribution describes a basic foundation for developing and analyzing such algorithms. Special attention is paid to the rationale behind the different algorithms, thus distinguishing between "optimal" algorithms and "ad hoc" algorithms. We also outline the basic approaches to performance analysis of adaptive algorithms.

Identification for robust control must deliver not only a nominal model, but also a reliable estimate of the uncertainty associated with the model. This paper addresses recent approaches to robust identification, that aim at dealing with contributions from the two main uncertainty sources: unmodeled dynamics and noise affecting the data. In particular, non-stationary Stochastic Embedding, Model Error Modeling based on prediction error methods and Set Membership Identification are considered. Moreover, we show how Set Membership Identification can be embedded into a Model Error Modeling framework. Model validation issues are easily addressed in the proposed framework. A discussion of asymptotic properties of all methods is presented. For all three methods, uncertainty is evaluated in terms of the frequency response, so that it can be handled by H8 control techniques. An example, where a nontrivial undermodeling is ensured by the presence of a nonlinearity in the system generating the data, is presented to compare these methods.

The sequence of estimates formed by the LMS algorithm for a standard linear regression estimation problem is considered. It is known since earlier that smoothing these estimates by simple averaging will lead to, asymptotically, the recursive least-squares algorithm. In this paper, it is first shown that smoothing the LMS estimates using a matrix updating will lead to smoothed estimates with optimal tracking properties, also in case the true parameters are slowly changing as a random walk. The choice of smoothing matrix should be tailored to the properties of the random walk. Second, it is shown that the same accuracy can be obtained also for a modified algorithm, SLAMS, which is based on averages and requires much less computations.

An identification method is described to determine a weighted combination of local linear state-space models from input and output data. Normalized radial basis functions are used for the weights, and the system matrices of the local linear models are fully parameterized. By iteratively solving a non-linear optimization problem, the centres and widths of the radial basis functions and the system matrices of the local models are determined. To deal with the non-uniqueness of the fully parameterized state-space system, a projected gradient search algorithm is described. It is pointed out that when the weights depend only on the input, the dynamical gradient calculations in the identification method are stable. When the weights also depend on the output, certain difficulties might arise. The methods are illustrated using serveral examples that have been studied in the literature before.

The standard machinery for system identification of linear time-invariant (LTI) models delivers a nominal model and a confidence (uncertainty) region around it, based on (second order moment) residual analysis and covariance estimation. In most cases this gives an uncertainty region that tends to zero as more and more data become available, even if the true system is nonlinear and/or time-varying. In this paper, the reasons for this are displayed, and a characterization of the limit LTI model is gh,en under quite general conditions. Various ways are discussed, and tested, to obtain a more realistic limiting model, with uncertainty. These should reflect the distance to the true possibly nonlinear, time-varying system, and also form a reliable basis for robust LTI control design.

Identification of nonlinear stochastic processes via differential neural networks is discussed. A new "dead-zone" type learning law for the weight dynamics is suggested. By a stochastic Lyapunov-like analysis the stability conditions for the identification error as well as for the neural network weights are established. The adaptive trajectory tracking using the obtained neural network model is realized for the subclass of stochastic completely controllable processes linearly dependent on control. The upper bounds for the identification and adaptive tracking errors are established.

It is well known that the output error and Box-Jenkins model structures cannot be used for prediction error identification of unstable systems. The reason for this is that the predictors in this case generically will be unstable. Typically, this problem is handled by projecting the parameter vector onto the region of stability, which gives erroneous results when the underlying system is unstable. The main contribution of this work is that we derive modified, but asymptotically equivalent, versions of these model structures that can also be applied in the case of unstable systems.

The problem of designing the identification experiments to make them maximally informative with respect to the intended model use is studied. The focus is on how to identify models that are good for control, so called `Identification for Control'. A main result is that we derive explicit expressions for the optimal controller and the optimal reference signal spectrum to use in the identification experiment for the case that only the misfit in the dynamics model is penalized and when a linear combination of the input and output variances is constrained.

The performance of terrain-aided navigation of aircraft depends on the size of the terrain gradient in the area. The point-mass filter (PMF) described in this work yields an approximate Bayesian solution that is well suited for the unstructured nonlinear estimation problem in terrain navigation. It recursively propagates a density function of the aircraft position. The shape of the point-mass density reflects the estimate quality; this information is crucial in navigation applications, where estimates from different sources often are fused in a central filter. Monte Carlo simulations show that the approximation can reach the optimal performance, and realistic simulations show that the navigation performance is very high compared with other algorithms and that the point-mass filter solves the recursive estimation problem for all the types of terrain covered in the test. The main advantages of the PMF is that it works for many kinds of nonlinearities and many kinds of noise and prior distributions. The mesh support and resolution are automatically adjusted and controlled using a few intuitive design parameters. The main disadvantage is that it cannot solve estimation problems of very high dimension since the computational complexity of the algorithm increases drastically with the dimension of the state space. The implementation used in this work shows real-time performance for 2D and in some cases 3D models, but higher state dimensions are usually intractable.

