Output Tracking Control for Nonlinear Affine Systems With Relative Degree Two Based on Control Barrier Functions

Nonlinear affine systems with relative degree two widely exist in the control field, and the unified output tracking control approach of these systems is still an open problem. This paper presents a formal framework for output tracking control design methodology for nonlinear affine systems with relative degree two. Specifically, a control barrier function is constructed to enforce the forward invariance or the stability of a set contained within a 0-sublevel set of the first-order derivative of pre-stablished Lyapunov function, thereby stabilizing the system output to the desired value. Additionally, by taking the first-order derivative of the elaborated control barrier function, control input terms with a nonzero coefficient are introduced into the stability condition, addressing the weakness of controller design using conventional control Lyapunov function approaches, which fails in states where the coefficient of the control input term is zero. Furthermore, the stability conditions are proposed in the form of linear inequality constraints involving control inputs, with which a collection of admissible controllers is provided. Then, by resorting to the pointwise minimum norm (PMN) technique, a numerical solution of an admissible controller can be derived in the context of a quadratic program. We finally demonstrate the effectiveness of the controller design method via two simulations, adaptive cruise control and vehicle lateral control.