Singular perturbation analysis of a flexible beam used in underwater exploration

This article focuses on the analysis of a flexible beam based on two approaches of singular perturbations and feedback linearisation. First, a model is obtained for a flexible beam consisting of flexible joints and actuators for underwater positioning and/or tracking control leading to a singularly perturbed structure based on the slow phenomena due to the rigid portion and fast phenomena due to the flexible part of the system. Next, two ways are given for introducing the small parameter responsible for time-scale behaviour, one using a separate small parameter for each beam joint and beam link and the second using a single small parameter with appropriate scaling. Using the singular perturbation approach, the original systems are decoupled into slow and fast subsystems. The main objective is to use a feedback linearisation approach and design a proportional-integral-derivative computed-torque, composite (slow and fast) controller for improving the tracking performance. An illustrative example is provided.