Wiener-Hammerstein System Identification applied to bioimpedance spectroscopy: Computationally fast model configuration

Research output: ThesisPhD Thesis

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Abstract

A popular class of nonlinear systems is the block-oriented structure. It is composed of series and/or parallel connections of both static nonlinearity (N)and linear dynamical blocks (L). The Wiener-Hammerstein (W-H) system is an L-N-L structure of two dynamical systems, separated by a nonlinearity. This thesis aims for a fast identification technique of the W-H system while keeping it sufficiently noise robust. Indeed, it is a computationally very intensive task ifthere is no prior information about the system structure to obtain the best model configuration among all possible L-N-L structures.
The W-H structure can be linearised by the Best Linear Approximation (BLA)which linearises the nonlinearity. It converts the whole system into a single cascaded transfer function and is often described by an Auto-Regressive-Moving-Average (ARMA) model. The AR-part and MA-part are the numerators of m zero(s) and the denominators of n pole(s) respectively. As two Linear Time Invariant (LTI) blocks are cascaded into the BLA, any pole/zero can be either from the first dynamic or the second dynamic. The standard technique makes 2m+n possible LTI configurations, proceeded by an estimate of the nonlinearity. The considered configuration and the nonlinearity should be further optimized by any nonlinear curve fitting algorithm to minimize the output error. But, the optimization of the 2m+n initialized model is a computationally intensive task. In this thesis, we propose a Spearman correlation based technique to speed-up the identification procedure significantly. The Spearman correlation measures the existence of a monotonic relationship between two random variables. Let, u(t) and y(t) be the measurable input and output signals respectively. Assume a guess of the LTI blocks from the BLA areG1andG2. Thus, the non-measurable internal signals are p=G1(u) and q=G−12(y). If there is a monotonic nonlinearity between p and q, the Spearman Correlation will be high. The selection of high Spearman correlated internal signals is done by its comparison to the standard error of the Spearman correlation estimation. Only selected models (with high Spearman correlation) require further optimization. It reduces the computational time from hours to minutes on SYSID’09 Benchmark data. The main question is whether configurations corresponding to a high Spearman correlation perform best after final optimization. Although an affirmative answer, its performance deteriorates with the noise level.
The random forest can further speedup the computation while remaining sufficiently noise robust. The set of the pole-zero from the BLA is the main feature set. Each possible model configuration is a tree in the forest. To validate each ‘Tree’, a training set and validation set are composed by partitioning the measured time series. The BLA and nonlinearity is estimated on the training sub-segment while validated on the validation segment by computing its Prediction Error (PE). After growing a user-defined number of trees in the Forest, half of the trees are removed, performing less than the median PE. If there is a B number of trees in the forest then this iterative procedure converges in terms of the PE which requires log2(B) + 1 iterations. For instance on the Benchmark data, it suffices to optimize only the best model in the forest in 99% of the trails.
Finally, the proposed theory is tested and assessed on a real-life electrosurgery application. The electrosurgery uses electric current to heat, coagulate and ligate tissue. The current flow induces a voltage due to the bio-impedance of the biological tissue. The mathematical formulation can answer the fundamental questions about bio-impedance in terms of time variation, diffusion, and nonlinearities. The parametric study of the W-H system can differentiate among femoral, mesentery and renal tissues. Although the nonlinearity is weak, the goodness of fit improves by 5.85% using a W-H system compared to a linear system. The predicted output by the W-H system is 99.86% accurate after model selection.
Original languageEnglish
QualificationDoctor in Medical Sciences
Awarding Institution
  • Vrije Universiteit Brussel
Supervisors/Advisors
  • Barbé, Kurt, Supervisor
Award date21 Sep 2022
Publication statusPublished - 2022

Keywords

  • Wiener-Hammerstein System Identification
  • spectroscopy

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