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For data-driven iterative learning control (ILC) methods, both the model estimation and controller design problems are converted to parameter estimation problems for some chosen model structures. It is well-known that if the model order is not chosen carefully, models with either large variance or large bias would be resulted, which is one of the obstacles to further improve the modeling and tracking performances of data-driven ILC in practice. An emerging trend in the system identification community to deal with this issue is using regularization instead of the statistical tests, e.g., AIC, BIC, and one of the representatives is the so-called kernel-based regularization method (KRM). In this paper, we integrate KRM into data-driven ILC to handle a class of repetitive linear time-varying systems, and moreover, we show that the proposed method has ultimately bounded tracking error in the iteration domain. The numerical simulation results show that in contrast with the least squares method and some existing data-driven ILC methods, the proposed one can give faster convergence speed, better accuracy and robustness in terms of the tracking performance., Comment: 17 pages

In the overview paper on nonlinear system identification Schoukens and Ljung (2019), it was indicated that reliable expressions to calculate the variance of an estimated nonlinear model are lacking, especially if the disturbing noise is entering the nonlinear regressors. In this study, we provide a better view on the driving mechanisms of the variability of estimated nonlinear models that is due to noise on the output. To do so, we follow a double approach. Firstly, a basic insight on the impact of disturbing noise on the monomial ynis studied. Next, these insights are used in a case study on the forced Duffing oscilator data, also called the Silver box (Schoukens and Noel, 2016). The following models are studied: Nonlinear autoregressive exogenous models (NARX) using a 2-layer Neural Net (NARX-NN) and a polynomial (NARX-poly) expansion; and polynomial nonlinear state space models (PNLSS). This limited study indicates that an output error criterion (PNLSS, and NARX used in a simulation mode) does better than minimizing the equation error (NARX-NN and NARX-poly in prediction mode). When the signal-to-noise ratio (SNR) drops below 20 dB, the reduction in the error is more than a factor 10. This is a strong indication that, just as for linear identification, it is very important to properly deal with the noise properties in the cost function whenever the SNR of the output measurements drops such that noise becomes more important than structural model errors.

International audience; 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 paper 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.

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

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

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

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

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

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

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

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

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

International audience; 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.

A promising method for estimation of the time-delay in continuous-time linear dynamical systems uses the phase of the all-pass part of a discrete-time model of the system. We have discovered that this method can sometimes fail totally and we suggest a method for avoiding such failures.

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.

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.

This paper treats direct identification of continuous-time autoregressive moving average (CARMA) time-series models. The main result is a method for estimating the continuous-time power spectral de ...

Closed loop subspace identification has become a focus of interest with several recent developments. Notably are the innovation estimation approach (Qin and Ljung, 2003), the state space approach with ARX pre-estimates (SSARX, (Jansson, 2003)), and the whitening filter approach (Chiuso and Picci, 2004). All these approaches use an extended future horizon to form the projection or regression from which an observable subspace is extracted. Yet there are other methods such as OKID of (Phan and Longman, 1992) and that of (Ljung and McKelvey, 1996) that do not use an extended horizon in the projection or regression step. Instead, a single high order ARX model is used. A natural question is whether the future horizon is necessary and if so what role does it play in these steps. In this paper we investigate the role of the future horizon using the whitening filter approach of (Chiuso and Picci, 2004), which works for both open-loop and closed-loop data. We conclude that the role of future horizon in this algorithm is merely extending the order of a bank of already high order ARX models. The difference from a single ARX model is insignificant if the ARX order or past horizon is sufficiently high. The role of future horizon is mainly in the model reduction step where it serves to elevate the order of the Hankel matrix. We complement the analysis with simulations.

International audience; A Wiener system is composed of a linear dynamic subsystem followed by a static nonlinearity. It is well known in the literature that the identification of the linear subsystem of a Wiener system can be separated from that of the output nonlinearity, if the input signal is a Gaussian noise. In order to deal with the non Gaussian input case, two new algorithms are proposed in this paper for direct identification of the linear subsystem, regardless of any parametrization of the output nonlinearity. The essential assumption required in this paper is the strict monotonousness of the output nonlinearity.

