Covariance Analysis, Positivity and the Yakubovich-Kalman-Popov Lemma

This paper presents theory and algorithms for covariance analysis and stochastic realization without any minimality condition imposed. Also without any minimality conditions, we show that several properties of covariance factorization and positive realness hold. The results are significant for validation in system identification of state-space models from finite input-output sequences. Using the Riccati equation, we have designed a procedure to provide a reduced-order stochastic model that is minimal with respect to system order as well as the number of stochastic inputs thereby avoiding several problems appearing in standard application of stochastic realization to the model validation problem. The case considered includes the problem of rank-deficient residual covariance matrices, a case which is encountered in applications with mixed stochastic-deterministic input-output properties as well as for cases where outputs are linearly dependent,thus extending previous results in covariance analysis.