Error-bounds for balanced model-reduction of linear time-varying systems

IEEE Transactions on Automatic Control

Volumn 48, Issue 6, 2003, Pages 946-956

  Lall, Sanjay ^a   Beck, Carolyn ^b  

^a Department of Aeronautics and Astronautics Department of Mechanical Engineering and Industrial for Computational and Mathematical Engineering   (United States)

^b University of Illinois at Urbana Champaign   (United States)

Author keywords

Balanced truncation; Model reduction; Time varying systems

Indexed keywords

COMPUTATIONAL METHODS; COMPUTER SIMULATION; MATHEMATICAL MODELS; PARTIAL DIFFERENTIAL EQUATIONS; SYSTEM STABILITY;

BALANCED TRUNCATION;

TIME VARYING CONTROL SYSTEMS;

EID: 0038576712     PISSN: 00189286     EISSN: None     Source Type: Journal    
DOI: 10.1109/TAC.2003.812779     Document Type: Article

References (37)

1. V. M. Adamjan, D. Z. Arov, and M. G. Krein, "Analytic properties of Schmidt pairs for a Hankel operator and the generalized Schur-Takagi problem," Math USSR Sbornik, vol. 15, pp. 31-73, 1971.
2. K. Glover, "All optimal Hankel-norm approximations of linear multivariable systems and their L-infinity error bounds," Int. J. Control, vol. 39, pp. 1115-1193, 1984.
3. C. T. Mullis and R. A. Roberts, "Synthesis of minimum roundoff noise fixed point digital filters," IEEE Trans. Circuits Syst., vol. CAS-23, pp. 551-562, Sept. 1976.
4. B. C. Moore, "Principal component analysis in linear systems: controllability, observability, and model reduction," IEEE Trans. Automat. Contr., vol. AC-26, pp. 17-32, Feb. 1981.
5. S. Y. Kung, "A new identification and model reduction algorithm via singular value decompositions," in Proc. 12th Asilomar Conf. Circuits, Signals, Computers, 1978, pp. 705-714.
6. L. Pernebo and L. M. Silverman, "Balanced systems and model reduction," in Proc. IEEE Conf. Decision Control, 1979.
7. D. F. Enns, "Model reduction for control system design," Ph.D. dissertation, Stanford Univ., Stanford, CA, 1984.
8. U. M. Al-Saggaf and G. F. Franklin, "An error-bound for a discrete reduced-order model of a linear multivariable system," IEEE Trans. Automat. Contr., vol. AC-32, pp. 815-819, Sept. 1987.
9. D. Hinrichsen and A. J. Pritchard, "An improved error estimate for reduced-order models of discrete-time systems," IEEE Trans. Automat. Contr., vol. 35, pp. 317-320, Mar. 1990.
10. C. L. Beck, "Model reduction and minimality for uncertain systems," Ph.D. dissertation, California Inst. Technol., Pasadena, CA, 1996.
11. C. L. Beck and J. C. Doyle, "A necessary and sufficient minimality condition for uncertain systems," IEEE Trans. Automat. Contr., vol. 44, pp. 1802-1813, Oct. 1999.
12. C. L. Beck, J. C. Doyle, and K. Glover, "Model reduction of multidimensional and uncertain systems," IEEE Trans. Automat. Contr., vol. 41, pp. 1466-1477, Oct. 1996.
13. J. Imae, J. E. Perkins, and J. B. Moore, "Toward time-varying balanced realization via Riccati equations," Math. Control, Signals, Syst., pp. 313-326, 1992.
14. S. Shokoohi, L. M. Silverman, and P. M. van Dooren, "Stability of balanced time-variable systems approximations," in Proc. IEEE Conf. Decision Control, 1980, pp. 500-505.
15. _, "Linear time-variable systems: Balancing and model reduction," IEEE Trans. Automat. Contr., pp. 810-822, Aug. 1983.
16. _, "Linear time-variable systems: Stability of reduced models," Automatica, vol. 20, pp. 59-57, 1984.
