| 51. |
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| 52. |
- Abd-Elrady, Emad
(författare)
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An adaptive grid point RPEM algorithm for harmonic signal modeling
- 2001
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- Periodic signals can be modeled as a real wave with unknown period in cascade with a piecewise linear function. In this report, a recursive Gauss-Newton prediction error identification algorithm for joint estimation of the driving frequency and the parameters of the nonlinear output function parameterized in a number of adaptively estimated grid points is introduced. The Cramer-Rao bound (CRB) is derived for the suggested algorithm. Numerical examples indicate that the suggested algorithm gives better performance than using fixed grid point algorithms and easily can be modified to track both the fundamental frequency variations and the time varying amplitude.
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| 53. |
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| 54. |
- Abd-Elrady, Emad, et al.
(författare)
-
Bias Analysis in Least Squares Estimation of Periodic Signals Using Nonlinear ODE's
- 2004
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- Periodic signals can be modeled by means of second-order nonlinear ordinary differential equations (ODE's). The right hand side function of the ODE is parameterized in terms of known basis functions. The least squares algorithm developed for estimating the coefficients of these basis functions gives biased estimates, especially at low signal to noise ratios. This is due to noise contributions to the periodic signal and its derivatives evaluated using finite difference approximations. In this paper an analysis for this bias is given.
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| 55. |
- Abd-Elrady, Emad
(författare)
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Convergence of the RPEM as applied to harmonic signal modeling
- 2000
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- Arbitrary periodic signals can be estimated recursively by exploiting the fact that a sine wave passing through a static nonlinear function generates a spectrum of overtones. The estimated signal model is hence parameterized as a real wave with unknown period in cascade with a piecewise linear function. The driving periodic wave can be chosen depending on any prior knowledge. The performance of a recursive Gauss-Newton prediction error identification algorithm for joint estimation of the driving frequency and the parameters of the nonlinear output function is therefore studied. A theoretical analysis of local convergence to the true parameter vector as well as numerical examples are given. Furthermore, the Cramer-Rao bound (CRB) is calculated in this report.
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| 56. |
- Abd-Elrady, Emad
(författare)
-
Harmonic signal modeling based on the Wiener model structure
- 2002
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- The estimation of frequencies and corresponding harmonic overtones is a problem of great importance in many situations. Applications can, for example, be found in supervision of electrical power transmission lines, in seismology and in acoustics. Generally, a periodic function with an unknown fundamental frequency in cascade with a parameterized and unknown nonlinear function can be used as a signal model for an arbitrary periodic signal. The main objective of the proposed modeling technique is to estimate the fundamental frequency of the periodic function in addition to the parameters of the nonlinear function.The thesis is divided into four parts. In the first part, a general introduction to the harmonic signal modeling problem and different approaches to solve the problem are given. Also, an outline of the thesis and future research topics are introduced.In the second part, a previously suggested recursive prediction error method (RPEM) for harmonic signal modeling is studied by numerical examples to explore the ability of the algorithm to converge to the true parameter vector. Also, the algorithm is modified to increase its ability to track the fundamental frequency variations.A modified algorithm is introduced in the third part to give the algorithm of the second part a more stable performance. The modifications in the RPEM are obtained by introducing an interval in the nonlinear block with fixed static gain. The modifications that result in the convergence analysis are, however, substantial and allows a complete treatment of the local convergence properties of the algorithm. Moreover, the Cramér–Rao bound (CRB) is derived for the modified algorithm and numerical simulations indicate that the method gives good results especially for moderate signal to noise ratios (SNR).In the fourth part, the idea is to give the algorithm of the third part the ability to estimate the driving frequency and the parameters of the nonlinear output function parameterized also in a number of adaptively estimated grid points. Allowing the algorithm to automatically adapt the grid points as well as the parameters of the nonlinear block, reduces the modeling errors and gives the algorithm more freedom to choose the suitable grid points. Numerical simulations indicate that the algorithm converges to the true parameter vector and gives better performance than the fixed grid point technique. Also, the CRB is derived for the adaptive grid point technique.
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| 57. |
- Abd-Elrady, Emad, 1970-
(författare)
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Nonlinear Approaches to Periodic Signal Modeling
- 2005
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Periodic signal modeling plays an important role in different fields. The unifying theme of this thesis is using nonlinear techniques to model periodic signals. The suggested techniques utilize the user pre-knowledge about the signal waveform. This gives these techniques an advantage as compared to others that do not consider such priors. The technique of Part I relies on the fact that a sine wave that is passed through a static nonlinear function produces a harmonic spectrum of overtones. Consequently, the estimated signal model can be parameterized as a known periodic function (with unknown frequency) in cascade with an unknown static nonlinearity. The unknown frequency and the parameters of the static nonlinearity are estimated simultaneously using the recursive prediction error method (RPEM). A treatment of the local convergence properties of the RPEM is provided. Also, an adaptive grid point algorithm is introduced to estimate the unknown frequency and the parameters of the static nonlinearity in a number of adaptively estimated grid points. This gives the RPEM more freedom to select the grid points and hence reduces modeling errors. Limit cycle oscillations problem are encountered in many applications. Therefore, mathematical modeling of limit cycles becomes an essential topic that helps to better understand and/or to avoid limit cycle oscillations in different fields. In Part II, a second-order nonlinear ODE is used to model the periodic signal as a limit cycle oscillation. The right hand side of the ODE model is parameterized using a polynomial function in the states, and then discretized to allow for the implementation of different identification algorithms. Hence, it is possible to obtain highly accurate models by only estimating a few parameters. In Part III, different user aspects for the two nonlinear approaches of the thesis are discussed. Finally, topics for future research are presented.
