On the bounds on the solutions of the algebraic Lyapunov and Riccati equations

Czornik, A.; Nawrat, A.

Artykuł - szczegóły

Tytuł artykułu

On the bounds on the solutions of the algebraic Lyapunov and Riccati equations

Autorzy

Czornik A. , Nawrat A.

Wybrane pełne teksty z tego czasopisma

https://journals.pan.pl/acs/

Identyfikatory

Warianty tytułu

Języki publikacji

Abstrakty

Different types of bounds for solutions of continuous and discrete Lyapunov and Riccati equations obtained up to now are summarized in this paper. Some new bounds are also presented. The efficiency of each bound is illustrated with three numerical examples. A discussion and comparison are given as well. The results may be particularly convenient to get the ready estimate of the solution while solving the equations numerically or to develop theoretical results that rely on these bounds.

Słowa kluczowe

Riccati equations Lyapunov equations eigenvalues bounds

Wydawca

Polish Academy of Sciences, Committee of Automatic Control and Robotics

Czasopismo

Archives of Control Sciences

Rocznik

2000

Tom

Vol. 10, no. 3/4

Strony

197--244

Opis fizyczny

Bibliogr. 77 poz., tab., wzory

Twórcy

autor

Czornik A.

Department of Automatic Control, Silesian Univerity of Technology, Akademicka 16, 44-100 Gliwice, Poland

autor

Nawrat A.

Department of Mathematic, Silesian Univerity of Technology, Kaszubska 23, 44-100 Gliwice, Poland