Direct prediction error identification of systems operating in closed loop may lead to biased results due to the correlation between the input and the output noise. The authors study this error, what factors affect it, and how it may be avoided. In particular, the role of the noise model is discussed and the authors show how the noise model should be parameterized to avoid the bias. Apart from giving important insights into the properties of the direct method, this provides a nonstandard motivation for the indirect method.

Model Validation is at the heart of the System Identification process. Recently, much renewed interest has been expressed in so called "identification for control". This means that the design variables associated with the identification process are tailored to achieve models that are well suited for control design purposes. A separate, but closely related issue is to devise validation tests that give information about the model's quality and suitability for control design. This paper shows and discusses how a basic and classical residual test gives such information.

Guo and Ljung (1995) established some general results on exponential stability of random linear equations, which can be applied directly to the performance analysis of a wide class of adaptive algorithms, including the basic LMS ones, without requiring stationarity, independency, and boundedness assumptions of the system signals. The current paper attempts to give a complete characterization of the exponential stability of the LMS algorithms by providing a necessary and sufficient condition for such a stability in the case of possibly unbounded, nonstationary, and non-φ-mixing signals. The results of this paper can be applied to a very large class of signals, including those generated from, e.g., a Gaussian process via a time-varying linear filter. As an application, several novel and extended results on convergence and the tracking performance of LMS are derived under various assumptions. Neither stationarity nor Markov-chain assumptions are necessarily required in the paper.

The so-called subspace methods for direct identification of linear models in state space form have drawn considerable interest recently. They have been found to work well in many cases but have one drawback — they do not yield consistent estimates for data collected under output feedback. The present paper points to the reasons for this. We stress how the basic idea is to focus on the estimation of the state-variable candidates — the k-step ahead output predictors. By recomputing these from a ‘non-parametric’ (or, rather, high order ARX) one-step ahead predictor model, closed loop data can be handled.

Two noniterative subspace-based algorithms which identify linear, time-invariant MIMO (multi-input/multioutput) systems from frequency response data are presented. The algorithms are related to the recent time-domain subspace identification techniques. The first algorithm uses equidistantly, in frequency, spaced data and is strongly consistent under weak noise assumptions. The second algorithm uses arbitrary frequency spacing and is strongly consistent under more restrictive noise assumptions, promising results are obtained when the algorithms are applied to real frequency data originating from a large flexible structure.

A multiple model least-squares method based on matrix decomposition is proposed. Compared with the conventional implementation of the least-squares method, the proposed method is simpler and more flexible in implementation and produces more information. An application example in parameter estimation is included. As a basic numerical tool, the proposed method can be used in many different application areas.

Tracking and adaptation algorithms are, from a formal point of view, nonlinear systems which depend on stochastic variables in a fairly complicated way. The analysis of such algorithms is thus quite complicated. A first step is to establish the exponential stability of these systems. This is of interest in its own right and a prerequisite for the practical use of the algorithm. It is also a necessary starting point to analyze the performance in terms of tracking and adaptation because that is how close the estimated parameters are to the time-varying true ones. In this paper we establish some general conditions for the exponential stability of a wide and common class of tracking algorithms. This includes least mean squares, recursive least squares, and Kalman filter based adaptation algorithms. We show how stability of an averaged (linear and deterministic) equation and stability of the actual algorithm are linked to each other under weak conditions on the involved stochastic processes. We also give explicit conditions for exponential stability of the most common algorithms. The tracking performance of the algorithms is studied in a companion paper.

In this paper we discuss the role of criterion minimization as a means for parameter estimation. Most traditional methods, such as maximum likelihood and prediction error identification are based on these principles. However, somewhat surprisingly, it turns out that it is not always ‘optimal’ to try to find the absolute minimum point of the criterion. The reason is that ‘stopped minimization’ (where the iterations have been terminated before the absolute minimum has been reached) has more or less identical properties as using regularization (adding a parametric penalty term). Regularization is known to have beneficial effects on the variance of the parameter estimates and it reduces the ‘variance contribution’ of the misfit. This also explains the concept of ‘overtraining’ in neural nets. How does one know when to terminate the iterations then? A useful criterion would be to stop iterations when the criterion function applied to a validation data set no longer decreases. However, in this paper, we show that applying this technique extensively may lead to the fact that the resulting estimate is an unregularized estimate for the total data set: estimation + validation data.

We discuss several aspects of the mathematical foundations of the nonlinear black-box identification problem. We shall see that the quality of the identification procedure is always a result of a certain trade-off between the expressive power of the model we try to identify (the larger the number of parameters used to describe the model, the more flexible is the approximation), and the stochastic error (which is proportional to the number of parameters). A consequence of this trade-off is the simple fact that a good approximation technique can be the basis of a good identification algorithm. From this point of view, we consider different approximation methods, and pay special attention to spatially adaptive approximants. We introduce wavelet and ‘neuron’ approximations, and show that they are spatially adaptive. Then we apply the acquired approximation experience to estimation problems. Finally, we consider some implications of these theoretical developments for the practically implemented versions of the ‘spatially adaptive’ algorithms.

