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  • 1.
    Fransson, Martin
    et al.
    Linköping University, Department of Computer and Information Science, PELAB - Programming Environment Laboratory. Linköping University, The Institute of Technology.
    Gerdin, Markus
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Fritzson, Peter
    Linköping University, Department of Computer and Information Science, PELAB - Programming Environment Laboratory. Linköping University, The Institute of Technology.
    Identification of a Nonlinear Cell Cycle System with Linear Models2005In: Proceedings of the Conference on Modeling and Simulation in Biology, Medicine and Biomedical EngineeringMay 26-27, 2005, Linköping, Sweden / [ed] Peter Fritzson, Linköping, 2005, p. 129-138Conference paper (Refereed)
    Abstract [en]

    A mathematical model by Chen et al. representing the cell cycle of the budding yeast cell (Saccharomyces cerevisiae) is used to simulate levels of 13 different cell biological variables. By using a linear systems approach on the generated data the goal is to identify the original equations of the mathematical model. Since this model is described by a nonlinear system the linear systems approach only gives correct estimations of the linear parts of the system, with one exception. A main issue thus becomes to identify which data that belongs to the linear equations and which does not. This problem is solved using two local linear models on different time intervals. Parameter estimations of coefficients from linear and time invariant equations should be the same over all time intervals.

  • 2.
    Gerdin, Markus
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Computation of a Canonical Form for Linear Differential-Algebraic Equations2004In: Proceedings of Reglermöte 2004, 2004Conference paper (Other academic)
    Abstract [en]

    This paper describes how a commonly used canonical form for linear differential-algebraic equations can be computed using numerical software from the linear algebra package LAPACK. This makes it possible to automate for example observer construction and parameter estimation in linear models generated by a modeling language like Modelica.

  • 3.
    Gerdin, Markus
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Computation of a Canonical Form for Linear Differential-Algebraic Equations2004Report (Other academic)
    Abstract [en]

    This paper describes how a commonly used canonical form for linear differential-algebraic equations can be computed using numerical software from the linear algebra package LAPACK. This makes it possible to automate for example observer construction and parameter estimation in linear models generated by a modeling language like Modelica.

  • 4.
    Gerdin, Markus
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Identification and Estimation for Models Described by Differential-Algebraic Equations2006Doctoral thesis, monograph (Other academic)
    Abstract [en]

    Differential-algebraic equations (DAEs) form the natural way in which models of physical systems are delivered from an object-oriented modeling tool like Modelica. Differential-algebraic equations are also known as descriptor systems, singular systems, and implicit systems. If some constant parameters in such models are unknown, one might need to estimate them from measured data from the modeled system. This is a form of system identification called gray box identification. It may also be of interest to estimate the value of time-varying variables in the model. This is often referred to as state estimation. The objective of this work is to examine how gray box identification and estimation of time-varying variables can be performed for models described by differential-algebraic equations.

    If a model has external stimuli that are not measured or uncertain measurements, it is often appropriate to model this as stochastic processes. This is called noise modeling. Noise modeling is an important part of system identification and state estimation, so we examine how well-posedness of noise models for differential-algebraic equations can be characterized. For well-posed models, we then discuss how particle filters can be implemented for estimation of time-varying variables. We also discuss how constant parameters can be estimated.

    When estimating time-varying variables, it is of interest to examine if the problem is observable, that is, if it has a unique solution. The corresponding property when estimating constant parameters is identifiability. In this thesis, we discuss how observability and identifiability can be determined for DAEs. We propose three approaches, where one can be seen as an extension of standard methods for state-space systems based on rank tests.

    For linear DAEs, a more detailed analysis is performed. We use some well-known canonical forms to examine well-posedness of noise models and to implement estimation of time-varying variables and constant parameters. This includes formulation of Kalman filters for linear DAE models. To be able to implement the suggested methods, we show how the canonical forms can be computed using numerical software from the linear algebra package LAPACK.

  • 5.
    Gerdin, Markus
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Local Identifiability and Observability of Nonlinear Differential-Algebraic Equations2006In: Proceedings of the 14th IFAC Symposium on System Identification, 2006, p. 802-807Conference paper (Refereed)
    Abstract [en]

    Identifiability is important to guarantee convergence in system identification applications, and observability is important in applications such as control and diagnosis. In this paper, recent results on analysis of nonlinear differential-algebraic equations are used to derive criteria for local identifiability and local weak observability for such models. The criteria are based on rank tests. Examples show the relationship between the new criteria and standard methods for state-space systems.

