Adaptive stabilization for a class of nonlinearly parameterized systems with multiple unknown control directions via switching linear controllers

In this paper, the adaptive control via switching linear controllers is studied for a class of nonlinearly parameterized systems with unknown control directions. For the studied system, the parametric uncertainties entering the state equations nonlinearly can be fast time-varying or jumping at unknown time instants and the bounds of the parametric uncertainties are not required to know a priori and the control directions can be unknown. First, sufficient conditions for designing an adaptive stabilizer are derived. Then, the adaptive stabilizer is a switching-type one, in which a linear controller with two undetermined design parameters to be tuned is recursively designed by backstepping, and a switching mechanism is proposed to tune these parameters online for compensating the unknown control directions and the unknown bounds of the parametric uncertainties. The undetermined design parameters are based on the upper bound estimate of the unknown parameters existing not only in the nonlinear functions but also in the control directions functions. The proposed adaptive controller globally asymptotically stabilizes the system in the sense that, for any initial conditions, the state converges to the origin while all the signals of the closed-loop system are bounded. Finally, an example is given to illustrate the effectiveness of the proposed method.