Low-rank iterative methods for periodic projected Lyapunov equations and their application in model reduction of periodic descriptor systems. (English) Zbl 1305.65158

Summary: We discuss the numerical solution of large-scale sparse projected periodic discrete-time Lyapunov equations in lifted form which arise in model reduction of periodic descriptor systems. We extend the alternating direction implicit method and the Smith method to such equations. Low-rank versions of these methods are also presented, which can be used to compute low-rank approximations to the solutions of projected periodic Lyapunov equations in lifted form with low-rank right-hand side. Moreover, we consider an application of the Lyapunov solvers to balanced truncation model reduction of periodic discrete-time descriptor systems. Numerical results are given to illustrate the efficiency and accuracy of the proposed methods.


65K10 Numerical optimization and variational techniques
93C55 Discrete-time control/observation systems
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[1] Amodio, P., Cash, J.R., Roussos, G., Wright, R., Fairweather, G., Gladwell, I., Kraut, G., Paprzycki, M.: Almost block diagonal linear systems: sequential and parallel solution techniques,and applications. Numer. Linear Algebra Appl. 7, 275-317 (2000) · Zbl 1051.65018 · doi:10.1002/1099-1506(200007/08)7:5<275::AID-NLA198>3.0.CO;2-G
[2] Benner, P., Faßbender, H.: On the numerical solution of large-scale sparse discrete-time Riccati equations. Adv. Comput. Math. 35(2-4), 119-147 (2011) · Zbl 1230.65070 · doi:10.1007/s10444-011-9174-7
[3] Benner, P., Hossain, M.S., Stykel, T.: Model reduction of periodic descriptor systems using balanced truncation. In: Benner, P., Hinze, M., ter Maten, J. (eds.) Model Reduction in Circuit Simulation, Lecture Notes in Electrical Engineering, vol. 74, pp. 187-200. Springer-Verlag, Berlin (2011)
[4] Benner, P., Kürschner, P., Saak, J.: An improved numerical method for balanced truncation for symmetric second order systems. Math. Comput. Model. Dyn. Syst. doi:10.1080/13.873954.2013.794363 (to appear) · Zbl 1305.93043
[5] Benner, P., Quintana-Ortí, E.: Model reduction based on spectral projection methods. In: Benner, P., Mehrmann, V., Sorensen, D. (eds.) Dimension Reduction of Large-Scale Systems, Lecture Notes in Computational Science and Engineering, vol. 45, pp. 5-45. Springer-Verlag, Berlin (2005) · Zbl 1106.93015
[6] Benner, P., Saak, J.: Numerical solution of large and sparse continuous time algebraic matrix Riccati and Lyapunov equations: a state of the art survey. GAMM-Mitteilungen 36(1), 32-52 (2013) · Zbl 1279.65044 · doi:10.1002/gamm.201310003
[7] Bittanti, S., Colaneri, P.: Invariant representations of discrete-time periodic systems. Automatica 36, 1777-1793 (2000) · Zbl 0963.93053 · doi:10.1016/S0005-1098(00)00087-X
[8] Calvetti, D., Levenberg, N., Reichel, L.: Iterative methods for X−\(AXB=CX-AXB=C\). J. Comput. Appl. Math. 86, 73-101 (1997) · Zbl 0890.65040 · doi:10.1016/S0377-0427(97)00149-0
[9] Chu, E.W., Fan, H.Y., Lin, W.W.: Reachability and observability of periodic descriptor systems. Preprint NCTS. National Tsing Hua University, Hsinchu (2005) · Zbl 1187.93008
[10] Chu, E.W., Fan, H.Y., Lin, W.W.: Projected generalized discrete-time periodic Lyapunov equations and balanced realization of periodic descriptor systems. SIAM J. Matrix Anal. Appl. 29(3), 982-1006 (2007) · Zbl 1140.93418 · doi:10.1137/040606715
[11] Fairweather, G., Gladwell, I.: Algorithms for almost block diagonal linear systems. SIAM Rev. 44(1), 49-58 (2004) · Zbl 1062.65031 · doi:10.1137/S003614450240506X
[12] Farhood, M., Beck, C., Dullerud, G.: Model reduction of periodic systems: a lifting approach. Automatica 41, 1085-1090 (2005) · Zbl 1091.93003 · doi:10.1016/j.automatica.2005.01.008
[13] Gugercin, S., Sorensen, D., Antoulas, A.: A modified low-rank Smith method for large-scale Lyapunov equations. Numer. Algoritm. 32(1), 27-55 (2003) · Zbl 1034.93020 · doi:10.1023/A:1022205420182
[14] Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, Series in Computational Mathematics, vol. 14. Springer-Verlag, Berlin (1996) · Zbl 1192.65097 · doi:10.1007/978-3-642-05221-7
[15] Harlow, F., Welch, J.: Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface. Phys. Fluids 8, 2182-2189 (1965) · Zbl 1180.76043 · doi:10.1063/1.1761178
