Non-linear and signal energy optimal asymptotic filter design

The paper studies some connections between the main results of the well known Wiener-Kalman-Bucy stochastic approach to filtering problems based mainly on the linear stochastic estimation theory and emphasizing the optimality aspects of the achieved results and the classical deterministic frequency domain linear filters such as Chebyshev, Butterworth, Bessel, etc. A new non-stochastic but not necessarily deterministic (possibly non-linear) alternative approach called asymptotic filtering based mainly on the concepts of signal power, signal energy and a system equivalence relation plays an important role in the presentation. Filtering error invariance and convergence aspects are emphasized in the approach. It is shown that introducing the signal power as the quantitative measure of energy dissipation makes it possible to achieve reasonable results from the optimality point of view as well. The property of structural energy dissipativeness is one of the most important and fundamental features of resulting filters. Therefore, it is natural to call them asymptotic filters. The notion of the asymptotic filter is carried in the paper as a proper tool in order to unify stochastic and non-stochastic, linear and nonlinear approaches to signal filtering

Solution of the System Structure Reconstruction Problem Based on Generalization of Tellegen's Principle

An extraordinary generality, conceptual simplicity and practical usefulness of the Tellegen's theorem is well known in the field of electrical engineering [1]. It is one of few general theoretical results that apply in non-linear and time-varying situations, too. For standard linear electrical network models with constant parameters many classical results of electrical circuits theory can be derived as direct consequences of it. In the paper a more general class of abstract strictly causal system representations is addressed. A new problem, that of the abstract state space system representation structure reconstruc-tion has been formulated in [3], and partially solved in [3] and [4]. In this paper a new approach based on a generalized form of the classical Tellegen's principle, providing an equivalence class of physically as well as mathematically correct solutions is developed and some well-known, as well as new results are shown to be straightforward consequences of the derived struc-ture. Some connections of dissipativity, conservativity, state and parameter minimality, instability and chaos with system representation structures are investigated from this point of view. Analytical results are illustrated by a number of typical examples and visualized by simulations