From time series to linear system - Part I. Finite dimensional linear time invariant systems
From time series to linear system - Part I. Finite dimensional linear time invariant systems
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DOI：
10.1016/0005-1098(86)90066-x
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发表时间：
1986-09
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影响因子：
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通讯作者：
J. Willems
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文献类型：
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作者：
J. Willems
在本文的第一部分中，将开发一个动力系统的定义，简单地由一个家庭的时间序列。在这种情况下，线性，时间不变性和有限维的概念将被引入。我们将证明，一个给定的时间序列族可以用一个（AR）方程组表示：Riw（t+1）+Rl− 1 w（t +l− 1）+...+ R 0 w（t）= 0，或者等价地，用一个有限维线性时不变系统表示：x（t+ 1）=Ax（t）+Bu（t）; y（t）=Cx（t）+Du（t）; w =（u，y），当且仅当这个族是线性的、平移不变的和完备的（或者等价地，在逐点收敛的拓扑中是闭的）。这就产生了一组非常高水平和优雅的公理，它们描述了这些熟悉的对象。然而，需要强调的是，对于w的哪些分量是输入，哪些分量是输出，没有优先选择。这种分离总是存在于任何特定的线性时不变模型中。从这些定义开始，这样的系统的结构指数进行了介绍，它示出了如何（AR）表示具有给定的行为的系统可以被构造。这些结果将用于模拟的情况下，在第二部分的文件。
In the first part of this paper the definition of a dynamical system as simply consisting of a family of time series will be developed. In this context the notions of linearity, time invariance, and finite dimensionality will be introduced. It will be shown that a given family of time series may be represented by a system of (AR) equations:Riw(t+l) +Rl− 1w(t+l− 1) + … +R0w(t) = 0, or, equivalently, by a finite dimensional linear time invariant system: x(t+ 1) =Ax(t) +Bu(t); y(t) =Cx(t) +Du(t); w = (u, y), if and only if this family islinear, shift invariantandcomplete(or, as is equivalent, closed in the topology of pointwise convergence). This yields a very high level and elegant set of axioms which characterize these familiar objects. It is emphasized, however, that noa priorichoice is made as to which components of w are inputs and which are outputs. Such a separation always exists in any specific linear time invariant model. Starting from these definitions, the structural indices of such systems are introduced and it is shown how an (AR) representation of a system having a given behaviour can be constructed. These results will be used in a modelling context in Part II of the paper.