The paper describes some sibjective aspects on some current research topics in system identification. A 'classical' standpoint is taken regarding complexity models. The need for specific tools for 'semi-physical-modeling' is also pointed out, and a discussion on different disturbance models is also included.

An analysis is given of the performance of the standard forgetting factor recursive least squares (RLS) algorithm when used for tracking time-varying linear regression models. Three basic results are obtained: (1) the ‘P-matrix’ in the algorithm remains bounded if and only if the (time-varying) covariance matrix of the regressors is uniformly non-singular; (2) if so, the parameter tracking error covariance matrix is of the order O(μ + γ²/μ), where μ = 1 - λ, λ is the forgetting factor and γ is a quantity reflecting the speed of the parameter variations; (3) this covariance matrix can be arbitrarily well approximated (for small enough μ) by an expression that is easy to compute.

Model quality and model accuracy are of prime interest in system identification. In this contribution we will review and discuss these concepts. In particular we will split model errors into contributions from a ‘random error’ and a ‘bias error’ and describe and discuss how to assess these two concepts.

The problem of deriving so-called hard-error bounds for estimated transfer functions is addressed. A hard bound is one that is sure to be satisfied, i.e. the true system Nyquist plot will be confined with certainty to a given region, provided that the underlying assumptions are satisfied. By blending a priori knowledge and information obtained from measured data, it is shown how the uncertainty of transfer function estimates can be quantified. The emphasis is on errors due to model mismatch. The effects of unmodeled dynamics can be considered as bounded disturbances. Hence, techniques from set membership identification can be applied to this problem. The approach taken corresponds to weighted least-squares estimation, and provides hard frequency-domain transfer function error bounds. The main assumptions used in the current contribution are: that the measurement errors are bounded, that the true system is indeed linear with a certain degree of stability, and that there is some knowledge about the shape of the true frequency response.

A reliable quality estimate of a given model is a prerequisite for any reasonable use of the model. The model error consists of two different contributions: the bias error and the random error. In this contribution, it is shown that the size (variance) of the random error can be reliably estimated in the case where a true system description cannot be achieved in the model structure used. This consistent error estimate can differ considerably from the conventionally used variance estimate, which could thus be misleading.

The purpose of this paper is to extend recent results of Ljung and Priouret to a more general class of regressors.

An approach is described how to incorporate knowledge of symbolic/logic character into a conventional framework of noisy observations in dynamical systems. The idea is based on approximating the optimal solution that could theoretically be computed if a complete Bayesian framework were known (and infinite computational power were available). The nature of the approximations, the deviations from optimality and the sensitivity to ad hoc parameters are specifically addressed. This merging of logic and numerics is essential in many problems of adaptation in control and signal processing.

This paper presents an algorithm for stabilizing a linear system with bounded disturbances, by a sampled-data regulator. The prior knowledge required is that the system has finite order and bounded disturbances. Bounds on the disturbances and on the order of the system need not be known.

The implications of this technical result on theoretical convergence issues in adaptive control are also discussed.

Members of IFAC's Technical Committee on Theory prepared this review of papers published during 1984–1986 for the participants of the 10th IFAC World Congress in Munich. The review is limited to the scope of the Theory Committee. The idea is to give a status report on Control Theory and to give some guidance into the extensive literature on this topic.

The propagation of single numerical errors is studied for several different adaptation algorithms for linear regression problems. The case where the regressors are poorly exciting (the information matrix is singular or close to singular) is of particular interest. It is shown how appropriate regularization algorithms may secure exponential stability of the error propagation (and of the propagation of initial conditions in general). Gradient and Gauss-Newton algorithms with forgetting factors as well as directional forgetting are studied.

The purpose of the design of identification experiments is to make the collected data maximally informative with respect to the intended use of the model, subject to constraints that might be at hand. When the true system is replaced by an estimated model, there results a performance degradation that is due to the error in the transfer function estimates. Using some recent asymptotic expressions for the bias and the variance of the estimated transfer function, it is shown how this performance degradation can be minimized by a proper experiment design. Several applications, where it is beneficial to let the experiment be carried out in closed loop, are highlighted.

The numerical properties of implementations of the recursive least-squares identification algorithm are of great importance for their continuous use in various adaptive schemes. Here we investigate how an error that is introduced at an arbitrary point in the algorithm propagates. It is shown that conventional LS algorithms, including Bierman's UD-factorization algorithm are exponentially stable with respect to such errors, i.e. the effect of the error decays exponentially. The base of the decay is equal to the forgetting factor. The same is true for fast lattice algorithms. The fast least-squares algorithm, sometimes known as the ‘fast Kalman algorithm’ is however shown to be unstable with respect to such errors.

