Velocity observer design via reducing to linear regression for the Euler-Lagrange systems class

Abstract

In this paper, we consider the problem of velocity estimation for Euler-Lagrange systems. For these systems, classical observers are quite limited because, in the general case, the equations have quadratic (bilinear) dependence on generalized velocities. The proposed approach is based on the existing state observer designed for bilinear systems. This observer requires that the unknown multiplier of the state is integrable. Of course, not every arbitrary system satisfies this condition. However, Euler-Lagrange equations naturally contain partial derivatives and, what is more important, full derivatives of the mass matrix elements, as well as more complicated expressions that come from them. An advantage of this bilinear systems observer is that it reduces the velocity estimation problem to a simple linear regression form. Thanks to this, we can use different adaptive, robust, and even finite-time estimation algorithms. In the paper, we formulate some requirements that guarantee the presence of the full derivative of the natural logarithm of the mass matrix determinant in the equations of motion. This makes explicit integration possible. For the sake of completeness, we partly reproduce the existing procedure for observer synthesis with minor modifications. Also, simulation results for the cart-pendulum system implemented in MATLAB/Simulink are shown. These results demonstrate that the proposed observer converges well and has better robustness to measurement noise compared to the dirty derivative approach.