Concurrent Learning Control Lyapunov and Barrier Functions for Uncertain Nonlinear Safety-Critical Systems With High Relative Degree Constraints

This paper studies the problem of safety-critical tracking control for uncertain nonlinear systems which are allowed to be with high relative degree constraints, parametric uncertainty in the control and drift vector fields. The key to our method is to construct an augmented stored matrix and an augmented estimation vector to simultaneously handle unknown parameters in the control and drift vector fields. By designing a parameter update law depending on stored data and instantaneous data, a concurrent learning control Lyapunov function (CL-CLF) is proposed to achieve exponential convergence of tracking errors and parameter estimation errors under a finite excitation condition. By developing a class
$\mathcal{K}$
function with respect to the augmented stored matrix, a CL high order control barrier function (CL-HOCBF) is constructed to ensure the safety of uncertain nonlinear systems with arbitrary relative degree constraints. Based on the CL-CLF and CL-HOCBF, the safety-critical controller can be obtained by solving quadratic programming (QP) without adding additional constraints. Theoretical analysis shows that the proposed CLF-CBF-QP framework can complete the challenge of transforming stability and safety requirements into verifiable linear constraints on control inputs in the presence of uncertainty in both the control and drift vector fields. In addition, the effectiveness of the developed approach is also verified by addressing the adaptive cruise control problem with uncertain vehicle dynamics.
Note to Practitioners
—In many control applications, the relative degrees of safety-critical constraints typically are larger than one, such as constraints regarding the configuration of mechanical systems. In addition, uncertainties in parameters including friction, control coefficients and damping widely exist in real-world systems. Motivated by these practical factors, this paper develops a novel CLF-CBF-QP framework for more general safety-critical systems with high relative degree constraints, parametric uncertainty in the control and drift vector fields, which can overcome the difficulty that transforms stability and safety requirements into verifiable linear constraints on control inputs. In particular, the practicability of presented method is demonstrated by solving the adaptive cruise control problem for automobiles with safety constraints and vehicle limitations, unknown parameters in aerodynamic drag term and control effectiveness.