Semi–Global and Robust Finite–Time Regulation of the Heisenberg System

In this article, a novel controller design is proposed to solve the finite–time regulation problem of the Heisenberg system, a prototypical nonholonomic system diffeomorphic to the Brockett integrator and many physical systems. Some inherent properties of the Heisenberg system, e.g., the skew–symmetric/diagonal structure, are exploited to obtain a simple–to–tune and bounded control that ensures the finite–time convergence to any point of the state space outside of a zero–measure set. In addition, the resulting controller is globally bounded, and this bound can be fixed independently of the set of initial conditions. Contrasting with other approaches, this design allows us to provide the estimation of an upper bound of the settling–time function. Moreover, for a particular set of parameters, the exact settling–time function is recovered, and a predefined convergence time can be selected. Furthermore, some robustness analysis is provided, considering nonvanishing matched perturbations, leading to the design conditions for ensuring exact finite–time convergence. The performance of the proposed controller is illustrated by numerical simulations and comparisons.