Robust High-Order Control Barrier Functions-Based Optimal Control for Constrained Nonlinear Systems With Safety-Stability Perspectives

In this article, we propose a robust high-order control barrier functions (HoCBFs)-based optimal control method for nonlinear systems with state constraints to achieve safety-stability perspectives. First, a kind of HoCBFs is presented for constrained nonlinear systems to address state constraints with high relative degrees. Second, the robustness property of the HoCBFs is analyzed based on the asymptotic stability of the forward invariant set. Specifically, a robust HoCBFs-based Lyapunov function is constructed to prove the uniform asymptotic stability of the set associated with the HoCBFs. In this way, a new sufficient condition is obtained for the stability analysis of the forward invariant set by using the inequalities of high-order derivatives of Lyapunov function. Third, a robust HoCBFs-based optimal control scheme is proposed for the constrained nonlinear system to achieve the safety-stability perspectives of constraints satisfaction and system stabilization, where the robust HoCBFs are combined with control Lyapunov functions (CLFs) to satisfy the small control property (SCP) in solving a quadratic program (QP). Furthermore, the proposed optimal control scheme is shown to be Lipschitz continuous and has no initial condition restrictions. Finally, two examples are presented to demonstrate the control performance of the proposed scheme. Note to Practitioners-The motivation of this article is that constraints exist widely in actual control systems, and the lack of constraint satisfaction in control systems may inevitably lead to safety defects, which usually degrade the control performances or even damage the entire system. In this article, a robust HoCBFs-based optimal control scheme is proposed for constrained nonlinear systems. The theoretical derivation demonstrates that the proposed control scheme can achieve safety-stability perspectives, which ensure system stabilization and task-oriented performance without violating the state constraints. The satisfactory control performances of the simulation on a constrained robotic manipulator show the potential practical application on a real robotic system.