Comparison of systems with complex behavior: spectral methods
Comparison of systems with complex behavior: spectral methods
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具有复杂行为的系统的比较：谱方法
DOI：
10.1109/cdc.2000.912022
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发表时间：
2000
期刊：
影响因子：
--
通讯作者：
A. Banaszuk
中科院分区：
文献类型：
--
作者：
I. Mezić;A. Banaszuk
我们提出了一个形式主义比较动力系统的渐近动力学与物理系统，他们的模型。通常不需要对一个动力系统和它所模拟的物理系统进行详细的比较，而只需要在统计意义上进行比较。为此，通常考虑不变测度。但是，不变测度通常不能在实验中直接观测到。因此，我们将我们的形式主义建立在从单个可观察到的时间平均值上。特别是，我们建设性地证明，一般来说，一个单一的观察是必要的，以恢复不变的遍历措施。动力系统空间上的伪度量可以使用这种形式主义来定义，以便比较它们的统计行为。我们还确定需要超越比较只有不变的遍历措施的系统，并引入遍历理论治疗一类谱泛函，允许这一点。将该方法推广到一类随机系统：离散随机动力系统。该方法可用于具有复杂行为的非线性模型的参数辨识和模型验证。作为一个例子，我们提供了一个例子，我们比较的渐近行为的燃烧系统的实验测量的渐近行为的模型，是一个随机控制动态系统。
We present a formalism for comparing the asymptotic dynamics of dynamical systems with the physical systems that they model. There is often no need for the detailed (trajectory-wise) comparison of a dynamical system and the physical system that it models, but only comparison in statistical sense. For that purpose, invariant measures are typically considered. But, invariant measures usually can not be observed directly in an experiment. Thus, we base our formalism on time-averages obtained from a single observable. In particular, we constructively prove that, generically, a single observable is needed in order to recover an invariant ergodic measure. Pseudometrics on the space of dynamical systems can be defined using this formalism in order to compare their statistical behavior. We also identify the need to go beyond comparing only invariant ergodic measures of systems and introduce an ergodic-theoretic treatment of a class of spectral functionals that allow for this. The formalism is extended for a class of stochastic systems: discrete random dynamical systems. The ideas introduced can be used for parameter identification and model validation of driven nonlinear models with complicated behavior. As an illustration we provide an example in which we compare the asymptotic behavior of a combustion system measured experimentally with the asymptotic behavior of the model that is a stochastic control dynamical system.