Backstepping Stabilization of Nonlinear Dynamical Systems under State Constraints

The problem of stabilizing the zero value of the state vector of constrained nonlineardynamical systems written in a special form is solved. The proposed control design accounts formagnitude constraints on the values of state variables and is based on the integrator backsteppingapproach using logarithmic Lyapunov barrier functions. The obtained stabilizing feedbacks, incontrast to similar known results, are based on the use of linear virtual stabilizing functions thatdo not grow unboundedly as the state variables approach boundary values. As an example, weconsider a state constraints aware solution of the control problem of positioning an autonomousunderwater vehicle at a given point in space.