Synthesis of Programmed Motion Based on Special Optimal Control

A method is proposed for the synthesis of a closed-loop system with controls that ensure the movement of an object with minimal deviations from a given trajectory of the output coordinate and its higher derivatives and a transition to this set. To solve the problem, the Pontryagin maximum principle is used to study special situations without analysis of auxiliary variables, supplemented by the apparatus of general position conditions for nonlinear systems in an extended coordinate space, taking into account the object, a functional that is nonlinear regarding deviations of the output coordinate and the explicit occurrence of time. The combined use of these methods allows us, firstly, to find special trajectories of coordinates that are higher derivatives of the output coordinate, and after excluding time, a special phase trajectory is found, which is a switching line for reaching the final state, a given programmed motion along which in a closed system is carried out by special control. Secondly, access to a special phase trajectory from the initial state is carried out for linear objects by relay control, and for nonlinear objects, under certain boundary conditions, relay control is supplemented by a special control of the speed problem. Examples of control of programmed motion with oscillatory and aperiodic processes of a given duration for linear and nonlinear objects are given. Taking into account the nature of equilibrium states, determined by the methods of the qualitative theory of differential equations, and restrictions on control and coordinates, topologies of trajectories are obtained for the implementation of a continuous special control or sliding mode. New algorithms and structures of control systems are obtained. The results are accompanied by modeling, illustrating the effectiveness of algorithms and structures of control systems according to the proposed synthesis method and confirming analytical materials. The results of the work can be used to control linear and nonlinear objects in mechatronics, robotics, thermal processes and other industries.

1. Filimonov A. B., Filimonov N. B. About the problem of synthesis of coordinating systems of automatic control, Izvestiya SFedU. Technical Science, 2012, no. 3 (128), pp. 172—180 (in Russian).

2. Boychuk L. M. Method of structural synthesis of nonlinear automatic control systems, Moscow, Energiya, 1971, 112 p. (in Russian).

3. Khoroshavin V. S., Grudinin V. S. Optimal programmed motion with variable control time, Radio Industry, 2020, vol. 30, no. 3, pp. 40—49, DOI: 10.21778/2413-9599-2020-30-3-40-49 (in Russian).

4. Ruderman M., Iwasaki M., Chen W. H. Motion-control techniques of today and tomorrow: a review and discussion of the challenges of controlled motion, IEEE Industrial Electronics Magazine, 2020, vol. 14, no. 1, рр. 41—55.

5. Pleshivtseva Yu. E., Dyakonov A. I., Popov A. V. Model two-dimensional problems of optimal control according to typical criteria of quality control of temperature modes of induction heating, Theoretical and Applied Aspects of Modern Science, 2015, no. 9-2, pp. 94—104 (in Russian).

6. Mokrushin S. A., Khoroshavin V. S., Okhapkin S. I., Zotov A. V., Grudinin V. S. Analysis of controllability and stability of an approximate model of heat transfer in an autoclave, Bulletin of the Mordovian University, 2018, vol. 28, no. 3, pp. 416—428, DOI: 10.15507/0236-2910.028.201803.416-428 (in Russian).

7. Kolesnikov A. A. Sequential optimization of nonlinear aggregated control systems, Moscow, Energoatomizdat, 1987, 160 p. (in Russian).

8. Pontryagin L. S., Boltyanskiy V. G., Gamkrelidze R. V., Mishchenko E. F. Mathematical theory of optimal processes, Moscow, Nauka, 1969, 384 p. (in Russian).

9. Oleinikov V. A. Optimal control of technological processes in the oil and gas industry, Leningrad, Nedra, 1982, 216 p. (in Russian).

10. Khoroshavin V. S., Zotov A. V. Special optimal control of nonlinear objects, Kirov, Scientific Publishing house of VyatSU, 2019, 219 p. (in Russian).

11. Wonham W. M., Johnson C. D. Optimal bang-bang control with quadratic performance index, Journal of Basic Engineering, 1964, vol. 86, no.1, pp. 107—115.

12. Clements D. J., Anderson B. D. Singular Optimal Control: The Linear-Quadratic Problem, Springer-Verlag, New York, 1978.

13. Khoroshavin V. S. Comparison of algorithms for controlling the thermal process in terms of speed and minimum resources, Izvestiya Tula State University. Technical Science, 2020, no. 7, pp. 211—216 (in Russian).

14. Bautin N. N., Leontovich E. A. Methods and methods of qualitative research of dynamical systems on a plane, Moscow, Nauka, 1976, 496 p. (in Russian).

15. Gutierrez Soto M., Adeli H. Recent advances in control algorithms for smart structures and machines, Expert Systems, 2017, vol. 34, no. 2, p. e12205.

16. Chernykh I. V. System of numerical and mathematical modeling MatLab. Simulink dynamic systems modeling system, Electronic Resource, URL: http://bourabai.kz/cm/simulink.htm (date of access: 25.02.2021) (in Russian).

17. Tabunshchikov Yu. A., Brodach M. M. Experimental study of optimal control of energy consumption, AVOK, 2006, no. 1, pp. 32—36 (in Russian).

18. Panferov V. I., Anisimova E. Yu., Nagornaya A. N. On optimal control of the thermal regime of buildings, Bulletin of the South Ural State University. Series: Energy, 2007, no. 20 (92), pp. 3—9 (in Russian).

19. Biyik E., Kahraman A. A. Predictive control strategy for optimal management of peak load, thermal comfort, energy storage and renewables in multi-zone buildings, Journal of building engineering, 2019, vol. 25, p. 100826.

20. Executive mechanisms: purpose, classification, design features, Electronic Resource, URL: https://webkonspect.com/?room=profile&id=4999&labelid=39160 (date of access: 25.02.2021) (in Russian).

21. Technical characteristics of control valves, Electronic Resource, URL: http://www.ktto.com.ua/kharakteristiki/krm (date of access: 25.02.2021) (in Russian).