Every unfalsified model must include the most powerful unfalsified model

ELEC seminar room, Building K, 6th floor

Speaker: Vikas Mishra

Abstract: Exact model identification refers to the problem of identifying the true data generating model from an observed trajectory of the model. For an infinite noise-free time series, the problem has been tackled by Jan C. Willems by introducing the notion of the most powerful unfalsified model (MPUM). The MPUM is defined as the least complex linear time-invariant model explaining a given time series, wherein the complexity is defined by the pair (number of inputs, model order) in lexicographic manner. It is known that the MPUM always exists and is unique, and under certain assumptions it coincides with the data generating model. However, for a finite time series, there are two issues: (i) The MPUM may not exist or be non-unique, and (ii) The MPUM always coincides with a finite dimensional space corresponding to an autonomous model (model with no inputs) due to the above definition of complexity. In this talk, these issues are resolved by assuming that the number of inputs is a priori known and minimizing the complexity by minimizing the order of the model. First, necessary and sufficient conditions are established for the existence and uniqueness of the MPUM in this setting. Then, in case of non-uniqueness of unfalsified models, it is shown that the set of unfalsified models admits an affine structure. More specifically, every unfalsified model is a sum of the most powerful unfalsified model and an autonomous model of bounded order. Finally, numerical examples are discussed to illustrate the results.

Best Linear Approximation of Nonlinear Continuous-Time Systems Subject to Process Noise

ELEC seminar room

Speaker: Rik Pintelon

Abstract: In many engineering applications the level of nonlinear distortions in frequency response function (FRF) measurements is quantified using specially designed periodic excitation signals called random phase multisines and periodic noise. The technique is based on the concept of the best linear approximation (BLA) and it allows one to check the validity of the linear framework with a simple experiment. Although the classical BLA theory can handle measurement noise only, in most applications the noise generated by the system – called process noise – is the dominant noise source. Therefore, there is a need to extend the existing BLA theory to the process noise case. In this presentation we study in detail the impact of the process noise on the BLA of nonlinear continuous-time systems. It is shown that the existing nonparametric estimation methods for detecting and quantifying the level of nonlinear distortions in FRF measurements are still applicable in the presence of process noise. All results are also valid for discrete-time systems and for systems operating in closed loop.

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