Damping parameters indentification of Lienard polynomial system

In many applications of physics, biology, and other sciences, an approach based on the concept of model equations is used as an approximate model of complex nonlinear processes. The basis of this concept is the provision that a small number of characteristic types movements of simple mathematical models inherent in systems give the key to understanding and exploring a huge number of different phenomena. In particular, it is well known that the complex oscillatory motion can be modeled by a system consisting of one or more coupled nonlinear oscillators that governs by differential equation of a second-order. A Lienard system, namely $ \ddot x(t)+f(x(t))\dot x(t)+g(x(t)) = 0$, is a generalization of the such models. Here $f(x(t))$ and $g(x(t))$ are functions that represent various nonlinear phenomena. The typical sources of nonlinearities in Lienard systems are as follows: large displacements of the structure provoking geometric nonlinearities, a nonlinear material behavior, complex law of damping dissipation, etc. In fact, parameter identification is the base of several engineering tasks: identification can be used for the following: (i) to gain knowledge about the process behavior, (ii) to validate theoretical models, (iii) to tune controller parameters, (iv) to design adaptive control algorithms, (v) to process supervision and fault detection, (vi) to on-line optimization. Hence, in order to represent these nonlinearities, identifying the parameters characterizing their behaviors is essential. The problem of constructing globally convergent identificator for polynomial representation of damping force in general Lienar oscillator is addressed. The method of invariant relations is used for identification scheme design. This aproach is based on dynamical extension of original system and construct of appropriate invariant relations, from which the unknowns parameters can be expressed as a functions of the known quantities on the trajectories of extended system. The final synthesis is carried out from the condition of obtaining asymptotic estimates of unknown parameters. It is shown that an asymptotic estimate of the unknown states can be obtained by rendering attractive an appropriately selected invariant manifold in the extended state space.