Backstepping-Based Lyapunov Function Construction Using Approximate Dynamic Programming and Sum of Square Techniques

In this paper, backstepping for a class of block strict-feedback nonlinear systems is considered. Since the input function could be zero for each backstepping step, the backstepping technique cannot be applied directly. Based on the assumption that nonlinear systems are polynomials, for each backstepping step, Lypunov function can be constructed in a polynomial form by sum of square (SOS) technique. The virtual control can be obtained by the Sontag feedback formula, which is equivalent to an optimal control - the solution of a Hamilton-Jacobi-Bellman equation. Thus, approximate dynamic programming (ADP) could be used to estimate value functions (Lyapunov functions) instead of SOS. Through backstepping technique, the control Lyapunov function (CLF) of the full system is constructed finally making use of the strict-feedback structure and a stabilizable controller can be obtained through the constructed CLF. The contributions of the proposed method are twofold. On one hand, introducing ADP into backstepping can broaden the application of the backstepping technique. A class of block strict-feedback systems can be dealt by the proposed method and the requirement of nonzero input function for each backstepping step can be relaxed. On the other hand, backstepping with surface dynamic control actually reduces the computation complexity of ADP through constructing one part of the CLF by solving semidefinite programming using SOS. Simulation results verify contributions of the proposed method.