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  • 1.
    Barenthin, Märta
    et al.
    Royal Institute of Technology, Sweden.
    Enqvist, Martin
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Wahlberg, Bo
    Royal Institute of Technology, Sweden.
    Hjalmarsson, Håkan
    Royal Institute of Technology, Sweden.
    Gain Estimation for Hammerstein Systems2006In: Proceedings of the 14th IFAC Symposium on System Identification, 2006, p. 784-789Conference paper (Refereed)
    Abstract [en]

    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.

  • 2.
    Enqvist, Martin
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    A Weighting Method for Approximate Nonlinear System Identification2007In: Proceedings of the 46th IEEE Conference on Decision and Control, 2007, p. 5104-5109Conference paper (Refereed)
    Abstract [en]

    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.

  • 3.
    Enqvist, Martin
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    A Weighting Method for Approximate Nonlinear System Identification2007Report (Other academic)
    Abstract [en]

    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.

  • 4.
    Enqvist, Martin
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Benefits of the Input Minimum Phase Property for Linearization of Nonlinear Systems2005In: Proceedings of the 2005 International Symposium on Nonlinear Theory and Its Applications, 2005, p. 618-Conference paper (Refereed)
    Abstract [en]

    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.

  • 5.
    Enqvist, Martin
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Benefits of the Input Minimum Phase Property for Linearization of Nonlinear Systems2005Report (Other academic)
    Abstract [en]

    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.

  • 6.
    Enqvist, Martin
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Identification of Block-oriented Systems Using the Invariance Property2010In: Block-Oriented Nonlinear System Identification / [ed] Fouad Giri and Er-Wei Bai, Springer London, 2010, p. 147-158Chapter in book (Refereed)
    Abstract [en]

    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.

  • 7.
    Enqvist, Martin
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Identification of Hammerstein Systems Using Separable Random Multisines2006In: Proceedings of the 14th IFAC Symposium on System Identification, 2006, p. 768-773Conference paper (Refereed)
    Abstract [en]

    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.

  • 8.
    Enqvist, Martin
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Identification of Hammerstein Systems Using Separable Random Multisines2005Report (Other academic)
    Abstract [en]

    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.

  • 9.
    Enqvist, Martin
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Linear Models of Nonlinear FIR Systems with Gaussian Inputs2002In: Proceedings of the 4th Conference on Computer Science and Systems Engineering, 2002, p. 147-151Conference paper (Other academic)
    Abstract [en]

    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.

  • 10.
    Enqvist, Martin
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Linear Models of Nonlinear FIR Systems with Gaussian Inputs2003In: Proceedings of the 13th IFAC Symposium on System Identification, 2003, p. 1910-1915Conference paper (Refereed)
    Abstract [en]

    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.

  • 11.
    Enqvist, Martin
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Linear Models of Nonlinear FIR Systems with Gaussian Inputs2002Report (Other academic)
    Abstract [en]

    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.

  • 12.
    Enqvist, Martin
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Linear Models of Nonlinear Systems2005Doctoral thesis, monograph (Other academic)
    Abstract [en]

    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.

  • 13.
    Enqvist, Martin
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Nonlinearity Detection and Impulse Response Estimation Using a Weighting Approach2009In: Proceedings of 15th IFAC Symposium on System Identification, 2009, p. 628-633Conference paper (Refereed)
    Abstract [en]

    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.

  • 14.
    Enqvist, Martin
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Separability of Scalar Random Multisine Signals2011In: Automatica, ISSN 0005-1098, E-ISSN 1873-2836, Vol. 47, no 9, p. 1860-1867Article in journal (Refereed)
    Abstract [en]

    Random multisines have successfully been used as input signals in many system identification experiments. In this paper, it is shown that scalar random multisine signals with a flat amplitude spectrum are separable of order one. The separability property means that certain conditional expectations are linear and it implies that random multisines can easily be used to obtain accurate estimates of the linear time-invariant part of a Hammerstein system. Furthermore, higher order separability is investigated.

  • 15.
    Enqvist, Martin
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Some Results on Linear Models of Nonlinear Systems2003Licentiate thesis, monograph (Other academic)
    Abstract [en]

    Linear time-invariant approximations of nonlinear systems are used in many a pplications. Such approximations can be obtained in many 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 general come from a nonlinear system. The main objective of this thesis is to explain some properties of such estimated models.

