Perception of sounds and speech involves structures in the auditory brainstem that rapidly process ongoing auditory stimuli. The role of these structures in speech processing can be investigated by measuring their electrical activity using scalp-mounted electrodes. However, typical analysis methods involve averaging neural responses to many short repetitive stimuli that bear little relevance to daily listening environments. Recently, subcortical responses to more ecologically relevant continuous speech were detected using linear encoding models. These methods estimate the temporal response function (TRF), which is a regression model that minimises the error between the measured neural signal and a predictor derived from the stimulus. Using predictors that model the highly non-linear peripheral auditory system may improve linear TRF estimation accuracy and peak detection. Here, we compare predictors from both simple and complex peripheral auditory models for estimating brainstem TRFs on electroencephalography (EEG) data from 24 participants listening to continuous speech. We also investigate the data length required for estimating subcortical TRFs, and find that around 12 minutes of data is sufficient for clear wave V peaks (>3 dB SNR) to be seen in nearly all participants. Interestingly, predictors derived from simple filterbank-based models of the peripheral auditory system yield TRF wave V peak SNRs that are not significantly different from those estimated using a complex model of the auditory nerve, provided that the nonlinear effects of adaptation in the auditory system are appropriately modelled. Crucially, computing predictors from these simpler models is more than 50 times faster compared to the complex model. This work paves the way for efficient modelling and detection of subcortical processing of continuous speech, which may lead to improved diagnosis metrics for hearing impairment and assistive hearing technology.

Unless a segregated airspace and the corresponding clearances can be afforded, flight testing of remotely piloted aircraft is often done near the ground and within visual line-of-sight. In addition to the increased exposure to turbulence, this setup also limits the available time for test manoeuvres on each pass, especially for subscale demonstrators with a relatively high wing loading and flight speed. A suitable testing procedure, efficient excitation signals and a robust system identification method are therefore fundamental. Here, the authors use ground-based flight control augmentation to inject multisine signals with low correlation between the different inputs. Focusing on initial flight-envelope expansion, where linear regression is common, this paper also describes the improvement of an existing frequency-domain method by using an instrumental variable (IV) approach to better handle turbulence and measurement noise and to enable real-time identification analysis. Both simulations and real flight tests on a subscale demonstrator are presented. The results show that the combination of multisine input signals and the enhanced frequency-domain method is an effective way of improving flight testing of remotely piloted aircraft in confined airspace.

This letter deals with the issue of obtaining consistent parameter estimators in nonlinear regression models where the regressors are second-order modulus functions, which is a structure that is often used in models of marine vessels. It is shown that the accuracy of an instrumental variable estimator can be improved by conducting experiments where the input signal has a static offset of sufficient amplitude and the instruments are forced to have zero mean. The proposed method is then evaluated in a simulation example.

In this paper the main goal is to compare the instrumental variables and the least squares methods applied to parameter estimation in continuous-time systems, avoiding any preliminary discretization of the process, and to analyse which method is more suitable for estimation in continuous-time under stochastic perturbations. A numerical example illustrates the effectiveness of the algorithms.

This brief presents a behavioral model structure and a model-based phase-only predistortion method that are suitable for outphasing RF amplifiers. The predistortion method is based on a model of the amplifier with a constant gain factor and phase rotation for each outphasing signal, and a predistorter with phase rotation only. The method has been used for enhanced data rates for GSM evolution (EDGE) and wideband code-division multiple-access (WCDMA) signals applied to a Class-D outphasing RF amplifier with an on-chip transformer used for power combining in 90-nm CMOS. The measured peak power at 2 GHz was +10.3 dBm with a drain efficiency and power-added efficiency of 39% and 33%, respectively. For an EDGE 8 phase-shift-keying (8-PSK) signal with a phase error of 3 degrees between the two input outphasing signals, the measured power at 400 kHz offset was -65.9 dB with predistortion, compared with -53.5 dB without predistortion. For a WCDMA signal with the same phase error between the input signals, the measured adjacent channel leakage ratio at 5-MHz offset was -50.2 dBc with predistortion, compared with -38.0 dBc without predistortion.

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.

