Realization of Stochastic Systems with Application to Subspace Identification Method

本研究通过扩展Akaike（1974）的时间序列随机实现算法，研究外生输入下随机系统的实现算法。在输出对输入没有反馈的假设下，导出了具有外生输入的平稳过程的最小状态空间模型族，并对其进行了刻画。我们引入了一个非常自然的块结构，它通常是最小的。这种模型结构导致了基于子空间的识别算法，该算法的结构比现有文献中的算法更简单。这些结果发表在信号处理（1996）和第13届国际会计师联合会世界大会（1996）的预印本中。我们解决了具有外生输入的线性离散系统的随机实现问题。状态向量的选择采用基于未来输入的过去和未来的典型相关分析（CCA）。然后，我们根据状态向量和未来输入推导出未来输出的最佳预测器的状态方程。这些方程导致了外生投入存在下产出过程的前向创新模型。实现过程的基本步骤是对未来输出和给定未来输入的过去数据的条件协方差矩阵进行因式分解。这种分解是基于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识别算法，使实现过程适应有限输入输出数据。数值结果也包括在内。

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.