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  • 1.
    Lyzell, Christian
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Initialization Methods for System Identification2009Licentiate thesis, monograph (Other academic)
    Abstract [en]

    In the system identification community a popular framework for the problem of estimating a parametrized model structure given a sequence of input and output pairs is given by the prediction-error method. This method tries to find the parameters which maximize the prediction capability of the corresponding model via the minimization of some chosen cost function that depends on the prediction error. This optimization problem is often quite complex with several local minima and is commonly solved using a local search algorithm. Thus, it is important to find a good initial estimate for the local search algorithm. This is the main topic of this thesis.

    The first problem considered is the regressor selection problem for estimating the order of dynamical systems. The general problem formulation is difficult to solve and the worst case complexity equals the complexity of the exhaustive search of all possible combinations of regressors. To circumvent this complexity, we propose a relaxation of the general formulation as an extension of the nonnegative garrote regularization method. The proposed method provides means to order the regressors via their time lag and a novel algorithmic approach for the \textsc{arx} and \textsc{lpv-arx} case is given.

     

    Thereafter, the initialization of linear time-invariant polynomial models is considered. Usually, this problem is solved via some multi-step instrumental variables method. For the estimation of state-space models, which are closely related to the polynomial models via canonical forms, the state of the art estimation method is given by the subspace identification method. It turns out that this method can be easily extended to handle the estimation of polynomial models. The modifications are minor and only involve some intermediate calculations where already available tools can be used. Furthermore, with the proposed method other a priori information about the structure can be readily handled, including a certain class of linear gray-box structures. The proposed extension is not restricted to the discrete-time case and can be used to estimate continuous-time models.

     

    The final topic in this thesis is the initialization of discrete-time systems containing polynomial nonlinearities. In the continuous-time case, the tools of differential algebra, especially Ritt's algorithm, have been used to prove that such a model structure is globally identifiable if and only if it can be written as a linear regression model. In particular, this implies that once Ritt's algorithm has been used to rewrite the nonlinear model structure into a linear regression model, the parameter estimation problem becomes trivial. Motivated by the above and the fact that most system identification problems involve sampled data, a version of Ritt's algorithm for the discrete-time case is provided. This algorithm is closely related to the continuous-time version and enables the handling of noise signals without differentiations.

  • 2.
    Lyzell, Christian
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Structural Reformulations in System Identification2012Doctoral thesis, monograph (Other academic)
    Abstract [en]

    In system identification, the choice of model structure is important and it is sometimes desirable to use a flexible model structure that is able to approximate a wide range of systems. One such model structure is the Wiener class of systems, that is, systems where the input enters a linear time-invariant subsystem followed by a time-invariant nonlinearity. Given a sequence of input and output pairs, the system identification problem is often formulated as the minimization of the mean-square prediction error. Here, the prediction error has a nonlinear dependence on the parameters of the linear subsystem and the nonlinearity. Unfortunately, this formulation of the estimation problem is often nonconvex, with several local minima, and it is therefore difficult to guarantee that a local search algorithm will be able to find the global optimum.

    In the first part of this thesis, we consider the application of dimension reduction methods to the problem of estimating the impulse response of the linear part of a system in the Wiener class. For example, by applying the inverse regression approach to dimension reduction, the impulse response estimation problem can be cast as a principal components problem, where the reformulation is based on simple nonparametric estimates of certain conditional moments. The inverse regression approach can be shown to be consistent under restrictions on the distribution of the input signal provided that the true linear subsystem has a finite impulse response. Furthermore, a forward approach to dimension reduction is also considered, where the time-invariant nonlinearity is approximated by a local linear model. In this setting, the impulse response estimation problem can be posed as a rank-reduced linear least-squares problem and a convex relaxation can be derived.

    Thereafter, we consider the extension of the subspace identification approach to include linear time-invariant rational models. It turns out that only minor structural modifications are needed and already available implementations can be used. Furthermore, other a priori information regarding the structure of the system can incorporated, including a certain class of linear gray-box structures. The proposed extension is not restricted to the discrete-time case and can be used to estimate continuous-time models.

    The final topic in this thesis is the estimation of discrete-time models containing polynomial nonlinearities. In the continuous-time case, a constructive algorithm based on differential algebra has previously been used to prove that such model structures are globally identifiable if and only if they can be written as a linear regression model. Thus, if we are able to transform the nonlinear model structure into a linear regression model, the parameter estimation problem can be solved with standard methods. Motivated by the above and the fact that most system identification problems involve sampled data, a discrete-time version of the algorithm is developed. This algorithm is closely related to the continuous-time version and enables the handling of noise signals without differentiations.

