Numerically robust delta-domain solutions to discrete-time Lyapunov equations. (English) Zbl 1106.65314
Summary: A problem of numerical conditioning of a special kind of discrete-time Lyapunov equations is considered. It is assumed that a discretization procedure equipped with the zero-order holder mechanism is utilized that leads to the data matrices that are affinely related to the sampling period and matrices that are independent or linearly related to the squared sampling period. It is shown that common forward shift operator techniques for solving these equations become ill-conditioned for a sufficiently small sampling period and that numerical robustness and reliability of computations can be significantly improved via utilizing the so-called delta operator form of the origin equations.
MSC:
| 65F35 | Numerical computation of matrix norms, conditioning, scaling |
| 93B40 | Computational methods in systems theory (MSC2010) |
| 93C55 | Discrete-time control/observation systems |
Keywords:
Lyapunov equations; Delta operator; Numerical conditioning; Discrete-time modelling; Reliability of computationsReferences:
| [1] | Anderson, E.; Bai, Z.; Bischof, C.; Blackford, S.; Demmel, J.; Dongarra, J.; Du Croz, J.; Greenbaum, A.; Hammarling, S.; Mckenney, A.; Sorensen, D., LAPACK. Users’ Guide (1999), SIAM: SIAM Philadelphia, PA · Zbl 0934.65030 |
| [2] | Barraud, A. Y., A numerical algorithm to solve \(A^T XA \)−\(X = Q\), IEEE Trans. Automat. Control, 22, 5, 883-885 (1977) · Zbl 0361.65022 |
| [3] | Bartels, H.; Stewart, G. W., Algorithm 432, Solution of the matrix equation \(AX + XB = C\), Comm. ACM, 15, 9, 820-826 (1972) · Zbl 1372.65121 |
| [4] | Byers, R., A LINPACK-style condition estimator for the equation \(AX − XB^T = C\), IEEE Trans. Automat. Control, 29, 10, 926-928 (1984) · Zbl 0544.65027 |
| [5] | Gahinet, P. M.; Laub, A. J.; Kenney, C. S.; Hewer, G. A., Sensitivity of the stable discrete-time Lyapunov equation, IEEE Trans. Automat. Control, 35, 11, 1209-1217 (1990) · Zbl 0721.93064 |
| [6] | Gajic, Z.; Qureshi, M., Lyapunov Matrix Equation in System Stability and Control (1995), Academic Press: Academic Press New York · Zbl 1153.93300 |
| [7] | Gevers, M.; Li, G., Parametrizations in Control, Estimation and Filtering Problems (1993), Springer: Springer London · Zbl 0929.93002 |
| [8] | Ghavimi, A. R.; Laub, A. J., Residual bounds for discrete-time Lyapunov equations, IEEE Trans. Automat. Control, 40, 7, 1244-1249 (1995) · Zbl 0923.93022 |
| [9] | Golub, G. H.; Nash, S.; Van Loan, C. F., A Hessenberg-Schur method for the problem \(AX + XB = C\), IEEE Trans. Automat. Control, 24, 6, 909-913 (1979) · Zbl 0421.65022 |
| [10] | Golub, G. H.; Van Loan, C. F., Matrix Computations (1996), The Johns Hopkins University Press: The Johns Hopkins University Press Baltimore, London · Zbl 0865.65009 |
| [11] | Goodwin, G. C.; Middleton, R. H.; Poor, H. V., High-speed digital signal processing and control, Proc. IEEE, 80, 2, 240-259 (1992) |
| [12] | Hammarling, S. J., Numerical solution of the stable, nonnegative definite Lyapunov equation, IMA J. Numer. Anal., 2, 303-323 (1982) · Zbl 0492.65017 |
| [13] | Hewer, G. A.; Kenney, C. S., The sensitivity of the stable Lyapunov equation, SIAM J. Control Optim., 26, 2, 321-344 (1988) · Zbl 0655.93050 |
| [14] | Higham, N. J., Perturbation theory and backward error for \(AX − XB = C\), BIT, 33, 1, 124-136 (1993) · Zbl 0781.65034 |
| [15] | Higham, N. J., Accuracy and Stability of Numerical Algorithms (1996), SIAM: SIAM Philadelphia, PA · Zbl 0847.65010 |