The problem to estimate transfer functions of linear systems is considered. The quality of the resulting estimate depends, among other things, on the input used during the identification experiment. We measure the quality using a quadratic norm in the frequency domain. The problem to determine optimal inputs, i.e. inputs that minimize the chosen norm, subject to constrained input variance, has long been studied. We point out that such procedures may involve a prejudice (that the system is to be found in a certain model set) that may have some surprising effects. We discuss how such a prejudice can be reduced by allowing the possibility that the true system cannot be exactly described in the chosen model set. We also calculate explicit expressions for the resulting “unprejudiced” optimal inputs. These expressions relate the signal-to-noise ratio (as a function of frequency) to the chosen weighting function in the quadratic norm. We also point out the role of the employed noise model for the design.

This paper addresses the following problem: Given a possibly non-stationary second-order or Gaussian stochastic process over a certain time interval, find all state-space realizations that produce the given process as their output. A constructive solution to this problem is described. In particular, the degrees of freedom left in this problem are explicitly displayed in terms of additional random variables and processes injected into the state-space description. Special attention is paid to output-induced, or internal, state-space realizations, that is, those for which the state vector is completely determined by the given process. The development is self-contained, the approach and proofs are fairly elementary, and explicit formulas are given for the various realizations.

A common approach to regulator design is to define an objective function, which is minimized with respect to adjustable regulator parameters. Here we discuss how such objective functions can be minimized on-line, thus providing adaptive control. Such an approach to adaptive control has its roots in early contributions to learning systems, and it is here further developed and discussed in the light of the recent development of the field. A general algorithm is given and special attention is paid to minimization of quadratic criteria. A key problem is to obtain information about the system dynamics in order to compute the derivatives of the criterion with respect to the regulator parameters. It is shown that the self-tuning regulator is obtained as a special case, corresponding to a particular quadratic criterion and a particular way of estimating the system dynamics. Explicit ways using instrumental variables techniques based on extra injected noise are also discussed. A specific feature of the algorithm is that it does not utilize specific knowledge about how to calculate the optimal regulator. The algorithm is the same for minimum phase as well as for non-minimum phase systems.

Parameter estimation problems that can be formulated as linear regressions are quite common in many applications. Recursive (on-line, sequential) estimation of such parameters can be performed using the recursive least squares (RLS) algorithm or a stochastic gradient version of this algorithm. In this paper the convergence properties of the gradient algorithm are analyzed under the assumption that the gain tends to zero. The technique is the same as the so-called ordinary differential equation approach, but the treatment here is self-contained and includes a proof of the boundedness of the estimates. A main result is that the convergence conditions for the gradient algorithm are the same as those for the recursive least squares algorithm.

We present a new approach to two filter smoothing formulae via diagonalization of the general time variant hamiltonian equations of the linear estimation problem. This approach shows the special role of the famous Mayno-Fraser two filter formulae and also provides insight into certain invariance properties of backwards Kalman filter estimates.

System Identification concerns the problem of building mathematical models of a dynamical system from measured input-output data. In this paper we examine some aspects of this problem. Typical sets of models are displayed and basic principles for fitting them to data are discussed. Asymptotic properties of the resulting model are also quoted.

A general class of algorithms for recursive identification of (stochastic) dynamical systems is studied. In this class, the discrepancy between the measured output and the output, predicted from previous data according to a candidate model (‘the prediction error’) is minimized over the model set using a stochastic approximation approach. It is proved that this class of methods has the same convergence properties as its off-line counterparts under mild and general assumptions. These assumptions do not, for example, include stationary conditions or conditions that the true system can be exactly represented within the model set.

The considered class of methods contains as special cases several well-known algorithms. Other common recursive identification methods, like the extended least squares method and the extended Kalman filter can be interpreted as approximate prediction error methods with simplified gradient calculations. Therefore, the approach taken here, may also serve as a basis for unified description of many recursive identification methods.

A general class of parameter estimation methods for stochastic dynamical systems is studied. The class contains the least squares method, output-error methods, the maximum likelihood method and several other techniques. It is shown that the class of estimates so obtained are asymptotically normal and expressions for the resulting asymptotic covariance matrices are given. The regularity conditions that are imposed to obtain these results, are fairly weak. It is, for example, not assumed that the true system can be described within the chosen model set, and, as a consequence, the results in this paper form a part of the so-called approximate modeling approach to system identification. It is also noteworthy that arbitrary feedback from observed system outputs to observed system inputs is allowed and stationarity is not required

The problem of solving linear equations, or equivalently of inverting matrices, arises in many fields. Efficient recursive algorithms for finding the inverses of Toeplitz or displacement-type matrices have been known for some time. By introducting a way of characterizing matrices in terms of their “distance” from being Toeplitz, a natural extension of these algorithms is obtained. Several new inversion formulas for the representation of the inverse of non-Toeplitz matrices are also presented.

The extended Kalman filter is an approximate filter for nonlinear systems, based on first-order linearization. Its use for the joint parameter and state estimation problem for linear systems with unknown parameters is well known and widely spread. Here a convergence analysis of this method is given. It is shown that in general, the estimates may be biased or divergent and the causes for this are displayed. Some common special cases where convergence is guaranteed are also given. The analysis gives insight into the convergence mechanisms and it is shown that with a modification of the algorithm, global convergence results can be obtained for a general case. The scheme can then be interpreted as maximization of the likelihood function for the estimation problem, or as a recursive prediction error algorithm.

