# 12. Nonparametric Estimation of Fast Time-varying Dynamics

## Background

In all engineering disciplines, mathematical models are required to describe the reality surrounding us. The goal of System identification is to compute these models, based on real-life measurements on the system of interest. Instead of making assumptions on the physics of the system, one relies on actual observations, which results in a very pragmatic way of working.

For many applications, one assumes that the system of interest is LTI (Linear and Time Invariant). Although this assumption has shown to perform well in many cases, in quite some applications the linearity and time-invariance hypotheses are only approximately true or not valid at all. Think, for example, of

• in vivo measurement of the electrical impedance of the heart tissue: due to the periodic contraction of the heart the electrical impedance changes periodically in time (see the figure),
• acoustical lung impedance measurements: due to the periodic expansion of the lungs the impedance changes periodically in time,
• the linearization of nonlinear dynamics around a periodic steady state solution, which results in a linear system with periodically time-varying coefficients. The left block diagram in the figure below gives a graphical representation of the corresponding direct model.  ## What will you learn?

At the department ELEC-VUB we have a lot of expertise in system identification in general and, since more recently, in the identification of linear time-varying systems. In this master thesis, you will learn:

• the basic concepts of time-variant dynamics (understand its spectral response, describe its dynamic behavior mathematically),
• to estimate non-parametrically time-varying systems via the indirect model
• to develop new estimation algorithms, analyze their stochastic properties, and adapt them to be applicable to real-life cases.
• to design and perform experiments on time-varying systems, for example, an electronic circuit with time-varying resonance frequency and quality factor.

## References

1. E. Louarroudi, R. Pintelon, and J. Lataire (2012). Nonparametric tracking of the time-varying dynamics of weakly nonlinear periodically time-varying systems using periodic inputs. IEEE Trans. Instrum. and Meas., vol. 61, no. 5, pp. 1384-1394.
2. R. Pintelon, E. Louarroudi, and J. Lataire (2015). Nonparametric time-variant frequency response function estimates using arbitrary excitations, Automatica, vol. 51, no. 1, pp. 308-317.

2019