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| 58. |
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| 59. |
- Abdalmoaty, Mohamed, 1986-
(författare)
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Identification of Stochastic Nonlinear Dynamical Models Using Estimating Functions
- 2019
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Data-driven modeling of stochastic nonlinear systems is recognized as a very challenging problem, even when reduced to a parameter estimation problem. A main difficulty is the intractability of the likelihood function, which renders favored estimation methods, such as the maximum likelihood method, analytically intractable. During the last decade, several numerical methods have been developed to approximately solve the maximum likelihood problem. A class of algorithms that attracted considerable attention is based on sequential Monte Carlo algorithms (also known as particle filters/smoothers) and particle Markov chain Monte Carlo algorithms. These algorithms were able to obtain impressive results on several challenging benchmark problems; however, their application is so far limited to cases where fundamental limitations, such as the sample impoverishment and path degeneracy problems, can be avoided.This thesis introduces relatively simple alternative parameter estimation methods that may be used for fairly general stochastic nonlinear dynamical models. They are based on one-step-ahead predictors that are linear in the observed outputs and do not require the computations of the likelihood function. Therefore, the resulting estimators are relatively easy to compute and may be highly competitive in this regard: they are in fact defined by analytically tractable objective functions in several relevant cases. In cases where the predictors are analytically intractable due to the complexity of the model, it is possible to resort to {plain} Monte Carlo approximations. Under certain assumptions on the data and some conditions on the model, the convergence and consistency of the estimators can be established. Several numerical simulation examples and a recent real-data benchmark problem demonstrate a good performance of the proposed method, in several cases that are considered challenging, with a considerable reduction in computational time in comparison with state-of-the-art sequential Monte Carlo implementations of the ML estimator.Moreover, we provide some insight into the asymptotic properties of the proposed methods. We show that the accuracy of the estimators depends on the model parameterization and the shape of the unknown distribution of the outputs (via the third and fourth moments). In particular, it is shown that when the model is non-Gaussian, a prediction error method based on the Gaussian assumption is not necessarily more accurate than one based on an optimally weighted parameter-independent quadratic norm. Therefore, it is generally not obvious which method should be used. This result comes in contrast to a current belief in some of the literature on the subject. Furthermore, we introduce the estimating functions approach, which was mainly developed in the statistics literature, as a generalization of the maximum likelihood and prediction error methods. We show how it may be used to systematically define optimal estimators, within a predefined class, using only a partial specification of the probabilistic model. Unless the model is Gaussian, this leads to estimators that are asymptotically uniformly more accurate than linear prediction error methods when quadratic criteria are used. Convergence and consistency are established under standard regularity and identifiability assumptions akin to those of prediction error methods.Finally, we consider the problem of closed-loop identification when the system is stochastic and nonlinear. A couple of scenarios given by the assumptions on the disturbances, the measurement noise and the knowledge of the feedback mechanism are considered. They include a challenging case where the feedback mechanism is completely unknown to the user. Our methods can be regarded as generalizations of some classical closed-loop identification approaches for the linear time-invariant case. We provide an asymptotic analysis of the methods, and demonstrate their properties in a simulation example.
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| 60. |
- Abdalmoaty, Mohamed, 1986-
(författare)
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Learning Stochastic Nonlinear Dynamical Systems Using Non-stationary Linear Predictors
- 2017
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- The estimation problem of stochastic nonlinear parametric models is recognized to be very challenging due to the intractability of the likelihood function. Recently, several methods have been developed to approximate the maximum likelihood estimator and the optimal mean-square error predictor using Monte Carlo methods. Albeit asymptotically optimal, these methods come with several computational challenges and fundamental limitations.The contributions of this thesis can be divided into two main parts. In the first part, approximate solutions to the maximum likelihood problem are explored. Both analytical and numerical approaches, based on the expectation-maximization algorithm and the quasi-Newton algorithm, are considered. While analytic approximations are difficult to analyze, asymptotic guarantees can be established for methods based on Monte Carlo approximations. Yet, Monte Carlo methods come with their own computational difficulties; sampling in high-dimensional spaces requires an efficient proposal distribution to reduce the number of required samples to a reasonable value.In the second part, relatively simple prediction error method estimators are proposed. They are based on non-stationary one-step ahead predictors which are linear in the observed outputs, but are nonlinear in the (assumed known) input. These predictors rely only on the first two moments of the model and the computation of the likelihood function is not required. Consequently, the resulting estimators are defined via analytically tractable objective functions in several relevant cases. It is shown that, under mild assumptions, the estimators are consistent and asymptotically normal. In cases where the first two moments are analytically intractable due to the complexity of the model, it is possible to resort to vanilla Monte Carlo approximations. Several numerical examples demonstrate a good performance of the suggested estimators in several cases that are usually considered challenging.
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