Very rapid initial convergence of the equalizer tap coefficients is a requirement of many data communication systems which employ adaptive equalizers to minimize intersymbol interference. As shown in recent papers by Godard, and by Gitlin and Magee, a recursive least squares estimation algorithm, which is a special case of the Kalman estimation algorithm, is applicable to the estimation of the optimal (minimum MSE) set of tap coefficients. It was furthermore shown to yield much faster equalizer convergence than that achieved by the simple estimated gradient algorithm, especially for severely distorted channels. We show how certain "fast recursive estimation" techniques, originally introduced by Morf and Ljung, can be adapted to the equalizer adjustment problem, resulting in the same fast convergence as the conventional Kalman implementation, but with far fewer operations per iteration (proportional to the number of equalizer taps, rather than the square of the number of equalizer taps). These fast algorithms, applicable to both linear and decision feedback equalizers, exploit a certain shift-invariance property of successive equalizer contents. The rapid convergence properties of the "fast Kalman" adaptation algorithm are confirmed by simulation.

A new implementation is presented for the optimum likelihood ratio detector for stationary Gaussian signals in white Gaussian noise that uses only two causal time-invariant filters. This solution also has the advantage that fast algorithms based on the Levinson and Chandrasekhar equations can he used for the determination of these time-invariant filters. By using a notion of "closeness to stationarity,' there is a natural extension of the above results for nonstationary signal processes.

In this paper five different recursive identification methods will be analyzed and compared, namely recursive versions of the least squares method, the instrumental variable method, the generalized least squares method, the extended least squares method and the maximum likelihood method. They are shown to be similar in structure and need of computer storage and time. Making use of recently developed theory for asymptotic analysis of recursive stochastic algorithms, these methods are examined from a theoretical viewpoint. Possible convergence points and their global and local convergence properties are studied. The theoretical analysis is illustrated and supplemented by simulations.

A sequence of vectors {x(t)} with dimension N is given, such that x(t+1) is obtained from x(t) by introducing p new elements, deleting p old ones, and shifting the others in some fashion. The sequence of vectors $ is sought. This paper presents a method of calculating these vectors with proportional-to-Npoperations and memory locations, in contrast to the conventional way which requires proportional-to-N ²operations and memory locations. A non-symmetric case is also treated.

The convergence with probability one of a recently suggested recursive identification method by Landau is investigated. The positive realness of a certain transfer function is shown to play a crucial role, both for the proof of convergence and for convergence itself. A completely analogous analysis can be performed also for the extended least squares method and for the self-tuning regulator of Åström and Wittenmark. Explicit conditions for convergence of all these schemes are given. A more general structure is also discussed, as well as relations to other recursive algorithms.

We give simple proofs of formulas for converting linear least-squares filtered and smoothed estimates derived for one set of initial conditions to estimates valid for some other set. These are then used to study the possible advantages of first deliberately mischoosing the initial conditions so as to allow computational benefits to be obtained by using certain fast algorithms. In the course of this application we also obtain a new "dual" set of Chandrasekhar equations that provide a fast algorithm for fixed-point smoothing.

The possibility of estimating the parameters of a dynamic system when it is operating in closed loop is examined. Earlier considered ways of designing regulators to achieve desired identifiability properties are unified and generalized. The result of the analysis of this short paper gives a simple criterion, which contains earlier known conditions as simple special cases.

A state-space model of a second-order random process is a representation as a linear combination of a set of state-variables which obey first-order linear differential equations driven by an input process that is both white and uncorrelated with the initial values of the state-variables. Such a representation is often called a Markovian representation. There are applications in which it is useful to consider time running backwards and to obtain corresponding backwards Markovian representations. Except in one very special circumstance, these backwards representations cannot be obtained simply by just reversing the direction of time in a forwards Markovian representation. We show how this problem can be solved, give some examples, and also illustrate how the backwards model can be used to clarify certain least squares smoothing formulas.

Reglerteknik förekommer numera i de flesta tekniska system: motorstyrning, antisladdsystem och farthållare i bilar; effektstyrning för mobiltelefoner; banföljning i industrirobotar; styrautomater i flygplan; styrning av allehanda kvalitetsvariabler i processindustrin liksom många tillämpningar inom konsumentelektronik...[Bokinfo]

This book provides signal processing exercises and can with advantage be used together with the text book Signal Processing by Fredrik Gustafsson, Lennart Ljung and Mille Millnert. The chapters of the books are aligned, which means that there are matching exercises to each theory chapter. The first part of the book treats classical digital signal processing based on transforms and filters, while model based digital processing is in focus in the second part. Some exercises are more theoretical and solved by hand, while others are intended for Matlab on a computer. The book material is inspired by real problems, and so are the exercises. This is emphasized by the use of data sets, both simulated and real. Most exercises have complete solutions, and a section with hints provides guidance to some exercises. Selected exercises also result in a Matlab function corresponding to specific signal processing algorithms. These functions are used to solve other exercises. Thereby, the reader gradually build up a signal processing toolbox during the studies of the material. The book homepage contains more information and links to access the matlab functions, data sets and examples used in the book. Main book Signal Processing

