Control of Dynamic Objects in the Conditions of Uncertainty in the Point Sliding Mode

There is development of the well-known sliding mode, which in the classical formulation didn’t find the development to be applied to control systems discussed. Alternatively, there is method of organizing one of the uniformity of the sliding mode called the "point sliding mode" proposed. The distinctive feature of this mode is that here the control gaps occur at time-equal points of the switching line (hyperplane) which allows the origin of coordinates for a finite number of switches. The possibility of changing the time interval between these points makes it possible to obtain various modes: a finite mode, in which a given point is reached from any initial state in one switch, and in this mode the switch line is "isochronous"; point sliding mode in which a given point is reached in a finite number of switchings; limit mode, when the length of time intervals tend to zero, and the switching frequency to infinity. Considering this feature the concept of "degree of slip" is introduced. It is shown that in the case of forced movement in the SPS, a sliding motion is observed, which does not allow for ensuring invariance with respect to external disturbances. There are two ways to eliminate the forced component of the movement offered. One of the advantages of using a point sliding mode is that, in order to improve performance, it is not necessary to use a boundary layer, which is realized by entering various logical conditions into the control algorithm. The practical significance of a point sliding mode lies in the fact that, with a small switching frequency, it is possible to maintain the quality indices of an undefined object within an acceptable interval. The studies were conducted for onedimensional second-order linear systems (SISO). Results can be generalized for higher order multidimensional systems. Solution of model problems on MATLAB / Simulink allows us to make a number of positive conclusions that are of great practical importance in terms of expanding the area of use of skipping modes, especially in relation to the management of undefined objects.

1. Filippov A. F. Application of differential equations with a discontinuous right side to nonlinear problems of automatic control, Trudi I Kongressa IFAC, Мoscow, Izd-vo AN SSSR, 1961 (in Russian).

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

3. Dekarlo R. A., Jak S. H., Mattews G. P. Control with variable structure of nonlinear multidimensional systems: An overview, 1988, ТIIEP, vol. 76, no. 3, pp. 4—27 (in Russian).

4. Levant A. Universal SISO sliding-mode controllers with finit-time convergence, IEEE Transactions on Automatic Control, 2001: 46(9), pp. 1447—1451.

5. Levant A. Principles of 2-sliding mode design, Automatica (Journal of IFAC), 2007, vol. 43, iss. 4, pp. 576—586.

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

7. Lagrat I., Ouakka H., Boumhidi I. Fuzzy Sliding mode PI control for Nonlinear Systems, Proceedings of the 6th WSEAS International Conference on Simulation. Modelling and Optimization, Lisbon-Portugal, September 22—24, 2006, pp. 534—539.

8. Rustamov G. A., Abdullayeva A. T., Elchuyev I. A. Realization of sliding motion in a nonlinear stabilization system based on the method of the Lyapunov function, Automatic Control and Computer Sciences, 2009, vol. 43, no. 6, pp. 336—341.

9. Mamedov G. A., Rustamov G. A., Gasanov Z. R. Elimination of the sliding mode breakdown in forced motion of controlled objects, Automatic Control and Computer Sciences, 2009, vol. 43, no. 2, pp. 74—79.

10. Beyranovich V. A. Invariant automatic control systems with a relay amplifier, Dokladi TUSURa, 2010, no. 1(21), Ch. 1, pp. 70—73 (in Russian).

11. Klinachov N. V. Theory of automatic control systems, Uch. Metodicheskiy kompleks, 800 faylov, Chelabinsk, 2010—2013 (in Russian).

12. Meshanov А. S., Sultanova А. F. To the design of varieties of slip and control in case of failure invariance conditions, Vestnik Kazanskogo tekhnologicheskogo universiteta, 2016, vol. 19, no. 10, pp. 113—117 (in Russian).

13. Blekhman I. I. What is vibration? About vibration mechanics and vibration technology. Мoscow, 2017, 216 p. (in Russian).

14. Sofiyev A. E., Shauro V. S. The use of vibration control for chemical-technological objects. Software: theory and applications, 2004, pp. 475—490 (in Russian).

15. Filimonov А. B., Filimonov N. B. Меkhatronika, Avtomatizatsiya, Upravleniye, 2009, no. 2 (95), pp. 2—10 (in Russian).

16. Filimonov N. B. Mekhatronika, Avtomatizatsija, Upravlenie, 2016, vol. 17, no. 5, pp. 291—299 (in Russian).

17. Rustamov G. A. Mekhatronika, Avtomatizatsiya, Upravlenie, 2018, vol. 19, no. 6, pp. 363—373 (in Russian).

18. Filimoniv A. B., Filimonov N. B. Mekhatronika, Avtomatizatsiya, Upravlenie, 2014, no. 12 (165), pp. 3—10 (in Russian).

19. Micumaki Т. Modified optimal control system, Trudi I kong. IFAC, vol. 2. Мoscow, Publishing house of АN SSSR, 1960, pp. 608—623 (in Russian).

20. Vostrikov А. S., Francuzova G. А. Theory of automatic control, Novosibirsk, Publishing house of NGTU, 2003, 364 p. (in Russian).

21. Govorov А. А. Methods and means of designing regulators with enhanced functionality for continuous technological processes, Tula, 2002, 500 p. (in Russian).

22. Rustamov G. А., Farkhadov V. G., Rustamov R. G. Mekhatronika, Avtomatizatsiya, Upravleniye, no. 11, vol. 19. 2018, pp. 699—706 (in Russian).

23. Namazov M. Fuzzy Control and Sliding mode Fuzzy Control of DS motor, Journal of Engineerign and Natural Scinces Mühendislik ve Fen Bilikleri Dedgisi. Sigma 32, 2014, pp. 97—108.