Based on simple Frequency Response Function (FRF) measurements, we give the user some guidance in the selection of an appropriate nonlinear block structure for the system to be modeled. The method consists in measuring the FRF using a Gaussian-like input signal and varying in a first experiment the root-mean-square (rms) value of this signal while maintaining the coloring of the power spectrum. Next, in a second experiment, the coloring of the power spectrum is varied while keeping the rms value constant. Based on the resulting behavior of the FRF, an appropriate nonlinear block structure can be selected to approximate the real system. The identification of the selected block-oriented model itself is not addressed in this paper. A theoretical analysis and two practical applications of this structure identification method are presented for some nonlinear block structures.

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]

Block-oriented Nonlinear System Identification deals with an area of research that has been very active since the turn of the millennium. The book makes a pedagogical and cohesive presentation of the methods developed in that time. These include: iterative and over-parameterization techniques; stochastic and frequency approaches; support-vector-machine, subspace, and separable-least-squares methods; blind identification method; bounded-error method; and decoupling inputs approach.The identification methods are presented by authors who have either invented them or contributed significantly to their development. All the important issues e.g., input design, persistent excitation, and consistency analysis, are discussed. The practical relevance of block-oriented models is illustrated through biomedical/physiological system modelling. The book will be of major interest to all those who are concerned with nonlinear system identification whatever their activity areas. This is particularly the case for educators in electrical, mechanical, chemical and biomedical engineering and for practising engineers in process, aeronautic, aerospace, robotics and vehicles control. Block-oriented Nonlinear System Identification serves as a reference for active researchers, new comers, industrial and education practitioners and graduate students alike.

The control system of industrial robots is often model-based, and the quality of the model of high importance. Therefore, a fast and easy-to-use process for finding the model parameters from a combination of prior knowledge and measurement data is required. It has been shown that the experiment design can be improved in terms of short experiment times and an accurate parameter estimate if the robot configurations for the identification experiments are selected carefully. Estimates of the information matrix can be generated based on simulations for a number of candidate configurations, and an optimization problem can be solved for finding the optimal configurations. This work shows that the proposed method for improved experiment design works with a real manipulator, i.e. it is demonstrated that the experiment time is reduced significantly and the accuracy of the parameter estimate can be maintained or reduced if experiments are conducted only in the optimal manipulator configurations. It is also shown that the model improvement is relevant for realizing accurate control. Finally, the experimental data reveals that, in order to further improve the model accuracy, a more advanced model structure is needed for taking into account the commonly present nonlinear transmission stiffness of the robotic joints.

This work deals with a class of nonlinear regression models called second-order modulus models. It is shown that the possibility of obtaining consistent parameter estimators for these models depends on how process disturbances enter the system. Two scenarios where consistency can be achieved for instrumental variable estimators despite non-additive system disturbances are demonstrated, both in theory and by simulation examples. Copyright (C) 2020 The Authors.

This paper concerns the estimation of a dynamic model from two measured signals when it is not clear which signal should be used as input to the model. In this case, both a forward and an inverse model can be estimated. Here, a basic instrumental variable approach is used and it is shown that the forward and inverse model estimators give identical parameter estimates provided that corresponding model structures have been used. Furthermore, it is shown that this scenario occurs when properties of a quadcopter are estimated from accelerometer and gyro signals and, hence, that it does not matter which signal is used as input.

In this work, we estimate a model of the vertical dynamics of a quadcopter and explain how this model can be used for mass estimation and diagnosis of system changes. First, a standard thrust model describing the relation between the calculated control signals of the rotors and the thrust that is commonly used in literature is estimated. The estimation results are compared to those using a refined thrust model and it turns out that the refined model gives a significant improvement. The combination of a nonlinear model and closed-loop data poses some challenges and it is shown that an instrumental variables approach can be used to obtain accurate estimates. Furthermore, we show that the refined model opens up for fault detection of the quadcopter. More specifically, this model can be used for mass estimation and also for diagnosis of other parameters that might vary between and during missions.

