| 1. |
- 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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| 2. |
- Abd-Elrady, Emad, et al.
(författare)
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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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| 3. |
- 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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| 4. |
- Abd-Elrady, Emad
(författare)
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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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| 5. |
- 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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| 6. |
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| 8. |
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| 9. |
- Söderström, Torsten, et al.
(författare)
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Maximum likelihood modeling of orbits of nonlinear ODEs
- 2004
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- This report treats a new approach to the problem of periodic signal estimation. The idea is to model the periodic signal as a function of the state of a second order nonlinear ordinary differential equation (ODE). This is motivated by Poincare theory which is useful for proving the existence of periodic orbits for second order ODEs. The functions of the right hand side of the nonlinear ODE are then parameterized, and a maximum likelihood algorithm is developed for estimation of the parameters of these unknown functions from the measured periodic signal. The approach is analyzed by derivation and solution of a system of ODEs that describes the evolution of the Cramer-Rao bound over time. The proposed methodology reduces the number of estimated unknowns at least in cases where the actual signal generation resembles that of the imposed model. This in turn is expected to result in an improved accuracy of the estimated parameters.
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| 10. |
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