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Das Buch behandelt die Systemidentifizierung in dem theoretischen Bereich, der direkte Auswirkungen auf Verstaendnis und praktische Anwendung der verschiedenen Verfahren zur Identifizierung hat. Das Buch ist so konzipiert, dass der Leser, der eine schnelle Anleitung benoetigt, nicht alle theoretischen Erlaeuterungen durcharbeiten muss. Im ersten Abschnitt ueber Systeme und Modelle wird auf zeitkonstante lineare Systeme und deren Modelle, auf Simulation, Vorhersage und Regelung sowie auf Modelle fuer zeitvariable und nichtlineare Systeme eingegangen. Die verschiedenen Methoden der Systemidentifizierung, wie z.B. die Fourier- und die Spektralanalyse sowie Parameterschaetzverfahren werden im zweiten Abschnitt behandelt. Der letzte Abschnitt ueber die Auswahl der Verfahren behandelt die Auswahl der Identifizierungskriterien, den Experimententwurf, die Auswahl der Modellstruktur, die Modellbestaetigung und weitere praktische Aspekte der Systemidentifizierung.

Within the class of nonlinear system models, the so-called block-oriented models have gained wide recognition and attention by the system identification and automatic control community. Typically, these models are constructed by joining linear dynamic system blocks with static nonlinear mappings in various forms of interconnection.

In this edited collection we commemorate the 60th birthday of Prof. Christopher Byrnes and the retirement of Prof. Anders Lindquist from the Chair of Optimization and Systems Theory at KTH. These papers were presented in part at a 2009 workshop in KTH, Stockholm, honoring the lifetime contributions of Professors Byrnes and Lindquist in various fields of applied mathematics.

For some reason I now forget, Keith Glover and I were asked to prepare a “discussion paper” on frequency and time domain methods in system identification by the organizers of the IFAC symposium on System Identification in Darmstadt 1979, [8]. I visited Keith in Cambridge in April 1979 to prepare our paper. This was my first “scientific” visit to Cambridge, and I had some very productive and pleasant days with Keith there. After the symposium we were asked to produce an Automatica version of the discussion paper, and this was eventually published as [9]. Although I have met Keith and had interesting and enjoyable discussions with him many, many times since then, these two papers remain our only joint publications.

The purpose of this contribution is to point out and exploit the kinship between identification of linear dynamic systems and classical curve fitting. For curve fitting we discuss both global and local parametric methods as well as non-parametric ones, such as local polynomial methods. We view system identification as the estimation of the frequency function curve. The empirical transfer function estimate is taken as the "observations" of this curve. In particular we discuss how this could be done for multi-variable systems. Local and non-parametric curve fitting methods lead to variants of classical spectral analysis, while the prediction error/maximum likelihood framework corresponds to global parametric methods. The role of the noise model in dynamic models is also illuminated from this perspective.

In this contribution we give an overview and discusison of the basic steps of System Identification. The four main ingredients of the process that tkes us from observed data to a validated model are: (1) The data itself, (2) The set of candidate models, (3) The criterion of fit and (4) The validation procedure. We discuss how these ingredients can be blended to a useful mix for model-building in practice.

The problem of identification is to determine a model that describes input–output data obtained from a certain system. In this chapter, strong consistency for general prediction error methods, including the maximum-likelihood (ML) method is considered. The results are valid for general process models: linear and nonlinear. An error identification method is discussed in the chapter along with a general model for stochastic dynamic systems. Different identifiability concepts are also introduced, where a procedure to prove consistency is outlined. Consistency is shown for a general system structure, as well as for linear systems. The application of the results to linear time-invariant systems is also discussed in the chapter.

Asymptotic theory for the regularized system identification has received increasing interests in recent years. In this paper, for the finite impulse response (FIR) model and filtered white noise inputs, we show the convergence in distribution of the Steins unbiased risk estimator (SURE) based hyper-parameter estimator and find factors that influence its convergence properties. In particular, we consider the ridge regression case to obtain closed-form expressions of the limit of the regression matrix and the variance of the limiting distribution of the SURE based hyper-parameter estimator, and then demonstrate their relation numerically.

Deep state space models (SSMs) are an actively researched model class for temporal models developed in the deep learning community which have a close connection to classic SSMs. The use of deep SSMs as a black-box identification model can describe a wide range of dynamics due to the flexibility of deep neural networks. Additionally, the probabilistic nature of the model class allows the uncertainty of the system to be modelled. In this work a deep SSM class and its parameter learning algorithm are explained in an effort to extend the toolbox of nonlinear identification methods with a deep learning based method. Six recent deep SSMs are evaluated in a first unified implementation on nonlinear system identification benchmarks. Copyright (C) 2021 The Authors.

