Identification of the quadratic kernel of the Volterra equation for modeling non-linear dynamic systems

In the last two decades, integral models have been used to describe the dynamics of stationary nonlinear systems in which the terms of the Volterra series are the kernels. The most commonly used are the linear term (the impulse transition function depends on one variable) and the quadratic term (depending on two variables). An active experiment in which a special combination of rectangular pulses is fed to the input of the system is carried out to select two of its components in the output signal of the identified system - the output of the linear "subsystem" and the output of the "quadratic" subsystem. After isolating the output of the "quadratic" subsystem, the identification of the quadratic term of the Volterra series is reduced to solving a two-dimensional integral equation of the first kind. In the literature, inversion formulas are given in which the quadratic kernel function is obtained as a result of arithmetic operations with second-order derivatives of the output signal. Differentiation of functions is an incorrectly posed problem, when small errors in the assignment of a function (measurement noise) cause large errors in derivatives (especially in second-order derivatives). The paper proposes the use of smoothing cubic splines for stable calculation of derivatives. To calculate the mixed second-order derivative, a spline with two variables is built - a smoothing bicubic spline. The main problem that arises in practice when processing the data of a real experiment is the selection of a smoothing parameter, on the value of which a smoothing error of noisy data depends. As a rule, the value of the variance of the measurement noise is unknown in the experiment. Therefore, in this work, it is proposed to use an algorithm based on the L-curve method to select the smoothing parameter in the constructed splines (especially in the bicubic one), which does not require setting the variance of the measurement noise. The proposed identification algorithm has a high computational efficiency. The performed computational experiment showed a small methodical error (about 1%) and good resistance to noise in measurements of the output signals of the identified system. To reduce the random component of the identification error, it is proposed to use post-processing with a local-spatial compound filter.