Convergence analysis of parameters for linearly parametric nonlinear strict feedback system with unknown control direction

In this paper, the adaptive tracking control and parameters identification problems are investigated for a class of linearly parametric strict feedback system with unknown control direction. By using backstepping design procedure, the adaptive tracking control scheme combined with Nussbaum gain function is proposed. In the controller, the adaptive law of estimated parameters is derived from Lyapunov stability theorem and Nussbaum-type function. And all the signals in closed-loop system are proved to be bounded. Secondly, the identification of estimated parameters in the strict feedback system with unknown control direction is studied. By constructing a novel Lyapunov function, a sufficient condition (PE condition), which can guarantee that the parameters estimation converge to the actual values of parameters, is obtained for the first time. Also, it is more simplified than the existing results on PE. Under the PE condition proposed here, it is shown that the parameters estimation errors are convergent to zero asymptotically by using Nussbaum function technique and Barbalat´s lemma. Finally, an illustrated example is given to demonstrate the main results.