The selections for the model orders and the number of controller parameters have not been discussed for many data-driven iterative learning control (ILC) methods. If they are not chosen carefully, the estimated model and designed controller will lead to either large variance or large bias. In this paper we try to use the kernel-based regularization method (KRM) to handle the model estimation problem and the controller design problem for unknown repetitive linear time-varying systems. In particular, we have used the diagonal correlated kernel and the marginal likelihood maximization method for the two problems. Numerical simulation results show that smaller mean square errors for each time instant are obtained by using the proposed ILC method in comparison with an existing data-driven ILC approach. Copyright (C) 2021 The Authors.

In this paper, we study the influence of ill-conditioned regression matrix on two hyper-parameter estimation methods for the kernel-based regularization method: the empirical Bayes (EB) and the Steins unbiased risk estimator (SURE). First, we consider the convergence rate of the cost functions of EB and SURE, and we find that they have the same convergence rate but the influence of the ill-conditioned regression matrix on the scale factor are different: for upper bounds, the scale factor for SURE contains one more factor cond(Phi(T)Phi) than that of EB, where Phi is the regression matrix and cond(.) denotes the condition number of a matrix. This finding indicates that when Phi is ill-conditioned, i.e., cond(Phi(T)Phi) is large, the cost function of SURE converges slower than that of EB. Then we consider the convergence rate of the optimal hyper-parameters of EB and SURE, and we find that they are both asymptotically normally distributed and have the same convergence rate, but the influence of the ill-conditioned regression matrix on the scale factor are different. In particular, for the ridge regression case, we show that the optimal hyper-parameter of SURE converges slower than that of EB with a factor of 1/n(2), as cond(Phi(T)Phi) goes to infinity, where n is the FIR model order.

We propose a generalization of the popular nonlinear ARX model structure by treating its parameters as varying over time. The parameters are considered generated by linear filters operating on the models regressors. The filters are expressed as a sum of atoms that are either sum of damped exponentials and sinusoids, or sinusoids with time varying frequencies. This form allows us to enforce stability of the parameter evolution as well as leverage the atomic norm minimization framework for inducing sparsity. It also facilitates easy incorporation of smoothness related priors that that making it possible to treat these models as nonlinear extensions of the familiar LPV models. (C) 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.

We explore the problem of identification of LPV models when the scheduling variables are not known in advance and the model parameters exhibit a dynamic dependence on them. We consider an affine ARX model structure whose parameters vary with time. We solve for the models parameters and scheduling variables in two steps. In the first step, we use the measured input-output data to realize a parameter trajectory by solving a regularized Hankel matrix rank minimization problem. The regularization penalty is guided by the prior knowledge regarding the nature of systems time variation. In the second step, the scheduling variables are estimated as parameters of a sparse ARX structure relating the models parameters to the measured input-output variables. The effectiveness of the proposed approach is illustrated with two practical examples. (C) 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.

In this paper, we study the asymptotic properties of the generalized cross validation (GCV) hyperparameter estimator and establish its connection with the Steins 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 GCV has the same asymptotic property as the SURE does and both of them converge to the best hyperparameter in the MSE sense. We illustrate the efficacy of the result by Monte Carlo simulations. (C) 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.

Because of the increased demand on fault detection, monitoring and predictive maintenance, online or recursive identification is playing a more important role in systems engineering. In the recent releases of System Identification Toolbox (TM) for MATLAB (R), this has been reflected in a more substantial support for online techniques. This contribution gives an account of these improvements. It covers the addition of nonlinear filtering algorithms, such as the extended Kalman filter, the unscented Kalman filter and particle filters. The traditional recursive estimation techniques for polynomial models have also been enhanced with a more versatile syntax. Several new Simulink (R) blocks have been developed to support Simulink (R) models with online estimation. (C) 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.

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.

This paper addresses reconstruction of linear dynamic networks from heterogeneous datasets. Those datasets consist of measurements from linear dynamical systems in multiple experiments subjected to different experimental conditions, e.g., changes/perturbations in parameters, disturbance or noise. A main assumption is that the Boolean structures of the underlying networks are the same in all experiments. The ARMAX model is adopted to parameterize the general linear dynamic network representation "Dynamical Structure Function" (DSF), which provides the Granger Causality graph as a special case. The network identification is performed by integrating all available datasets and promote group sparsity to assure both network sparsity and the consistency of Boolean structures over datasets. In terms of solving the problem, a treatment by the iterative reweighted l(1) method is used, together with its implementations via proximal methods and ADMM for large-dimensional networks. (C) 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.

The continuous-time subspace identification using state-variable filtering has been investigated for a long time. Due to the simple orthogonal basis functions that were adopted by the existing methods, the identification performance is quite sensitive to the selection of the system-dynamic parameter associated with an orthogonal basis. To cope with this problem, a subspace identification method using generalized orthonormal(Takenaka-Malmquist) basis functions is developed, which has the potential to perform better than the existing state-variable filtering methods since the adopted Takenaka-Malmquist basis has more degree of freedom in selecting the system-dynamic parameters. As a price for the flexibility of the generalized orthonormal bases, the transformed state-space model is time-varying or parameter-varying which cannot be identified using traditional subspace identification methods. To this end, a new subspace identification algorithm is developed by exploiting the structural properties of the time-variant system matrices, which is then validated by numerical simulations.