Bibliografia

[1] J. C. Allwright: A lower bound lor the solution of the algebraic Riccati equation of optimal control and a geometric convergence rate for the Kleinman algorithm. IEEE Transactions on Automatic Control, 25 (1980), 826-829.
[2] S. Barnett and C. Storey: Matrix methods in stability theory. Barness and Nobel, New York, 1970.
[3] S. Białas: On the Lyapunov matrix equation. IEEE Transactions on Automatic Control, 25 (1980), 813-814.
[4] S. Bittani, A. Laub and J. C. Willems: The Riccati equation. Springer-Verlag, Berlin 1991.
[5] R. Canales: A lower bound on the performance of optimal regulators. IEEE Transactions on Automatic Control, 15 (1970), 409-415.
[6] A. Czornik and A. Nawrat: On upper bound for the unified algebraic Riccati equation. System & Control Letters, 32 (1997), 235-239.
[7] A. Czornik: Comments on bounds for the eigenvalues of the algebraic Riccati equation. International Journal of Systems Science, 31 (2000).
[8] M. M. Fahmy and A. A. R. Hanaey: Further corrections to: Comments on "On the Lyapunov matrix equation. IEEE Transactions on Automatic Control, 26 (1981), 619.
[9] Y. Fang, K. A. Loparo and X. Feng: New estimates for solutions of Lyapunov equations. IEEE Transactions on Automatic Control, 42 (1997), 408-411.
[10] J. Garloff: Bounds lor the eigenvalues of the solution of the discrete Riccati and Lyapunov equation and the continuous Lyapunov equation. International Journal of Control, 37 (1986), 423-431.
[11] J C. Geromel and J. Bernussou: On bounds of Lyapunov's matrix equation. IEEE Transactions on Automatic Control, 24 (1979), 482-483
[12] A. Hmamed: Discrete Lyapunov equation: Simultaneous eigenvalue lower bounds. International Journal of System and Science, 22 (1991), 1121-1126.
[13] A. Hmamed: Riccati equation: simultaneous eigenvalue lower bounds. Report LESSI 05/89.
[14] A. Hmamed and M. Mrabti: Riccati and Lyapunov matrix equations: simultaneous eigenvalue upper bounds. Report LESSI 02/91.
[15] A. Hmamed and M. Mrabti: On the determinant of the Lyapunov differentia and difference equations. Report LESSI 03/91.
[16] A. Hmamed and M. Mrabti: Simultaneous eigenvalue bounds for differentia Lyapunov equation. Report LESSI 04/91.
[17] T. Kang, B. S. Kim and J. G. Lee: Spectral norm and trace bounds of algebraic matrix Riccati equations. IEEE Transactions on Automatic Control, 41 (1996), 1828-1830.
[18] V. R. Karanam: Lower hounds on the solution of Lyapunov matrix and algebraic Riccati equation. IEEE Transactions on Automatic Control 26 (1981), 1288-1290.
[19] V. R. Karanam: Eigenvalue bounds for algebraic Riccati and Lyapunov equations. IEEE Transactions on Automatic Control, 27 (1982), 461-463.
[20] V. R. Karanam: A note on eigenvalue bounds in algebraic Riccati equation. IEEE Transactions on Automatic Control, 28 (1983), 109-111.
[21] J. H. Kim and Z. Bien: Some bounds of the solution of the algebraic Riccati equation. IEEE Transactions on Automatic Control, 37 (1992), 1209-1210.
[22] N. Komaroff: Simultaneous eigenvalue lower hounds for the Lyapunov matrix equation. IEEE Transactions on Automatic Control, 33 (1988), 126 128.
[23] N. Komaroff: Simultaneous eigenvalue lower bounds for the Riccati matrix equation. IEEE Transactions on Automatic Control, 34 (1989), 175-177.
|24] N. Komaroff: Bounds on eigenvalues of matrix products with an applications to the algebraic Riccati equation. IEEE Transactions on Automatic Control, 35 (1990), 348-350.
[25] N. Komaroff: Upper bounds for the eigenvalues of the solution of the Lyapunov matrix equation. IEEE Transactions on Automatic Control. 35 (1990), 737-739.
[26] (1) N. Komaroff: Lower bounds for the solution of the discrete algebraic Lyapunov equation. IEEE Transactions on Automatic Control, 37 (1992), 1017 1018.
[27] N. Komaroff: Upper summation and products bounds for solution eigenvalues of the Lyapunov matrix equation. IEEE transactions on Automatic Control, 37 (1992), 1040 1042.
[28] (2) N. Komaroff: Upper bounds for Ihe Solution of the discrete Riccati equation. IEEE Transactions on Automatic Control, 37 (1992), 1370-1372.
[29| (1) N. Komaroff: Diverse bounds for the eigenvalues of the continuous algebraic Riccati eguation. IEEE Transactions on Automatic Control, 39 (1994), 532-534.
[30] (2) N. Komaroff: Iterative matrix hounds and computational solution to the discrete algebraic Riccati equation. IEEE Transactions on Automatic Control, 39 (1994), 1676-1678.
[31] N. Komaroff: On bounds for the Riccati equation for discrete-time control systems. Control and Dynamics Systems, 78 (1996), 275-311.
[32] N. Komaroff and B. Shahian: Lower summation bounds for the discrete Riccati and Lyapunov equations. IEEE Transactions on Automatic Control. 37 (1992). 1078-1080.
[33] W. H. Kwon and A. E. Pearson: A note on the algebraic matrix Riccati equation. IEEE Transactions on Automatic Control, 22 (1977), 143-144.
[34] B. H. Kwon, M. J. Youn and Z. Bien: On bounds of the Riccati and Lyapunov equations. IEEE Transactions on Automatic Control, 30 (1985), 1134-1135.
[35] B. H. Kwon and M. J. Youn: Comments on "A note on the discrete Lyapunov and Riccati matrix equations. International Journal of Control, 41 (1985), 1633-1644.
[36] B. H. Kwon and M.J. Youn: Further Comments on "A note on the discrete Lyapunov and Riccati matrix equations. International Journal of Control, 42 (1986), 1731-1744.
[37] B.H Kwon and M. J. Youn: Coments on "On some bounds in the algebraic Riccati and Lyapunov equations". IEEE Transactions on Automatic Control, 31 (1986), 591.
[38] W. H. Kwon, Y. S. Moon and S. C. Ahn: Bounds in algebraic Riccati and Lyapunov equations: a survey and some new results. International Journal of Control, 64(1996), 377-389.
[39] P. Lancaster and L. Rodman: Algebraic Riccati equation. Oxford Univ. Press, 1995.
