A modified H2 algorithm for improved frequency response function and nonlinear parameter estimation

H1, H2 and Hv are the most commonly used algorithms for frequency response function estimation. H1 and Hv can be defined for multiple input–multiple output systems, whereas the H2 estimator requires that the number of inputs equals the number of outputs for a unique solution. Otherwise, a non-square matrix needs to be inverted leading to numerical problems. These numerical problems can affect the Nonlinear Identification through Feedback of the Outputs (NIFO) nonlinear parameter estimation algorithm in which nonlinearities are treated as internal feedback forces (or inputs), which can lead to more inputs than outputs. In this paper, a modified form of the H2 algorithm is presented that enables accurate estimates using NIFO. The modification to the H2 algorithm is the addition of a correlated output, based on the nature of the nonlinearity in the system being identified, in order to make the matrix to be inverted square. Analytical models with multiple nonlinearities are used to show that this modification to H2 leads to accurate estimates that are robust to noise on both the input and the output and that the modified algorithm is more robust to simultaneous noise on the input and output measurements than the H1 and the Hv algorithms. Mathematical reasoning is used to explain the greater robustness of the modified H2 algorithm over the traditional H1 algorithm.