This paper concerns the estimation of approximate (linear) inverse models of block-oriented systems and the presented results give an improved understanding of these approximations. The estimated inverse is intended to be used as a pre- or postdistorter of the original system and a good inverse model would thus be one that, when used in series with the original system, produces a signal that resembles the original input. An inverse model of a nonlinear system can either be estimated in the standard way (from input to output) and then inverted, or directly (from output to input). This choice will affect the model. In the general case, the two modeling approaches will lead to different models, which will be shown for Hammerstein and Wiener systems. However, for a noise-free Hammerstein system with a white Gaussian input, the two approaches will result in the same model, up to a constant. When the two models are not equal, and the goal is to use the inverse as described above, it can be beneficial to estimate an approximate inverse directly. It will also be illustrated in an example how the inverse estimate can be used to get a nonparametric estimate of the nonlinearity in a block-oriented system. 

A ship's roll dynamics is sensitive to the mass and mass distribution. Changes in these physical properties might introduce unpredictable behavior of the ship and a worst-case scenario is that the ship will capsize. In this paper, a recently proposed approach for online estimation of mass and center of mass is validated using experimental data. The experiments were performed using a scale model of a ship in a wave basin. The data were collected in free run experiments where the rudder angle was recorded and the ship's motion was measured using an inertial measurement unit. The motion measurements are used in conjunction with a model of the roll dynamics to estimate the desired properties. The estimator uses the rudder angle measurements together with an instrumental variable method to mitigate the influence of disturbances. The experimental study shows that the properties can be estimated with quite good accuracy but that variance and robustness properties can be improved further.

In vehicle load detection and estimation, having an accurate estimate of the height of the Center of Gravity (CG) and the Roll Center (RC) is crucial. In this paper, taking advantage of the availability of Inertial Measurement Units (IMUs) in automotive vehicles, the nominal CG and RC heights are estimated and used for the main purpose of roof load detection. RC estimation is formulated as an Errors-in-Variables regression model. The Total and Corrected Least Squares (TLS and CLS) approaches are applied and a comparison is performed. Using the knowledge of the covariance matrix, CLS provides consistent estimates of the RC height. The estimated RC and CG heights are used to estimate the roof load using a greybox model and the method is applied to measurements from normal driving conditions.

In this paper, a method for estimating physical parameters using limited sensors is investigated. As a case study, measurements from an IMU are used for estimating the change in mass and the change in center of mass of a ship. The roll motion is studied and an instrumental variable method estimating the parameters of a transfer function from the tangential acceleration to the angular velocity is presented. It is shown that only a subset of the unknown parameters are identifiable simultaneously. A multi-stage identification approach is presented as a remedy for this. A limited simulation study is also presented to show the properties of the estimator. This shows that the method is indeed promising but that more work is needed to reduce the variance of the estimator.

The performance and design of lateral stability systems in cars depend on the ratio between the height of the center of gravity and the wheel base. This ratio is car specific, but a roof load can affect this and decrease the stability margins. We investigate the use of vehicle roll dynamics to detect and estimate changes in the overall sprung mass as well as the load positioned on the roof. It is assumed that the vehicle is equipped with a lateral accelerometer and a roll gyro, and a second order physical model is derived. The parameters in this model are partly unknown, and here estimated with a greybox and an ARMAX approach. The changes in load distribution can be detected and the approach is supported by experimental data in a lab environment. 

The estimation of nonlinear models can be a challenging problem, in particular when the number of available data points is small or when the dimension of the regressor space is high. To meet these challenges, several dimension reduction methods have been proposed in the literature, where a majority of the methods are based on the framework of inverse regression. This allows for the use of standard tools when analyzing the statistical properties of an approach and often enables computationally efficient implementations. The main limitation of the inverse regression approach to dimension reduction is the dependence on a design criterion which restricts the possible distributions of the regressors. This limitation can be avoided by using a forward approach, which will be the topic of this paper. One drawback with the forward approach to dimension reduction is the need to solve nonconvex optimization problems. In this paper, a reformulation of a well established dimension reduction method is presented, which reveals the structure of the optimization problem, and a convex relaxation is derived.

