Backstepping time-scale separation in control of a class of nonlinear systems with uncertainty

In this paper an adaptive control approach for completely non-affine pure-feedback systems with linear parameterization is investigated. In this approach, the idea of positive function of linearly connected parameters is coupled effectively with combination of backstepping and time scale separation. Subsequently, a fast dynamical equation is derived from the original subsystem, where the solution is sought to approximate the corresponding ideal virtual/actual control inputs. Furthermore, the adaptation law of unknown parameters can be derived based on Lyapunov theory in the backstepping technique. The closed loop stability and the state regulation of these systems are all proved. Finally the simulation results for an electromechanical system are provided to demonstrate the effectiveness of the proposed approach.