A projection approach to covariance equivalent realizations of discrete systems

Covariance equivalent realization theory has been used recently in continuous systems for model reduction [1]-[4] and controller reduction [2], [5]. In model reduction, this technique produces a reduced-order model that matches
q + 1
output covariances and
q
Markov parameters of the full-order model. In controller reduction, it produces a reduced controller that is "close" to matching
q + 1
output covariances of the full-order controller, and
q
Markov parameters of the closed-loop system. For discrete systems, a method was devised to produce a reduced-order model that matches the
q + 1
covariances [6], but not any Markov parameters. This method requires a factorization to obtain the input matrix, and since the dimension of this matrix factor depends upon rank properties not known a priori, this method may not maintain the original dimension of the input vector. Hence, this method [6] is obviously not suitable for controller reduction. A new projection method is described that matches
q
covariances and
q
Markov parameters of the original system while maintaining the correct dimension of the input vector.