Robustness of Controlled Lagrangian Method to the Structured Uncertainties

Controlled Lagrangian method uses the inherent geometric structure of the energy of the mechanical systems to provide a stabilizing algorithm for underactuated mechanical systems. The presented method belongs to a larger family of nonlinear control algorithms, namely energy shaping methods in which the controller is designed by providing necessary modifications in the mechanical energy of the system. This paper presents a sensitivity analysis of Controlled Lagrangian method. It is shown that the method presents a suitable performance under the effect of structured (or parametric) uncertainties such as masses values, their positions and their influence on the inertia tensor. Then, the sequel investigates the robustness level of the designed controller in the presence of structured uncertainties. A detailed robustness proof of the scheme is established in this paper. Simulations are provided for a linear inverted pendulum cart system to validate analytical results of robustness to parametric uncertainties. Simulation results confirm that the designed controller for the inverted pendulum, which is unstable and underactuated, is well robust against parametric uncertainties as the analytical studies predicted. The method was also compared with the sliding mode approach, which showed a superior robustness against parametric uncertainties and a more practical control input value.