[16] Jbilou, K., Riquet, A.: Projection methods for large Lyapunov matrix equations. Linear Algebra Appl. 415(2-3), 344-358 (2006) · Zbl 1094.65039 · doi:10.1016/j.laa.2004.11.004
[17] Kressner, D.: Large periodic Lyapunov equations: algorithms and applications. In: Proceedings of ECC03, Cambridge, UK (2003)
[18] Li, J.R., White, J.: Low rank solution of Lyapunov equations. SIAM J. Matrix Anal. Appl. 24(1), 260-280 (2002) · Zbl 1016.65024 · doi:10.1137/S0895479801384937
[19] Lu, A., Wachspress, E.: Solution of Lyapunov equations by alternating direction implicit iteration. Comput. Math. Appl. 21, 43-58 (1991) · Zbl 0724.65041 · doi:10.1016/0898-1221(91)90124-M
[20] Mehrmann, V.: A step toward a unified treatment of continuous and discrete-time control problems. Linear Algebra Appl. 241-243, 749-779 (1996) · Zbl 0859.93013 · doi:10.1016/0024-3795(95)00257-X
[21] Misra, P.: Time-invariant representation of discrete periodic systems. Automatica 32(2), 267-272 (1996) · Zbl 0844.93023 · doi:10.1016/0005-1098(96)85558-0
[22] Park, B., Verriest, E.: Canonical forms of discrete linear periodically time-varying systems and a control application. In: Proceedings of the 28th Conference on Decision and Control, Tampa, FL, pp. 1220-1225 (1989)
[23] Peaceman, D., Rachford, H.: The numerical solution of elliptic and parabolic differential equations. J. Soc. Indust. Appl. Math. 3(xix,40), 28-41 (1955) · Zbl 0067.35801 · doi:10.1137/0103003
[24] Penzl, T.: A cyclic low rank Smith method for large sparse Lyapunov equations. SIAM J. Sci. Comput. 21, 1401-1418 (2000) · Zbl 0958.65052 · doi:10.1137/S1064827598347666
[25] Saad, Y.: Iterative Methods for Sparse Linear Systems. PWS Publishing Company, Boston (1996) · Zbl 1031.65047
[26] Simoncini, V.: A new iterative method for solving large-scale Lyapunov matrix equations. SIAM J. Sci. Comput. 29(3), 1268-1288 (2007) · Zbl 1146.65038 · doi:10.1137/06066120X
[27] Smith, R.: Matrix equation XA+\(BX=CXA+BX=C\). SIAM J. Appl. Math. 16, 198-201 (1968) · Zbl 0157.22603 · doi:10.1137/0116017
[28] Sreedhar, J., Dooren, P.V.: Periodic descriptor systems: solvability and conditionability. IEEE Trans. Automat. Control 44(2), 310-313 (1999) · Zbl 0958.93061 · doi:10.1109/9.746256
[29] Stykel, T.: Gramian-based model reduction for descriptor systems. Math. Control Signals Syst. 16, 297-319 (2004) · Zbl 1067.93011 · doi:10.1007/s00498-004-0141-4
[30] Stykel, T.: On some norms for descriptor systems. IEEE Trans. Automat. Control 51(5), 842-847 (2006) · Zbl 1366.93064 · doi:10.1109/TAC.2006.875010
[31] Stykel, T.: Low-rank iterative methods for projected generalized Lyapunov equations. Electron. Trans. Numer. Anal. 30, 187-202 (2008) · Zbl 1171.65385
[32] Stykel, T., Simoncini, V.: Krylov subspace methods for projected Lyapunov equations. Appl. Numer. Math. 62, 35-50 (2012) · Zbl 1258.65044 · doi:10.1016/j.apnum.2011.09.007
[33] Varga, A.: Periodic Lyapunov equations: some applications and new algorithms. Internat. J. Control 67(1), 69-87 (1997) · Zbl 0873.93057 · doi:10.1080/002071797224360
[34] Varga, A.: Balancing related methods for minimal realization of periodic systems. Syst. Control Lett. 36, 339-349 (1999) · Zbl 0915.93021 · doi:10.1016/S0167-6911(98)00107-8
[35] Varga, A.: Balanced truncation model reduction of periodic systems. In: Proceedings of CDC’2000, Sydney, Australia (2000) · Zbl 1067.93011
[36] Varga, A.: Computation of Kronecker-like forms of periodic matrix pairs. In: Proceedings of Mathematical Theory of Networks and Systems (MTNS 2004, Leuven, Belgium, 5-9 July 2004) (2004) · Zbl 1366.93064
[37] Varga, A.: A Periodic Systems Toolbox for MATLAB. In: Proceedings of IFAC’05 World Congress, Prague, Czech Republik, 3-8 July 2005 · Zbl 0958.93061
[38] Varga, A.: An overview of recent developments in computational methods for periodic systems. In: Proceedings of the Third IFAC Workshop on Periodic Control Systems (St. Petersburg, Russia, 29-31 August 2007) (2007) · Zbl 0157.22603
[39] Varga, A., Van Dooren, P.: Computing the zeros of periodic descriptor systems. Syst. Control Lett. 50, 371-381 (2003) · Zbl 1157.93379 · doi:10.1016/S0167-6911(03)00194-4
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