Journal of the Institute of Electronics Engineers of Korea SC (전자공학회논문지SC)

The Institute of Electronics and Information Engineers (대한전자공학회)
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Guaranteed Cost Controller Design Method for Singular Systems with Time Delays using LMI

선형행렬부등식을 이용한 시간지연 특이시스템의 보장비용 제어기 설계방법

  • 김종해 (선문대학교 전자정보통신공학부)
  • Published : 2003.05.01
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Abstract

This paper is concerned with the problem of designing a guaranteed cost state feedback controller for singular systems with time-varying delays. The sufficient condition for the existence of guaranteed cost controller, the controller design method, and the optimization problem to get the upper bound of guaranteed cost function are proposed by LMI(linear matrix inequality), singular value decomposition, Schur complements, and change of variables. Since the obtained sufficient conditions can be changed to LMI form, all solutions including controller gain and the upper bound of guaranteed cost function can be obtained simultaneously. Moreover, the proposed controller design method can be extended to the problem of robust guaranteed cost controller design method for singular systems with parameter uncertainties and time-varying delays. The validity of the proposed design algorithm is investigated through a numerical example.

본 논문에서는 시변 시간지연을 가지는 특이시스템에 대한 보장비용 상태제환 제어기 설계방법을 제시한다. 보장비용 제어기가 존재할 충분조건과 보장비용 제어기 설계방법 및 보장비용 함수의 상한치를 구하는 최적화 문제를 선형행렬부등식, 특이치 분해(singular value decomposition), 슈어 여수(Schur complements) 정리, 변수 치환 등에 의하여 제시한다. 구한 충분조건은 선형행렬부등식의 형태로 되기 때문에 보장비용 제어기의 이득과 보장비용 함수의 상한치를 포함하는 충분조건의 모든 해를 동시에 구할 수 있다, 또한, 제안한 알고리듬을 이용하면 변수 불확실성과 시변 시간지연을 동시에 가지는 특이시스템에 대한 강인 보장비용 제어기 설계문제에도 쉽게 확장됨을 보인다. 마지막으로, 제안한 알고리듬의 타당성을 수치예제를 통하여 확인한다.

Keywords

References

  1. I. Masubuchi, Y. Kamitane, A. Ohara, and N. Suda, '$H_{infty}$ control for descriptor systems: A matrix Inequalities approach,' Automatica, vol. 33, pp. 669-673, 1997 https://doi.org/10.1016/S0005-1098(96)00193-8
  2. H. S. Wang, C. F. Yung, and F. R. Chang, 'Bounded real lemma and $H_{infty}$ control for descriptor systems,' IEE Proc.-Control Theory Appl., vol.145, pp. 316-322, 1998 https://doi.org/10.1049/ip-cta:19982048
  3. D. J. Bender and A. J. Laub, 'The linearquadratic optimal regulator for descriptor systems,' IEEE Trans. Automat. Control, vol. 32, pp. 672-688, 1987 https://doi.org/10.1109/TAC.1987.1104694
  4. J. D. Cobb, 'Controllability, observability, and duality in singular systems,' IEEE Trans. Automat. Control, vol. 29, pp. 1076-1082, 1984 https://doi.org/10.1109/TAC.1984.1103451
  5. F. L. Lewis, 'Preliminary notes on optimal control for singular systems,' Proc. 24th Conf. on Decision and Control, pp. 262-272, 1985
  6. A. Rehm and F. Allgower, '$H_{infty}$ control of descriptor systems with norm-bounded uncertainties m the system matrices,' Proc. Amreican Control conference, pp. 3244-3248, 2000 https://doi.org/10.1109/ACC.2000.879164
  7. K. Takaba and T. Katayama, 'Robust $H_2$ performance of uncertain descriptor systems,' Proc. European Control Conf., WE-E-B-2, 1997
  8. K. Takaba, N. Morihara, and T. Katayama, 'A generalized Lyapunov theorem for descriptor system,' Systems and Control Lett., vol. 24, pp. 49-51, 1995 https://doi.org/10.1016/0167-6911(94)00041-S
  9. S. S. L. Chang and T. K. C. Peng, 'Adaptive guaranteed cost control of systems with uncertain parameters,' IEEE Trans. Automat. Control, vol. 17, no. 4, pp. 474-483, 1972 https://doi.org/10.1109/TAC.1972.1100037
  10. I. R. Petersen, 'Guaranteed cost LQG control of uncertain linear systems,' IEE proceedings-D, vol. 142, no. 2, pp. 95-102, 1995 https://doi.org/10.1049/ip-cta:19951732
  11. P. L. D. Peres, J. C. Geromel, and S. R. Souza, '$H_{infty}$ guaranteed cost control for uncertain continuous-time linear systems,' Systems and Control Letters, vol. 20, no. 4, pp. 413-418, 1993 https://doi.org/10.1016/0167-6911(93)90102-C
  12. I. R. Petersen and D. C. McFarlane, 'Optimal guaranteed cost control and filtering for uncertain linear systems,' IEEE Trans. Automat. Control, vol. 39, pp. 1971-1977, 1994 https://doi.org/10.1109/9.317138
  13. H. H. Choi and M. J. Chung, 'Memoryless Boo controller design for linear systems with delayed state and control,' Automatica, vol. 31, pp. 917-919, 1995 https://doi.org/10.1016/0005-1098(95)00001-D
  14. J. H. Kim and H. B. Park, '$H_{infty}$ state feedback control for generalized continuous/discrete time-delay system,' Automatica, vol. 35, pp. 1443-1451, 1999 https://doi.org/10.1016/S0005-1098(99)00038-2
  15. L. Yu and J. Chu, 'An LMI approach to guaranteed cost control of linear uncertain time-delay systems,' Automatica, vol. 35, pp. 1155-1159, 1999 https://doi.org/10.1016/S0005-1098(99)00007-2
  16. J. H. Kim, 'Guaranteed cost control of parameter uncertain systems with time delays,' Transactions on Control, Automation and Systems Engineering, vol. 2, no. 1, pp. 19-23, 2000
  17. J. H. Kim, J. H. Lee, and H. B. Park, 'Robust $H_{infty}$ control of singular systems with time delays and uncertainties,' Proc, American Control Conference in Anchorage, Alaska, USA, pp. 620-621, 2002 https://doi.org/10.1109/ACC.2002.1024878
  18. P. Gahinet, A. Nemirovski, A. J. Laub, and M. Chilali, LMI Control Toolbox, The Math Works Inc., 1995