Positive invariance tests with efficient Hessian matrix eigenvalues bounds
Positive invariance tests with efficient Hessian matrix eigenvalues bounds
复制标题
具有有效 Hessian 矩阵特征值界限的正不变性测试
DOI：
--
复制
发表时间：
2008
期刊：
影响因子：
--
通讯作者：
M. Mönnigmann
中科院分区：
文献类型：
--
作者：
M. Mönnigmann
摘要：我们研究了 n 维非线性自治离散时间系统域中集合的正不变性的两个简单充分准则。这些标准源自具有线性和二次余数项的精确泰勒展开式。通过一个简单的例子，我们证明存在可以用二阶准则建立正不变性的系统，但不能用一阶准则建立正不变性。由于二阶准则需要模型方程的 Hessian 矩阵，因此该准则的计算量很大。然而，我们表明，二阶准则可以以惊人的低计算成本进行评估。具体来说，我们表明计算复杂度比 Hessian 矩阵的计算低一个数量级。
Abstract We investigate two simple sufficient criteria for positive invariance of sets in the domain of n-dimensional nonlinear autonomous discrete time systems. These criteria are derived from the exact Taylor expansion with linear and quadratic remainder terms. By a simple example we demonstrate that systems exist for which positive invariance can be established with the second order criterion but not with the first order criterion. Since the second order criterion requires the Hessian matrices of the model equations, this criterion is computationally expensive. We show, however, that the second order criterion can be evaluated at a surprisingly low computational cost. Specifically, we show that the computational complexity is an order of magnitude lower than the calculation of the Hessian matrices.