Abstract-Parameter estimation in structured models is generally considered a difficult problem. For example, the prediction error method (PEM) typically gives a non-convex optimization problem, while it is difficult to incorporate structural information in subspace identification. In this contribution, we revisit the idea of iteratively using the weighted least-squares method to cope with the problem of non-convex optimization. The method is, essentially, a three-step method. First, a high order least-squares estimate is computed. Next, this model is reduced to a structured estimate using the least-squares method. Finally, the structured estimate is re-estimated, using weighted least-squares, with weights obtained from the first structured estimate. This methodology has a long history, and has been applied to a range of signal processing problems. In particular, it forms the basis of iterative quadratic maximum likelihood (IQML) and the Steiglitz-McBride method. Our contributions are as follows. Firstly, for output-error models, we provide statistically optimal weights. We conjecture that the method is asymptotically efficient under mild assumptions and support this claim by simulations. Secondly, we point to a wide range of structured estimation problems where this technique can be applied. Finally, we relate this type of technique to classical prediction error and subspace methods by showing that it can be interpreted as a link between the two, sharing favorable properties with both domains.

In identification of dynamical systems, the prediction error method using a quadratic cost function provides asymptotically efficient estimates under Gaussian noise and additional mild assumptions, but in general it requires solving a non-convex optimization problem. An alternative class of methods uses a non-parametric model as intermediate step to obtain the model of interest. Weighted null-space fitting (WNSF) belongs to this class. It is a weighted least-squares method consisting of three steps. In the first step, a high-order ARX model is estimated. In a second least-squares step, this high-order estimate is reduced to a parametric estimate. In the third step, weighted least squares is used to reduce the variance of the estimates. The method is flexible in parametrization and suitable for both open-and closedloop data. In this paper, we show that WNSF provides estimates with the same asymptotic properties as PEM with a quadratic cost function when the model orders are chosen according to the true system. Also, simulation studies indicate that WNSF may be competitive with state-of-the-art methods.

High-order ARX models can be used to approximate a quite general class of linear systems in a parametric model structure, and wellestablished methods can then be used to retrieve the true plant and noise models from the ARX polynomials. However, this commonly used approach is only valid when the plant is stable or if the unstable poles are shared with the true noise model. In this contribution, we generalize this approach to allow the unstable poles not to be shared, by introducing modifications to correctly retrieve the noise model and noise variance.

In system identification, it is often difficult to use a physical intuition when choosing a noise model structure. The importance of this choice is that, for the prediction error method (PEM) to provide asymptotically efficient estimates, the model orders must be chosen according to the true system. However, if only the plant estimates are of interest and the experiment is performed in open loop, the noise model can be over-parameterized without affecting the asymptotic properties of the plant. The limitation is that, as PEM suffers in general from non-convexity, estimating an unnecessarily large number of parameters will increase the risk of getting trapped in local minima. Here, we consider the following alternative approach. First, estimate a high-order ARX model with least squares, providing non-parametric estimates of the plant and noise model. Second, reduce the highorder model to obtain a parametric model of the plant only. We review existing methods to do this, pointing out limitations and connections between them. Then, we propose a method that connects favorable properties from the previously reviewed approaches. We show that the proposed method provides asymptotically efficient estimates of the plant with open-loop data. Finally, we perform a simulation study suggesting that the proposed method is competitive with state-of-the-art methods.

For identification of systems embedded in dynamic networks, applying the prediction error method (PEM) to a correct tailor-made parametrization of the complete network provided asymptotically efficient estimates. However, the network complexity often hinders a successful application of PEM, which requires minimizing a non-convex cost function that in general becomes more difficult for more complex networks. For this reason, identification in dynamic networks often focuses in obtaining consistent estimates of particular network modules of interest. A downside of such approaches is that splitting the network in several modules for identification often costs asymptotic efficiency. In this paper, we consider the particular case of a dynamic network with the individual systems connected in a serial cascaded manner, with measurements affected by sensor noise. We propose an algorithm that estimates all the modules in the network simultaneously without requiring the minimization of a non-convex cost function. This algorithm is an extension of Weighted Null-Space Fitting (WNSF), a weighted least-squares method that provides asymptotically efficient estimates for single-input single-output systems. We illustrate the performance of the algorithm with simulation studies, which suggest that a network WNSF may also be asymptotically efficient estimates when applied to cascade networks, and discuss the possibility of extension to more general networks affected by sensor noise.