Practical Realization of Implicit Homogeneous Controllers for Linearized Systems

This article deals with the practical implementation of implicit homogeneous controllers (IHCs) for linearized mechanical systems. The control design includes the methodology to get gains of the IHC based on the linearized approximation of the system. If the approximation error enforced by the linearization is vanishing with the state, locally measurable and bounded, the IHC can lead the state to the origin in finite-time. This IHC control allows accelerating the convergence rate of the states. A semiexplicit algorithm is provided to exert the digital implementation of the controller. The application of the bisection method estimates the controller gain ensuring the finite-time convergence of the state to the origin. A complementary analysis provides a simplified algorithm with a reduced number of computation stages but equally efficient gain estimation. The proposed IHC is applied to a rotary inverted pendulum QUBE$^{\text{{{}}}}$ Servo 2 platform of Quanser$^\text{{{{}}}}$. The obtained results for the state convergence are compared with other classical feedback controllers to validate the effectiveness of the proposed scheme. The comparative analysis of the state convergence provides evidence of the faster convergence for the state trajectory to a zone centered on the origin with a smaller hypervolume than the one gotten with the classical controllers.