System identification based on characteristic curves: a mathematical connection between power series and Fourier analysis for first-order nonlinear systems

Abstract

Recently, the sinosoidal output response in power series (SORPS) formalism was presented for system identification and simulation. Based on the concept of characteristic curves (CCs), it establishes a mathematical connection between power series and Fourier series for a first-order nonlinear system (Gonzalez in Sci Rep 13:1955, 2023). However, the system identification procedure discussed there, based on fast Fourier transform (FFT), presents the limitations of requiring a sinusoidal single tone for the dynamical variable and equally spaced time steps for the input–output dataset (DS). These limitations are here addressed by introducing a different approach: we use a power series-based model (referred to as model 1) for system modeling instead of FFT, where two hyperparameters A^0" role="presentation"> and A^1" role="presentation"> are optimally defined depending on the DS. Subsequently, two additional models are obtained from parameters obtained in model 1: another power series-based model (model 2) and a Fourier analysis-based model (model 3). These models are useful for comparing parameters obtained from different DSs. Through an illustrative example, we show that while the predicted values from the models are the same due to a mathematical equivalence, the parameters obtained for each model vary to a greater or lesser extent depending on the DS used for system estimation. Hence, the parameters of the Fourier analysis-based model exhibit notably less variation compared to those of the power series-based model, highlighting the reliability of using the Fourier analysis-based model for comparing model parameters obtained from different DSs. Overall, this work expands the applicability of the SORPS formalism to system identification from arbitrary input–output data and represents a groundbreaking contribution relying on the concept of CCs, which can be straightforwardly applied to higher-order nonlinear systems. The method of CCs can be considered as complementary to the commonly used approach (such as NARMAX-models and sparse regression techniques) that emphasizes the estimation of the individual parameter values of the model. Instead, the CCs-based methods emphasize the computation of the CCs as a whole. CCs-based models present the advantages that the system identification is uniquely defined, and that it can be applied for any system without additional algebraic operations. Thus, the parsimonious principle defined by the NARMAX-philosophy is extended from the concept of a model with as few parameters as possible to the concept of finding the lowest model order that correctly describe the input–output data. This opens up a wide variety of potential applications, covering areas such as vibration analysis, structural dynamics, viscoelastic materials, design and modeling of nonlinear electric circuits, voltammetry techniques in electrochemistry, structural health monitoring, and fault diagnosis.