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

The Institute of Electronics and Information Engineers (대한전자공학회)
권호 표지

(Robust Non-fragile $H^\infty$ Controller Design for Parameter Uncertain Systems)

파라미터 불확실성 시스템에 대한 견실 비약성 $H^\infty$ 제어기 설계

  • Jo, Sang-Hyeon (School of Electronic & Electrical Engineering, Kyungpook National University) ;
  • Kim, Gi-Tae (School of Electronic & Electrical Engineering, Kyungpook National University) ;
  • Park, Hong-Bae (School of Electronic & Electrical Engineering, Kyungpook National University)
  • 조상현 (경북대학교 전자전기공학부) ;
  • 김기태 (경북대학교 전자전기공학부) ;
  • 박홍배 (경북대학교 전자전기공학부)
  • Published : 2002.05.01
Download PDF
⟨ Previous Next ⟩

Abstract

This paper describes the synthesis of robust and non-fragile H$\infty$ state feedback controllers for linear varying systems with affine parameter uncertainties, and static state feedback controller with structured uncertainty. The sufficient condition of controller existence, the design method of robust and non-fragile H$\infty$ static state feedback controller, and the set of controllers which satisfies non-fragility are presented. The obtained condition can be rewritten as parameterized Linear Matrix Inequalities(PLMls), that is, LMIs whose coefficients are functions of a parameter confined to a compact set. However, in contrast to LMIs, PLMIs feasibility problems involve infinitely many LMIs hence are inherently difficult to solve numerically. Therefore PLMls are transformed into standard LMI problems using relaxation techniques relying on separated convexity concepts. We show that the resulting controller guarantees the asymptotic stability and disturbance attenuation of the closed loop system in spite of controller gain variations within a degree.

본 논문에서는 구조화된 어파인(affine) 파라미터 불확실성을 가지는 시변 선형시스템과 구조적 불확실성을 가지는 상태궤환 제어기에 대한 견실 비약성 H∞ 제어기 설계방법을 다루었다. 또한 견실 비약성 H∞ 제어기가 존재할 충분조건, 제어기 설계방법 및 비약성을 만족하는 제어기의 꽉찬 집합(compact set)을 제시하였다. 이 때 제시한 조건은 변수치환과 슈어 여수(Schur complement)정리를 통하여 선형행렬부등식 (LMI : Linear Matrix Inequality)의 계수가 꽉찬 집합 내의 파라미터의 함수로 정의되는 파라미터화 선형 행렬부등식(PLMls: parameterized Linear Matrix Inequalities)으로 표현되므로 분리 볼록개념 (separated convexity concepts)에 기초한 완화기법을 이용하여 유한개의 LMI로 변환하였다. 그리고 본론문에서 제시한 견실 비약성 H∞ 제어기가 제어기이득의 변화에도 불구하고 폐루프시스템의 점근적 안정성 (asymptotic stability)과 외란감쇠 성능을 보장함을 보였다.

Keywords

References

  1. L. H. Keel and S. P. Bhattacharyya, 'Digital implementation of fragile controllers,' Proc. Amer. Contr. Conf. in Philadelphia, Pennsylvania, pp. 2852-2856, 1998 https://doi.org/10.1109/ACC.1998.688377
  2. L. H. Keel and S. P. Bhattacharyya, 'Robust, fragile, or optimal,' IEEE Trans. Automat. Contr., vol. 42, no. 8, pp. 1098-1105, 1997 https://doi.org/10.1109/9.618239
  3. J. R. Corrado and W. M. Haddad, 'Static output feedback controllers for systems with parametric uncertainty and controller gain variation,' Proc. Amer. Contr. Conf. in San Diego, California, pp. 915-919, 1999 https://doi.org/10.1109/ACC.1999.783173
  4. P. Dorato, 'Non-fragile controller design : An overview,' Proc. Amer. Contr. Conf. in Philadelphia, Pennsylvania, pp. 2829-2831, 1998 https://doi.org/10.1109/ACC.1998.688371
  5. P. Dorato, C. T. Abdallah, and D. Famularo, 'On the design of non-fragile compensators via symbolic quantifier elimination,' World Automation Congress in Anchorage, Alaska, pp. 9-14, 1998
  6. D. Famularo, C. T. Abdallah, A. Jadbabaie, P. Dorato, and W. M. Haddad, 'Robust non-fragile LQ controllers : The static state feedback case,' Proc. Amer. Contr. Conf. in Philadelphia, Pennsylvania, pp. 1109-1113, 1998 https://doi.org/10.1109/ACC.1998.703583
  7. W. M. Haddad and J. R. Corrado, 'Robust resilient dynamic controller for systems with parametric uncertainty and controller gain variations,' Proc. Amer. Contr. Conf. in Philadelphia, Pennsylvania, pp. 2837-2841, 1998 https://doi.org/10.1109/ACC.1998.688373
  8. A. Jadbabie, C. T. Abdallah, D. Famularo, and P. Dorato, 'Robust, non-fragile and optimal controller via linear matrix inequalities,' Proc. Amer. Contr. Conf. in Philadelphia, Pennsylvania, pp. 2842-2846, 1998 https://doi.org/10.1109/ACC.1998.688374
  9. J. H. Kim, S. K. Lee, and H. B. Park, 'Robust and non-fragile $H^{\infty}$ control of parameter uncertain time-varying delay systems,' SICE in Morioka, pp. 927-932, July 1999 https://doi.org/10.1109/SICE.1999.788673
  10. P. Apkarian and H. D. Tuan, 'Parameterized LMIs in control theory,' Proc. IEEE Conf. Dec. Contr. in Florida, pp. 152-157, 1998 https://doi.org/10.1109/CDC.1998.760607
  11. H. D. Tuan and P. Apkarian, 'Relaxations of parameterized LMIs with control applications,' Int. J. of Robust Nonlinear Control, vol. 9, pp. 59-84, 1999 https://doi.org/10.1002/(SICI)1099-1239(199902)9:2<59::AID-RNC392>3.0.CO;2-O
  12. S. Boyd, L. E. Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, SIAM, 1994
  13. P. Gahinet and P. Apkarian, 'A linear matrix inequality approach to $H^{\infty}$ control,' Int. J. of Robust Nonlinear Control, vol. 4, pp. 421-448, 1994 https://doi.org/10.1002/rnc.4590040403