参数不确定系统$α$ -置信可达集的凸估计
Convex Estimation of the $α$-Confidence Reachable Sets of Systems with Parametric Uncertainty
摘要：
准确地建模和验证在动态环境中交互的系统的正确操作是具有挑战性的。通过利用模型描述中的参数不确定性，可以放松对精确描述与环境的相互作用的要求;然而，尽管存在不确定性，仍然必须保证模型的行为是可以接受的。本文提出了一种凸优化方法来有效地计算一个多项式动力系统的配置，能够安全地达到用户定义的目标集，尽管在模型中的参数不确定性。由于规划中存在的不确定性可能会导致不良的保守性，本文计算的不确定非线性系统的轨迹是$α$ -可能达到期望的配置。所提出的方法使用占用措施的概念来描述一个非线性系统与参数不确定性的轨迹的演变为一个线性方程的措施，其支持下的轨迹相一致的调查。这个线性方程近似与消失保守性使用层次的半定的程序，其中每个被证明是计算一个近似的初始条件是$α$ -可能达到用户定义的目标集安全，尽管不确定性。该方法的有效性示出了四个系统的参数不确定性。
英文摘要：
Accurately modeling and verifying the correct operation of systems interacting in dynamic environments is challenging. By leveraging parametric uncertainty within the model description, one can relax the requirement to describe exactly the interactions with the environment; however, one must still guarantee that the model, despite uncertainty, behaves acceptably. This paper presents a convex optimization method to efficiently compute the set of configurations of a polynomial dynamical system that are able to safely reach a user defined target set despite parametric uncertainty in the model. Since planning in the presence of uncertainty can lead to undesirable conservatives, this paper computes those trajectories of the uncertain nonlinear systems which are $α$-probable of reaching the desired configuration. The presented approach uses the notion of occupation measures to describe the evolution of trajectories of a nonlinear system with parametric uncertainty as a linear equation over measures whose supports coincide with the trajectories under investigation. This linear equation is approximated with vanishing conservatism using a hierarchy of semidefinite programs each of which is proven to compute an approximation to the set of initial conditions that are $α$-probable of reaching the user defined target set safely in spite of uncertainty. The efficacy of this method is illustrated on four systems with parametric uncertainty.
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