Based on the sliding mode control methodology, this paper presents a robust control strategy for underactuated systems with mismatched uncertainties. The system consists of a nominal system and the mismatched uncertainties. Since the nominal system can be considered to be made up of several subsystems, a hierarchical structure for the sliding surfaces is designed. This is achieved by taking the sliding surface of one of the subsystems as the first-layer sliding surface and using this sliding surface and the sliding surface of another subsystem to construct the second-layer sliding surface. This process continues till the sliding surfaces of all the subsystems are included. A lumped sliding mode compensator is designed at the last-layer sliding surface. The asymptotic stability of all of the layer sliding surfaces and the sliding surface of each subsystem is proven. Simulation results show the validity of this robust control method through stabilization control of a system consisting of two inverted pendulums and mismatched uncertainties.

In this paper, the vibration modes of the structural model of a nearly cyclic multiply mono-coupled multi-degree-ofifreedom (multi-DOF) component assembly are examined. The attention is focused on the mode similarity phenomenon among the vibration deformation shapes of all the components for the disordered assemblies with relatively weak coupling. By means of this particular characteristic of the normal modes of such a system, an effective solution method based on the analysis of a single component system is developed, which can be used to find any specified structural frequency within any given frequency band and its corresponding mode of the system directly. This solution scheme, in a new perspective, provides a direct physical insight into the relation between the imperfection effects and the various structural and imperfection parameters. Numerical examples are given at the end.