论文标题
椭圆形的有限几何形状总和上的广义外边界
Generalized Outer Bounds on the Finite Geometric Sum of Ellipsoids
论文作者
论文摘要
审查了凸体的一般结果，并用于为$ n $ diblesemensional Euclidean Space的几何（Minkowski）总和提供$ k $椭圆形的几何（Minkowski）总和的精确封闭形式参数公式。以前是通过迭代算法完成的，其中将每个新的椭圆形添加到先前椭球总和的椭圆形近似中。在这里，我们提供一个镜头公式，直接添加$ k $椭圆形，而无需中间近似值。这使我们能够观察几何椭圆形界限家族的新自由度。我们演示了这些工具的应用来计算可触时的离散时间动力系统。
General results on convex bodies are reviewed and used to derive an exact closed-form parametric formula for the boundary of the geometric (Minkowski) sum of $k$ ellipsoids in $n$-dimensional Euclidean space. Previously this was done through iterative algorithms in which each new ellipsoid was added to an ellipsoid approximation of the sum of the previous ellipsoids. Here we provide one shot formulas to add $k$ ellipsoids directly with no intermediate approximations required. This allows us to observe a new degree of freedom in the family of ellipsoidal bounds on the geometric sum. We demonstrate an application of these tools to compute the reachable set of a discrete-time dynamical system.
