Abstract
A popular class of nonlinear systems is the blockoriented structure. It is composed of series and/or parallel connections of both static nonlinearity (N)and linear dynamical blocks (L). The WienerHammerstein (WH) system is an LNL structure of two dynamical systems, separated by a nonlinearity. This thesis aims for a fast identification technique of the WH 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 LNL structures.
The WH 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 AutoRegressiveMovingAverage (ARMA) model. The ARpart and MApart 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 speedup 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 nonmeasurable 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 polezero 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 subsegment while validated on the validation segment by computing its Prediction Error (PE). After growing a userdefined 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 reallife electrosurgery application. The electrosurgery uses electric current to heat, coagulate and ligate tissue. The current flow induces a voltage due to the bioimpedance of the biological tissue. The mathematical formulation can answer the fundamental questions about bioimpedance in terms of time variation, diffusion, and nonlinearities. The parametric study of the WH system can differentiate among femoral, mesentery and renal tissues. Although the nonlinearity is weak, the goodness of fit improves by 5.85% using a WH system compared to a linear system. The predicted output by the WH system is 99.86% accurate after model selection.
The WH 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 AutoRegressiveMovingAverage (ARMA) model. The ARpart and MApart 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 speedup 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 nonmeasurable 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 polezero 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 subsegment while validated on the validation segment by computing its Prediction Error (PE). After growing a userdefined 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 reallife electrosurgery application. The electrosurgery uses electric current to heat, coagulate and ligate tissue. The current flow induces a voltage due to the bioimpedance of the biological tissue. The mathematical formulation can answer the fundamental questions about bioimpedance in terms of time variation, diffusion, and nonlinearities. The parametric study of the WH system can differentiate among femoral, mesentery and renal tissues. Although the nonlinearity is weak, the goodness of fit improves by 5.85% using a WH system compared to a linear system. The predicted output by the WH system is 99.86% accurate after model selection.
Original language  English 

Qualification  Doctor in Medical Sciences 
Awarding Institution 

Supervisors/Advisors 

Award date  21 Sep 2022 
Publication status  Published  2022 
Keywords
 WienerHammerstein System Identification
 spectroscopy