Representations of jointly stationary stochastic feedback processes†

Stable constant linear closed-loop systems relating an input vector u to an output vector u and vice versa produce a jointly stationary (y, u)-procoss. On the other hand it is often natural to split up a stationary vector random process z into component vectors yand u, and to examine the closed-loop relations between y and u. This paper presents a number of new results on the spectral factorization and the closed-loop representation of a jointly stationary vector (y, u)-proccss. Conditions are derived on the closed-loop models for the joint process model to bo oE minimal degree, stable and minimum-phase. Relations between different joint process models producing the same spectrum φyu(z)are established.