Methods for recovery input signals of nonlinear nonstationary dynamic systems

The problem of input signals recovery is one of the key problems in many branches of science and technology: in measurement technology for dynamic measurements; in control tasks, where the control is based on the input signal (stabilization task); in tasks of image restoration and filtering, etc., which determines the relevance of the development of methods of input signal recovery. The relevance of development of input signal recovery methods is increasing every year and becomes the most pronounced at operation of measuring and control systems in extreme and harsh operating conditions. Non-stationarity and nonlinearity in these conditions are the most pronounced, accounting of which is a prerequisite for the creation of new and improvement of existing information-measuring and control systems. The paper proposes methods for determining the input signal of nonlinear dynamic systems described by the Volterra functional series. The methods are based on the solving of a nonlinear integral equation, which is defined by a finite segment of the Volterra series. The input signal recovery is carried out under the assumption that the integral transforms of Volterra kernels possess factorization based on Borel's theorem, which leads to a nonlinear algebraic equation. Recovery methods of continuous nonlinear dynamic systems described by a finite Volterra series and their discrete analog are considered. When recovering the input signal, for continuous systems the integral Laplace transform is applied, for discrete systems the Z-transform is applied. An example of a mathematical solution to the problem of restoring a discrete one-dimensional signal is given, illustrating the efficiency and effectiveness of the developed methods. The results of studies on restoring signals of nonlinear dynamic systems will be useful to specialists engaged in theoretical research and mathematical modeling in the field of digital signal processing and vector analysis of electrical circuits.

1. Granovskii V. A. Dynamic measurements: theory and metrological assurance at yesterday and tomorrow. Sensors & Systems, (3(201)), 57–72 (2016). (In Russ.) https://www.elibrary.ru/xhfkcr

2. Sizikov V. S. Direct and inverse tasks of image reconstruction, spectroscopy and tomography with MatLab. The training manual. Lan’, St. Petersburg (2017). (In Russ.) https://elibrary.ru/ytyjex

3. Kusaykin D. V., Porshnev S. V., Safiullin N. T. Methods of recovery of discrete signals. Fundamentals of theory, software tools, accuracy analysis. Lan’, St. Petersburg (2021). (In Russ.)

4. Khurgin Ya. I., Yakovlev V. P. Finite functions in physics and engineering. LIBROKOM, Moskva (2019). (In Russ.)

5. Boikov, I. V., Krivulin, N. P. Analytical and numerical methods for the identification of dynamical systems. Monograph. PGU, Penza (2016). (In Russ.)

6. Boikov I. V., Krivulin N. P., Abramov S. V., Malanin V. P. Kikot V. V. Recovery of the input signals of eddy-current displacement transducers under thermal-shock actions. Measurement Techniques, 61(11), 1118–1125 (2019). https://doi.org/10.1007/s11018-019-01558-5

7. Shcherbakov M. A. Iteration method of optimal nonlinear image filtering. University proceedings. Volga region. Technical sciences, (4(20)), 43–56 (2011). (In Russ.) https://elibrary.ru/ovyzzp

8. Bychkov Yu. A., Solov’ev E. B., Shcherbakov S. V. Continuous and discrete nonlinear models of dynamical systems. Lan’, St. Petersburg (2018). (In Russ.)

9. Krasnov M. L., Kiselev A. I., Makarenko G. I. Operational calculus. Theory of stability. Tasks and examples with detailed solutions. Lenand, Moscow (2018). (In Russ.)

10. Ganicheva A. V. Fundamentals of the theory of the function of a complex variable. Operational calculus. Lan’, St. Petersburg (2023). (In Russ.)

11. Boikov I. V., Krivulin N. P. Identification of parameters of nonlinear dynamical systems simulated by the Volterra polynomials. Journal of Applied and Industrial Mathematics, 12(2), 220–233 (2018). https://doi.org/10.1134/S1990478918020035

12. Boikov I. V., Krivulin N. P. Recovery of characteristics of non–stationary dynamic systems from three test signals. Measurement Techniques, 63(3), 158–165 (2020). https://doi.org/10.1007/s11018-020-01766-4

13. Tikhonov A. N., Arsenin V. Ya. Methods for solving incorrect problems. Nauka, Moscow (2022). (In Russ.)

14. Proskuryakov I. V. Collection of linear algebra problems. Lan’, St. Petersburg (2010). (In Russ.)