Adaptive robust stabilization for a class of uncertain nonlinear systems with unknown virtual control coefficients

The problem of robust stabilization is considered for a class of nonlinear systems in the presence of uncertain system parameters, external disturbances, and unknown time-varying virtual control coefficients. It is supposed that the upper bounds of external disturbances, and the upper and lower bounds of unknown time-varying virtual control coefficients are unknown. Some improved adaptation laws with σ-modification are employed to estimate the values which contain all the unknown bounds of such uncertainties. Then, by making use of the updated values of these unknown bounds, a class of backstepping approach based continuous adaptive robust state feedback controllers is synthesized for such uncertain dynamical systems. Furthermore, by employing the improved Lyapunov-Krasovskii functional method, it is also shown that the proposed adaptive robust backstepping controller can guarantee the uniform asymptotic stability of such uncertain dynamical systems. Finally, a numerical example is given to demonstrate the validity of the results.