  • 6.
    Gerdin, Markus
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Local Identifiability and Observability of Nonlinear Differential-Algebraic Equations2005Report (Other academic)
    Abstract [en]

    Identifiability is important to guarantee convergence in system identification applications, and observability is important in applications such as control and diagnosis. In this paper, recent results on analysis of nonlinear differential-algebraic equations are used to derive criteria for local identifiability and local weak observability for such models. The criteria are based on rank tests. Examples show the relationship between the new criteria and standard methods for state-space systems.

  • 7.
    Gerdin, Markus
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, Department of Electrical Engineering, Communication Systems. Linköping University, The Institute of Technology.
    Parameter Estimation in Linear Descriptor Systems2004Licentiate thesis, monograph (Other academic)
    Abstract [en]

    Linear descriptor systems form the natural way in which linear models of physical systems are delivered from an object-oriented modeling tool like Modelica. Linear descriptor systems are also known as linear differential-algebraic equations in the continuous-time case. If some parameters in such models are unknown, one might need to estimate them from measured data from the modeled system. This is a form of system identification called gray box identification. The objective of t his work is to examine how gray box identification can be performed for linear descriptor systems.

    To solve this problem, we use some well-known canonical forms to examine how to transform the descriptor systems into state-space form. In general, the input must be redefined to make the transformation into statespace form possible. To be able to implement the suggested identification methods, we examine how the transformations can be computed using numerical software from the linear algebra package LAPACK.

    Noise modeling is an important part of parameter estimation and system identification, so we also examine how a noise model can be added to linear descriptor systems. The result is that white noise in general cannot be added to all equations of a linear continuous-time descriptor system, since this could lead to differentiation of the noise which is not well defined. It is also noted that a Kalman filter can be implemented if the model is transformed into state-space form.

    We also discuss the problem of finding initial values for the paramet er search. We show how to formulate a biquadratic polynomial, that gives initial values for the parameter search when minimized.

  • 8.
    Gerdin, Markus
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Parameter Identification in Linear Time Invariant descriptor Systems2003Report (Other academic)
    Abstract [en]

    This report treats some basics on parameter identification inlinear time invariant descriptor systems, both time continuous and time discrete. It is shown how identification methods fornormal state space systems can be used by transforming the descriptor system to state space form. It is also examined how identification can be performed in the frequency domain.

  • 9.
    Gerdin, Markus
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Using DAE Solvers to Examine Local Identifiability for Linear and Nonlinear Systems2006In: Proceedings of the 14th IFAC Symposium on System Identification, 2006, p. 808-813Conference paper (Refereed)
    Abstract [en]

    If a model structure is not identifiable, then it is not possible to uniquely identify its parameters from measured data. This contribution describes how solvers for differential-algebraic equations (DAE) can be used to examine if a model structure is locally identifiable. The procedure can be applied to both linear and nonlinear systems. If a model structure is not identifiable, it is also possible to examine which functions of the parameters that are locally identifiable.

  • 10.
    Gerdin, Markus
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Using DAE Solvers to Examine Local Identifiability for Linear and Nonlinear Systems2005Report (Other academic)
    Abstract [en]

    If a model structure is not identifiable, then it is not possible to uniquely identify its parameters from measured data. This contribution describes how solvers for differential-algebraic equations (DAE) can be used to examine if a model structure is locally identifiable. The procedure can be applied to both linear and nonlinear systems. If a model structure is not identifiable, it is also possible to examine which functions of the parameters that are locally identifiable.

  • 11.
    Gerdin, Markus
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Fransson, Martin
    Linköping University, Department of Computer and Information Science, PELAB - Programming Environment Laboratory. Linköping University, The Institute of Technology.
    Fritzon, Peter
    Linköping University, Department of Computer and Information Science, PELAB - Programming Environment Laboratory. Linköping University, The Institute of Technology.
    Identification of a Nonlinear Cell Cycle System with Linear Models2005Report (Other academic)
    Abstract [en]

    A mathematical model by Chen et al. representing the cell cycle of the budding yeast cell (Saccharomyces cerevisiae) is used to simulate levels of 13 different cell biological variables. By using a linear systems approach on the generated data the goal is to identify the original equations of the mathematical model. Since this model is described by a nonlinear system the linear systems approach only gives correct estimations of the linear parts of the system, with one exception. A main issue thus becomes to identify which data that belongs to the linear equations and which does not. This problem is solved using two local linear models on different time intervals. Parameter estimations of coefficients from linear and time invariant equations should be the same over all time intervals.

