Brockett’s First Example: An FAS Approach Treatment

doi:10.1007/s11424-022-2090-8

In this note, the well-known Brockett’s first example system is treated with the fully actuated system (FAS) approach. Firstly, it is shown that the system can be exponentially substabilized by a smooth controller in the sense that, except those starting from initial values on the z₀-axis of the initial value space, all trajectories of the designed system as well as the control signals decay to zero exponentially. Secondly, global stabilization is realized through a way of enabling the trajectories starting from initial values on the z₀-axis also to go to the origin. The idea is to firstly move an initial point on the z₀-axis away from the axis using a pre-controller, and then to take over by the designed exponentially sub-stabilizing controller.

Key words: Exponential sub-stabilization, feedback stabilization, fully actuated system approach, region of attraction, singularity sets

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