为摆杆倒立系统提供从摆起到稳定的全程可达性保证
Reachability Guarantees for Cart-Pole Swing-Up and Stabilization
- 基于能量塑造与LQR的切换控制，利用相空间几何设计摆起策略
- 理论证明几乎全局收敛，且摆起后状态进入稳定器的吸引域
- 首次实现从摆起到稳定的闭环可达性形式化验证，适合控制理论研究者
Cart-pole摆起是欠驱动非线性控制系统中的经典基准问题，但端到端的摆起与稳定衔接的可达性保证极少被形式化。本文提出一种切换能量型/LQR控制器的可达性分析，证明系统可从紧致初始条件集收敛至直立平衡点。摆起设计利用保守摆的相空间几何：直立平衡点位于同宿轨道上，能量塑造律使能量误差归零，引导摆杆沿该轨道运动；收敛性由LaSalle不变性原理保证。通过构造增广Lyapunov函数，进一步调节稳态小车速度为零，证明闭环系统几乎全局收敛。局部LQR控制器具有已认证的椭球吸引域，数值验证表明摆起阶段可将状态送入该区域，完成手递手交接。数值模拟支持理论分析。
原文摘要 · Abstract (English)
The cart-pole swing-up is a canonical benchmark for nonlinear control of underactuated systems, yet an end-to-end guarantee linking the global swing-up maneuver to the local stabilizer is seldom formalized. We present a reachability analysis of a switched energy-based/LQR controller that certifies convergence to the upright equilibrium from a compact set of initial conditions. The swing-up design exploits the phase-space geometry of the conservative pendulum: the upright equilibrium lies on the homoclinic orbit, and an energy-shaping law drives the energy error to zero, steering the pendulum onto this orbit; convergence follows from LaSalle's invariance principle. An augmented Lyapunov function additionally regulates the steady-state cart velocity to zero, and we prove almost-global convergence of the resulting closed-loop system. A local LQR with a certified ellipsoidal region of attraction stabilizes the upright equilibrium, and we verify numerically that the swing-up phase delivers the state into this region, formalizing the handoff. Numerical simulations corroborate the theoretical analysis.
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