×

Solution formulas for differential Sylvester and Lyapunov equations. (English) Zbl 1432.15015

Summary: The differential Sylvester equation and its symmetric version, the differential Lyapunov equation, appear in different fields of applied mathematics like control theory, system theory, and model order reduction. The few available straight-forward numerical approaches when applied to large-scale systems come with prohibitively large storage requirements. This shortage motivates us to summarize and explore existing solution formulas for these equations. We develop a unifying approach based on the spectral theorem for normal operators like the Sylvester operator \(\mathcal{S}(X) = AX + XB\) and derive a formula for its norm using an induced operator norm based on the spectrum of \(A\) and \(B\). In view of numerical approximations, we propose an algorithm that identifies a suitable Krylov subspace using Taylor series and use a projection to approximate the solution. Numerical results for large-scale differential Lyapunov equations are presented in the last sections.

MSC:

15A24 Matrix equations and identities
65F45 Numerical methods for matrix equations

Software:

MESS; mftoolbox
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Abou-Kandil, H.; Freiling, G.; Ionescu, V.; Jank, G., Matrix Riccati Equations in Control and Systems Theory (2003), Basel: Birkhäuser, Basel · Zbl 1027.93001
[2] Amato, F.; Ambrosino, R.; Ariola, M.; Cosentino, C.; De Tommasi, G., Finite-Time Stability and Control (2014), London: Springer, London · Zbl 1297.93001
[3] Antoulas, Ac, Approximation of Large-Scale Dynamical Systems (2005), Philadelphia, PA: SIAM Publications, Philadelphia, PA · Zbl 1112.93002
[4] Benner, P., Köhler, M., Saak, J.: M.E.S.S.—the matrix equations sparse solvers library. https://www.mpi-magdeburg.mpg.de/projects/mess. Accessed 14 Nov 2019
[5] Benner, P.; Li, Rc; Truhar, N., On the ADI method for Sylvester equations, J. Comput. Appl. Math., 233, 4, 1035-1045 (2009) · Zbl 1176.65050 · doi:10.1016/j.cam.2009.08.108
[6] Benner, P., Mena, H.: BDF methods for large-scale differential Riccati equations. In: De Moor, B., Motmans, B., Willems, J., Van Dooren, P., Blondel, V. (eds.) Proceedings of the 16th International Symposium on Mathematical Theory of Network and Systems, MTNS 2004 (2004)
[7] Benner, P.; Mena, H., Rosenbrock methods for solving Riccati differential equations, IEEE Trans. Autom. Control, 58, 11, 2950-2957 (2013) · Zbl 1369.65088 · doi:10.1109/TAC.2013.2258495
[8] Benner, P.; Saak, J.; Benner, P.; Mehrmann, V.; Sorensen, D., A semi-discretized heat transfer model for optimal cooling of steel profiles, Dimension Reduction of Large-Scale Systems, 353-356 (2005), Berlin: Springer, Berlin · Zbl 1170.80341
[9] Brocket, Ra, Finite Dimensional Linear Systems (1970), New York: Wiley, New York · Zbl 0216.27401
[10] Byers, R.; Nash, S., On the singular “vectors” of the Lyapunov operator, SIAM J. Algebraic Discrete Methods, 8, 1, 59-66 (1987) · Zbl 0633.65042 · doi:10.1137/0608003
[11] Gajić, Z.; Qureshi, M., Lyapunov Matrix Equation in System Stability and Control (1995), San Diego, CA: Academic Press, San Diego, CA · Zbl 1153.93300
[12] Güldoǧan, Y., Hached, M., Jbilou, K., Kurulay, M.: Low rank approximate solutions to large-scale differential matrix Riccati equations. Technical report. arXiv:1612.00499v2, arXiv (2017). Math.NA
[13] Hached, M.; Jbilou, K., Numerical solutions to large-scale differential Lyapunov matrix equations, Numer. Algorithms, 79, 741-757 (2017) · Zbl 1416.65116 · doi:10.1007/s11075-017-0458-y
[14] Hached, M., Jbilou, K.: Approximate solutions to large nonsymmetric differential Riccati problems. Technical report. arXiv:1801.01291v1, arXiv (2018). Math.NA · Zbl 1416.65116