17. E. I. Verriest and T. Kailath, "On generalized balanced realizations," in Proc. IEEE Conf. Decision Control, 1981, pp. 504-505.
18. _, "On generalized balanced realizations," IEEE Trans. Automat. Contr., pp. 833-844, Aug. 1983.
19. S. Shokoohi and L. M. Silverman, "Identification and model reduction of time-varying discrete-time systems," Automatica, vol. 23, no. 4, pp. 509-521, 1987.
20. S. Shokoohi, L. M. Silverman, and P. M. van Dooren, "Model reduction of discrete time-variable systems via balancing," in Proc. IEEE Conf. Decision Control, 1981, pp. 676-681.
21. X. Yang, M. Kawamata, and T. Higuchi, "Balanced realizations and model reduction of periodically time-varying state-space digital filters," IEE Proc. Visual Image Signal Processing, pp. 370-376, Dec. 1996.
22. F. J. Clara and L. M. Silverman, "Two-dimensional discrete space varying systems: Identification, balancing and model reduction," in Proc. 6th IFAC Symp., 1982, p. 1304.
23. R. A. R and V. L. Syrmos, "Balancing of systems with periodic jumps," Kybernetika, vol. 32, pp. 575-590, 1996.
24. P. Dewilde and A.-J. van der Veen, "On the Hankel-norm approximation of upper-triangular operators and matrices," Int. Equal. Oper. Theory, vol. 17, pp. 1-45, 1993.
25. K. Glover, P. J. Goddard, and Y. C. Chu, "Model reduction for classes of uncertain, multi-dimensional, parameter varying and nonlinear systems," in Lecture Notes in Control and Information Sciences. New York; Springer-Verlag, 1999, pp. 269-280.
26. G. D. Wood, P. J. Goddard, and K. Glover, "Approximation of linear parameter-varying systems," in Proc. IEEE Conf. Decision Control, 1996, pp. 406-411.
27. S. Lall, C. Beck, and G. Dullerud, "Guaranteed error bounds for model reduction of linear time-varying systems," in Proc. Amer. Control Conf., 1998, pp. 634-638.
28. S. Longhi and G. Orlando, "Balanced reduction of linear periodic systems," Kybernetika, vol. 35, no. 6, pp. 737-751, 1999.
29. A. Varga, "Balanced truncation model reduction of periodic systems," in Proc. IEEE Conf. Decision Control, 2000, pp. 2379-2384.
30. G. Dullerud and S. Lall, "A new approach for analysis and synthesis of time-varying systems," IEEE Trans. Automat. Contr., vol. 44, pp. 1486-1497, Aug. 1999.
31. D. Alpay, P. Dewilde, and H. Dym, "Lossless inverse scattering and reproducing kernels for upper triangular operators," in Extension and Interpolation of Linear Operators and Matrix Functions. Boston, MA: Birkhaüser, 1990, vol. 47, Operator Theory: Advances and Applications, pp. 61-135.
32. J. A. Ball, I. Gohberg, and M. A. Kaashoek, "A frequency response function for linear time-varying systems," Math. Control, Signals, Syst., vol. 8, pp. 334-351, 1995.
33. P. Dewilde and A.-J. V. D. Veen, Time-Varying Systems and Computations. Norwell, MA: Kluwer, 1998.
34. A. Feintuch, "Realization theory for linear time-varying, discrete-time linear systems," Adv. Appl. Math., vol. 9, pp. 211-225, 1988.
35. E. W. Kamen, P. P. Khargonekar, and K. R. Poolla, "A transfer-function approach to linear time-varying discrete-time systems," SIAM J. Control Optim., vol. 23, no. 4, pp. 550-565, 1985.
36. M. A. Peters and P. A. Iglesias, "Spectral test for observability and reach-ability of time-varying systems," SIAM J. Control Optim., vol. 37, no. 5, pp. 1330-1345, 1999.
37. B. D. O. Anderson, "Internal and external stability of linear time-varying systems," SIAM J. Control Optim., vol. 20, no. 3, pp. 408-413, 1982.