14. Nonparametric Estimation of Parameter-varying Dynamics


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

  • civil engineering structures (bridges, buildings, …) subject to increasing damage, or to varying loads
  • the impedance of metals, subject to (pit) corrosion,
  • aging/mortification in biological systems,
  • or systems with a varying set point (e.g. flight flutter analysis), or varying system (scheduling) parameters (e.g. pick-and-place robot with extendable gripper arm, a crane lifting a load, see the figure).

The common problem in these examples is that the system dynamics change during the measurements! For example, the oscillation frequency of a swinging mass is varying when it is lifted by a crane, viz.


In some of the examples the time-variation is due to a changing parameter, called scheduling parameter, that can be measured accurately. For example, the height and speed of the airplane in flight flutter, or the length of an extendable gripper arm, or the length of the rope of a crane (see the figure). In these cases, one speaks of parameter-variant dynamics rather than time-variant dynamics. A parameter-variant model is more general than a time-variant model because it can predict the response of the dynamical system for other trajectories of the scheduling parameter(s).


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- and parameter-varying systems. In this master thesis, you will learn:

  • ·the basic concepts of time- and parameter-variant dynamics,
  • to estimate non-parametrically time- and parameter-varying systems (understand its spectral response, describe its dynamic behavior mathematically, the state-of-the -art of the identification tools, ...)
  • 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 parameter-varying systems, for example, an electronic circuit with parameter-varying resonance frequency and quality factor.



  1.  J. Goos (2016). Modeling and Identification of LPV systems, PhD thesis, Vrije Universiteit Brussel, dept. ELEC.
  2.  J. Lataire, R. Pintelon, and E. Louarroudi (2012). Non-parametric estimate of the system function of a time-varying system, Automatica, vol. 48, no. 4, pp. 666-672.
  3. R. Tóth, P. S. C. Heuberger, and P. M. J. Van den Hof (2009). Asymptotically optimal orthonormal basis functions for LPV system identification, Automatica, vol. 45, no. 6, pp. 1359-1370
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