Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)


For citation:

Ivanov V. M. Estimation of Quality of Non-Stationary Systems on the Return Frequency Characteristic Plane. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2019, vol. 19, iss. 2, pp. 207-216. DOI: 10.18500/1816-9791-2019-19-2-207-216, EDN: QOSZHK

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
28.05.2019
Full text:
(downloads: 145)
Language: 
Russian
Heading: 
Article type: 
Article
UDC: 
681.5.037
EDN: 
QOSZHK

Estimation of Quality of Non-Stationary Systems on the Return Frequency Characteristic Plane

Autors: 
Ivanov Vladimir M., Ulyanovsk State Technical University
Abstract: 

Direct quality parameters, such as time of regulation, overshoot, damping decrement are widely used for estimation of linear systems quality. Alongside with direct parameters indirect estimations of quality are used. One of such quality parameters for nonlinear systems is the degree of stability or response speed. A number of research studies show that properties of nonlinear systems investigation is reduced to the analysis of absolute stability of processes. The study considers structural representation of non-stationary linearized system, which allows to present additional evidence the statement of problem and to prove transition to the system with the generalized non-linearity. In general, the non-stationary parametrical characteristic caused by a multiplying part, can be present in four quadrants. However, in most practical problems the characteristic of the multiplying part can be presented as two quadrants, because one of the variables, describing the current value of the parameter is represented by a positive value. The basic features of the block diagram are defined by the fact that a change of $\Delta k$ factor is equivalent to $\Delta kx_{0}$ revolting influence caused by entry conditions. Non-stationary properties of the interfaced contour define the free process characterized by transition from the initial state to a steady status of the balance. Entry conditions are defined by an initial contour and are equivalent to the input impact. Thus, the system with a multiplying part can be generally presented as a system with the generalized non-linearity. We studied return the frequency characteristic plane that allows to simplify analytical problems of the systems with two-dimensional non-linearity of multiplying parts. Practical applications demonstrate the algorithm of calculation and analysis of the frequency characteristics for the purpose of their graphic representation and definition of stability.

References: 
  1. Naumov B. N., Tsypkin Ya. Z. Frequency criterion for process absolute stability in non-linear automatic control system. Avtomat. i Telemekh. [Automation and Remote Control], 1964, vol. 25, iss. 6, pp. 852–867 (in Russian).
  2. Naumov B. N. An investigation of absolute stability of the equilibrium state in nonlinear automatic control systems by means of logarithmic frequency characteristics. Autom. Remote Control, 1965, vol. 26, iss. 4, pp. 593–601.
  3. Lur’e A. I., Postnikov V. N. K teorii ustoichivosti reguliruemykh sistem [To the theory of stability of adjustable systems]. Prikladnaya matematika i mekhanika [Journal of Applied Mathematics and Mechanics], 1944, vol. 8, no. 3, pp. 246–248 (in Russian).
  4. Letov A. M. Ustoichivost’ nelineinykh reguliruemykh system [Stability of nonlinear adjustable systems]. Moscow, Gosenergoizdat, 1955. 312 p. (in Russian).
  5. Ajzerman M. A., Gantmaher F. R. Absolyutnaya ustoichivost’ reguliruemykh system [Absolute stability of adjustable systems]. Moscow, Izd-vo AN SSSR, 1963. 140 p. (in Russian).
  6. Megretski A., Rantzer A. System analysis via integral quadratic constraints. IEEE Trans. Automat. Contr. 1997, vol. 42, no. 6, pp. 819–830. DOI: https://doi.org/10.1109/9.587335
  7. Liu Z., Lü S., Zhong S., Ye M. Improved Robust Stability Criteria of Uncertain Neutral Systems with Mixed Delays. Abstr. Appl. Anal., 2009, vol. 2009, pp. 1–18. DOI: https://doi.org/10.1155/2009/294845
  8. Wu M., He Y., She J.-H. Stability Analysis, Robust Control of Time-Delay Systems. Beijing, Science Press; London, Springer, 2010. 335 p.
  9. Shatyrko A., Khusainov D. On the Interval Stability of Weak-Nonlinear Control Systems with Aftereffect. Sci. World J., 2016, vol. 2016, pp. 1–8. DOI: https://doi.org/10.1155/2016/6490826
  10. Popov V. M. On absolute stability of non-linear automatic control systems. Avtomat. i Telemekh. [Automation and Remote Control], 1961, vol. 22, iss. 8, pp. 961–979 (in Russian).
  11. Popov V. M. Giperustoichivost’ avtomaticheskikh system [Hyperstability of Automatic Systems]. Moscow, Nauka, 1970. 454 p. (in Russian).
  12. Haddad W. M., Kapila V. Absolute stability criteria for multiple slope-restricted monotonic nonlinearities. Proceedings of American Control Conference, 1994, vol. 1, pp. 1020–1021. DOI: https://doi.org/10.1109/ACC.1994.751901
  13. D’yakonov V. MATLAB 6: uchebnyj kurs [MATLAB 6: training course]. St. Petersburg, Piter, 2001. 592 p. (in Russian).
  14. D’yakonov V. P. Mathematica 5.1/5.2/6. Programmirovanie i matematicheskie vychisleniya [Mathematica 5.1/5.2/6. Programming and mathematical calculations]. Moscow, DMK Press, 2008. 576 p. (in Russian).
  15. Sledyaschie privody : v 2 t. [The watching drives : in 2 vols.]. Vol. 1 / ed. by V. K. Chemodanov. Moscow, Energiya, 1976. 480 p. (in Russian).
  16. Besekerskij V. A., Popov E. P. Teoriya sistem avtomaticheskogo regulirovaniya [Theory of systems of automatic control]. Moscow, Nauka, 1979. 768 p. (in Russian).
  17. Tsypkin Ya. Z. Osnovy teorii avtomaticheskikh system [Fundamentals of the theory of automatic systems]. Moscow, Nauka, 1977. 560 p. (in Russian).
  18. Voronov A. A. Ustoichivost’, upravlyaemost’, nablyudaemost’ [Stability, controllability, observability]. Moscow, Nauka, 1979. 336 p. (in Russian).
Received: 
05.09.2018
Accepted: 
09.02.2019
Published: 
28.05.2019
Short text (in English):
(downloads: 85)