Tom Oomen

Research on Numerically Reliable Identification of Complex Systems

We envisage that next-generation motion systems are lightweight. Consequently, these will exhibit dominant lightly damped flexible dynamics. In the future, we will actively compensate these dynamics through many actuators and sensors. In addition, a distinction will be made between measured variables and performance variables, as is explained on the inferential control page. As is argued in Oomen et al., 2014, model-based control is essential to systematically design controllers for such highly complex systems. The main challenge lies in accurately identifying the relevant models of such systems, which have very high order and large input-output dimensionality.

Identification of models for mechanical systems is often done in the frequency domain, since this enables an efficient data reduction, combination of experiments for multivariable systems, visual inspection of the model fit, and control-relevant modeling.

A main challenge lies in the numerically reliable identification of such models. Already for the two input-two output case reported in Oomen et al., 2014, traditional approaches using monomial basis functions often lead to a numerical breakdown (= red text in matlab). An initial solution to address this problem is to use so-called orthonormal polynomials with respect to a data-based discrete inner product. These orthonormal polynomials essentially obtain condition number kappa = 1, i.e., optimal conditioning. The comparison of traditional monomial basis with the new orthonormal polynomials is reported in

Besides the above developments to enhance numerical conditioning, we have also investigated enhancements of the underlying algorithm. Interestingly, instrumental variable-based algorithms turn out to have significantly better convergence properties compared to the Sanathanan-Koerner iteration that is used in the references above. The main drawback of these instrumental variable-based algorithms is twofold. First, the conditioning is approximately quadratically worse (!) compared to the Sanathanan-Koerner algorithm. Second, the orthonormal polynomials with respect to a data-based discrete inner product that lead to kappa = 1 cannot be used due to the lack of an appropriate inner product. We have recently developed a new framework that we call bi-orthonormal polynomials with respect to a data-based bilinear form. Initial steps in this direction have been reported in

In recent years, we have further developed the theory. The underlying fundamental theory is presented in the following two identical documents

Application of the developed theory to instrumental variable system identification, requiring a orthonormal vector extension of the fundamental theory mentioned above, and experimental validation is presented in the following two identical documents

Recently, we have benchmarked the developed approach and compared it to alternative approaches in the literature. In addition, we have made some interesting extensions to pre-existing approaches. The results are documented in


The success of all the above work is due to the hard and excellent work of many people involved in this research, including

  • Active researchers at TU/e-ME-CST: Robbert Voorhoeve, Maarten Steinbuch

  • Previous reseachers at TU/e-ME-CST: Okko Bosgra, Robbert van Herpen

  • Industrial collaborators from Philips/ASML: Marc van de Wal, Wouter Aangenent
    and many others

Note that all figures shown on this page can be found in the mentioned papers. Please follow the guidelines regarding copyright and references when citing these.