Problem 9.2
Stability of a nonlinear adaptive system
for filtering and parameter estimation
Masoud Karimi-Ghartemani
Department of Electrical and Computer Engineering
University of Toronto
Toronto, Ontario
Canada M5S 3G4
Alireza K. Ziarani
Department of Electrical and Computer Engineering
Clarkson University
Potsdam, NY
USA 13699-5720
1 DESCRIPTION OF THE PROBLEM
2 MOTIVATION AND HISTORY OF THE PROBLEM
The dynamical system presented by (1) was proposed by the authors to devise a system for the extraction of a sinusoidal component with time-varying parameters when it is corrupted by other sinusoids and noise [1, 2]. This is of significant interest in power system applications, for instance [3]. Estimation of the basic parameters of the extracted sinusoid, namely the amplitude, phase, and frequency, was another object of the work. These parameters provide important information useful in electrical engineering applications. Some applications of the system in biomedical engineering are presented in [2, 4]. This dynamical system presents an alternative structure for the well-known phase-locked loop (PLL) system with significantly advantageous features.
3 AVAILABLE RESULTS
Theorem 1, corresponding to the case of 
has been proved by the authors in [1] where the existence, local uniqueness and stability of a To- periodic orbit are shown
using the Poincare map theorem as stated in [5, page 70]. Extensive computer simulations verified by laboratory experimental
results are presented in [1, 2]. Some of the wide-ranging applications of the dynamical system are presented in [2, 3, 4].
The algorithm governed by the proposed dynamical system presents a powerful signal processing method of analysis/synthesis
of nonstationary signals. Alternatively, it may be thought of as a nonlinear adaptive notch filter capable of estimation of
parameters of the output signal.