Sliding mode control of the systems with uncertain direction of control vector

The sliding mode control approach is used for a class of nonlinear systems x˙=f(t,x)+B(t,x)u with uncertain matrix B. The control design is based on the partition of the extended system state space onto the cells with fixed control vector inside each cell. That results in multiple stable equilibriums for the extended system. In general, these points are different for different values of B. Each equilibrium corresponds to the stability of the origin for the given system, If areas of attraction for the multiple stability points covers the entire state space the proposed control allows to stabilize the system even if the direction of the actuation is unknown