# zbMATH — the first resource for mathematics

Faster and more accurate computation of the $$\mathcal{H}_\infty$$ norm via optimization. (English) Zbl 1401.93087

##### MSC:
 93B36 $$H^\infty$$-control 93C05 Linear systems in control theory 93C55 Discrete-time control/observation systems 93A15 Large-scale systems 93B40 Computational methods in systems theory (MSC2010) 93D09 Robust stability 93-04 Software, source code, etc. for problems pertaining to systems and control theory
##### Software:
Algorithm 961; DPA_TDEFL; HIFOO; PSAPSR; QDPA
Full Text:
##### References:
 [1] N. Aliyev, P. Benner, E. Mengi, P. Schwerdtner, and M. Voigt, Large-scale computation of $$\mathcal{L}_∞$$-norms by a greedy subspace method, SIAM J. Matrix Anal. Appl., 38 (2017), pp. 1496–1516. · Zbl 1379.65020 [2] A. C. Antoulas, Approximation of Large-Scale Dynamical Systems, Adv. Des. Control 6, SIAM, Philadelphia, 2005. · Zbl 1112.93002 [3] M. N. Belur and C. Praagman, An efficient algorithm for computing the $${H}_∞$$ norm, IEEE Trans. Automat. Control, 56 (2011), pp. 1656–1660. · Zbl 1368.93143 [4] P. Benner, R. Byers, V. Mehrmann, and H. Xu, Numerical computation of deflating subspaces of skew-Hamiltonian/Hamiltonian pencils, SIAM J. Matrix Anal. Appl., 24 (2002), pp. 165–190. · Zbl 1035.49022 [5] P. Benner, A. Cohen, M. Ohlberger, and K. Willcox, Model Reduction and Approximation: Theory and Algorithms, Comput. Sci. Eng. 15, SIAM, Philadelphia, 2017. · Zbl 1378.65010 [6] P. Benner and T. Mitchell, Extended and Improved Criss-Cross Algorithms for Computing the Spectral Value Set Abscissa and Radius, arXiv:1712.10067, 2017. [7] P. Benner, V. Sima, and M. Voigt, $$\mathcal{L}_∞$$-norm computation for continuous-time descriptor systems using structured matrix pencils, IEEE Trans. Automat. Control, 57 (2012), pp. 233–238. · Zbl 1369.93174 [8] P. Benner, V. Sima, and M. Voigt, Algorithm 961: Fortran 77 subroutines for the solution of skew-Hamiltonian/Hamiltonian eigenproblems, ACM Trans. Math. Software, 42 (2016). [9] P. Benner and M. Voigt, On the computation of particular eigenvectors of Hamiltonian matrix pencils, Proc. Appl. Math. Mech., 11 (2011), pp. 753–754. [10] P. Benner and M. Voigt, A structured pseudospectral method for $$\mathcal{H}_∞$$-norm computation of large-scale descriptor systems, Math. Control Signals Systems, 26 (2014), pp. 303–338. · Zbl 1290.93083 [11] S. Boyd and V. Balakrishnan, A regularity result for the singular values of a transfer matrix and a quadratically convergent algorithm for computing its $${L}_∞$$-norm, Systems Control Lett., 15 (1990), pp. 1–7. · Zbl 0704.93014 [12] S. Boyd, V. Balakrishnan, and P. Kabamba, A bisection method for computing the $$\mathcal{H}_{∞}$$ norm of a transfer matrix and related problems, Math. Control Signals Systems, 2 (1989), pp. 207–219. · Zbl 0674.93020 [13] N. A. Bruinsma and M. Steinbuch, A fast algorithm to compute the $$H{∞}$$-norm of a transfer function matrix, Systems Control Lett., 14 (1990), pp. 287–293. · Zbl 0699.93021 [14] J. V. Burke, D. Henrion, A. S. Lewis, and M. L. Overton, HIFOO—A MATLAB package for fixed-order controller design and $$H_∞$$ optimization, in 5th IFAC Symposium on Robust Control Design ROCOND 2006, IFAC Proc. Vol. 39, 2006, pp. 339–344. [15] R. Byers, A bisection method for measuring the distance of a stable to unstable matrices, SIAM