    More specifically, linear time-invariant models that are optimal approximations in the mean-square error sense are studied. Although this is a classic field of research, relatively few results exist about the properties of such models when they are based on signals from nonlinear systems. In this thesis, some interesting, but in applications usually undesirable, properties of linear approximations of nonlinear systems are pointed out. It is shown that the 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.

    In order to improve the models, conditions are given on the input signal implying various useful properties of the linear approximations. It is shown, for instance, that minimum phase filtered white noise in many senses is a good choice of input signal. Furthermore, some special properties of Gaussian signals are discussed. These signals turn out to be especially useful for approximations of generalized Hammerstein or Wiener systems. Using a Gaussian input, it is possible to estimate the denominator polynomial of the linear part of such a system without compensating for the nonlinearities. In addition, some theoretical results about almost linear systems and about separable input processes are presented. Linear models, both with and without a noise description, are studied.

  • 16.
    Enqvist, Martin
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Variance-Bias Tradeoff in Finite Impulse Response Estimates Obtained by Correlation Analysis2002Report (Other academic)
    Abstract [en]

    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.

  • 17.
    Enqvist, Martin
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Gunnarsson, Svante
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Norrlöf, Mikael
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Wernholt, Erik
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Hansson, Anders
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    The CDIO Initiative from an Automatic Control Project Course Perspective2005In: Proceedings of the 16th IFAC World Congress, 2005, p. 2283-2283Conference paper (Refereed)
    Abstract [en]

    The CDIO (Conceive Design Implement Operate) Initiative is explained, and some of the results at the Applied Physics and Electrical Engineering program at Linköping University, Sweden, are presented. A project course in Automatic Control is used as an example. The projects within the course are carried out using the LIPS (Linköping interactive project steering) model. An example of a project, the golf playing industrial robot, and the results from this project are also covered.

  • 18.
    Enqvist, Martin
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Gunnarsson, Svante
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Norrlöf, Mikael
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Wernholt, Erik
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Hansson, Anders
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    The CDIO Initiative from an Automatic Control Project Course Perspective2004Report (Other academic)
    Abstract [en]

    The CDIO (Conceive Design Implement Operate) Initiative is explained, and some of the results at the Applied Physics and Electrical Engineering program at Linköping University, Sweden, are presented. A project course in Automatic Control is used as an example. The projects within the course are carried out using the LIPS (Linköping interactive project steering) model. An example of a project, the golf playing industrial robot, and the results from this project are also covered.

  • 19.
    Enqvist, Martin
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Ljung, Lennart
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Approximation of Non-Linear Systems in a Neighborhood of LTI Systems2002In: Proceedings of Reglermöte 2002, 2002, p. 289-291Conference paper (Other academic)
  • 20.
    Enqvist, Martin
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Ljung, Lennart
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Estimating Nonlinear Systems in a Neighborhood of LTI-approximants2002In: Proceedings of the 41st IEEE Conference on Decision and Control, 2002, p. 1005-1010 vol.1Conference paper (Refereed)
    Abstract [en]

    The estimation of Linear Time Invariant (LTI) models is a standard procedure in system identification. Any real-life system will however be nonlinear and time-varying, and the estimated model will converge to the LTI second order equivalent (LTI-SOE) of the true system. In this paper we consider some aspects of this convergence and the distance between the true system and its LTI-SOE. We show that there may be cases where even the slightest nonlinearity may cause big differences in the LTI-SOE. We also show a result that gives conditions that guarantee that the LTI-SOE is close to "the natural" LTI approximant. Finally, an upper bound on the distance between the LTI-SOE of a nonlinear FIR system with a white input signal and the linear part of the system is derived.

  • 21.
    Enqvist, Martin
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Ljung, Lennart
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Estimating Nonlinear Systems in a Neighborhood of LTI-approximants2002Report (Other academic)
    Abstract [en]

    The estimation of Linear Time Invariant (LTI) models is a standard procedure in system identification. Any real-life system will however be nonlinear and time-varying, and the estimated model will converge to the LTI second order equivalent (LTI-SOE) of the true system. In this paper we consider some aspects of this convergence and the distance between the true system and its LTI-SOE. We show that there may be cases where even the slightest nonlinearity may cause big differences in the LTI-SOE. We also show a result that gives conditions that guarantee that the LTI-SOE is close to "the natural" LTI approximant. Finally, an upper bound on the distance between the LTI-SOE of a nonlinear FIR system with a white input signal and the linear part of the system is derived.