| [16] | B. Kågström, P. Poromaa, LAPACK-style algorithms and software for solving the generalized Sylvester equation and estimating the separation between regular matrix pairs, Report UMINF 93.23, 1993, Institute of Information Processing, University of Umeå, Umeå, Sweden.; B. Kågström, P. Poromaa, LAPACK-style algorithms and software for solving the generalized Sylvester equation and estimating the separation between regular matrix pairs, Report UMINF 93.23, 1993, Institute of Information Processing, University of Umeå, Umeå, Sweden. · Zbl 0884.65031 |
| [17] | B. Kågström, P. Poromaa, Computing eigenspaces with specified eigenvalues of a regular matrix pair \((AB\); B. Kågström, P. Poromaa, Computing eigenspaces with specified eigenvalues of a regular matrix pair \((AB\) |
| [18] | Kågström, B.; Westin, L., Generalized Schur methods with condition estimators for solving the generalized Sylvester equation, IEEE Trans. Automat. Control, 34, 7, 745-751 (1989) · Zbl 0687.93025 |
| [19] | D. Kreßner, V. Mehrmann, T. Penzl, CTLEX—a collection of benchmark examples for continuous-time Lyapunov equations, SLICOT Working Note, 1999, 6, Katholieke University of Leuven, Belgium.; D. Kreßner, V. Mehrmann, T. Penzl, CTLEX—a collection of benchmark examples for continuous-time Lyapunov equations, SLICOT Working Note, 1999, 6, Katholieke University of Leuven, Belgium. |
| [20] | D. Kreßner, V. Mehrmann, T. Penzl, DTLEX—a collection of benchmark examples for discrete-time Lyapunov equations’, SLICOT Working Note, 1999, 7, Katholieke University of Leuven, Belgium.; D. Kreßner, V. Mehrmann, T. Penzl, DTLEX—a collection of benchmark examples for discrete-time Lyapunov equations’, SLICOT Working Note, 1999, 7, Katholieke University of Leuven, Belgium. |
| [21] | Laub, A. J.; Patel, R. V.; Van Dooren, P. M., Numerical and computational issues in linear control and system theory, (Levine, W. S., The Control Handbook (1996), CRC Press, IEEE Press: CRC Press, IEEE Press Boca Raton, FL), 399-414 |
| [22] | Li, G.; Gevers, M., Roundoff noise minimization using delta-operator realizations, IEEE Trans. Signal Process., 41, 2, 629-637 (1993) · Zbl 0825.93877 |
| [23] | Middleton, R. H.; Goodwin, G. C., Improved finite word length characteristics in digital control using delta operators, IEEE Trans. Automat. Control, 31, 11, 1015-1021 (1986) · Zbl 0608.93053 |
| [24] | Middleton, R. H.; Goodwin, G. C., Digital Control and Estimation (1990), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ · Zbl 0636.93051 |
| [25] | Neuman, C. P., Properties of the delta operator model of dynamic physical systems, IEEE Trans. Systems, Man, and Cybernet., 23, 1, 296-301 (1993) · Zbl 0775.93172 |
| [26] | Petkov, P. H.; Christow, N. D.; Konstantinov, M. M., Computational Methods for Linear Control Systems (1991), Prentice-Hall: Prentice-Hall New York · Zbl 0790.93001 |
| [27] | P.H. Petkov, D.W. Gu, M.M. Konstantinov, V. Mehrmann, Condition and error estimates in the solution of Lyapunov and Riccati equations, NICONET Report, 2000, 1, Katholieke University of Leuven, Belgium.; P.H. Petkov, D.W. Gu, M.M. Konstantinov, V. Mehrmann, Condition and error estimates in the solution of Lyapunov and Riccati equations, NICONET Report, 2000, 1, Katholieke University of Leuven, Belgium. |
| [28] | Suchomski, P., Robust design in delta domain for SISO plantsphase advance and phase lag controllers, Systems Anal. Modelling Simulation, 41, 3, 503-549 (2001) · Zbl 1042.93025 |
| [29] | P. Suchomski, A \(JH_∞\); P. Suchomski, A \(JH_∞\) |
| [30] | Van Loan, C. F., Computing integrals involving the matrix exponential, IEEE Trans. Automat. Control, 23, 3, 395-404 (1978) · Zbl 0387.65013 |
| [31] | Wahlberg, B., The effects of rapid sampling in system identification, Automatica, 26, 1, 167-170 (1990) · Zbl 0714.93055 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
