Polynomial-time algorithms for probabilistic solutions of parameter-dependent linear matrix inequalities

Automatica

Volumn 43, Issue 3, 2007, Pages 538-545

  Oishi, Yasuaki ^a  

^a UNIVERSITY OF TOKYO   (Japan)

Author keywords

Computational complexity; Optimization; Parameter dependent linear matrix inequalities; Probabilistic solutions; Randomized algorithms; Stopping rules

Indexed keywords

COMPUTATIONAL COMPLEXITY; ITERATIVE METHODS; MATRIX ALGEBRA; OPTIMIZATION; POLYNOMIALS; PROBABILITY;

PARAMETER-DEPENDENT LINEAR MATRIX INEQUALITIES; PROBABILISTIC SOLUTIONS; RANDOMIZED ALGORITHMS; STOPPING RULES;

ALGORITHMS;

EID: 33846781128     PISSN: 00051098     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.automatica.2006.09.020     Document Type: Article

References (24)

1. Bland R.G., Goldfarb D., and Todd M.J. The ellipsoid method: A survey. Operations Research 29 6 (1981) 1039-1091
2. Boyd S., El Ghaoui L., Feron E., and Balakrishnan V. Linear Matrix Inequalities in Systems and Control Theory (1994), SIAM, Philadelphia, USA
3. Calafiore G.C., and Campi M.C. Uncertain convex programs: Randomized solutions and confidence levels. Mathematical Programming 102 1 (2005) 25-46
4. Calafiore G., and Campi M.C. The scenario approach to robust control design. IEEE Transactions on Automatic Control 51 5 (2006) 742-753
5. Calafiore, G., Dabbene, F., & Tempo, R. (2003). Randomized algorithms in robust control. Proceedings of the 42nd IEEE Conference on Decision and Control (pp. 1908-1913), Maui, USA.
6. Calafiore G., and Polyak B. Stochastic algorithms for exact and approximate feasibility of robust LMIs. IEEE Transactions on Automatic Control 46 11 (2001) 1755-1759
7. Hiriart-Urruty J.-B., and Lemaréchal C. Fundamentals of Convex Analysis (2001), Springer, Berlin, Germany
8. Kanev S., De Schutter B., and Verhaegen M. An ellipsoid algorithm for probabilistic robust controller design. Systems & Control Letters 49 5 (2003) 365-375
9. Khargonekar, P., & Tikku, A. (1996). Randomized algorithms for robust control analysis and synthesis have polynomial complexity. Proceedings of the 35th IEEE Conference on Decision and Control (pp. 3470-3475), Kobe, Japan.
10. Liberzon D., and Tempo R. Common Lyapunov functions and gradient algorithms. IEEE Transactions on Automatic Control 49 6 (2004) 990-994
11. Monteiro R.D.C. Primal-dual path-following algorithms for semidefinite programming. SIAM Journal on Optimization 7 3 (1997) 663-678
12. Nemirovskii A. Several NP-hard problems arising in robust stability analysis. Mathematics of Control, Signals, and Systems 6 2 (1993) 99-105
13. Oishi, Y. (2003). Probabilistic design of a robust state-feedback controller based on parameter-dependent Lyapunov functions. Proceedings of the 42nd IEEE Conference on Decision and Control (pp. 1920-1925), Maui, USA.
14. Oishi, Y. (2004a). Polynomial-time algorithms for probabilistic solutions of parameter-dependent linear matrix inequalities. Technical Report METR 2004-23, Department of Mathematical Informatics, The University of Tokyo.
15. Oishi, Y. (2004b). Implementation of a randomized algorithm for solving parameter-dependent linear matrix inequalities. Proceedings of the 2004 IEEE Conference on Control Applications (pp. 1183-1188), Taipei, Taiwan.
16. Oishi, Y. (2006). Probabilistic design of a robust controller using a parameter-dependent Lyapunov function. In G. Calafiore, & F. Dabbene (Eds.), Probabilistic and Randomized Methods for Design under Uncertainty (pp. 303-316). London, UK: Springer.
17. Oishi Y., and Kimura H. Computational complexity of randomized algorithms for solving parameter-dependent linear matrix inequalities. Automatica 39 12 (2003) 2149-2156
18. Polyak, B. T. (2001). Random algorithms for solving convex inequalities. In D. Butnariu, Y. Censor, & S. Reich (Eds.), Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications (pp. 409-422). Amsterdam, The Netherlands: Elsevier.
19. Polyak B.T., and Tempo R. Probabilistic robust design with linear quadratic regulators. Systems & Control Letters 43 5 (2001) 343-353
20. Rugh W.J., and Shamma J.S. Research on gain scheduling. Automatica 36 10 (2000) 1401-1425
21. Shiryaev A.N. Probability. (2nd ed.) (1996), Springer, New York, USA
22. Takagi K., and Nishimura H. Gain-scheduled control of a tower crane considering varying load-rope length. JSME International Journal, Series C 42 4 (1999) 914-921
23. Tempo R., Bai E.W., and Dabbene F. Probabilistic robustness analysis: Explicit bounds for the minimum number of samples. Systems & Control Letters 30 5 (1997) 237-242
24. Tempo R., Calafiore G., and Dabbene F. Randomized Algorithms for Analysis and Control of Uncertain Systems (2005), Springer, London, UK