The ADI method for bounded real and positive real Lur’e equations. (English) Zbl 1361.65026
An algorithm (whose idea is based on the ADI iteration technique employed for algebraic Riccati equations), is proposed for the numerical solution of Lur’e equations. These equations are much more general than the well-studied algebraic Riccati equations. The algorithm yields approximate solutions in low-rank factored form. Each iteration consists of the solution of a linear system involving a shift parameter. The sequence of approximations is shown to be monotonically increasing and when one chooses the shift parameters appropriately, then it converges to the minimal solution of Lur’e equations. An illustrative numerical example is included to demonstrate the applicability of the developed technique.
Reviewer: Kanakadurga Sivakumar (Chennai)
MSC:
| 65F30 | Other matrix algorithms (MSC2010) |
| 15A24 | Matrix equations and identities |
| 93B52 | Feedback control |
| 65K10 | Numerical optimization and variational techniques |
| 65F10 | Iterative numerical methods for linear systems |
Keywords:
Lur’e equations; algebraic Riccati equations; approximate solutions; low-rank factored form; projected optimal control; alternating direction implicit (ADI); algorithm; numerical exampleReferences:
| [1] | Benner, P., Kürschner, P., Saak, J.: Self-generating and efficient shift parameters in ADI methods for large Lyapunov and Sylvester equations. Electron. Trans. Numer. Anal. 43, 142-162 (2014) · Zbl 1312.65068 |
| [2] | 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-Mitt. 36(1), 32-52 (2013) · Zbl 1279.65044 · doi:10.1002/gamm.201310003 |
| [3] | Curtain, R.F.: Linear operator inequalities for strongly stable weakly regular linear systems. Math. Control Signals Syst. 14(4), 299-338 (1997) · Zbl 1114.93029 · doi:10.1007/s498-001-8039-4 |
| [4] | Gantmacher, F.R.: The Theory of Matrices, vol. II. Chelsea, New York (1959) · Zbl 0085.01001 |
| [5] | Golub, G.H., Van Loan, C.F.: Matrix Computations, 3rd edn. The John Hopkins University Press, Baltimore (1996) · Zbl 0865.65009 |
| [6] | Ilchmann, A., Reis, T.: Outer transfer functions of differential-algebraic systems. ESAIM Control Optim. Calc. Var. doi:10.1051/cocv/2015051 (2016) · Zbl 1358.93051 |
| [7] | 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 |
| [8] | Lin, Y., Simoncini, V.: A new subspace iteration method for the algebraic Riccati equation. Numer. Linear Algebra Appl. 22(1), 26-47 (2015) · Zbl 1363.65076 · doi:10.1002/nla.1936 |
| [9] | Lu, A., Wachspress, E.L.: Solution of Lyapunov equations by alternating direction implicit iteration. Comput. Math. Appl. 21(9), 43-58 (1991) · Zbl 0724.65041 · doi:10.1016/0898-1221(91)90124-M |
| [10] | Massoudi, A., Opmeer, M.R., Reis, T.: Analysis of an iteration method for the algebraic Riccati equation. SIAM. J. Matrix Anal. Appl. https://preprint.math.uni-hamburg.de/public/papers/hbam/hbam2014-16 (2016, accepted for publication) · Zbl 1339.15010 |
| [11] | Opdenacker, P.C., Jonckheere, E.A.: A contraction mapping preserving balanced reduction scheme and its infinity norm error bounds. IEEE Trans. Circuits Syst. I Regul. Pap. 35(2), 184-189 (1988) · Zbl 0659.93013 |
| [12] | Guiver, C., Opmeer, M.R.: Error bounds in the gap metric for dissipative balanced approximations. Linear Algebra Appl. 439(12), 3659-3698 (2013) · Zbl 1280.93017 · doi:10.1016/j.laa.2013.09.032 |
| [13] | Penzl, T.: A cyclic low-rank Smith method for large sparse Lyapunov equations. SIAM J. Sci. Comput. 21(4), 1401-1418 (1999/2000) · Zbl 0958.65052 |
| [14] | Opmeer, M.R., Reis, T., Wollner, W.: Finite-rank ADI iteration for operator Lyapunov equations. SIAM J. Control Optim. 51(5), 4084-4117 (2013) · Zbl 1279.93022 · doi:10.1137/120885310 |
| [15] | Poloni, F., Reis, T.: A deflation approach for large-scale Lur’e equations. SIAM J. Matrix Anal. Appl. 33(4), 1339-1368 (2012) · Zbl 1263.15015 · doi:10.1137/120861679 |
| [16] | Poloni, F., Reis, T.: A structured doubling algorithm for Lur’e equations. Numer. Linear Algebra Appl. 23(1), 169-186 (2016) · Zbl 1413.65102 · doi:10.1002/nla.2019 |
| [17] | Reis, T.: Lur’e equations and even matrix pencils. Linear Algebra Appl. 434, 152-173 (2011) · Zbl 1208.15016 · doi:10.1016/j.laa.2010.09.005 |
| [18] | Reis, T., Stykel, T.: Positive real and bounded real balancing for model reduction of descriptor systems. Int. J. Control 83(1), 74-88 (2010) · Zbl 1184.93026 · doi:10.1080/00207170903100214 |
| [19] | Saad, Y.: Numerical Methods for Large Eigenvalue Problems. Manchester University Press, Manchester (1992) · Zbl 0991.65039 |
| [20] | Sabino, J.: Solution of Large-Scale Lyapunov Equations via the Block Modified Smith Method. Ph.D. thesis, Rice University (2006) |
| [21] | Wachspress, E.L.: Iterative solution of the Lyapunov matrix equation. Appl. Math. Lett. 1, 87-90 (1988) · Zbl 0631.65037 · doi:10.1016/0893-9659(88)90183-8 |
| [22] | Weidmann, J.: Linear Operators in Hilbert Spaces. Springer, New York (1980) · Zbl 0434.47001 · doi:10.1007/978-1-4612-6027-1 |
| [23] | Weiss, M., Weiss, G.: Optimal control of stable weakly regular linear systems. Math. Control Signals Syst. 10(4), 287-330 (1997) · Zbl 0884.49021 · doi:10.1007/BF01211550 |
| [24] | Willems, J.C.: Least squares stationary optimal control and the algebraic Riccati equation. IEEE Trans. Autom. Control 16, 621-634 (1971) · doi:10.1109/TAC.1971.1099831 |
| [25] | Willems, J.C.: Dissipative dynamical systems. Part I: general theory. Arch. Ration. Mech. Anal. 45, 321-351 (1972) · Zbl 0252.93002 · doi:10.1007/BF00276493 |
| [26] | Willems, J.C.: Linear systems with quadratic supply rates. Part II: dissipative dynamical systems. Arch. Ration. Mech. Anal. 45, 352-393 (1972) · Zbl 0252.93003 · doi:10.1007/BF00276494 |
| [27] | Zhou, K., Doyle, J.D., Glover, K.: Robust and Optimal Control. Prentice-Hall, Princeton (1996) · Zbl 0999.49500 |
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.
