Some Feasibility Problems in the Exact Solution of Сontrol Exercises

There is studied the mathematical foundations of the synthesis methodology in the engineering interpretation of a number of popular feedback control systems and the reasons for the impracticability of the results due to the appearance in the synthesis equations of pure differentiation operators and sources of various types of roughness violation. The global reason for the increasingly accelerated divergence of control theory from practice is associated with the impact on creative thinking such as mutation, incompatibility, randomness, fuzziness, asymmetry which underlies the evolution of synergetic systems. Both the "methodological crisis" and a number of seemingly insignificant engineering inconsistencies lead to a decrease in the planned efficiency of the developed control systems. There is a tendency to solve this practical problem through its excessive mathematization. As a result, there is nonsense — "the more mathematics, the worse", which leads to a "mathematical labyrinth", to exit from which the mathematical apparatus becomes more and more complicated until the creation of a new theory. It is shown that the use of even "correct" mathematical relations, which are the basis of the synthesis method, often leads to a violation of feasibility and rudeness. It is cited that the neglect of important poorly formalized technical indicators and the conditions of rudeness (robustness) when setting the problem does not allow us to obtain a constructive solution and is one of the main reasons for the discrepancy between theoretical results and practice. A number of popular directions of the classical theory of feedback control are considered: an inverse approach-compensation method, which forms the basis for constructing astatic, invariant, robust and other compensation systems; synthesis methods for systems with a finite settling time; assessment and control methods based on the concept of " inverse dynamics problems"; high gain limit systems. Violation of various types of feasibility and rudeness is demonstrated by specific examples tested on Matlab / Simulink. Computer research has made it possible to draw a number of positive conclusions that have important applied value.

1. Filimonov N. B. Methodological Crisis of the "All Winning Mathematization" of the Modern Control Theory, Mekhatronika, Avtomatizatsiya, Upravlenie, 2016, vol. 17, no. 5, pp. 291—301 (in Russian).

2. Vladimirov I. Q., Kurdyukov A. P., Semyonov A. V. Anisotropy of signals and entropy of linear stationary systems, Reports of R AS, 1995, vol. 342, no. 3, pp. 87—96 (in Russian).

3. Andronov А. А., Pontryagin L. S. Rough systems, Reports of USSR’s AOS, 1937, vol. 14, no. 5, pp. 247—250 (in Russian).

4. Andronov А. А., Vitt А. А., Khaykin S. E. Oscillation theory, Moscow, Nauka, 1981, pp. 341—359 (in Russian).

5. Aleksandrov А. G. Synthesis of regulators of multidimensional systems, Moscow, Mashinostroyeniye, 1986, 272 p. (in Russian).

6. Doyle J. C., Stein G. Multivariable feedback Design: Concepts for a Classical Modern Synthesis, IEEE Traus. Auto. Control, 1981, vol. AC26, no. 1.

7. Petrov B. N. About feasibility of invariance conditions. Book of The theory of invariance and its application in automatic devices, Proceedings of the 1st A ll-Union Conference on Invariance Theory, Kiev, 16—20 October, 1958, Мoscow, Publishing House of USSR’s AoS. 1959, pp. 59—80 (in Russian).

8. Ushakovskaya Y. D. Synergetics and causes of the evolution of the universe, available at: http:/spkurdyumov.narod.ru/Ushakovskaya.htm (in Russian).

9. Krutko P. D. Inverse problems of dynamics in the theory of automatic control, Мoscow, Мashinostroyeniye, 2004, 576 p. (in Russian).

10. Utkin V. I. Sliding Modes in Optimization and Control Problems, Springer Verlag, New York, 1992, 387 p.

11. DeCarlo R. A., Zak S. H., Matthews G. P. Variable structure control of nonlinear multivariable systems: a tutorial, Journal IEEE, 1988, 76 (3), pp. 212—232.

12. Yemelyanov S. V., Korovin S. K. New types of feedback: Control in conditions of uncertainty, Мoscow, Nauka,1997, 352 p. (in Russian).

13. Rustamov G. A., Namazov M. V., Gasimov A. Y., Rustamov R. G. Control of dynamic objects in the conditions of uncertainty in a point sliding mode, Mekhatronika, Avtomatizasiya, Upravleniye, 2019, vol. 20, no. 12, pp. 714—722 (in Russian).

14. Goodwin G. C., Graebe S. F., Salgado M. E. ControlSystem Design, Prentice Hall, Upper Saddle River, New Jersey 07458, 2002, pp. 908.