  • 22.
    Enqvist, Martin
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Ljung, Lennart
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Linear Approximations of Nonlinear FIR Systems for Separable Input Processes2005Report (Other academic)
    Abstract [en]

    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 in ut should be separable of a certain order, i.e., that certain conditional expectations should be,P 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.

  • 23.
    Enqvist, Martin
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Ljung, Lennart
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Linear Approximations of Nonlinear FIR Systems for Separable Input Processes2005In: Automatica, ISSN 0005-1098, E-ISSN 1873-2836, Vol. 41, no 3, p. 459-473Article in journal (Refereed)
    Abstract [en]

    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 in ut should be separable of a certain order, i.e., that certain conditional expectations should be,P 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.

  • 24.
    Enqvist, Martin
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Ljung, Lennart
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    LTI Approximations of Slightly Nonlinear Systems: Some Intriguing Examples2004In: Proceedings of the 6th IFAC Symposium on Nonlinear Control Systems, 2004, p. 639-Conference paper (Refereed)
    Abstract [en]

    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.

  • 25.
    Enqvist, Martin
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Ljung, Lennart
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    LTI Approximations of Slightly Nonlinear Systems: Some Intriguing Examples2004In: Proceedings of Reglermöte 2004, 2004Conference paper (Other academic)
    Abstract [en]

    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.

  • 26.
    Enqvist, Martin
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Ljung, Lennart
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    LTI Approximations of Slightly Nonlinear Systems: Some Intriguing Examples2005Report (Other academic)
    Abstract [en]

    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.

  • 27.
    Enqvist, Martin
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Schoukens, Johan
    Vrije Universiteit Brussel, Belgium.
    Pintelon, Rik
    Vrije Universiteit Brussel, Belgium.
    Detection of Unmodeled Nonlinearities Using Correlation Methods2007In: Proceedings of the 2007 IEEE Instrumentation and Measurement Technology Conference, 2007, p. 1-6Conference paper (Refereed)
    Abstract [en]

    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.

  • 28.
    Escobar, Jesica
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Enqvist, Martin
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    On the Detection of Nonlinearities in Sampled Data2012In: Proceedings of the 16th IFAC Symposium on System Identification, 2012, p. 1587-1592Conference paper (Refereed)
    Abstract [en]

    Here we deal with the choice of the sampling rate in nonlinear system identification applications. In particular, we focus on the effect of the sampling rate when the prediction-error method is used. On one hand, a high sampling rate is advantageous since it enables the measurement of high-frequent nonlinear components in the output signal of the system without aliasing. However, a high sampling rate might also make it harder to realize that the system is nonlinear, since the nonlinearities cannot be detected in the residuals from a linear model in some cases. Here, this phenomenon is illustrated in a couple of numerical examples and a way to avoid it is proposed.

  • 29.
    Fritzin, Jonas
    et al.
    Linköping University, Department of Electrical Engineering, Electronic Devices. Linköping University, The Institute of Technology.
    Jung, Ylva
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Landin, Per Niklas
    Royal Institute of Technology, Sweden.
    Handel, Peter
    Royal Institute of Technology, Sweden.
    Enqvist, Martin
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Alvandpour, Atila
    Linköping University, Department of Electrical Engineering, Electronic Devices. Linköping University, The Institute of Technology.
    Phase Predistortion of a Class-D Outphasing RF Amplifier in 90 nm CMOS2011In: IEEE Transactions on Circuits and Systems - II - Express Briefs, ISSN 1549-7747, E-ISSN 1558-3791, Vol. 58, no 10, p. 642-646Article in journal (Refereed)
    Abstract [en]

    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.

  • 30.
    Glad, Torkel
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Helmersson, Anders
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Enqvist, Martin
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Ljung, Lennart
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Controllers for Amplitude Limited Model Error Models2005In: Proceedings of the 16th IFAC World Congress, 2005, p. 662-662Conference paper (Refereed)
    Abstract [en]

    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.

  • 31.
    Glad, Torkel
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Helmersson, Anders
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Enqvist, Martin
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Ljung, Lennart
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Controllers for Amplitude Limited Model Error Models2005Report (Other academic)
    Abstract [en]

    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.

