Adaptive neural identification and non-singular control of pure-feedback nonlinear systems

This paper proposes a new constructive identification and adaptive control method for nonlinear pure-feedback systems, which remedies the 'explosion of complexity' and potential control singularity encountered in the traditional adaptive backstepping controllers. First, to avoid using the backstepping recursive design, alternative state variables and the corresponding coordinate transformation are introduced to reformulate the pure-feedback system into an equivalent canonical model. Then, a high-order sliding mode (HOSM) observer is used to reconstruct the unknown states for this canonical model. To remedy the potential singularity in the control, the unknown system dynamics are lumped to derive an alternative identification structure and one-step control synthesis, where two radial basis function neural networks (RBFNN) are adopted to online estimate these lumped dynamics. In this framework, the online estimation of control gain is not in the denominator of controller, and thus the division by zero in the controllers is avoided. Finally, a new online learning algorithm is constructed to obtain the RBFNNs' weights, ensuring the convergence to the neighborhood of true values and allowing accurate identification of unknown dynamics. Theoretical analysis elaborates that the convergence of both the tracking error and the estimation error is obtained simultaneously. Simulations and practical experiments on a hydraulic servo test-rig verify the effectiveness and utility of the suggested methods.