# Non-linear identification of periodic signals

Periodic signal modeling plays an important role in signal processing e.g. 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, feedforward control and biomedical engineering.
- 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 to use 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 proposed nonlinear technique 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) of 8.

Limit cycle oscillation problems 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. A number of results have been achieved, including:

- Conditions that imply when a second-order nonlinear ODE model is sufficient to model a given periodic signals has been derived, in the form of a loop-criterion (1). The loop-criterion expresses the requirement that the phase plane plot of the modeled periodic signal is not allowed to have any self-intersections. The loop-criterion has been generalized to higher dimnesions as well.
- Different off-line (least squares, Markov and maximum likelihood) and on-line (Kalman filter and extended Kalman filter) algorithms have been developed for the identification of the ODE model. See the papers 2-7 for details.
- The CRB has been derived for the suggested ODE model in the paper 2.
- An alternative ODE model based on the Lienard's equation has been introduced in 5.
- Parts of the work is summarized in 3.

### References

1. T. Wigren and T. Söderström, A second order ODE is sufficient for modeling of many periodic signals, Int. J. Contr. , vol. 78, no. 13, pp. 982-996, 2005.

2. 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, 2005.

3. E. Abd-El Rady, "Nonlinear approaches to periodic signal modeling", Ph. D. Dissertation, Department of Information Technology, Uppsala University, Uppsala, Sweden, January 26, 2005.

4. 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.

5. E. Abd-Elrady, T. Söderström and T. Wigren. Periodic signal modeling based on Liénard's equation, IEEE Trans. Automat. Contr., vol. 49, no. 10, pp. 1773-1778, 2004.

6. 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.

7. 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, Honkong, China, April 6-10, 2003.

8. T. Wigren and P. Händel, Harmonic signal modeling using adaptive nonlinear function estimation, Proc. ICASSP, Atlanta, Georgia, U.S.A., pp. 2952-2955, 1996.