On the Estimation of Time-varying Parameters in Continuous-time Nonlinear Systems

The estimation of time-varying parameters in continuous-time nonlinear systems is considered under the framework of the modulating functions method. The parameter is approximated as a finite Fourier series, which is reconstructed from the estimated Fourier spectral coefficients. Unlike the popular polynomial approximation, this approach is general enough for piecewise smooth parameter changes. The locations of abrupt jumps are accurately identified by the presence of Gibbs phenomenon. The global Fourier spectral coefficients are then used to extract local finite Gegenbauer polynomial series to recover smooth parameter variations between the jumps. This method of resolution of the Gibbs phenomenon avoids the necessity of estimating a large number of Fourier coefficients for series convergence. A van der Pol oscillator simulation example is included to demonstrate the performance of the approach.