  • 32.
    Ho, Du
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, Faculty of Science & Engineering.
    Linder, Jonas
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, Faculty of Science & Engineering.
    Hendeby, Gustaf
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, Faculty of Science & Engineering.
    Enqvist, Martin
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, Faculty of Science & Engineering.
    Mass estimation of a quadcopter using IMU data2017In: 2017 International Conference on Unmanned Aircraft Systems (ICUAS), June 13-16, 2017, Miami, FL, USA, Institute of Electrical and Electronics Engineers (IEEE), 2017, p. 1260-1266Conference paper (Refereed)
    Abstract [en]

    In this paper, an approach to estimate the mass of a quadcopter using only inertial measurements and pilot commands is presented. For this purpose, a lateral dynamic model describing the relation between the roll rate and the lateral acceleration is formulated. Due to the quadcopter’s inherent instability, a controller is used to stabilize the system and the data is collected in closed loop. Under the effect of feedback and disturbances, the inertial measurements used as input and output are correlated with the disturbances, which complicates the parameter estimation. The parameters of the model are estimated using several methods. The simulation and experimental results show that the instrumental-variable method has the best potential to estimate the mass of the quadcopter in this setup.

  • 33.
    Ho, Du
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, Faculty of Science & Engineering.
    Linder, Jonas
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, Faculty of Science & Engineering.
    Hendeby, Gustaf
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, Faculty of Science & Engineering.
    Enqvist, Martin
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, Faculty of Science & Engineering.
    Vertical modeling of a quadcopter for mass estimation and diagnosis purposes2017In: Proceedings of the Workshop on Research, Education and Development on Unmanned Aerial Systems, RED-UAS, Linköping, Sweden, 3-5 October, 2017, Institute of Electrical and Electronics Engineers (IEEE), 2017Conference paper (Refereed)
    Abstract [en]

    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.

  • 34.
    Jung, Ylva
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Enqvist, Martin
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Estimating models of inverse systems2013In: Proceedings of the 52nd IEEE Conference on Decision and Control, 2013, p. 7143-7148Conference paper (Refereed)
    Abstract [en]

    This paper considers the problem of how to estimate a model of the inverse of a system. The use of inverse systems can be found in many applications, such as feedforward control and power amplifier predistortion. The inverse model is here estimated with the purpose of using it in cascade with the system itself, as an inverter. A good inverse model in this setting would be one that, when used in series with the original system, reconstructs the original input. The goal here is to select suitable inputs, experimental conditions and loss functions to obtain a good input estimate. Both linear and nonlinear systems will be discussed.

    For nonlinear systems, one way to obtain a linearizing prefilter is by Hirschorn’s algorithm. It is here shown how to extend this to the postdistortion case, and some formulations of how the pre- or postinverter could be estimated are also presented. 

  • 35.
    Jung, Ylva
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, Faculty of Science & Engineering.
    Enqvist, Martin
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, Faculty of Science & Engineering.
    On estimation of approximate inverse models of block-oriented systems2015In: Proceedings of the 17th IFAC Symposium on System Identification, 2015, p. 1226-1231Conference paper (Refereed)
    Abstract [en]

    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. 

  • 36.
    Jung, Ylva
    et al.
    Linköping University, Department of Electrical Engineering, Electronic Devices. Linköping University, The Institute of Technology.
    Fritzin, Jonas
    Linköping University, Department of Electrical Engineering, Electronic Devices. Linköping University, The Institute of Technology.
    Enqvist, Martin
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Alvandpour, Atila
    Linköping University, Department of Electrical Engineering, Electronic Devices. Linköping University, The Institute of Technology.
    Least-Squares Phase Predistortion of a +30dBm Class-D Outphasing RF PA in 65nm CMOS2013In: IEEE Transactions on Circuits and Systems Part 1: Regular Papers, ISSN 1549-8328, E-ISSN 1558-0806, Vol. 60, no 7, p. 1915-1928Article in journal (Refereed)
    Abstract [en]

    This paper presents a model-based phase-only predistortion method suitable for outphasing radio frequency (RF) power amplifiers (PA). 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. Exploring the structure of the outphasing PA, the problem can be reformulated from a nonconvex problem into a convex least-squares problem, and the predistorter can be calculated analytically. The method has been evaluted for 5MHz Wideband Code-Division Multiple Access (WCDMA) and Long Term Evolution (LTE) uplink signals with Peak-to-Average Power Ratio (PAPR) of 3.5 dB and 6.2 dB, respectively, applied to a fully integrated Class-D outphasing RF PA in 65nm CMOS. At 1.95 GHz for a 5.5V supply voltage, the measured output power of the PA was +29.7dBm with a power-added efficiency (PAE) of 26.6 %. For the WCDMA signal with +26.0dBm of channel power, the measured Adjacent Channel Leakage Ratio (ACLR) at 5MHz and 10MHz offsets were -46.3 dBc and -55.6 dBc with predistortion, compared to -35.5 dBc and -48.1 dBc without predistortion. For the LTE signal with +23.3dBm of channel power, the measured ACLR at 5MHz offset was -43.5 dBc with predistortion, compared to -34.1 dBc without predistortion.