J. Sci. Statist. Comput., 9 (1988), pp. 875–881. · Zbl 0658.65044 [16] M. A. Freitag, A. Spence, and P. Van Dooren, Calculating the $$H_∞$$-norm using the implicit determinant method, SIAM J. Matrix Anal. Appl., 35 (2014), pp. 619–635. · Zbl 1305.65161 [17] Y. Genin, P. Van Dooren, and V. Vermaut, Convergence of the calculation of $$\mathcal{H}_∞$$-norms and related questions, in Proceedings of MTNS-98, 1998, pp. 429–432. [18] G. H. Golub and C. F. Van Loan, Matrix Computations, 4th ed., Johns Hopkins Stud. Math. Sci., Johns Hopkins University Press, Baltimore, 2013. [19] N. Guglielmi, M. Gürbüzbalaban, T. Mitchell, and M. L. Overton, Approximating the real structured stability radius with Frobenius-norm bounded perturbations, SIAM J. Matrix Anal. Appl., 38 (2017), pp. 1323–1353. · Zbl 1379.65040 [20] N. Guglielmi, M. Gürbüzbalaban, and M. L. Overton, Fast approximation of the $$H_∞$$ norm via optimization over spectral value sets, SIAM J. Matrix Anal. Appl., 34 (2013), pp. 709–737. · Zbl 1271.93057 [21] N. Guglielmi and M. L. Overton, Fast algorithms for the approximation of the pseudospectral abscissa and pseudospectral radius of a matrix, SIAM J. Matrix Anal. Appl., 32 (2011), pp. 1166–1192. · Zbl 1248.65034 [22] C. He and G. A. Watson, An algorithm for computing the distance to instability, SIAM J. Matrix Anal. Appl., 20 (1999), pp. 101–116. · Zbl 0927.65055 [23] D. Hinrichsen and A. J. Pritchard, Mathematical Systems Theory I: Modelling, State Space Analysis, Stability and Robustness, Springer-Verlag, Berlin, 2005. · Zbl 1074.93003 [24] D. Hinrichsen and N. K. Son, Stability radii of linear discrete-time systems and symplectic pencils, Internat. J. Robust Nonlinear Control, 1 (1991), pp. 79–97. · Zbl 0754.93060 [25] T. Kato, A Short Introduction to Perturbation Theory for Linear Operators, Springer-Verlag, New York, 1982. · Zbl 0493.47008 [26] P. Lancaster, On eigenvalues of matrices dependent on a parameter, Numer. Math., 6 (1964), pp. 377–387. · Zbl 0133.26201 [27] F. Leibfritz, $$COMPl_{e}ib$$: COnstrained Matrix-optimization Problem library—a collection of test examples for nonlinear semidefinite programs, control system design and related problems, (2004). [28] T. Mitchell and M. L. Overton, Hybrid expansion-contraction: a robust scaleable method for approximating the $$H_∞$$ norm, IMA J. Numer. Anal., 36 (2016), pp. 985–1014. · Zbl 1433.93100 [29] M. L. Overton and R. S. Womersley, Second derivatives for optimizing eigenvalues of symmetric matrices, SIAM J. Matrix Anal. Appl., 16 (1995), pp. 697–718. · Zbl 0832.65036 [30] A. P. Popov, H. Werner, and M. Millstone, Fixed-structure discrete-time $$\mathcal{H}_∞$$ controller synthesis with HIFOO, in Proceedings of the 49th IEEE Conference on Decision and Control, 2010, pp. 3152–3155. [31] J. Rommes and N. Martins, Computing transfer function dominant poles of large-scale second-order dynamical systems, IEEE Trans. Power Syst., 21 (2006), pp. 1471–1483. [32] J. Sreedhar, P. Van Dooren, and A. Tits, A fast algorithm to compute the real structured stability radius, in Proceedings of the Conference Centennial Hurwitz on Stability Theory, Ticino, 1995. [33] A. Varga, Computation of irreducible generalized state-space realizations, Kybernetika (Prague), 26 (1990), pp. 89–106. · Zbl 0715.93030
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.