Adaptive Backstepping Control Design for Uncertain Non-smooth Strictfeedback Nonlinear Systems with Time-varying Delays

This paper is concerned with the problem of adaptive neural tracking control for uncertain non-smooth nonlinear time-delay systems with a class of lower triangular form. Based on Filippov’s theory, the bounded stability and asymptotic stability are extended to the ones for the considered systems, which provides the theory foundation for the subsequent adaptive control design. In the light of Cellina approximate selection theorem and smooth approximation theorem for Lipschitz functions, the system under investigation is first transformed into an equivalent system model, based on which, two types of controllers are designed by using adaptive neural network (NN) algorithm. The first designed controller can guarantee the system output to track a target signal with bounded error. In order to achieve asymptotic tracking performance, the other type of controller with proportional-integral(PI) compensator is then proposed. It is also noted that by exploring a novel Lyapunov-Krasovskii functional and designing proper controllers, the singularity problem frequently encountered in adaptive backstepping control methods developed for time-delay nonlinear systems with lower triangular form is avoided in our design approach. Finally, a numerical example is given to show the effectiveness of our proposed control schemes.