Preview

Mekhatronika, Avtomatizatsiya, Upravlenie

Advanced search

Synthesis Robust Hinfinity-Regulator of the Low Order by using of Linear Matrix Inequalities and Projective Lemmas

Abstract

In this article the direct method of synthesis of a robust regulator of the low order is considered. For synthesis robust Нinfinity-regulator of the low order for plant with polytopic uncertainty are using bounded real lemma for linear matrix inequalities and two procedures of projection: 1) the projective lemma for linear matrix inequalities and 2) projection of nonnegative matrixes to reduced space also nonnegative matrixes. At the first stage of design the weakened problem with a convex linear matrix inequality is solved. For performance of not convex rank condition a procedure of orthogonal projection of singular value decomposition of a matrix and by rejection zero singular values is used. The order reduction a regulator is carried out by rejection small singular values. The submitted algorithm of synthesis of the reduced regulator is considered on an example of synthesis of robust regulator for plant with polytopic uncertainty. The plant is a satellite connected by a flexible boom with the sensor package (two-mass system). It is necessary to control angular position of the sensor package on which there is a star sensor and the sensor of angular position of the package, and the actuator control by angular position of the satellite. In view of no rigid connections inconsistency of movements of the actuator and the sensor of angular position of the sensor package takes place, i.e. there is a noncollocated system. Synthesis of a robust regulator for the weak damping plant of the fourth order with polytopic uncertainty is in detail considered. It is shown, that the order of a regulator it is possible to lower with initial the fourth to the second at insignificant deterioration of performance. specifications.

About the Author

V. I. Krasnoshchechenko
Kaluga Branch of the Bauman Moscow State Technical University
Russian Federation


References

1. Chilali M., Gahinet P. Hinf Design with Pole Placement Constraints: An LMI Approach // IEEE Trans. Automat. Contr. 1996. Vol. 41. P. 358-367.

2. Gahinet P., Apkarian P. A Linear Matrix Inequality Approach to Hinf Control // Intern. J. Robust & Nonlinear Control. 1994. Vol. 4, N. 4. P. 421-448.

3. Iwasaki T., Skelton R. E. All Controllers for General Hinf Control Problem: DLMI Existence and State Space Formulas // Automatica. 1994. V. 30, N. 8. P. 1307-1317.

4. Баландин Д. В., Коган М. М. Применение линейных матричных неравенств в синтезе законов управления. Н. Новгород: Изд-во Нижегородского госуниверситета, 2010. 93 с.

5. Баландин Д. В., Коган М. М. Линейные матричные неравенства в задаче робастного ^„-управления по выходу // ДАН. 2004. Т. 396. № 6. С. 759-761.

6. Gu D. W., Petkov P. Hr., Konstantinov M. M. Robust Control Design with Matlab. London: Springer, 2005. 389 p.

7. Поляк Б. Т., Щербаков П. С. Робастная устойчивость и управление. М.: Наука, 2002. 303 с.

8. Grigoriadis K. M., Skelton R. E. Low-order Control Design for LMI Problems Using Alternating Projection Methods // Automatica. 1996. V. 32, N. 8. P. 1117-1125.

9. Sun X., Mao J. Low-order Controller Design Based on LMI Using Projection Method // Proceedings 14th World IFAC Congress. 1999. Paper N. G-2e-12-5.

10. Gu D. W., Choi B. W., Postlethwaite I. Low-Order Stabilizing Controllers // IEEE Trans. AC. 1994. V. 38, N. 11. P. 1713-1717.

11. Brasch F. M., Pearson J. B. Pole Placement Using Dynamic Compensator // IEEE Trans. AC. 1970. V. 15, N. 1. P. 34-43.

12. Boyd S., Ghaoui E., Feron E., Balakrishnan V. Linear Matrix in System and Control Theory. Philadelphia: SIAM, 1994. 193 p.

13. Hermann G., Turner M. C., Postlethwaite I. Linear Matrix Inequalities in Control. In: Mathematical Methods for Robust and Nonlinear Control / Eds M. C. Turner et al. Berlin: Springer, 2007. P. 123-142.

14. Finsler P. Uber das Vorkommen definiter und semi-definiter Formen in Scharen quadratischer Formen // Comentari mathematici Helvetici. 1937. V. 9. P. 192-199.

15. Laub A. J. Matrix Analysis for Scientists and Engineers. Philadelphia: SIAM, 2005. 157 p.

16. Zhang F. Matrix Theory. Basic Results and Techniques. NY: Springer, 2011. 399 p.

17. Higham N. J. Computing the Nearest Symmetric Positive Semidefinite Matrix // Lin. Algebra Aspplics. 1988. V. 103. P. 103-118.

18. Franclin G. F., Powell J. D., Emami-Naeini A. Feedback Control of Dynamic Systems. Fourth Edition. New Jersey: Prentice-Hall, 2002. 887 p.


Review

For citations:


Krasnoshchechenko V.I. Synthesis Robust Hinfinity-Regulator of the Low Order by using of Linear Matrix Inequalities and Projective Lemmas. Mekhatronika, Avtomatizatsiya, Upravlenie. 2018;19(4):219-231. (In Russ.)

Views: 590


Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 License.


ISSN 1684-6427 (Print)
ISSN 2619-1253 (Online)