The estimation of nonlinear functions can be challenging when the number of independent variables is high. This difficulty may, in certain cases, be reduced by first projecting the independent variables on a lower dimensional subspace before estimating the nonlinearity. In this paper, a statistical nonparametric dimension reduction method called sliced inverse regression is presented and a consistency analysis for dynamically dependent variables is given. The straightforward system identification application is the estimation of the number of linear subsystems in a Wiener class system and their corresponding impulse response.

Performing tests on complicated systems can be very expensive and having a good model that describes the true system well can significantly reduce cost. This is certainly true for testing of a highly maneuverable fighter aircraft. A real-time method could be useful during testing to help in the decision process for safety reasons and for monitoring the amount of excitation in the collected data, and thus making good post-flight model identification possible. Here, an existing frequency domain method is described and improvements, using the correct finite Fourier transformation of the system equations together with an Instrumental Variable approach to handle atmospheric turbulence as system noise, are suggested. Results from simulations as well as real flight tests are presented.

In order to get a good simulation environment where flight mechanical system design can be carried out the physics has to be modeled to certain accuracy, i.e., the effects of the forces and moments acting on the aircraft under investigation have to be modeled in the best possible way. There are three main contributors to these forces and moments. The first, mass and inertia determined by weighing the aircraft, and the second, engine effects usually taken from static test, are often considered as being "true". This leaves the third and final component, the aerodynamics, which often is harder to determine. Usually it is modeled using handbook methods, numerical calculations and wind tunnel tests prior to flight test. All of these methods have their limitations such as geometrical and physical simplifications. Flight tests have traditionally been looked upon as verification of the simulator models, but lately more and more tests are being dedicated to investigate and estimate aerodynamic parameters. Since flight tests are expensive it is crucial to get out as much information as possible from them. One way to do this is to monitor the behavior of the aerodynamic model vs. the flight test online. This is being done today at Saab, but since the model usually is complex and depends on many parameters it is not easy to determine which parameters should be updated in the model in order to get better agreement between flight test and simulation data. Another important issue is whether the test contains enough information to answer this question! An online method that estimates aerodynamic parameters in real-time could be useful during tests for surveying the amount of excitation in the collected data, and thus making good model identification possible. In this paper, a frequency domain method for this purpose is described. The method has been tested in a simulator environment where the parameters are known in order to investigate its accuracy and usefulness. Different maneuvers and turbulence settings have been used in the simulations. The method could be especially useful on demonstrator aircrafts which usually are not modeled and tested so thoroughly before flight testing and which may have unusual geometrical configurations where it can be hard to predict the level of excitation needed for identification purposes. So far the results in the simulator look promising. If the tests in the simulator are successful the algorithm will be implemented in the real flight test environment at Saab. Copyright © 2009 Roger Larsson.

In this paper, two applications of a particular weighting method are described. The first application concerns a nonlinearity test for system identification problems. Starting with a validated linear model, it turns out to be possible to detect nonlinearities from the variations in the linear approximations that are obtained for different input signal distributions. The method is computationally efficient and can be used on a single dataset. The second topic of the paper is an improved weighting method for identification of the linear subsystems in Wiener or Hammerstein systems.

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.

This paper concerns the validation of linear models with respect to unmodeled nonlinearities. Two versions of a method for nonlinearity detection based on estimators of higher order cross-correlations are described and evaluated. Unlike many existing approaches, the proposed method seems to be applicable to a wide range of systems and input signals. It can also distinguish between even and odd nonlinearities.

In this paper, we discuss and compare three different approaches for L2-gain estimation of Hammerstein systems. The objective is to find the input signal that maximizes the gain. A fundamental difference between two of the approaches is the class, or structure, of the input signals. The first approach involves describing functions and therefore the class of input signals is sinusoids. In this case we assume that we have a model of the system and we search for the amplitude and frequency that give the largest gain. In the second approach, no structure on the input signal is assumed in advance and the system does not have to be modelled first. The maximizing input is found using an iterative procedure called power iterations. In the last approach, a new iterative procedure tailored for memoryless nonlinearities is used to find the maximizing input forthe unmodelled nonlinear part of the Hammerstein system. The approaches are illustrated by numerical examples.