  • 3.
    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.

  • 4.
    Lyzell, Christian
    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.
    Inverse Regression for the Wiener Class of Systems2012In: Proceedings of the 16th IFAC Symposium on System Identification, 2012, p. 476-481Conference paper (Refereed)
    Abstract [en]

    The concept of inverse regression has turned out to be quite useful for dimension reduction in regression analysis problems. Using methods like sliced inverse regression (SIR) and directional regression (DR), some high-dimensional nonlinear regression problems can be turned into more tractable low-dimensional problems. Here, the usefulness of inverse regression for identification of nonlinear dynamical systems will be discussed. In particular, it will be shown that the inverse regression methods can be used for identification of systems of the Wiener class, that is, systems consisting of a number of parallel linear subsystems followed by a static multiple-input single-output nonlinearity. For a particular class of input signals, including Gaussian signals, the inverse regression approach makes it possible to estimate the linear subsystems without knowing or estimating the nonlinearity.

  • 5.
    Lyzell, Christian
    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.
    Inverse Regression for the Wiener Class of Systems2011Report (Other academic)
    Abstract [en]

    The concept of inverse regression has turned out to be quite useful for dimension reduction in regression analysis problems. Using methods like sliced inverse regression (SIR) and directional regression (DR), some high-dimensional nonlinear regression problems can be turned into more tractable low-dimensional problems. Here, the usefulness of inverse regression for identification of nonlinear dynamical systems will be discussed. In particular, it will be shown that the inverse regression methods can be used for identification of systems of the Wiener class, that is, systems consisting of a number of parallel linear subsystems followed by a static multiple-input single-output nonlinearity. For a particular class of input signals, including Gaussian signals, the inverse regression approach makes it possible to estimate the linear subsystems without knowing or estimating the nonlinearity.

  • 6.
    Lyzell, Christian
    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.
    Sliced Inverse Regression for the Identification of Dynamical Systems2012In: Proceedings of the 16th IFAC Symposium on System Identification, 2012, p. 1575-1580Conference paper (Refereed)
    Abstract [en]

    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.

  • 7.
    Lyzell, Christian
    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.
    Sliced Inverse Regression for the Identification of Dynamical Systems2011Report (Other academic)
    Abstract [en]

    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.

  • 8.
    Lyzell, Christian
    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.
    Ljung, Lennart
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Handling Certain Structure Information in Subspace Identification2009Report (Other academic)
    Abstract [en]

    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.

  • 9.
    Lyzell, Christian
    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.
    Ljung, Lennart
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Handling Certain Structure Information in Subspace Identification2009In: Proceedings of the 15th IFAC Symposium on System Identification, 2009, p. 90-95Conference paper (Refereed)
    Abstract [en]

    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.

  • 10.
    Lyzell, Christian
    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.
    Toth, Roland
    Delft University of Technology, The Netherlands.
    Heuberger, Peter S. C.
    Delft University of Technology, The Netherlands.
    Van den Hof, Paul M. J.
    Delft University of Technology, The Netherlands.
    Order and Structural Dependence Selection of LPV-ARX Models using a Nonnegative Garrote Approach2009In: Proceedings of the 48th IEEE Conference on Decision and Control held jointly with the 28th Chinese Control Conference, 2009, p. 7406-7411Conference paper (Refereed)
    Abstract [en]

    In order to accurately identify Linear Parameter-Varying (LPV) systems, order selection of LPV linear regression models has prime importance. Existing identification approaches in this context suffer from the drawback that a set of functional dependencies needs to be chosen a priori for the parametrization of the model coefficients. However in a black-box setting, it has not been possible so far to decide which functions from a given set are required for the parametrization and which are not. To provide a practical solution, a nonnegative garrote approach is applied. It is shown that using only a measured data record of the plant, both the order selection and the selection of structural coefficient dependence can be solved by the proposed method.

  • 11.
    Lyzell, Christian
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Glad, Torkel
    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.
    Difference Algebra and System Identification2011In: Automatica, ISSN 0005-1098, E-ISSN 1873-2836, Vol. 47, no 9, p. 1896-1904Article in journal (Refereed)
    Abstract [en]

    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.

