Realization of Stochastic Systems with Application to Subspace Identification Method

本文对Akaike(1974)提出的时间序列的随机实现算法进行了扩展，研究了具有外生输入的随机系统的实现算法。在输出到输入没有反馈的假设下，我们导出并刻画了具有外生输入的平稳过程的极小状态空间模型族。我们已经介绍了一种非常自然的块结构，它通常是最小的。这种模型结构导致了基于子空间的辨识算法，它的结构比文献中存在的算法更简单。这些结果发表在《信号处理》(1996年)和第13届世界联合会大会预印本(1996年)上。我们解决了具有外生输入的线性离散时间系统的随机实现问题。通过对过去和未来的典型相关分析(CCA)选择状态向量，条件是未来的输入。然后，我们根据状态向量和未来输入推导出未来输出的最优预测器的状态方程。这些方程导致了存在外生投入的产出过程的正向新息模型。实现程序的基本步骤是对未来产出和给定未来投入的过去数据的条件协方差矩阵进行因式分解。这种因式分解是基于CCA的，可以很容易地适应有限的输入输出数据。我们推导了四种随机子空间辨识算法，使实现过程适用于有限的输入输出数据。文中还给出了数值结果。

This study is concerned with the realization algorithms for the stochastic system subjected to exogenous input by extending the stochastic realization algorithm for time-series due to Akaike (1974).1. Under the assumption that there is no feedback from the output to the input, we have derived and characterized the family of minimal state-space models of the stationary process with exogenous inputs. We have introduced a very natural block structure which is generically minimal. This model structure leads to subspace-based identification algo-rithms which have a simpler structure of those existing in the literarture. These results are published in Signal Processing (1996) and in the Preprints of 13th IFAC World Congress (1996).2. We have solved the stochastic realization problem for a linear discrete-time system with an exogenous input. The state vector is chosen by using the canonical correlation analysis (CCA) of past and future conditioned on the future inputs. We then derive the state equations of the optimal predictor of the future outputs in terms of the state vector and the future inputs. These equations lead to a forward innovation model for the output process in the presence of exogenous inputs. The basic step of the realization procedure is a factorization of the conditional covariance matrix of future outputs and past data given future inputs. This factorization is based on CCA and can be easily adapted to finite input-output data. We derive four stochastic subspace identification algorithms which adapt the realization procedure to finite input-output data. Numerical results are also included.