Linear approximations of nonlinear systems can be obtained by fitting a linear model to data from a nonlinear system, for example, using the prediction-error method. In many situations, the type of linear model and the model orders are selected after estimating several models and evaluating them using various validation techniques. Two commonly used validation methods for linear models are spectral and residual analysis. Unfortunately, these methods will not always work if the true system is nonlinear. However, if the input can be viewed as if it has been generated by filtering white noise through a minimum phase filter, spectral and residual analysis can be used for validation of linear models of nonlinear systems. Furthermore, it can be shown that the input minimum phase property guarantees that a certain optimality result will hold. Here, the benefits of using minimum phase instead of non-minimum phase filters for input design will be shown both theoretically and in numerical experiments.

In this paper, a nonparametric method based on quadratic programming (QP) for identification of nonlinear autoregressive systems with exogenous inputs (NARX systems) is presented. We consider a mixed parametric/nonparametric model structure. The output is assumed to be the sum of a parametric linear part and a nonparametric Lipschitz continuous part. The consistency of the estimator is shown assuming only that an upper bound on the true Lipschitz constant is given. In addition, different types of prior knowledge about the system can easily be incorporated. Examples show that the method can give accurate estimates also for small data sets and that the estimate of the linear part sometimes can be improved compared to the linear least squares estimate.

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.

Linear time-invariant approximations of nonlinear systems are used in many applications and can be obtained in several ways. For example, using system identification and the prediction-error method, it is always possible to estimate a linear model without considering the fact that the input and output measurements in many cases come from a nonlinear system. One of the main objectives of this thesis is to explain some properties of such approximate models.

More specifically, linear time-invariant models that are optimal approximations in the sense that they minimize a mean-square error criterion are considered. Linear models, both with and without a noise description, are studied. Some interesting, but in applications usually undesirable, properties of such optimal models are pointed out. It is shown that the optimal linear model can be very sensitive to small nonlinearities. Hence, the linear approximation of an almost linear system can be useless for some applications, such as robust control design. Furthermore, it is shown that standard validation methods, designed for identification of linear systems, cannot always be used to validate an optimal linear approximation of a nonlinear system.

In order to improve the models, conditions on the input signal that imply various useful properties of the linear approximations are given. It is shown, for instance, that minimum phase filtered white noise in many senses is a good choice of input signal. Furthermore, the class of separable signals is studied in detail. This class contains Gaussian signals and it turns out that these signals are especially useful for obtaining approximations of generalized Wiener-Hammerstein systems. It is also shown that some random multisine signals are separable. In addition, some theoretical results about almost linear systems are presented.

In standard methods for robust control design, the size of the model error is assumed to be known for all input signals. However, in many situations, this is not a realistic assumption when a nonlinear system is approximated with a linear model. In this thesis, it is described how robust control design of some nonlinear systems can be performed based on a discrete-time linear model and a model error model valid only for bounded inputs.

It is sometimes undesirable that small nonlinearities in a system influence the linear approximation of it. In some cases, this influence can be reduced if a small nonlinearity is included in the model. In this thesis, an identification method with this option is presented for nonlinear autoregressive systems with external inputs. Using this method, models with a parametric linear part and a nonparametric Lipschitz continuous nonlinear part can be estimated by solving a convex optimization problem.

The interest for system identification in dynamic networks has increased recently with a wide variety of applications. In many cases, it is intractable or undesirable to observe all nodes in a network and thus, to estimate the complete dynamics. If the complete dynamics is not desired, it might even be challenging to estimate a subset of the network if key nodes are unobservable due to correlation between the nodes. In this contribution, we will discuss an approach to treat this problem. The approach relies on additional measurements that are dependent on the unobservable nodes and thus indirectly contain information about them. These measurements are used to form an alternative indirect model that is only dependent on observed nodes. The purpose of estimating this indirect model can be either to recover information about modules in the original network or to make accurate predictions of variables in the network. Examples are provided for both recovery of the original modules and prediction of nodes.

