线性时变系统可达集椭球近似的Hamilton-Jacobi-Bellman方法
A Hamilton-Jacobi-Bellman Approach to Ellipsoidal Approximations of Reachable Sets for Linear Time-Varying Systems
AI总结：
针对线性时变系统可达集计算的维数灾难问题，基于Hamilton-Jacobi-Bellman方程粘性解推导椭球集并交的动力学，生成多项式复杂度的可达集上下近似，提升高维系统的计算可行性。
摘要：
动力系统的可达集描述了在有限时间内、遵循系统动力学能够到达的系统状态集合。它们可用于在控制器设计中保证目标满足性，或验证系统将避开不安全区域。然而，计算这些集合的通用方法存在维数灾难问题，这通常导致它们即使对于线性系统，也无法应用于状态数稍多的系统。本文中，我们证明Hamilton-Jacobi-Bellman方程的粘性上解和下解可分别用于生成时变非线性系统可达集的下近似和上近似。基于这一观察，我们推导了椭球集的并集与交集的动力学，它们分别对受椭球输入约束和椭球终端（或初始）集约束的线性时变系统的可达集进行下近似和上近似。我们证明，可以选择这些椭球的动力学，以确保它们的边界沿系统的一条解轨迹与精确可达集的边界重合。这类椭球集的生成计算复杂度与状态数呈多项式关系，使得我们的近似方案对于维数相对较高的连续时间线性时变系统具有计算可处理性。
英文摘要：
Reachable sets for a dynamical system describe collections of system states that can be reached in finite time, subject to system dynamics. They can be used to guarantee goal satisfaction in controller design or to verify that unsafe regions will be avoided. However, general-purpose methods for computing these sets suffer from the curse of dimensionality, which typically prohibits their use for systems with more than a small number of states, even if they are linear. In this paper, we demonstrate that viscosity supersolutions and subsolutions of a Hamilton-Jacobi-Bellman equation can be used to generate, respectively, under-approximating and over-approximating reachable sets for time-varying nonlinear systems. Based on this observation, we derive dynamics for a union and intersection of ellipsoidal sets that, respectively, under-approximate and over-approximate the reachable set for linear time-varying systems subject to an ellipsoidal input constraint and an ellipsoidal terminal (or initial) set. We demonstrate that the dynamics for these ellipsoids can be selected to ensure that their boundaries coincide with the boundary of the exact reachable set along a solution of the system. The ellipsoidal sets can be generated with polynomial computational complexity in the number of states, making our approximation scheme computationally tractable for continuous-time linear time-varying systems of relatively high dimension.
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