论文标题
动态系统状态空间模型的解决方案属性的教程
A Tutorial on Solution Properties of State Space Models of Dynamical Systems
论文作者
论文摘要
状态空间模型的分析的起点是研究存在，唯一性和解决方案特性，例如semigroup属性以及解决方案的各种公式。几个概念，例如状态过渡矩阵，矩阵指数，常数公式的变化（Cauchy公式），Peano-Baker系列和PICARD迭代用于表征解决方案。在本说明中，给出了教程处理，其中所有这些概念都被证明是单个抽象方法的各种表现，即使用涉及前向整合的Volterra操作员的操作员Neumann系列求解方程。矩阵指数，Peano-Baker系列，Picard Iteration和Cauchy公式可以自然地从Neumann系列中“发现”。该系列的收敛性和迭代是Volterra操作员渐近衰变的关键特性的结果。该属性是严格的三角形矩阵的nilpotence属性的渐近版本。
The starting point of analysis of state space models is investigating existence, uniqueness and solution properties such as the semigroup property, and various formulas for the solutions. Several concepts such as the state transition matrix, the matrix exponential, the variations of constants formula (the Cauchy formula), the Peano-Baker series, and the Picard iteration are used to characterize solutions. In this note, a tutorial treatment is given where all of these concepts are shown to be various manifestations of a single abstract method, namely solving equations using an operator Neumann series involving the Volterra operator of forward integration. The matrix exponential, the Peano-Baker series, the Picard iteration, and the Cauchy formula can be "discovered" naturally from this Neumann series. The convergence of the series and iterations is a consequence of the key property of asymptotic nilpotence of the Volterra operator. This property is an asymptotic version of the nilpotence property of a strictly-lower-triangular matrix.
