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In this paper we reconsider the analysis and implementation of weighted Yule Walker or instrumental variable methods for estimating the AR parameters of ARMA signals. We present a simplified analysis and propose a new estimate of the optimal weighting matrix leading to more accurate parameter estimates compared to previous approaches.

In this paper we present a new identification method that points at the close relationship between high order ARX modeling and subspace identification. A high order ARX model is utilized to obtain initial estimates of certain Markov parameters. These parameters are then used to restructure the data model used for subspace identification to facilitate the estimation of the state sequence. Based on the estimated state sequence, the system parameters are estimated by linear regression. The method is shown to be competitive to existing subspace methods by a simulation example. The method can also be used, without modification, on closed loop data in contrast to most previously published subspace identification methods.

It is well known that the accuracy of instrumental variable (IV) methods for parameter estimation can be significantly improved by weighting the estimating equations. However, the optimal weight typically depends on unknown parameters and needs to be estimated before it can be used in the IV estimator. In many parameter estimation problems the optimal weighting matrix is equal to the DC power of a certain vector process. It is known to be a difficult task to consistently estimate power in a nonparametric manner at any frequency (e.g., the periodogram cannot achieve this). However, for the case considered in this paper we show that there exists a natural nonparametric consistent estimator of the DC power matrix or, equivalently, the optimal weighting matrix. This result is expected to increase the interest in using optimally weighted parameter estimation methods of instrumental variable or related type.

The class of subspace algorithms for system identification is an interesting complement to the maximum likelihood or prediction error methods, especially for multivariable systems. The statistical analysis of the subspace methods is difficult since the estimates depend on the data in a rather complicated manner. Previous results include proofs of generic consistency and asymptotic normality of the estimates. However, no explicit transparent expression for the covariance matrix of the limiting distribution has so far been reported because of the aforementioned difficulties. The main objective of this paper is to provide a methodology that simplifies the asymptotic analysis of subspace based estimation algorithms. The basic idea is illustrated by deriving the asymptotic covariance matrix corresponding to the estimates of the state-space matrices (or the transfer function estimate) of a quite general subspace algorithm.

System identification of linear dynamical systems using so-called subspace methods consists of two main steps, First, a signal subspace estimate is found. This usually corresponds to estimating the range space of the extended observability matrix. Then the system parameters are estimated from the subspace estimate. The main result of this note is explicit excitation conditions on the input signals, which guarantee consistent estimates of the range space of the extended observability matrix.

Subspace methods for identification of linear dynamical systems consist of two main steps. First, a so-called subspace estimate is constructed. This usually corresponds to estimating the range space of the extended observability matrix. Then, the system parameters are estimated from the subspace estimate. The main objective of this paper is to show that subspace methods for the identification of combined deterministic and stochastic systems may fail to provide a consistent estimate of the subspace. Hence, for certain systems, the transfer function model estimated using the subspace identification algorithms is not a consistent estimate of the true system.