用泰勒-拉格朗日方法实现任意阶非线性系统的安全控制
Taylor-Lagrange Control for Safety-Critical Systems
- 基于李导数和泰勒展开，将控制输入引入高阶项以保证安全
- 相比高阶CBF更宽松，无需安全集交集前向不变性条件
- 适用于自动驾驶与机器人控制，适合追求安全性的工程应用
本文提出一种新型的泰勒-拉格朗日控制（TLC）方法，用于非线性控制系统，在保证安全与稳定的同时利用带拉格朗日余项的泰勒定理。通过沿系统动力学对安全或稳定性函数进行时间展开，并结合李导数与泰勒展开，使控制输入出现在与函数相对阶数等价的高阶项中。所提TLC方法给出了系统安全的充要条件，适用于任意相对阶的系统与约束。TLC与现有控制屏障函数（CBF）和控制李雅普诺夫函数（CLF）方法存在联系，并可扩展至复数域，尤其适用于高阶情形。相较于高阶CBF（HOCBF），TLC更宽松，不要求多个安全集交集的前向不变性。我们采用TLC将约束最优控制问题转化为一系列具有零阶保持实现的二次规划问题，并利用事件触发控制方法处理采样间效应，确保零阶保持TLC的安全性。最后通过自适应巡航控制与机器人控制案例验证了TLC的有效性，并与现有CBF方法进行了对比。
原文摘要 · Abstract (English)
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.
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