This paper considers the identification of stochastic Wiener dynamic systems, that is linear dynamic systems with process noise, where the measurable output signal is a nonlinear function of the output from the linear system corrupted with additive measurement noise. It is shown how stochastic Wiener system identification can be viewed as a particular non-linear model errors-in-variables problem, for which there exists a large literature. We compare the maximum likelihood method with prediction error minimization methods based on the conditional mean predictor for Wiener systems. Related methods have previously been studied in the framework of identification of non-linear error-in-variables models. We extend these results by taking the input signal to the Wiener system into consideration. For example, the input will affect the variance of the prediction errors. Hence, a prediction error method with a variance weighting is derived to obtain more reliable parameter estimates. An advantage with the prediction error method is that for certain special cases we can avoid numerical integration. We also discuss how the unscented transform can be used to obtain an approximate predictor for the prediction error method. The numerical evaluation of these methods is performed on a simple first order FIR system with a cubic nonlinearity, for which some illustrative analytic properties are derived.

Nonlinear systems can be approximated by linear time-invariant (LTI) models in many ways. Here, LTI models that are optimal approximations in the mean-square error sense are analyzed. A necessary and sufficient condition on the input signal for the optimal LTI approximation of an arbitrary nonlinear finite impulse response (NFIR) system to be a linear finite impulse response (FIR) model is presented. This condition says that the input should be separable of a certain order, i.e., that certain conditional expectations should be linear. For the special case of Gaussian input signals, this condition is closely related to a generalized version of Bussgang's classic theorem about static nonlinearities. It is shown that this generalized theorem can be used for structure identification and for the identification of generalized Wiener-Hammerstein systems.

In this paper, we analyze two recently proposed closed-loop subspace identification methods, referred to as innovation estimation method and whitening filter approach respectively. The similarity and difference between them are investigated in detail. It turns out that all closed-loop subspace identification methods can be classified as one-step, two-step, or multi-stage projection methods. A SISO closed-loop simulation shows that to identify a consistent model the whitening filter approach might require longer future and past horizons than the innovation estimation method.

Approximations of slightly nonlinear systems with linear time-invariant (LTI) models are often used in applications. Here, LTI models that are optimal approximations in the mean-square error sense are studied. It is shown that these models can be very sensitive to small nonlinearities. Furthermore, the significance of the distribution of the input process is discussed. From the examples studied here, it seems that LTI approximations for inputs with distributions that are Gaussian or almost Gaussian are less sensitive to small nonlinearities.

We present a result that can be viewed as a generalization of Bussgang's classical theorem about static nonlinearities with Gaussian inputs. This result is used to characterize the best linear approximation of a nonlinear finite impulse response (NFIR) system with a Gaussian input. The best linear approximation is here defined as the causal and stable LTI system that minimizes the mean-square error. Furthermore, we discuss how this characterization can be used for structure identification and for identification of generalized Hammerstein and Wiener systems.

This report describes how parameter estimation can be performed in linear DAE systems. Both time domain and frequency domain identification are examined. The results are exemplified on a small syst ...

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, ...

The purpose of this paper is to show that the concept of differential neural networks (DNN) can be successfully applied to the identification and adaptive control design for stochastic nonlinear continuous time systems.

Asymptotic variance expressions are analyzed for models that are identified on the basis of closed-loop data. The considered methods comprise the classical 'direct' method, as well as the more recently developed indirect methods, employing coprime factorized models, dual Youla/Kucera parametrizations and the two-stage approach. The variance expressions are compared with the open-loop situation, and evaluated in terms of their relevance for subsequent model-based control design.

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 given 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.

Model validation and estimating the size of a possible model error is a central aspect of System Identification. In this contribution we discuss the model error concepts and model error modeling for control design. Of special interest is how to make use of periodic inputs, and how to deal with non-linear error models. The discussion is limited to SISO models and stability robustness issues.

A new method for closed-loop identification that allows fitting the model to the data with arbitrary frequency weighting is described and analyzed. Just as the direct method, this new method is applicable to systems with arbitrary feedback mechanisms. This is in contrast to other methods, such as the indirect method and the two-stage method, that assume linear feedback. The finite sample behavior of the proposed method is illustrated in a simulation study.

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.

Identification of systems operating in closed loop has long been of prime interest in industrial applications. The problem offers many possibilities, and also some fallacies, and a wide variety of approaches have been suggested, many quite recently. The purpose of the current contribution is to place most of these approaches in a coherent framework, thereby showing their connections and display similarities and differences in the asymptotic properties of the resulting estimates. The common framework is created by the basic prediction error method, and it is shown that most of the common methods correspond to different parameterizations of the dynamics and noise models. The so-called indirect methods, e.g., are indeed ''direct'' methods employing noise models that contain the regulator. The asymptotic properties of the estimates then follow from the general theory and take different forms as they are translated to the particular parameterizations. We also study a new projection approach to closed-loop identification with the advantage of allowing approximation of the open-loop dynamics in a given, and user-chosen frequency domain norm, even in the case of an unknown, nonlinear regulator.

