# Identification of nonlinear systems and modeling of periodic signals

Periodic signal modeling plays an important role in signal processing for the following reasons:

- In many applications of signal processing, it is desirable to eliminate or extract sine waves from observed data or to estimate their unknown frequencies.
- Tracking of time-varying parameters of sinusoids in additive noise is of great importance from both theoretical and practical point of view. It arises in many engineering applications such as radar, communications, control, biomedical engineering and others.
- Frequency domain analysis and design of power systems is complicated in the presence of harmonics. Presence of harmonics can lead to unpredicted interaction between components in power networks, which in worst case can lead to instability.

The unifying theme of this project is using nonlinear techniques to model periodic signals. The suggested techniques utilize the user pre-knowledge about the signal waveform. This gives the suggested techniques an advantage, provided that the assumed model structure is suitable, as compared to other approaches that do not consider such priors.

The *first* nonlinear technique studied and developed in this project 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 (chosen as the nonlinear function values in a set of *fixed* grid points) are estimated simultaneously using the recursive prediction error method (RPEM).

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 the *second* nonlinear technique of this project, a second-order nonlinear ODE model is used to model the periodic signal as a limit cycle oscillation. The right hand side of the suggested 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. Also, this is conditioned on the fact that the signal model is suitable.

The project was sponsored by The Swedish Research Council for Engineering Sciences as a part of a multiproject on Statistical Signal Processing.

The members of the project group are Emad Abd-Elrady, Torbjörn Wigren and Torsten Söderström

## Results achieved

### (a) Modeling periodic signals based on the Wiener model structure

Two approaches were introduced:

- The first approach relies on using a fixed grid points technique which is obtained by introducing an interval in the nonlinear block with fixed gain. The modification in the convergence analysis is, however, substantial and allows a complete treatment of the local convergence properties of the algorithm. Furthermore, the Cramer-Rao Bound (CRB) is calculated for the suggested algorithm. Moreover, the algorithm was modified to increase the ability to track both the fundamental frequency and the damped amplitude variations.

- The second approach uses an
*adaptive*grid point technique resulting in automatic grid adaptation. Also, the CRB is calculated for the new algorithm.

- Both the two approaches were compared by numerical examples which indicates that the second approach gives better performance.

### (b) Modeling periodic signals using second-order nonlinear ODE's

- It was proved that the second-order nonlinear ODE model is sufficient to model many periodic signals.

- Different off-line (least squares, Markov and maximum likelihood) and on-line (Kalman filter and extended Kalman filter) algorithms were developed for the estimation of the ODE model.

- The CRB was derived for the suggested ODE model.

- An alternative ODE model based on the Lienard's equation was introduced.

- A bias analysis in least squares estimation of periodic signals was presented.

The research so far has lead to the following publications:

1- E. Abd-Elrady. Convergence of the RPEM as applied to harmonic signal modeling, Tech. Rep. 2000-027, Information Technology, Uppsala University, Uppsala, Sweden. October 2000.

2- E. Abd-Elrady. Study of a nonlinear recursive method for harmonic signal modeling, Proc. of Proc of 20:th IASTED International Conference on Modeling, Identification and Control, Innsbruck, Austria, Feb. 19-22, 2001.

3- E. Abd-Elrady. A nonlinear approach to harmonic signal modeling, *Signal Processing*, vol. 84, no 1, pp. 163-195, January 2004.

4- E. Abd-Elrady. An adaptive grid point algorithm for harmonic signal modeling, Tech. Rep. 2001-018, Information Technology, Uppsala University, Uppsala, Sweden. August 2001.

5- E. Abd-Elrady. An adaptive grid point algorithm for harmonic signal modeling, Proc. of *Proc of 15:th IFAC World Congress on Automatic Control*, Barcelona, Spain, July 21-26, 2002.

6- E. Abd-Elrady. An adaptive grid point algorithm for harmonic signal modeling, Preprint of Reglermöte, Linköping, Sweden, May 29-30, 2002.

7- E. Abd-Elrady. Harmonic signal modeling based on the Wiener model structure, Licentiate Thesis 2002-003, Information Technology, Uppsala University, Uppsala, Sweden. May 2002.

8 - T. Wigren, E. Abd-Elrady and T. Söderström. Harmonic signal analysis with Kalman filters using periodic orbits of nonlinear ODEs, *Proc. of the IEEE International Conference on Acoustics, Speech and Signal Processing*, April 6-10, 2003, Hongkong.

9- T. Wigren, E. Abd-Elrady and T. Söderström. Least squares harmonic signal analysis using periodic orbits of ODEs, *Proc. of the IFAC Symposium on System Identification*, Rotterdam, The Netherlands, August 27-29, 2003.

10- E. Abd-Elrady, T. Söderström and T. Wigren. Periodic signal modeling based on Liénard's equation, *IEEE Transaction on Automatic Control*, vol. 49, no. 10, pp. 1773-1778, October 2004.

11- E. Abd-Elrady, T. Söderström and T. Wigren. Periodic signal analysis using periodic orbits of nonlinear ODEs based on the Markov estimate, *Proc. of 2nd IFAC Workshop on Periodic Control Systems (PSYCO 04)*, Yokohama, Japan, 2004.

12- T. Söderström, T. Wigren and E. Abd-Elrady. Periodic signal analysis by maximum likelihood modeling of orbits of nonlinear ODEs, *Automatica*, vol. 41, no. 5, pp. 793-805, May 2005.

13- E. Abd-Elrady and J. Schoukens. Least squares periodic signal modeling using orbits of nonlinear ODE's and fully automated spectral analysis, *Automatica*, vol. 41, no. 5, pp. 857-862, May 2005.

14- E. Abd-Elrady and T. Söderström: Bias analysis in periodic signals modeling using nonlinear ODE's, *Proc. of 16th IFAC World Congress on Automatic Control*, Prague, Czech Republic, 2005. (to appear)

15- E. Abd-Elrady: Nonlinear Approaches to Periodic Signal Modeling, Serie Acta Universitatis Uppsaliensis, Ph. D. Thesis, 2005-59.