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. Note to Practitioners—The motivation for this work originates from the control requirements of nonlinear systems with relative degree two. Many practical systems, such as vehicle motion systems, exhibit characteristics of a relative degree two. To address this, various output tracking control methods have been developed. This paper utilizes control barrier functions to design an optimization-based output tracking control method that does not rely on the structural analysis of the system itself. Additionally, for systems with relative degree two, the traditional control Lyapunov method may struggle to ensure the feasibility of solutions to the quadratic programming problem due to the possibility of control gain coefficients being zero, which in turn cannot guarantee the convergence of system output. The method proposed in this paper focuses on solving this issue, and the two simulation experiments demonstrated that the proposed method can be effectively applied to vehicle control systems with relative degree two.