A ship's roll dynamics is very sensitive to changes in the loading conditions and a worst-case scenario is the loss of stability.  This paper proposes an approach for online estimation of a ship's mass and center of mass. Instead of focusing on a sensor-rich environment where all possible signals on a ship can be measured and a complete model of the ship can be estimated, a minimal approach is adopted. A model of the roll dynamics is derived from a well-established model in literature and it is assumed that only motion measurements from an inertial measurement unit together with measurements of the rudder angle are available. Furthermore, identifiability properties and disturbance characteristics of the model are presented. Due to the properties of the model, the parameters are estimated with an iterative instrumental variable approach to mitigate the influence of the disturbances and it uses multiple datasets simultaneously to overcome identifiability issues. Finally, a simulation study is presented to investigate the sensitivity to the initial conditions and it is shown that there is a low sensitivity for the desired parameters.

A Wiener model is a fairly simple, well known, and often used nonlinearblock-oriented black-box model. A possible generalization of the class ofWiener models lies in the parallel Wiener model class. This paper presents amethod to estimate the linear time-invariant blocks of such parallel Wienermodels from input/output data only. The proposed estimation methodcombines the knowledge obtained by estimating the best linear approxima-tion of a nonlinear system with a dimension reduction method to estimatethe linear time-invariant blocks present in the model. The estimation of thestatic nonlinearity is fairly easy once the linear blocks are known.

The estimation of nonlinear functions can be challenging when the number of independent variables is high. This difficulty may, in certain cases, be reduced by first projecting the independent variables on a lower dimensional subspace before estimating the nonlinearity. In this paper, a statistical nonparametric dimension reduction method called sliced inverse regression is presented and a consistency analysis for dynamically dependent variables is given. The straightforward system identification application is the estimation of the number of linear subsystems in a Wiener class system and their corresponding impulse response.

The prediction-error approach to parameter estimation of linear models often involves solving a non-convex optimization problem. In some cases, it is therefore difficult to guarantee that the global optimum will be found. A common way to handle this problem is to find an initial estimate, hopefully lying in the region of attraction of the global optimum, using some other method. The prediction-error estimate can then be obtained by a local search starting at the initial estimate. In this paper, a new approach for finding an initial estimate of certain linear models utilizing structure and the subspace method is presented. The polynomial models are first written on the observer canonical state-space form, where the specific structure is later utilized, rendering least-squares estimation problems with linear equality constraints.

Many approximation results in nonlinear system identification concern particular signal distributions. This seems to limit the applicability of these results to cases where the relevant signals have these distributions. However, by using a weighting method that modifies the cost function used in the identification method, the available approximation results can be used also for rather general classes of signal distributions. The purpose of this paper is to describe this weighting approach and to point at some interesting application areas within nonlinear system identification. In particular, it will be described how the impulse response of a Hammerstein system can be estimated consistently for an arbitrary input signal.

The choice of input signal is very important in identification of nonlinear systems. In this paper, it is shown that random multisines with a flat amplitude spectrum are separable. The separability property means that certain conditional expectations are linear and it implies that random multisines easily can be used to obtain accurate estimates of the linear time-invariant part of a Hammerstein system. This is illustrated in a numerical example.

In this paper, systems where information about model accuracy is contained in a model error model are considered. The validity of such a model is typically restricted to input signals that are limited in amplitude. It is then natural to require the same amplitude restriction when designing controllers. The resulting implications for controller design are investigated in both the continuous and the discrete time case.

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 show a result that can be viewed as a generalization of Bussgang's classic theorem about static non-linearities with Gaussian inputs. This new result is used to characterize the best linear approximation of a non-linear finite impulse response (NFIR) system with a Gaussian input. The best linear approximation is here defined as the causal LTI system that minimizes the expected squared prediction error. Furthermore, we discuss how this characterization can be used for structure identication and for identication of generalized Hammerstein and Wiener systems.

Correlation analysis can in some cases produce better identification results than an ordinary least squares approach. This is for example the case when a Finite Impulse Response system is estimated from ill-conditioned input-output measurements. In this report, the correlation analysis method is rewritten as a regularized least squares algorithm and the performance of the method is discussed in this context. It turns out that the fact that correlation analysis can be viewed as a kind of regularization explains why and in what sense this method sometimes produces more accurate estimates than the ordinary least squares approach.