Synthesis of smoothing cubic spline in non-parametric identification technical systems’ algorithm

Many technical systems are described by mathematical models in the form of an integral Voltaire equation of the first kind with a difference kernel. The nonparametric identification problem for such systems will be reduced to constructing an estimate of the impulse transition function of the identified system based on the measured (with noise) the input and output signals’ values. A rectangular signal of constant amplitude is applied to the input of the identified system at some point in time in some identification schemes that are widely used in practice. For such an input signal, the pulse transition function is defined as the first derivative of the system output signal. However, the derivative calculation belongs to the class of incorrectly posed problems, and an essential feature of this problem is the calculated derivative instability to the errors in recording the output signal. For stable computation, various algorithms for smoothing experimental data are used. The most effective of them are smoothing cubic splines. Boundary conditions are specified to uniquely determine the coefficients of these splines. Unfortunately, traditional natural boundary conditions (zero second derivatives of the spline) do not allow taking into account the features of the identification problem. Therefore, we propose a smoothing algorithm, based on smoothing cubic splines, which allows one taking into account information about the identifiable impulse transition function in sufficient (to increase the accuracy of identification) in this research paper. The studies show the effectiveness of the proposed smoothing algorithm and the entire identification procedure as a whole.