基于网格的紧界非线性系统仿射抽象
Mesh-Based Affine Abstraction of Nonlinear Systems with Tighter Bounds
摘要：
本文考虑非线性系统的分段仿射抽象问题，即，通过一对分段仿射函数对其非线性动力学的过度近似，该分段仿射函数“包括”原始系统的动力学特性。因此，基于仿射抽象的控制器或估计器的保证也适用于原始非线性系统。我们的方法包括解决一个线性规划（LP）的问题，过近似的非线性函数在一个网格的网格点与给定的分辨率，然后占整个域通过适当的校正项。为了达到所需的近似精度，我们还迭代地细分到子区域的域。我们的方法适用于具有不同光滑度的非线性函数，包括Lipschitz连续函数，并通过使用更严格的边界来改进现有方法。最后，我们比较了我们的方法与现有的基于优化的方法在模拟的有效性，并说明其适用性估计设计。
英文摘要：
In this paper, we consider the problem of piecewise affine abstraction of nonlinear systems, i.e., the overapproximation of its nonlinear dynamics by a pair of piecewise affine functions that "includes" the dynamical characteristics of the original system. As such, guarantees for controllers or estimators based on the affine abstraction also apply to the original nonlinear system. Our approach consists of solving a linear programming (LP) problem that over-approximates the nonlinear function at only the grid points of a mesh with a given resolution and then accounting for the entire domain via an appropriate correction term. To achieve a desired approximation accuracy, we also iteratively subdivide the domain into subregions. Our method applies to nonlinear functions with different degrees of smoothness, including Lipschitz continuous functions, and improves on existing approaches by enabling the use of tighter bounds. Finally, we compare the effectiveness of our approach with existing optimization-based methods in simulation and illustrate its applicability for estimator design.
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