Research of K∞-Robust Systems with Constrained Control

At the present, there are no satisfactory engineering solutions related to the synthesis of robust regulators taking into account the constraints on control. In this connection, it is important to study the influence of saturation effect of the controller on the robust properties of systems. In this paper, this problem is considered in connection with K∞-robust control systems with a high gain. It is shown that in control limit systems, in particular, K∞-robust systems at the initial instant of time, the control assumes an excessively large value. This ensures the robustness of the dynamic mode. The reason for this feature is related to the fact that at the initial instant of time the initial conditions has a great importance. The main reason for the deterioration of robust properties is due to the tight control in the initial time interval. Provision of robustness of the static mode does not require great control efforts. For the first time, using computer graphics in a three-dimensional coordinate system, taking into account the time, a visual representation of the sections of phase trajectories pertaining to different types of movements is given: rapid motion from an arbitrary initial state that ensures hitting the imaging point into the degenerate trajectory; slow motion along this trajectory; steady-state motion within the specified accuracy. For the limit control systems, an integral robustness estimate is proposed, which consists in calculating the integral of the absolute value of the slow motion trajectory. This indicator characterizes the discrepancy (dispersion) of the real trajectory with respect to the limiting trajectory. The reliability of theoretical reasoning is confirmed by solving a model problem in the block-vision environment of Matlab/Simulink.

uncertainty, robust control system, Lyapunov function, high gain coefficient, attractor, constraints, saturation, estimation robustness

1. Filimonov A. B., Filimonov N. B. Metod bolshih kojefficientov usilenija i jeffekt lokalizacii dvizhenija v zadachah sinteza sistem avtomaticheskogo upravlenija (Large gain coefficients method and effect of the motion localization in the problem of synthesis of the automatic control system), Mekhatronika, Avtomatizatsiya, Upravlenie, 2009, no. 2, pp. 2—10 (in Russian).

2. Furtat I. B. Upravlenie s kompensaciej vozmushhenij (Control with compensation of disturbances), SPb, Publishing house of Universitet ITMO, 2017, pp. 65 (in Russian).

3. Rustamov G. A. Robastnaja sistema upravlenija c povyshennym potencialom (Robust Control System with High Potential), Izvestija Tomskogo Politehnicheskogo Universiteta, 2014, vol. 324, no. 5, pp. 13—20 (in Russian).

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

5. Rustamov G. A. Absolutely robust control systems, Automatic Control and Computer Sciences, 2013, vol. 47, no. 5, pp. 227—241.

6. Rustamov G. A., Rustamov G. A. Sistema robastnogo upravlenija (Robust control system), Evrazijskij patent № 025475 ot 30.12.2016 g. (in Russian).

7. Rustamov G. A. Analiz metodov postroenija robastnyh sistem upravlenija s bol’shim kojefficientom usilenija (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).

8. Meerov M. V. Sistemy avtomaticheskogo upravlenija, ustojchivye pri beskonechno bol’shih kojefficientah usilenija (Automatic control system, stable at infinitely large gain factors), Avtomatika i Telemehanika, 1947, vol. 8, no. 4, pp. 225—243 (in Russian).

9. Meerov M. V. Sintez struktur sistem avtomaticheskogo uprav lenija vysokoj tochnosti (Synthesis of structures of automatic control systems with high precision), Moscow, Nauka, 1967, 424 p. (in Russian).

10. Vostrikov A. S. Sintez sistem regulirovanija metodom lokalizacii (Synthesis of control systems by localization method), Novosibirsk, Publishing house of NGTU, 2007, 252 p. (in Russian).

11. Vostrikov A. S. Starshaja proizvodnaja i bol’shie kojefficienty usilenija v zadache upravlenija nelinejnymi nestacionarnymi obektami (The highest derivative and large coefficients in controlling the linear non-stationary objects), Mekhatronika, Avtomatizatsiya, Upravlenie, 2008, no. 5, pp. 2—7 (in Russian).

12. Vostrikov A. S. Problema sinteza reguljatorov dlja sistem avtomatiki: sostojanie i perspektivy (The problem of the synthesis of regulators for automation systems: state and prospects), Avtometrija, 2010, vol. 46, no. 2, pp. 3—19 (in Russian).

13. Vostrikov A. S., Francuzova A. G. Sintez PID-reguljatorov dlja nelinejnyh nestacionarnyh obektov (Synthesis of PID regulators for nonlinear nonstationary objects), Avtometrija, 2015, vol. 51, no. 5, pp. 53—60 (in Russian).

14. Filimonov A. B., Filimonov N. B. Robastnaja korrekcija dinamicheskih obektov v sistemah avtomaticheskogo upravlenija (Robust correction of dynamic objects in automatic control systems), Avtometrija, 2015, vol. 51, no. 5, pp. 61—68 (in Russian).

15. Tararykin S. V., Apolonskij V. V., Terehov A. I. Issledovanie vlijanija struktury i parametrov polinomial’nyh regulyatorov "vhoda-vyhoda" na robastnye svojstva sintezirovannyh system (Research of the influence of the structure and parameters of polynomial "input-output" regulators on the robust features of synthesized systems), Mekhatronika, Avtomatizatsiya, Upravlenie, 2013, vol. 152, no. 11, pp. 2—9 (in Russian).

16. Rustamov G. A. Sinhronizacija neidentichnyh dinamicheskih sistem s pomoshh’ju jekvivalentnogo upravlenija (Synchronization of non-identical dynamic systems using equivalent control), Izvestija Tomskogo Politehnicheskogo Universiteta, 2014, vol. 325, no. 5, pp. 33—42 (in Russian).

17. Utkin V. I. Sliding Mode in Optimization and Control Problems, New York, Springer-Verlag, 1992, 387 p.

18. Arnold V. I. Obyknovennye differencial’nye uravnenija (Ordinary differential equations), Moscow, Nauka, 1984, 272 p. (in Russian).