A subspace identification of closed-loop system by orthogonal decomposition

In the framework of a joint input-output approach, we show that under the assumption that the exogenous input is feedback free and persistently exciting, the closed-loop identification of deterministic part, i.e. the plant and controller can be decoupled from the stochastic part, i.e. the noise filter by the orthogonal decomposition. The state space models of the plant and controller can be obtained by applying some standard subspace methods to the deterministic component of the joint process, followed by a model reduction