Backstepping Stabilization of Nonlinear Dynamical Systemsunder State Constraints

The problem of stabilizing the zero value of the state vector of constrained nonlinear dynamical systems written in a special form is solved. The proposed control design accounts for magnitude constraints on the values of state variables and is based on the integrator backstepping approach using logarithmic Lyapunov barrier functions. The obtained stabilizing feedbacks, in contrast to similar known results, are based on the use of linear virtual stabilizing functions that do not grow unboundedly as the state variables approach boundary values. As an example, we consider a state constraints aware solution of the control problem of positioning an autonomous underwater vehicle at a given point in space.