Subspace identification of combined deterministic-stochastic systems by LQ decomposition

In this paper, we revisit the realization-based subspace identification problem for combined deterministic-stochastic discrete-time LTI systems. Under the assumption that the past horizon is sufficiently large, we reveal a block lower triangular structure of an L-factor related to the stochastic component in the LQ decomposition of data matrix based on an asymptotic analysis and covariance factorization. Adapting the theoretical result to finite input-output data, we then develop a method of computing the steady state Kalman gain and the covariance of innovation process, where K is obtained by a method similar to computing (B, D) parameters in the MOESP method. Thus, under the assumption that the input is persistently exciting (PE) of sufficiently high order, we can compute all the systems parameters from L-factors of a single LQ decomposition of the data matrix.