Miguel Galrinho

Miguel Galrinho, Cristian R. Rojas, Hakan Hjalmarsson

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 closed-loop 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.

Miguel Galrinho, Riccardo Prota, Mina Ferizbegovic et al.

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.

Niklas Everitt, Miguel Galrinho, Håkan Hjalmarsson

In system identification, it is often difficult to find a physical intuition to choose 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 may 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 chances of getting trapped in local minima. To avoid this, a high order ARX model can first be estimated by least squares, providing non-parametric estimates of the plant and noise model. Then, model order reduction can be used to obtain a parametric model of the plant only. We review existing methods to perform 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, which suggests that the proposed method is competitive with PEM and other similar methods.

Miguel Galrinho, Cristian R. Rojas, Hakan Hjalmarsson

Standard system identification methods often provide inconsistent estimates with closed-loop data. With the prediction error method (PEM), this issue is solved by using a noise model that is flexible enough to capture the noise spectrum. However, a too flexible noise model (i.e., too many parameters) increases the model complexity, which can cause additional numerical problems for PEM. In this paper, we consider the weighted null-space fitting (WNSF) method. With this method, the system is first modeled using a non-parametric ARX model, which is then reduced to a parametric model of interest using weighted least squares. In the reduction step, a parametric noise model does not need to be estimated if it is not of interest. Because the flexibility of the noise model is increased with the sample size, this will still provide consistent estimates in closed loop and asymptotically efficient estimates in open loop. In this paper, we prove these results, and we derive the asymptotic covariance for the estimation error obtained in closed loop, which is optimal for an infinite-order noise model. For this purpose, we also derive a new technical result for geometric variance analysis, instrumental to our end. Finally, we perform a simulation study to illustrate the benefits of the method when the noise model cannot be parametrized by a low-order model.

Miguel Galrinho, Niklas Everitt, Håkan Hjalmarsson

High-order ARX models can be used to approximate a quite general class of linear systems in a parametric model structure, and well-established 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.