Taylor-Lagrange Control for Safety-Critical Systems

Abstract: This paper proposes a novel Taylor-Lagrange Control (TLC) method for nonlinear control systems to ensure the safety and stability through Taylor's theorem with Lagrange remainder. To achieve this, we expand a safety or stability function with respect to time along the system dynamics using the Lie derivative and Taylor's theorem. This expansion enables the control input to appear in the Taylor series at an order equivalent to the relative degree of the function. We show that the proposed TLC provides necessary and sufficient conditions for system safety and is applicable to systems and constraints of arbitrary relative degree. The TLC exhibits connections with existing Control Barrier Function (CBF) and Control Lyapunov Function (CLF) methods, and it further extends the CBF and CLF methods to the complex domain, especially for higher order cases. Compared to High-Order CBFs (HOCBFs), TLC is less restrictive as it does not require forward invariance of the intersection of a set of safe sets while HOCBFs do. We employ TLC to reformulate a constrained optimal control problem as a sequence of quadratic programs with a zero-order hold implementation method, and demonstrate the safety of zero-order hold TLC using an event-triggered control method to address inter-sampling effects. Finally, we illustrate the effectiveness of the proposed TLC method through an adaptive cruise control system and a robot control problem, and compare it with existing CBF methods.