1. Introduction
Flexible structures are preferred in many applications such as flexible link robots, spacecrafts, surgicaloperating equipments (medical field) and variety of mechanical systems. Dynamics of elastic body system is governed by combination of highly non-linear, ordinary and/or partial differential equations (ODE's and/orPDE's). See for example [1]–[12] and references therein. Challenges basically arise due to the fact that flexible structure has infinite degrees of freedom [1], [9]. Mathematically capturing the dynamics of these distributed plus lumped parameter systems, in which gravity impact plays a major role, is extremely difficult. It further complicates the control design and implementation [1], [2], [4], [5], [13]. In this paper, the system of flexible inverted pendulum (with bob mass) on cart, representing the flexible structured systems, is considered. Typical schematic of this system is shown in Fig. 1. Due to elasticity, even slight movement/jerk to the body or fluctuation in bob/beam mass, causes deflection of flexible link with bending and vibrations, in single or multi-modes. Appropriately modeling and developing a reliable efficient controller for such flexible body/link systems would hence be a significant contribution.