  • 12.
    Gerdin, Markus
    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.
    On Identifiability of Object-Oriented Models2006In: Proceedings of the 14th IFAC Symposium on System Identification, 2006, p. 820-825Conference paper (Refereed)
    Abstract [en]

    When estimating unknown parameters, it is important that the model is identifiable so that the parameters can be estimated uniquely. For nonlinear differential-algebraic equation models with polynomial equations, a differential algebra approach to examine identifiability is available. This approach can be slow, so the present paper describes how this method can be modularized for object-oriented models. A characteristic set of equations is computed for components in model libraries, and stored together with the components. When an object-oriented model is built using such models, identifiability can be examined using the stored equations.

  • 13.
    Gerdin, Markus
    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.
    On Identifiability of Object-Oriented Models2005Report (Other academic)
    Abstract [en]

    When estimating unknown parameters, it is important that the model is identifiable so that the parameters can be estimated uniquely. For nonlinear differential-algebraic equation models with polynomial equations, a differential algebra approach to examine identifiability is available. This approach can be slow, so the present paper describes how this method can be modularized for object-oriented models. A characteristic set of equations is computed for components in model libraries, and stored together with the components. When an object-oriented model is built using such models, identifiability can be examined using the stored equations.

  • 14.
    Gerdin, Markus
    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.
    Ljung, Lennart
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Global Identifiability of Complex Models, Constructed from Simple Submodels2007Report (Other academic)
    Abstract [en]

    It is a typical situation in modern modeling that a total model is built up from simpler submodels, or modules, for example residing in a model library. The total model could be quite complex, while the modules are well understood and analysed. A procedure to decide global parameter identifiability for such a collection of model equations of differential-algebraic nature is suggested. It is shown how to make use of the natural modularization of the model structure. Basically, global identifiability is obtained if and only if each module is identifiable, and the connecting signals can be retrieved from the external signals, without knowledge of the values of the parameters.

  • 15.
    Gerdin, Markus
    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.
    Ljung, Lennart
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Parameter Estimation in Linear Differential-Algebraic Equations2003In: Proceedings of the 13th IFAC Symposium on System Identification, 2003, p. 1530-Conference paper (Refereed)
    Abstract [en]

    This report describes how parameter estimation can be performed in linear DAE systems. Both time domain and frequency domain identification are examined. The results are exemplified on a small system. A potential application for the algorithms is to make parameter estimation in models generated by a modeling language like Modelica.

  • 16.
    Gerdin, Markus
    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.
    Ljung, Lennart
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Parameter Estimation in Linear Differential-Algebraic Equations2003Report (Other academic)
    Abstract [en]

    This report describes how parameter estimation can be performed in linear DAE systems. Both time domain and frequency domain identification are examined. The results are exemplified on a small system. A potential application for the algorithms is to make parameter estimation in models generated by a modeling language like Modelica.

  • 17.
    Gerdin, Markus
    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.
    Ljung, Lennart
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Well-Posedness of Filtering Problems for Stochastic Linear DAE Models2005In: Proceedings of the 44th IEEE Conference on Decision and Control and European Control Conference, 2005, p. 350-355Conference paper (Refereed)
    Abstract [en]

    Modern modeling tools often give descriptor or DAE models, i.e., models consisting of a mixture of differential and algebraic relationships. The introduction of stochastic signals into such models in connection with filtering problems raises several questions of well-posedness. The main problem is that the system equations may contain hidden relationships affecting variables defined as white noise. The result might be that certain physical variables get infinite variance or contain formal differentiations of white noise. The paper gives conditions for well-posedness in terms of certain subspaces defined by the system matrices.

  • 18.
    Gerdin, Markus
    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.
    Ljung, Lennart
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Well-Posedness of Filtering Problems for Stochastic Linear DAE Models2005Report (Other academic)
    Abstract [en]

    Modern modeling tools often give descriptor or DAE models, i.e., models consisting of a mixture of differential and algebraic relationships. The introduction of stochastic signals into such models in connection with filtering problems raises several questions of well-posedness. The main problem is that the system equations may contain hidden relationships affecting variables defined as white noise. The result might be that certain physical variables get infinite variance or contain formal differentiations of white noise. The paper gives conditions for well-posedness in terms of certain subspaces defined by the system matrices.

  • 19.
    Gerdin, Markus
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Schön, Thomas
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Noise Modeling, State Estimation and System Identification in Linear Differential-Algebraic Equations2004In: Proceedings of the 5th Conference on Computer Science and Systems Engineering, 2004, p. 153-163Conference paper (Other academic)
    Abstract [en]

    This paper treats how parameter estimation and Kalman filtering can be performed using a Modelica model. The procedures for doing this have been developed earlier by the authors, and are here exemplified on a physical system. It is concluded that the parameter and state estimation problems can be solved using the Modelica model, and that the parameters estimation and observer construction to a large extent could be automated with relatively small changes to a Modelica environment.

