Backstepping model-free adaptive control for a class of second-order nonlinear systems

This paper proposes a backstepping model-free adaptive control (BS-MFAC) algorithm for the trajectory tracking control problem of a class of second-order nonlinear systems. Firstly, the model-free adaptive control (MFAC) is employed to track the virtual desired velocity. Then, based on the measured state data at the current moment, a virtual desired velocity is designed, combining MFAC with backstepping control in a clever manner. By designing a discrete form of Lyapunov function, it is proven that the system velocity error under this algorithm converges asymptotically and stably. Furthermore, a nonhomogeneous difference equation solution related to the system position is obtained through calculations. The analysis of the solution indicates that the system position output error will slide into a neighborhood around zero. This algorithm possesses advantages such as simple structure, model independence, and strong robustness. Additionally, a planar two-link space robot arm model is constructed using the Mbdyn platform, a multibody dynamics simulation software. Trajectory tracking control of joint angles is achieved under the proposed control algorithm in this paper. The simulation outcomes unequivocally establish the efficacy of the algorithm, which adeptly exploits the pseudo partial derivatives (PPD) estimation for second-order nonlinear systems in MFAC. The algorithm yields expedited and superior trajectory tracking control of said systems.