[15] Hairer, E.; Wanner, G., Solving Ordinary Differential Equations. I (1987), Berlin: Springer, Berlin · Zbl 0638.65058
[16] Heiland, J.: Decoupling and optimization of differential-algebraic equations with application in flow control. Dissertation, TU Berlin (2014)
[17] Higham, Nj, Functions of Matrices: Theory and Computation (2008), Philadelphia, PA: SIAM Publications, Philadelphia, PA · Zbl 1167.15001
[18] Jameson, A., Solution of the equation \(AX+XB=C\) by inversion of an \(M\times M\) or \(N\times N\) matrix, SIAM J. Appl. Math., 16, 1020-1023 (1968) · Zbl 0169.35202 · doi:10.1137/0116083
[19] Knobloch, Hw; Kwakernaak, H., Lineare Kontrolltheorie (1985), Berlin: Springer, Berlin · Zbl 0574.93001
[20] Kohaupt, L., Solution of the matrix eigenvalue problem \(VA+A^*V=\mu V\) with applications to the study of free linear dynamical systems, J. Comput. Appl. Math., 213, 1, 142-165 (2008) · Zbl 1142.65036 · doi:10.1016/j.cam.2007.01.001
[21] Köhler, M.; Lang, N.; Saak, J., Solving differential matrix equations using parareal, Proc. Appl. Math. Mech., 16, 1, 847-848 (2016) · doi:10.1002/pamm.201610412
[22] Konstantinov, M.; Mehrmann, V.; Petkov, P., On properties of Sylvester and Lyapunov operators, Linear Algebra Appl., 312, 1-3, 35-71 (2000) · Zbl 0962.15006 · doi:10.1016/S0024-3795(00)00082-3
[23] Konstantinov, Mm; Gu, Dw; Mehrmann, V.; Petkov, Ph, Perturbation Theory for Matrix Equations (2003), Amsterdam: Elsevier, Amsterdam · Zbl 1025.15017
[24] Koskela, A., Mena, H.: A structure preserving Krylov subspace method for large scale differential Riccati equations. e-print arXiv:1705.07507, arXiv (2017). Math.NA
[25] Lang, N.: Numerical methods for large-scale linear time-varying control systems and related differential matrix equations. Dissertation, Technische Universität Chemnitz, Germany (2017)
[26] Lang, N.; Mena, H.; Saak, J., On the benefits of the \(LDL^T\) factorization for large-scale differential matrix equation solvers, Linear Algebra Appl., 480, 44-71 (2015) · Zbl 1320.65110 · doi:10.1016/j.laa.2015.04.006
[27] Lang, N.; Saak, J.; Stykel, T., Towards practical implementations of balanced truncation for LTV systems, IFAC-PapersOnLine, 48, 1, 7-8 (2015) · doi:10.1016/j.ifacol.2015.05.135
[28] Lang, N.; Saak, J.; Stykel, T., Balanced truncation model reduction for linear time-varying systems, Math. Comput. Model. Dyn. Syst., 22, 4, 267-281 (2016) · Zbl 1417.93089 · doi:10.1080/13873954.2016.1198386
[29] Lang, N., Saak, J., Stykel, T.: LTV-BT for MATLAB (2017)
[30] Locatelli, A., Optimal Control: An Introduction (2001), Basel: Birkhäuser, Basel · Zbl 1096.49500
[31] Mena, H.: Numerical solution of differential Riccati equations arising in optimal control of partial differential equations. Dissertation, Escuela Politécnica Nacional, Ecuador (2007)
[32] Palmer, Tw, Banach Algebras and the General Theory of \(*\)-Algebras (2001), Cambridge: Cambridge University Press, Cambridge · Zbl 0983.46040
[33] Polderman, Jw; Willems, Jc, Introduction to Mathematical Systems Theory (1998), New York: Springer, New York
[34] Rome, H., A direct solution to the linear variance equation of a time-invariant linear system, IEEE Trans. Autom. Control, 14, 5, 592-593 (1969) · doi:10.1109/TAC.1969.1099271
[35] Stewart, Gw, Matrix Algorithms (2001), Philadelphia, PA: SIAM Publications, Philadelphia, PA · Zbl 0984.65031
[36] Stillfjord, T., Low-rank second-order splitting of large-scale differential Riccati equations, IEEE Trans. Autom. Control, 60, 10, 2791-2796 (2015) · Zbl 1360.65192 · doi:10.1109/TAC.2015.2398889
[37] Werner, D., Funktionalanalysis (2000), Berlin: Springer, Berlin · Zbl 0964.46001
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.