15. Solodovnikov V. V., Filimonov N. B. Dynamic Quality of Automatic Control Systems, Moscow, Publishing house of BMSTU, 1987, 84 р. (in Russian).

16. Filimonov A. B., Filimonov N. B. Control of Zeros and Poles in Problems of Synthesis of Regulation Systems. Part I. Compensation Approach, Mekhatronika, Avtomatizatsiya, Upravlenie, 2020, vol. 21, no. 8, pp. 453—460.

17. Stopakevich А. А. Design of robust controllers with a large delay, East European Journal of Advanced Technology, 2016, vol. 1, no. 2(79), pp. 48—56 (in Russian).

18. Rustamov Q. A. Robust Control Design For Uncertain Objects with Time Delay on the State, Automatic Control and Computer Sciences, vol. 50, no. 3, pp. 133—140.

19. Krasovskiy N. N. Motion control theory, Mockow, Nauka, 1968, pp. 121—230 (in Russian).

20. Gavrilyako V. M., Korobkov V. I., Sklyar G. M. The synthesis of limited control of dynamic systems in the whole space using the controllability function, Avtomatika i Telemekhanika, 1986, no. 11, pp. 30—36 (in Russian).

21. Korobkov V. I. General approach to solving the problem of synthesis of bounded controls in the controllability problem, Math collection, 1979, vol. 109, no. 4, pp. 582—606 (in Russian).

22. Korobkov V. I. Solving the synthesis problem using the controllability function, Reports of USSR’s AoS, 1979, vol. 248, no. 5, pp. 1051—1055 (in Russian).

23. Korobkov V. I., Sklyar G. M. Solution of the synthesis problem using the controllability functional for systems in infinitedimensional spaces, Reports of USSR’s AOS, 1983, no. 5, pp. 11—14 (in Russian).

24. Kritun V. I. Stabilization of the output signals of linear controlled systems in a finite time, Avtomatika i Telemekhanika, 1988, no. 10, pp. 63—67 (in Russian).

25. Rustamov G. A., Namazov M. B., Misrikhanov L. M. Rudeness violation in automatic control systems, The 1st International Conference on Control and Optimization with Industrial Applikations (COLA — 2005), Baku, A zerbaijan.

26. Vostrikov A. S. The problem of the synthesis of regulators for automation systems: state and prospects, Avtometrija, 2010, vol. 46, no. 2, pp. 3—19 (in Russian).

27. Vostrikov A. S., Francuzova A. G. Synthesis of PID regulators for nonlinear nonstationary objects, Avtometrija, 2015, vol. 51, no. 5, pp. 53—60 (in Russian).

28. Vostrikov A. S. The highest derivative and large coefficients in controlling the linear non-stationary objects, Mekhatronika, Avtomatizatsiya, Upravlenie, 2008, no. 5, pp. 2—7 (in Russian).

29. Но Н. F., Wong Y. K., Rad A. B. Adaptive Fuzzy Sliding Mode Control Design: Lyapunov Approach, Proc. IEEE International Conference on Fuzzy System, 2001, pp. 6—11.

30. Lee H., Tomizuka M. Adaptive Traction Control. University of California, Berkeley, Depertament of Mechanical Engineering, September, 1995, pp. 95—32.

31. Meerov M. V. Synthesis of structures of automatic control systems with high precision, Moscow, Nauka, 1967, 424 p. (in Russian).

32. Filimonov A. B., Filimonov N. B. Large gain coefficients method and effect of the motion localization in the problem of synthesis of the automatic control systems, Mekhatronika, Avtomatizatsiya, Upravlenie, 2009, vol. 95. no. 2, pp. 2—10 (in Russian).

33. Rustamov G. A. K∞-robust control systems, Mekhatronika, Avtomatizatsiya, Upravlenie, 2015, vol. 16, no. 7, pp. 435—442 (in Russian).

34. Rustamov G. A. Analysis of methods of design of robust control systems with high gain coefficient, Mekhatronika, Avtomatizatsiya, Upravlenie, 2018, vol. 19, no. 6, pp. 363—373 (in Russian).

35. Tararikyin S. V., Apolonskiy V. V., Terekhov A. I. Investigation of the inf luence of the structure and parameters of polynomial input-output controllers on the robust properties of synthesized systems, Mekhatronika, Avtomatizatsiya, Upravleniye, 2013, vol. 152, no. 11, pp. 2—9 (in Russian).

36. Furtat I. B., Nekhoroshikh A. N. Robust control of linear multi-agent systems using left differences to estimate derivatives, UBS, 2017, iss. 65, pp. 41—49 (in Russian).