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.

"Identification for Control" has drawn significant interest the past few years. The objective is to achieve a model that is suited for robust control design. Thus one has to tailor the experiment a ...

The trade-off between tracking ability and noise sensitivity inadaptive algorithms is a classical research topic. In this paper we will revisit some of the main results. An input/output view of the ...

Asymptotic variance expressions are analysed for models that are identified on the basis of closed-loop data. The considered methods comprise the classical ’direct’ and ’indirect’ method, as well as the more recently developed indirect methods, employing coprime factorized models and model parametrizations based on the dual Youla/Kucera parametrization. The variance expressions are compared with the open-loop situation, and evaluated in terms of their relevance for subsequent model-based control design. Additionally it is specified what is the optimal experimental situation in identification (open-loop or closed-loop), in view of the variance of the resulting model-based controller.

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-/spl phi/-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 problem of tracking time-varying properties of a signal is studied. The somewhat contradictory notion of "time-varying spectrum" and how to estimate the "current" spectrum in an on-line fashion is discussed. The traditional concepts and relations between time and frequency resolution are crucial for this problem. We introduce two definitions for the time resolution of filters, essentially measuring the effective number of past data that are used to form the estimate. In, for example, wavelet transform techniques, frequency-dependent time resolutions are used so that fewer data are used at higher frequencies, thus enabling faster tracking of high-frequency components (at the price of worse frequency resolution). The main contribution of the paper is to show how this same feature can be introduced when estimating spectra via a time-varying, autoregressive model of the signal. This is achieved by a special choice of nominal covariance matrix for the underlying parameter changes.

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.

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.

In this contribution aspects of inter-sample input signal behavior are examined. The starting point is that parametric identification always is performed on basis of discrete-time data. This is valid for identification of discrete-time models as well as continuous-time models. The usual assumptions on the input signal are; i) it is band-limited, ii) it is piecewise constant or iii) it is piecewise linear. One point made in this paper is that if a discrete-time model is used, the best possible (in the model structure) adjustment to data is made. This is independent of the assumption on the input signal. However, a transformation of the obtained discrete model to a continuous one is not possible without additional assumptions on the input signal. The other point made is that the frequency functions of the discrete models very well coincides with the frequency functions of the discretized continuous time models and the continuous time transfer function fitted in the frequency domain.

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 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.

Many tracking problems are split into two sub-problems, first a smooth reference trajectory is generated that meet the control design objectives, and then a closed loop control system is designed to follow this reference trajectory as well as possible. Applications of this kind include (autonomous) vehicle navigation systems and robotics. Typically, a spline model is used for trajectory generation and another physical and dynamical model is used for the control design. Here we propose a direct approach where the dynamical model is used to generate a control signal that takes the state trajectory through the waypoints specified in the design goals. The strength of the proposed formulation is the methodology to obtain a control signal with compact representation and that changes only when needed, something often wanted in tracking. The formulation takes the shape of a constrained least-squares problem with sum-of-norms regularization, a generalization of the l 1 -regularization. The formulation also gives a tool to, e.g. in model predictive control, prevent chatter in the input signal, and also select the most suitable instances for applying the control inputs.

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. >

By "leakage" in adaptive control and adaptive signal processing algorithm is understood that a pull term towards a given parameter value is introduced. Leakage has been introduced both as trick to ...

A prediction error method for parameter estimation in a dynamical system is studied. $$ \hat \vartheta = \arg {\mkern 1mu} \mathop {\min }\limits_\vartheta \mathop {\lim }\limits_{N \to \infty } \frac{1}{N}\sum\limits_{t = 1}^N {{\text{E}}l\left( {\varepsilon \left( {t,\vartheta } \right)} \right)} $$ where e are the prediction errors of a linear regression. A quadratic norm l is zero within an interval [−c, c]. This kind of a dead zone (DZ) criterion is very common in robust adaptive control. The following problems are treated in this chapter: When is the DZ estimate inconsistent, and what is the set of parameters which minimizes the criterion in the case of inconsistency? What happens to the variance of the estimate as the DZ is introduced? Does the DZ give a better estimate than least squares (LS) when there are unmodeled deterministic disturbances present? What are the relations between identification with a dead zone criterion and so called set membership identification?

In this contribution we consider the situation where no disturbances act on the system to be identified, but where - as always - it is not possible to find a model structure that exactly accommodates the true systems. The transient and asymptotic properties of the obtained estimates in this situation are considered. Special attention is paid to the practical problem of obtaining good adaptation properties by appropriate choices of forgetting factors and prefilters.