Stabilization of Chain of Integrators Under State Constraints

The problem of stabilizing the origin for dynamical systems written as a chain of integrators of arbitrary length is solved, taking into account constraints on absolute values of the state variables. It is shown that linear stabilizing state feedback based on the modal control principle ensures the fulfillment of the specified constraints on the state variables when choosing roots of the characteristic polynomial of the closed-loop system to satisfy nonlinear inequalities. The derived conditions on the roots of the characteristic polynomial are based on the results obtained using the integrator backstepping design combined with logarithmic barrier Lyapunov functions. As an illustrative example, the solution to the problem of stabilizing a given angular position of a single-link manipulator is considered. It is guaranteed that magnitude constraints on values of the angular coordinate, angular velocity, angular acceleration and jerk of the manipulator end-point are satisfied.