An Adaptive State Observer for a Class of Time-variant Bilinear Systems Under Condition of Partial Parametric Uncertainty

Abstract

An adaptive observer for a bilinear time-varying dynamic system under condition of partial parametric uncertainty is proposed. The problem is solved under the assumption that unknown parameters are contained in the control matrix/vector. The state matrix of the system is assumed to be of a certain form. The relevance of such assumption is demonstrated via example of the real technical system (robotic leg) described by the Euler-Lagrange equation. The key idea of the proposed algorithm is a new parameterization of the object. Two functions are defined based on the state vector of the matrix. The crucial point of this algorithm is that one of these functions can be found analytically using known and measured system signals. The proper solution for this function allows to eliminate the nonlinear part of the equation. After that application of linear filters transforms the initial system to a linear static regression model containing unknown constant parameters. The unknown parameters are estimated using a gradient algorithm, and estimates of unknown state vector components are calculated based on the obtained estimates. The proposed method is applicable to a wide class of bilinear systems, including Euler-Lagrange systems describing many real technical objects and robotic systems.