Abstract
In this paper we find some linear dependencies on the matrices used for B and D estimation by the Van Overschee and De Moore non-biased versions of the Combined Deterministic-Stochastic Subspace Identification algorithms (CDSSI). These dependencies allow us to formulate algorithms that significantly improve the numerical efficiency on estimating these parameters without loss of accuracy. Experiences performed on practical data sets showed that the robust versions of these algorithms are twice as fast as the robust version proposed by Van Overschee and De Moore.

Abstract
In this paper we analyse the estimates of the matrices produced by the non-biased deterministic-stochastic subspace identification algorithms (NBDSSI) proposed by Van Overschee and De Moor ( 1996). First, an alternate expression is derived for the A and C estimates. It is shown that the Chiuso and Picci result ( Chiuso and Picci 2004) stating that the A and C estimates delivered by this algorithm robust version and by the Verhaegen's MOESP (Verhaegen and Dewilde 1992a, Verhaegen and Dewilde 1992b, Verhaegen 1993, Verhaegen 1994) are equal, can be obtained from this expression. An alternative approach for the estimation of matrices B and D in subspace identification is also described. It is shown that the least squares approach for the estimation of these matrices estimation can be just expressed as an orthogonal projection of the future outputs on a lower dimension subspace in the orthogonal complement of the column space of the extended observability matrix. Since this subspace has a dimension equal to the number of outputs, a simpler and numerically more efficient ( but equally accurate) new subspace algorithm is provided.

Abstract
In this paper we provide a different way to estimate matrices B and D, in subspace identification algorithms. The starting point was the method proposed by Van Overschee and De Moor [10] - the only one applying subspace ideas to the estimation of those matrices. We have derived new (and simpler) expressions and we found that the method proposed by Van Overschee and De Moor [10] can be rewritten as a weighted least squares problem, involving the future outputs and inputs.

Abstract
A subspace-based on-line identification algorithm based on one specific technique, based on Van Overschee and De Moor's results, but can be adapted to other similar methods since they all recover from the state sequence and the observability matrix is presented. These results relate an estimated Kalman filter sequence with an oblique projection. With further improvements, the algorithm can adapt to the identification of time-variant systems.

Abstract
In this paper, two approaches, for the estimation of matrices A and C in CV A-type subspace identification algorithms are compared and the differences between the two obtained estimates are analysed. One of the methods, "least squares" based, was proposed in the original CVA algorithm. The other method, inspired in the techniques of the classical Realization Theory and proposed by Verhaegen, is far more efficient. Therefore, although the two methods produce two different estimates, a replacement is proposed and an expression for correction of the estimates is obtained, in order to reduce the loss of accuracy.

Abstract
In this paper, a recursive algorithm is presented, based on the CVA subspace identification algorithm. The main idea was to explore the relations between the orthogonal and oblique projections involved and to provide simpler expressions that allowed a recursive version of the algorithm - guaranteeing most of the advantages of this kind of methods, and still improving the numerical efficiency.