Division controllers for homogeneous dyadic bilinear systems

When a homogeneous bilinear system is dyadic, it permits the use of the so-called division controller for global stabilization. However, due to singularities in the control law, a dead-zone must be cascaded with the division controller in practical implementation. The contribution of this note is to show that the division controller, when cascaded with a dead-zone, can asymptotically drive the bilinear system state into a bounded set around the origin with the size of the bounded set proportional to the size of dead-zone. This note further proposes a C⁰ modification and a C^∞ modification of the conventional discontinuous dead-zone so that the control signals from the modified division controllers are continuous and smooth, respectively.