  • 37.
    Larsson, Roger
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Enqvist, Martin
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Real-time Aerodynamic Model Parameter Identification2009In: Proceedings of the 40th Annual International Symposium of the Society of Flight Test Engineers, 2009, p. 113-122Conference paper (Refereed)
    Abstract [en]

    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.

  • 38.
    Larsson, Roger
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Enqvist, Martin
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Sequential Aerodynamic Model Parameter Identification2012In: Proceedings of the 16th IFAC Symposium on System Identification, 2012, p. 1413-1418Conference paper (Refereed)
    Abstract [en]

    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.

  • 39.
    Larsson, Roger
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Sjanic, Zoran
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Enqvist, Martin
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Ljung, Lennart
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Direct Prediction-Error Identification of Unstable Nonlinear Systems Applied to Flight Test Data2009In: Proceedings of the 15th IFAC Symposium on System Identification, 2009, p. 144-149Conference paper (Refereed)
    Abstract [en]

    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.

  • 40.
    Lauwers, Lieve
    et al.
    Vrije Universiteit Brussel, Belgium.
    Schoukens, Johan
    Vrije Universiteit Brussel, Belgium.
    Pintelon, Rik
    Vrije Universiteit Brussel, Belgium.
    Enqvist, Martin
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    A Nonlinear Block Structure Identification Procedure using Frequency Response Function Measurements2008In: IEEE Transactions on Instrumentation and Measurement, ISSN 0018-9456, E-ISSN 1557-9662, Vol. 57, no 10, p. 2257-2264Article in journal (Refereed)
    Abstract [en]

    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.

  • 41.
    Lauwers, Lieve
    et al.
    Vrije Universiteit Brussel, Belgium.
    Schoukens, Johan
    Vrije Universiteit Brussel, Belgium.
    Pintelon, Rik
    Vrije Universiteit Brussel, Belgium.
    Enqvist, Martin
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Nonlinear Structure Analysis Using the Best Linear Approximation2006In: Proceedings of the 2006 International Conference on Noise and Vibration Engineering, 2006, p. 2751-2760Conference paper (Refereed)
    Abstract [en]

    In this paper, we propose a method to distinguish between some nonlinear system structures using the bestlinear approximation (BLA) of the system in order to select an appropriate model structure. The main ideaof the method is to apply a Gaussian input signal and to vary the root mean square (rms) value and thepower spectrum of this signal. Depending on the resulting changes of the amplitude and phasecharacteristics of the BLA, an appropriate model structure for the Device Under Test can be selected. Atheoretical analysis of the method is presented for some block-oriented structures.

  • 42.
    Linder, Jonas
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Enqvist, Martin
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    On Indirect Input Measurements2015Report (Other academic)
    Abstract [en]

    A common issue with many system identification problems is that the true input to the system is unknown. In this paper, a framework, based on indirect input measurements, is proposed to solve the problem when the input is partially or fully unknown, and cannot be measured directly. The approach relies on measurements that indirectly contain information about the unknown input. The resulting indirect model formulation, with both direct- and indirect input measurements as inputs, can be used to estimate the desired model of the original system. Due to the similarities with closed-loop system identification, an iterative instrumental variable method is proposed to estimate the indirect model. To show the applicability of the proposed method, it is applied to data from an inverted pendulum experiment with good results.

  • 43.
    Linder, Jonas
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, Faculty of Science & Engineering.
    Enqvist, Martin
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, Faculty of Science & Engineering.
    On Indirect Input Measurements2015In: Proceedings of the 17th IFAC Symposium on System Identification, 2015, p. 104-109Conference paper (Refereed)
    Abstract [en]

    A common issue with many system identification problems is that the true input to the system is unknown. In this paper, a framework, based on indirect input measurements, is proposed to solve the problem when the input is partially or fully unknown, and cannot be measured directly. The approach relies on measurements that indirectly contain information about the unknown input. The resulting indirect model formulation, with both direct and indirect input measurements as inputs, can be used to estimate the desired model of the original system. Due to the similarities with closed-loop system identification, an iterative instrumental variable method is proposed to estimate the indirect model. To show the applicability of the proposed method, it is applied to data from an inverted pendulum experiment with good results. 