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.

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.

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.

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.

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 MATLAB object-oriented programming. Two main features are (1) that the toolbox commands and plots are seamlessly integrated with the other MATLAB toolboxes that deal with linear dynamic systems and (2) several new features and model objects. The toolbox now supports multi-input--multi-output (MIMO) systems across all model objects, and more emphasis is placed on continuous-time models. Also a new model object, IDTF covers MIMO transfer function models in both continuous and discrete time.

This paper deals with the issue of estimating the parameters in a continuous-time nonlinear dynamical model from sampled data. We focus on the issue of bias-variance trade-offs. In particular, we show that the bias error can be significantly reduced by using a particular form of sampled data model based on truncated Taylor series. This model retains the conceptual simplicity of models based on Euler integration but has much improved accuracy as a function of the sampled period.

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.

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.

We present a novel clustering method, formulated as a convex optimization problem. The method is based on over-parameterization and uses a sum-of-norms (SON) regularization to control the trade-off between the model fit and the number of clusters. Hence, the number of clusters can be automatically adapted to best describe the data, and need not to be specified a priori. We apply SON clustering to cluster the particles in a particle filter, an application where the number of clusters is often unknown and time varying, making SON clustering an attractive alternative.

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.

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.

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 on a model for a whole batch of data. This paper shows that horizon estimation may also be used for joint parameter estimation and state estimation, as long as a bias correction based on the Kalman filter is included. A procedure how to approximate the bias correction for nonlinear systems is outlined.

High-dimensional gray-box identification is a fairly unexplored part of system identification. Nevertheless, system identification problems tend to be more high-dimensional nowadays. In this paper we deal with high-dimensional regression with regressors constrained to some manifold. A recent technique in this class is weight determination by manifold regularization (WDMR). WDMR, however, is a black-box identification method. We show how WDMR can be extended to a gray-box method and illustrate the scheme with some examples.

This paper describes a benchmark for nonlinear system identification. A Wiener-Hammerstein system is selected as test object. In such a structure there is no direct access to the static nonlinearity starting from the measured input/output, because it is sandwiched between two unknown dynamic systems. The signal-to-noise ratio of the measurements is quite high, which puts the focus of the benchmark on the ability to identify the nonlinear behaviour, and not so much on the noise rejection properties. The benchmark is not intended as a competition, but as a tool to compare the possibilities of different methods to deal with this specific nonlinear structure.

Identification of time-continuous models from sampled data is a long standing topic of discussion, and many approaches have been suggested. The Maximum Likelihood method is asymptotically and theoretically superior to other methods. However, it may suffer from numerical inaccuracies at fast sampling and it also requires reliable initial parameter values. A number of efficient and useful alternatives to the maximum-likelihood method have been developed over the years. The most important of these are State-Variable filters, combined with Instrumental Variable methods, including the simplified refined IV method. In this contribution we perform unpretentious numerical experiments to comment on these methods, and their mutual benefits.

Control system design for advanced, highly agile fighter aircraft, with unstable nonlinear aerodynamic characteristics, rely heavily on flight mechanical simulations. This makes the accuracy of the aerodynamic model in the simulators very important. Here, two methods for estimating parameters of nonlinear unstable systems where the control system is unknown are presented. Both approaches are direct prediction-error methods, either with a directly parametrized observer or with an Extended Kalman Filter as a predictor. These methods have been validated on simulated data, as well as on real flight test data and all approaches show promising results.

In system identification, the challenge of parameter estimation often lies in solving a non-convex optimization problem. In many cases, this implies that it is difficult to guarantee that the global optimum will be found. The tools of differential algebra, for example, Gr\"{o}bner bases and Ritt's algorithm, have turned out to be quite useful when dealing with certain nonlinear model structures. Some examples of successful applications are the determination of controllability, observability and global identifiability of these model structures. In this paper, difference algebraic techniques, which mimics the differential algebraic methods, will be presented. Besides making it possible to handle discrete-time systems, this opens up the possibility of dealing with noise. It turns out that the classical instrumental variables method plays a role.

Studierektorsområdet Reglersystem vid LiTH (Tekniska Högskolan vid Linköpings universitet) tilldelades 2007 utmärkelsen Framstående Utbildningsmiljö av Högskoleverket. Utmärkelsen gavs till sammanlagt fem utbildningsmiljöer, varav två är verksamma inom ingenjörsutbildning. I detta bidrag avser vi att redogöra för denna process och de faktorer som främst bidrog till att området Reglersystem fick denna utmärkelse. De faktorer som lyfts fram som centrala för att åstadkomma en god utbildningskvalité är framför allt att ha tydliga mål för utbildningen, en gedigen ämnesmässig grund, en väl fungerande organisation och positiv attityd bland alla medverkande samt former för lärande och examination som är anpassade till utbildningens mål.

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.

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 practise 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.