Nonlinear Boundary Control and Estimation of Distributed States in a Flexible Manipulator

Abstract

This paper presents a novel method for partial differential equation (PDE) endpoint control of a flexible manipulator, without using any form of mathematical simplification in PDE-based calculations. It simultaneously addresses the issue of the unavailability of distributed states in an infinite-dimensional flexible dynamic system. This two-fold approach allows for the incorporation of estimated data describing the pose of the flexible manipulator into a model-based PDE control scheme, facilitating convenient stability analysis and model-based control design. This is achieved while maintaining the easily attainable measurement requirements of standard boundary control techniques, which involve the installation of sensors solely at the manipulator's endpoint. To this end, initially a practical, stability-proofed distributed state observer is presented which is capable of estimating the deflection of the link throughout the length of the flexible arm based on a nonlinear PDE model. To reduce design complexity and the intensity of implementation, the observation is conducted using a novel partial state observer, which relies on a reduced number of system equations corresponding to the distributed state measurement error. Subsequently, the estimated states are used to construct a boundary controller for state tracking and the elimination of endpoint deflection, resulting in precise endpoint control. Furthermore, inevitable deviations from the nominal model are considered, ensuring the boundedness of the closed-loop system response in the presence of parametric uncertainties through an adaptive design. A rigorous stability analysis of all elements of the closed-loop systems incorporating PDE models is conducted based on Lyapunov stability theory and the extended LaSalle's invariance principle for infinite-dimensional systems. Numerical simulations and analysis demonstrate the precision and efficiency of the proposed solutions.