Nonparametric Identification of Linear Time-Varying Systems using Gaussian processes

Linear time-varying  (LTV) systems are a class of systems with dynamics evolving in time. For these systems we can define a time-varying transfer function. Accordingly, for each frequency a time-varying gain can be attributed to the system as opposed to a fixed gain in the case of time-invariant systems. The figure below shows a time-varying transfer function with the time-varying gain at 1 Hz indicated in red.

time-varying transfer function

The goal of this study is to estimate this time-varying transfer function based on observed data. There already exist methods to identify linear time-varying systems nonparametrically. The problem boils down to capturing the time-varying gains, which is solved in the literature by imposing basis functions and projecting the time-varying gains on the space spanned by these basis functions. In this research, however, we don’t use basis functions but rather Gaussian processes to capture the time-variation. Models based on Gaussian processes are very flexible and have the particular property that their model structure is continuously tunable. Furthermore, the model structure can be estimated during the identification process and not afterwards as opposed to classical basis function approaches. Hence, the use of Gaussian processes is of great advantage when modelling linear time-varying systems. The main advantage of this new identification method using Gaussian processes is the fact that the user should not choose a predefined model structure. Hence, it contributes to the use of system identification by non-experts.

Additionally a noise estimation technique has been developed using the Gaussian process approach. The figure below shows the measured output spectrum of an LTV system, with the estimated output spectrum, the residuals and the estimated noise standard deviation.

measured output spectrum of an LTV system

References:

  1. Christopher KI Williams and Carl Edward Rasmussen. Gaussian Processes for Machine Learning, volume 2. MIT Press Cambridge, MA, 2006.
  2. Rik Pintelon and Johan Schoukens. System Identification: a Frequency Domain Approach. John Wiley & Sons, 2012.
  3. John Lataire and Rik Pintelon. Estimating a nonparametric colored-noise model for linear slowly timevarying systems. IEEE Transactions on Instrumentation and Measurement, 58(5):1535–1545, 2009.
  4. John Lataire, Rik Pintelon, and Ebrahim Louarroudi. Non-parametric estimate of the system function of a time-varying system. Automatica, 48(4):666–672, 2012.
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