  • 44.
    Linder, Jonas
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Enqvist, Martin
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Fossen, Thor I.
    Centre for Autonomous Marine Operations and Systems (AMOS), Department of Engineering Cybernetics, Norwegian University of Science and Technology, Trondheim, Norway.
    Johansen, Tor Arne
    Centre for Autonomous Marine Operations and Systems (AMOS), Department of Engineering Cybernetics, Norwegian University of Science and Technology, Trondheim, Norway.
    Gustafsson, Fredrik
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Modeling for IMU-based Online Estimation of a Ship's Mass and Center of Mass2015Report (Other academic)
    Abstract [en]

    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.

  • 45.
    Linder, Jonas
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Enqvist, Martin
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Fossen, Thor I.
    Centre for Autonomous Marine Operations and Systems (AMOS), Department of Engineering Cybernetics, Norwegian University of Science and Technology, Trondheim, Norway.
    Johansen, Tor Arne
    Centre for Autonomous Marine Operations and Systems (AMOS), Department of Engineering Cybernetics, Norwegian University of Science and Technology, Trondheim, Norway.
    Gustafsson, Fredrik
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Modeling for IMU-based Online Estimation of a Ship's Mass and Center of Mass2015In: Proceedings of the 10th Conference on Manoeuvring and Control of Marine Craft, 2015, , p. 16Conference paper (Refereed)
    Abstract [en]

    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 the sensitivity is low for the desired parameters.

  • 46.
    Linder, Jonas
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Enqvist, Martin
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Fossen, Thor I.
    Centre for Autonomous Marine Operations and Systems (AMOS), Department of Engineering Cybernetics, Norwegian University of Science and Technology, Trondheim, Norway.
    Johansen, Tor Arne
    Centre for Autonomous Marine Operations and Systems (AMOS), Department of Engineering Cybernetics, Norwegian University of Science and Technology, Trondheim, Norway.
    Gustafsson, Fredrik
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Online Estimation of Ship's Mass and Center of Mass Using Inertial Measurements2015Report (Other academic)
    Abstract [en]

    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 was 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.

  • 47.
    Linder, Jonas
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Enqvist, Martin
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Fossen, Thor I.
    Centre for Autonomous Marine Operations and Systems (AMOS), Department of Engineering Cybernetics, Norwegian University of Science and Technology, Trondheim, Norway.
    Johansen, Tor Arne
    Centre for Autonomous Marine Operations and Systems (AMOS), Department of Engineering Cybernetics, Norwegian University of Science and Technology, Trondheim, Norway.
    Gustafsson, Fredrik
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Online Estimation of Ship's Mass and Center of Mass Using Inertial Measurements2015In: Proceedings of the 10th Conference on Manoeuvring and Control of Marine Craft, 2015, , p. 16Conference paper (Refereed)
    Abstract [en]

    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.

  • 48.
    Linder, Jonas
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Enqvist, Martin
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Gustafsson, Fredrik
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    A Closed-loop Instrumental Variable Approach to Mass and Center of Mass Estimation Using IMU Data2014In: Proceedings of the 53rd Conference on Decision and Control, 2014, p. 283-289Conference paper (Refereed)
    Abstract [en]

    In this paper, an instrumental variable (IV) method for estimating the mass and center of mass (CM) of a ship using IMU data has been further investigated. Here, this IV method, which was proposed in an earlier paper, has been analyzed from a closed-loop point of view. This new perspective reveals the properties of the system and dependencies of the signals used in the estimation procedure. Due to similarities with closed-loop identification, previous results in the closed-loop identification field have been used as an inspiration to improve the IV estimator. Since the roll dynamics of a ship is well described by a pendulum model, a pendulum experiment has been carried out to validate the performance both of the original and the improved IV estimators. The experiments gave good results for the improved IV estimator with significantly lower variances and relative errors than the previous IV estimator.

  • 49.
    Linder, Jonas
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Enqvist, Martin
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Gustafsson, Fredrik
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Sjöberg, Johan
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Identifiability of physical parameters in systems with limited sensors2014In: Proceedings of the 19th IFAC World Congress, 2014, p. 6454-6459Conference paper (Refereed)
    Abstract [en]

    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.

  • 50.
    Lyzell, Christian
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Andersen, Martin
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Enqvist, Martin
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    A Convex Relaxation of a Dimension Reduction Problem Using the Nuclear Norm2012In: Proceedings of the 51st IEEE Conference on Decision and Control, 2012, p. 2852-2857Conference paper (Refereed)
    Abstract [en]

    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.

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