Generating initial estimates for Wiener-Hammerstein systems using phase coupled multisines

Block oriented nonlinear models capture the dynamics of a nonlinear system with linear dynamic sub-systems (L), the nonlinear behavior is modelled using static nonlinear sub-blocks (N). In this paper we study the generation of initial estimates for the linear dynamic blocks of a Wiener-Hammerstein system that has a cascaded LNL structure. While it is very easy to identify the product of the transfer functions of the first and last dynamic block using linear system identification methods, it turns out to be very difficult to split the global dynamics over these individual blocks. In this paper a method is proposed that allows the poles of the best linear approximation to be assigned to the first or second linear block. Once this split is made, it is shown in the literature that the remaining initialization problem can be solved much easier than the original one. The first step of the method is the design of a special random phase multisine excitation, using pair-wise coupled random phases. Next, a modfied best linear approximation will be estimated on a shifted frequency grid. It will be shown that this procedure shifts the poles and zeros of the first linear sub-block with a known frequency offset, while those of the second sub-block are not changed. The shifted poles and zeros result in a transfer function with complex coefficients that can be identified using a modified frequency domain estimation method. This results in a simple initialization method, based on a linear system identification step.