[40] G. Langholz: A new lower-bound on the cost of optimal regulators. IEEE Transactions on Automatic Control, 24 (1979), 353-354.
[41] C. H. Lee Upper and lower bounds of the solution for the discrete Lyapunov equation. IEEE Transactions on Automatic Control, 41 (1996), 1338-1341.
[42] (1) C. H. Lee: New results for the hounds of the solution for the continuous Riccati and Lyapunov equations. IEEE Transactions on Automatic Control, 42 (1997), 118-122.
[43] (2) C. H. Lee: Eigenvalues upper bounds of the solution of the continuous Riccati eguations. IEEE Transactions on Automatic Control, 42 (1997), 1268-1271.
[44] (3) C. H. Lee: On the upper and lower hounds of the solution for the continuous Riccati matrix equation. International Journal on Control, 66 (1997), 105-118.
[45] (2) C. H. Lee: Upper and lower bounds of the solution of the discrete algebraic Riccati and Lyapunov matrix equation. International Journal on Control, 68 (1997), 579-598.
[46] A. W. Marshall and I. Olkin: Inequalities: Theory of majorization and its applications. New York: Acasemic, 1979.
[47] R. H. Middleton and G. C. Goodwin; Digital control and estimation A unified approach. Englewood Cliffs. NJ: Prentice-Hall, 1990.
[48] J. J. Montemayor and B. F. Womack: Comments on "On the Lyapunov matrix equation". IEEE Transactions on Automatic Control. 20 (1975), 814-815.
[49] T. Mori: A note on bounds for the solution to the algebraic matrix Riccati and Lyapunov equations. Transactions IEEE of Japan, 62 (1979), 760-761.
[50] T. Mori: Another bounds lor the solution to the Lyapunov matrix equation. Transactions IECE, 63A, (1980), 38-40.
[51] T. Mori: Note on the existence regions for eigenvalues of the solution to the discrete Lyapunov matrix equation. International Journal of Systems Science, 14 (1983), 463-466.
[52] T. Mori: On some bounds in the algebraic Riccaii and Lyapunov equations. IEEE Transactions on Automatic Control, 30 (1985), 162-164.
[53] T. Mori: Coments on ' A matrix inequality associated with bounds on solutions of algehraie Riccati and Lyapunov equation. IEEE Transactions on Automatic Control, 29 (1988), 1088.
[54] T. Mori: Recovered bounds for the solution to the discrete Lyapunov matrix equation. IEICE Transactions on Fundamentals of Electronics. Communications and Computer Sciences, E77A, (1994), 571-572.
[55] T. Mori and A. Derese: A brief summary of the bounds on the solution of the algebraic matrix equations in control theory. International Journal of Control, 39 (1984), 247-256.
[56] T. Mori, N. Fukuma and M. Kuwahara: On the discrete Lyapunov matrix equation. IEEE Transactions on Automatic Control, 27 (1982), 463-464.
[57] T. Mori, N. Fukuma and M. Kuwahara: Explicit solution and eigenvalue bounds in the Lyapunov matrix equation. IEEE Transactions on Automatic Control. 31 (1986), 656-658.
[58] (1) T. Mori, N. Fukuma and M. Kuwahara: Bounds in Lyapunov matrix differential equation. IEEE Transactions on Automatic Control, 32 (1987), 55-57.
[59] (2) T. Mori, N. Fukuma and M. Kuwahara: On the discrete Riccati equation. IEEE Transactions on Automatic Control. 32 (1987), 828-829.
[60] T. Mori and H. Kokame: On bounds for the solution to the discrete Lyapunov matrix equation. Transactions SICE, 15 (1979), 986-987.
[61] T. Mori, M. Kuwahara and E. Noldus: A way 10 stabilizable linear systems with delayed. Automatica, 19 (1983), 571-573.
[62] M. Mrabti and A. Hmamed: Bounds tor the solution of the Lyapunov matrix equation. A unified approach. System & Control Letters, 18 (1992), 73-81.
[63] M. Mrabti and M. Benseddik: Unified type non-stacionary Lyapunov matrix equation- Simultaneous eigenvalue hounds. Systems A Control Letters, 24 (1995), 53-59.
[64] M. Mrabti and M. Benseddik: Bounds fori the eigenvalues of the solution of the unified algebraic Riccati matrix equation. Systems & Control Letters, 24 (1995), 345-349.
[65] D. W. Nicholson: Eigenvalue bounds in the Lyapunov and Riccati matrix equations. IEEE Transactions on Automatic Control, 26 (1981), 1290-1291.
[66] P. Park On the trace bound of matrix product. IEEE Transactions on Automatic Control, 41 (1996), 1799-1802.
[67] R. V. Patel and M. Toada: On norm bounds for algebraic Riccati and Lyapunov equations. IEEE Transactions on Automatic Control, 23 (1978), 87-88.
[68] E. Y. Shapiro: On the Lyapunov matrix equation. IEEE Transactions on Automatic Control, 19 (1974), 594-595.
[69] M. Toda and R. Patel: Performance bounds for continuous-time filters in the presence of modeling errors. IEEE Transactions on Aerospace and Electronic Systems, 14 (1978), 912-919.
[70] M. T. Tran and M. E. Sawan: A nole on the discrete Lyapunov and Riccati matrix equations. International Journal of Control, 39 (1984), 337-341.
[71] M. T. Tran and M. E. Sawan: Correction to "A note on the discrete Lyapunov and Riccati matrix equations". International Journal of Control, 46 (1987), 1119.
[72] I. Troch: Improved Bounds for the Eigenvalues of Solution of Lyapunov Equations. IEEE Transactions on Automatic Control, 32 (1987), 744-747.
[73] S. Wang, T. Kuo and C. Hsu: Trace bounds on the solution of the algebraic matrix Riccati and Lyapunov equations. IEEE Transactions on Automatic Control, 31 (1986), 654-656.
[74] J. C. Willems: Least Squares stationary optimal control and the algebraic Riccati equation. IEEE Transactions on Automatic Control. 16 (1971), 621-634.
[75] K. Yasuda and K. Hirai: Upper and lower hounds on the solution of the algebraic Riccati equation. IEEE Transactions on Automatic Control, 24 (1979), 483-487.
[76] E. Yaz: Bounds for the eigenvalues of the solution matrix of the algebraic Riccati equation. International Journal of Systems Science, 16 (1985), 815-820.
[77] E. Yaz: Bounds on the solution of a matrix equation. International Journal on Control, 50 (1989), 693-697.

Typ dokumentu

Bibliografia

Identyfikator YADDA

bwmeta1.element.baztech-article-BSW9-0007-1320