  • 12.
    Lyzell, Christian
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Glad, Torkel
    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.
    Identification Aspects of Ritt's Algorithm for Discrete-Time Systems2009Report (Other academic)
    Abstract [en]

    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.

  • 13.
    Lyzell, Christian
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Glad, Torkel
    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, The Institute of Technology. Linköping University, Department of Electrical Engineering, Automatic Control.
    Identification Aspects of Ritt's Algorithm for Discrete-Time Systems2009In: Proceedings of the 15th IFAC Symposium on System Identification, 2009, p. 681-686Conference paper (Refereed)
    Abstract [en]

    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.

  • 14.
    Lyzell, Christian
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Roll, Jacob
    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.
    The Use of Nonnegative Garrote for Order Selection of ARX Models2008Report (Other academic)
    Abstract [en]

    Order selection of linear regression models has been thoroughly researched in the statistical community for some time. Different shrinkage methods have been proposed, such as the Ridge and Lasso regression methods. Especially the Lasso regression has won fame because of its ability to set less important parameters exactly to zero. However, these methods do not take dynamical systems into account, where the regressors are ordered via the time lag. To this end, a modified variant of the nonnegative garrote method will be analyzed.

  • 15.
    Lyzell, Christian
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Roll, Jacob
    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.
    The Use of Nonnegative Garrote for Order Selection of ARX Models2008In: Proceedings of the 47th IEEE Conferance on Decision and Control, 2008, , p. 1974-1979p. 1974-1979Conference paper (Refereed)
    Abstract [en]

    Order selection of linear regression models has been thoroughly researched in the statistical community for some time. Different shrinkage methods have been proposed, such as the Ridge and Lasso regression methods. Especially the Lasso regression has won fame because of its ability to set less important parameters exactly to zero. However, these methods do not take dynamical systems into account, where the regressors are ordered via the time lag. To this end, a modified variant of the nonnegative garrote method will be analyzed.

  • 16.
    Schoukens, Maarten
    et al.
    Vrije Universiteit Brussel, Belgium.
    Lyzell, Christian
    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.
    Combining the best linear approximation and dimension reduction to identify the linear blocks of parallel Wiener systems2013In: Proceedings of the 11th IFAC International Workshop on Adaptation and Learning in Control and Signal Processing, 2013, p. 372-377Conference paper (Refereed)
    Abstract [en]

    A Wiener model is a fairly simple, well known, and often used nonlinear block- oriented black-box model. A possible generalization of the class of Wiener models lies in the parallel Wiener model class. This paper presents a method to estimate the linear time-invariant blocks of such parallel Wiener models from input/output data only. The proposed estimation method combines the knowledge obtained by estimating the best linear approximation of a nonlinear system with the MAVE dimension reduction method to estimate the linear time- invariant blocks present in the model. The estimation of the static nonlinearity boils down to a standard static nonlinearity estimation problem starting from input-output data once the linear blocks are known. 

  • 17.
    Schoukens, Maarten
    et al.
    Vrije Universiteit Brussel ,Faculty of Engineering, Department of Fundamental Electricity and Instrumentation.
    Lyzell, Christian
    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.
    Combining the best linear approximation and dimension reduction to identify thelinear blocks of parallel Wiener systems2012Report (Other academic)
    Abstract [en]

    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.

  • 18.
    Toth, Roland
    et al.
    Delft University of Technology, The Netherlands.
    Lyzell, Christian
    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.
    Heuberger, Peter S.C.
    Delft University of Technology, The Netherlands.
    Van den Hof, Paul M. J.
    Delft University of Technology, The Netherlands.
    Order and Structural Dependence Selection of LPV-ARX Models using a Nonnegative Garrote Approach2010Report (Other academic)
    Abstract [en]

    In order to accurately identify Linear Parameter-Varying (LPV) systems, order selection of LPV linear regression models has prime importance. Existing identification approaches in this context suffer from the drawback that a set of functional dependencies needs to be chosen a priori for the parametrization of the model coefficients. However in a black-box setting, it has not been possible so far to decide which functions from a given set are required for the parametrization and which are not. To provide a practical solution, a nonnegative garrote approach is applied. It is shown that using only a measured data record of the plant, both the order selection and the selection of structural coefficient dependence can be solved by the proposed method.

1 - 18 of 18
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