Stabilization of State Feedback Linearizable Dynamical Systemsunder State Constraints

The problem of stabilizing the origin is solved for dynamical systems written in a form that admits state feedback linearization, taking into account the magnitude constraints on the state variable values. Based on known results on the possibility of obtaining identical control laws when using the integrator backstepping and the state feedback linearization methods to design stabilizing feedbacks, sufficient conditions are proposed for the gain coefficients and roots of the characteristic polynomial of the closed-loop system that ensure the validity of the specified constraints. The resulting conditions guaranteeing that the constraints hold are based on the results obtained using the integrator backstepping method combined with logarithmic barrier Lyapunov functions. As an example, a solution of the problem of controlling a generalized coordinate is considered for a mechanical system whose dynamics with respect to the selected generalized variable can be represented as a chain of fourth-order integrators, taking into account the constraints on the values of the generalized coordinate, velocity, acceleration, and jerk.