  • 20.
    Gerdin, Markus
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Schön, Thomas
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Noise Modeling, State Estimation and System Identification in Linear Differential-Algebraic Equations2004Report (Other academic)
    Abstract [en]

    This paper treats how parameter estimation and Kalman filtering can be performed using a Modelica model. The procedures for doing this have been developed earlier by the authors, and are here exemplified on a physical system. It is concluded that the parameter and state estimation problems can be solved using the Modelica model, and that the parameters estimation and observer construction to a large extent could be automated with relatively small changes to a Modelica environment.

  • 21.
    Gerdin, Markus
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Schön, Thomas
    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.
    Gustafsson, Fredrik
    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.
    On Parameter and State Estimation for Linear Differential-Algebraic Equations2007Report (Other academic)
    Abstract [en]

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

  • 22.
    Gerdin, Markus
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Schön, Thomas
    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.
    Gustafsson, Fredrik
    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.
    On Parameter and State Estimation for Linear Differential-Algebraic Equations2007In: Automatica, ISSN 0005-1098, E-ISSN 1873-2836, Vol. 43, no 3, p. 416-425Article in journal (Refereed)
    Abstract [en]

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

  • 23.
    Gerdin, Markus
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Electrical Engineering, Automatic Control.
    Sjöberg, Johan
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Nonlinear Stochastic Differential-Algebraic Equations with Application to Particle Filtering2006In: Proceedings of the 45th IEEE Conference on Decision and Control, 2006, p. 6630-6635Conference paper (Refereed)
    Abstract [en]

    Differential-algebraic equation (DAE) models naturally arise when modeling physical systems from first principles. To be able to use such models for state estimation procedures such as particle filtering, it is desirable to include a noise model. This paper discusses well-posedness of differential-algebraic equations with noise models, here denoted stochastic differential-algebraic equations. Since the exact conditions are rather involved, approximate implementation methods are also discussed. It is also discussed how a particle filter can be implemented for DAE models, and how the approximate implementation methods can be used for particle filtering. Finally, the particle filtering methods are exemplified by implementation of a particle filter for a DAE model.

  • 24.
    Gerdin, Markus
    et al.
    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.
    Nonlinear Stochastic Differential-Algebraic Equations with Application to Particle Filtering2007Report (Other academic)
    Abstract [en]

    Differential-algebraic equation (DAE) models naturally arise when modeling physical systems from first principles. To be able to use such models for state estimation procedures such as particle filtering, it is desirable to include a noise model. This paper discusses well-posedness of differential-algebraic equations with noise models, here denoted stochastic differential-algebraic equations. Since the exact conditions are rather involved, approximate implementation methods are also discussed. It is also discussed how a particle filter can be implemented for DAE models, and how the approximate implementation methods can be used for particle filtering. Finally, the particle filtering methods are exemplified by implementation of a particle filter for a DAE model.

  • 25.
    Schön, Thomas
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Gerdin, Markus
    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.
    Gustafsson, Fredrik
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    A Modeling and Filtering Framework for Linear Differential-Algebraic Equations2003In: Proceedings of the 42th IEEE Conference on Decision and Control, 2003, p. 892-897 vol.1Conference paper (Refereed)
    Abstract [en]

    General approaches to modeling, for instance using object-oriented software, lead to differential-algebraic equations (DAE). As the name reveals, it is a combination of differential and algebraic equations. For state estimation using observed system inputs and outputs in a stochastic framework similar to Kalman filtering, we need to augment the DAE with stochastic disturbances ("process noise"), whose covariance matrix becomes the tuning parameter. We will determine the subspace of possible causal disturbances based on the linear DAE model. This subspace determines all degrees of freedom in the filter design, and a Kalman filter algorithm is given. We illustrate the design on a system with two interconnected rotating masses.

  • 26.
    Schön, Thomas
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Gerdin, Markus
    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.
    Gustafsson, Fredrik
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    A Modeling and Filtering Framework for Linear Implicit Systems2003Report (Other academic)
    Abstract [en]

    General approaches to modeling, for instance using object-oriented software, lead to differential algebraic equations (DAE), also called implicit systems. For state estimation using observed system inputs and outputs in a stochastic framework similar to Kalman filtering, we need to augment the DAE with stochastic disturbances (process noise), whose covariance matrix becomes the tuning parameter. We will determine the subspace of possible causal disturbances based on the linear DAE model. This subspace determines all degrees of freedom in the filter design, and a Kalman filter algorithm is given.We illustrate the design on a system with two interconnected rotating masses.

1 - 26 of 26
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