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Consistency and efficiency in continuous-time system identification

Time: Thu 2020-06-04 10.00

Location: Sal Harry Nyquist, Malvinas väg 10, Stockholm (English)

Subject area: Electrical Engineering

Doctoral student: Rodrigo A. González , Reglerteknik

Opponent: Professor Hugues Garnier, Université de Lorraine

Supervisor: Associate Professor Cristian R. Rojas, Reglerteknik; Bo Wahlberg, Signaler, sensorer och system, Optimeringslära och systemteori, Reglerteknik

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Continuous-time system identification deals with the problem of building continuous-time models of dynamical systems from sampled input and output data. In this field, there are two main approaches: indirect and direct. In the indirect approach, a suitable discrete-time model is first determined, and then it is transformed into continuous-time. On the other hand, the direct approach obtains a continuous-time model directly from the sampled data. In both approaches there exists a dichotomy between discrete-time data and continuous-time models, which can induce robustness issues and complications in the theoretical analysis of identification algorithms. These difficulties are addressed in this thesis.

First, we consider the indirect approach to continuous-time system identification. For a zero-order hold sampling mechanism, this approach usually leads to a transfer function estimate with relative degree one, independent of the relative degree of the strictly proper true system. Inspired by the indirect prediction error method, we propose an indirect-approach estimator that enforces the desired number of poles and zeros in the continuous-time transfer function estimate, and show that the estimator is consistent and asymptotically efficient. A robustification of this method is also developed, by which the estimates are also guaranteed to deliver stable models.

In the second part of the thesis, we analyze asymptotic properties of the Simplified Refined Instrumental Variable method for Continuous-time systems (SRIVC), which is one of the most popular direct identification methods. This algorithm applies an adaptive prefiltering to the sampled input and output that requires assumptions on the intersample behavior of the signals. We present a comprehensive analysis on the consistency and asymptotic efficiency of the SRIVC estimator while taking into account the intersample behavior of the input signal. Our results show that the SRIVC estimator is generically consistent when the intersample behavior of the input is known exactly and subsequently used in the implementation of the algorithm, and we give conditions under which consistency is not achieved. In terms of statistical efficiency, we compute the asymptotic Cramér-Rao lower bound for an output error model structure with Gaussian noise, and derive the asymptotic covariance of the SRIVC estimates. We conclude that the SRIVC estimator is asymptotically efficient under mild conditions, and that this property can be lost if the intersample behavior of the input is not carefully accounted for in the SRIVC procedure.

Moreover, we propose and analyze the statistical properties of an extension of SRIVC that is able to deal with input signals that cannot be interpolated exactly via hold reconstructions. The proposed estimator is generically consistent for any input reconstructed using zero or first-order-hold devices, and we show that it is generically consistent for continuous-time multisine inputs as well. Comparisons with the Maximum Likelihood technique and an analysis of the iterations of the method are provided, in order to reveal the influence of the intersample behavior of the output and to propose new robustifications to the SRIVC algorithm.