Initial Estimates for the LFR Nonlinear Model Structure via the Best Linear Approximation

In this paper, a novel method is proposed for the identification of a fairly general nonlinear structure called Linear Fractional Representation (LFR), also related to Lur'e type systems. It consists of one static nonlinearity (SNL) connected to the input and the output via a multiple-input-multiple-output (MIMO) linear time-invariant (LTI) block. The nonlinear LFR structure encompasses, e.g., Wiener-Hammerstein and nonlinear feedback models. The procedure starts from 2 state-space models corresponding to the best linear approximation at 2 input variance levels. The MIMO LTI block is estimated by exploiting the approximate structural relationships, taking the state transformations carefully into account. Using the measured input and output, the input and the output of the SNL block are then reconstructed, yielding a nonparametric estimate of the SNL, which is finally converted into a parametric estimate. In the whole procedure, the internal signals, the linear and the nonlinear part need not be known. No stability requirement was imposed on the linear models used. The functional form of the SNL is not needed to find the MIMO LTI block and the nonparametric SNL estimate.The simulation results, supporting the theory, show the superior quality of the obtained initial estimate of the LFR nonlinear model compared to both linear models. This shows that the method is a very promising approach in the field of block-oriented nonlinear modelling.