Exponentially stabilizing division controllers for dyadic bilinear systems

It is difficult to asymptotically stabilize a dyadic bilinear system with only multiplicative control inputs when the open-loop dynamics are unstable. The previous approach of cascading a division controller with a constant-size dead zone can only stabilize but not asymptotically stabilize the system. This note proposes a new control design which cascades a division controller with a modified dead zone whose size is proportional to the modulus of the system state. It is shown that this new division controller can globally and exponentially stabilize any open-loop unstable dyadic bilinear system as long as it is controllable.