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A structured pseudospectral method for $$\mathcal {H}_{\infty}$$-norm computation of large-scale descriptor systems. (English) Zbl 1290.93083
Summary: In this paper, we discuss the problem of computing the $$\mathcal {H}_{\infty}$$-norm of transfer functions associated to large-scale descriptor systems. We exploit the relationship between the $$\mathcal {H}_{\infty}$$-norm and the structured complex stability radius of a corresponding matrix pencil. To compute the structured stability radius we consider so-called structured pseudospectra. Namely, we have to find the pseudospectrum touching the imaginary axis. Therefore, we set up an iteration over the real part of the rightmost pseudo-eigenvalue. For that, we use a new fast iterative scheme which is based on certain rank-1 perturbations of a matrix pencil. Finally, we analyze the performance of our algorithm by using real-world examples. In particular we compare our method with different other algorithms including a recently and independently derived method from Guglielmi, Gürbüzbalaban and Overton.

##### MSC:
 93B60 Eigenvalue problems 93B36 $$H^\infty$$-control 93A15 Large-scale systems 93C15 Control/observation systems governed by ordinary differential equations 93C05 Linear systems in control theory
##### Software:
Eigtool; benchmodred; Seigtool; PSAPSR; PABTEC
Full Text:
##### References:
 [1] Benner, P; Byers, R; Mehrmann, V; Xu, H, Numerical computation of deflating subspaces of skew-Hamiltonian/Hamiltonian pencils, SIAM J Matrix Anal Appl, 24, 165-190, (2002) · Zbl 1035.49022 [2] Benner P, Saak J, Schieweck F, Skrzypacz P, Weichelt HK (2012) A non-conforming composite quadrilateral finite element pair for feedback stabilization of the Stokes equations. J Numer Math. Accepted, also available as Max Planck Institute Magdeburg, Preprint MPIMD/12-19 · Zbl 1303.65095 [3] Benner P, Sima V, Voigt M (2012) $${\cal L}_∞$$-norm computation for continuous-time descriptor systems using structured matrix pencils. IEEE Trans Automat Control 57:233-238 · Zbl 1369.93174 [4] Benner P, Sima V, Voigt M (2012) Robust and efficient algorithms for $${\cal L}_∞$$-norm computation for descriptor systems. In: Proceedings of the 7th IFAC Symposium on Robust Control Design, Aalborg, Denmark, IFAC, pp 195-200 · Zbl 0805.15005 [5] Benner, P; Voigt, M, $${\cal {H}}_∞$$-norm computation for large and sparse descriptor systems, Proc Appl Math Mech, 12, 797-800, (2012) [6] Benner P, Voigt M (2012) Numerical computation of structured complex stability radii od large-scale matrices and pencils. In: Proceedings of the 51st IEEE Conference on Decision and Control. Maui, Hawaii, pp 6560-6565 [7] Boyd, S; Balakrishnan, V, A regularity result for the singular values of a transfer matrix and a quadratically convergent algorithm for computing its $$L_{∞ }$$-norm, Syst Control Lett, 15, 1-7, (1990) · Zbl 0704.93014 [8] Boyd, S; Balakrishnan, V; Kabamba, P, A bisection method for computing the $$H_∞$$ norm of a transfer matrix and related problems, Math Control Signals Syst, 2, 207-219, (1989) · Zbl 0674.93020 [9] Bruinsma, NA; Steinbuch, M, A fast algorithm to compute the $$H_{∞ }$$-norm of a transfer function matrix, Syst Control Lett, 14, 287-293, (1990) · Zbl 0699.93021 [10] Burke, JV; Lewis, AS; Overton, ML, Robust stability and a criss-cross algorithm for pseudospectra, IMA J Numer Anal, 23, 359-375, (2003) · Zbl 1042.65060 [11] Byers, R, A bisection method for measuring the distance of a stable matrix to the unstable matrices, SIAM J Sci Stat Comput, 9, 875-881, (1988) · Zbl 0658.65044 [12] Byers R, Nichols NK (1993) On the stability radius of generalized state-space systems. Linear Algebra Appl 188-189:113-134 · Zbl 0783.65056 [13] Chahlaoui Y, Gallivan K, Van Dooren P (2004) $${\cal H}_∞$$-norm calculations of large sparse systems. In: Proceedings of the International Symposium of Mathematical Theory of Networks and Systems. Leuven, Belgium · Zbl 1359.93134 [14] Chahlaoui Y, Gallivan K, Van Dooren P (Oct. 2007) Calculating the $${\cal H}_∞$$ norm of a large sparse system via Chandrasekhar iterations and extrapolation. In: ESAIM Proceedings, vol 20. Rabat, Algeria · Zbl 1359.93134 [15] Y. Chahlaoui, Van Dooren P (2002) A collection of benchmark examples for model reduction of linear time invariant dynamical systems. Technical report, Feb. 2002. SLICOT Working Note 2002-2 · Zbl 1100.93006 [16] Dai L (1989) Singular control systems, vol. 118 of Lecture Notes in Control and Inform. Sci. Springer, Heidelberg · Zbl 0669.93034 [17] Du, NH, Stability radii of differential algebraic equations with structured perturbations, Syst Control Lett, 57, 546-553, (2008) · Zbl 1140.93027 [18] Du NH, Linh VH, Mehrmann V (2013) Robust stability of differential-algebraic equations. In Surveys in Differential-Algebraic Equations I. In: Ilchmann A, Reis T (eds) Differential-Algebraic Equations Forum. Springer, Berlin, Heidelberg, ch. 2, pp 63-95 · Zbl 1281.34002 [19] Du, NH; Thuan, DD; Liem, NC, Stability radius of implicit dynamic equations with constant coefficients on time scales, Syst Control Lett, 60, 596-603, (2011) · Zbl 1236.93119 [20] Eich-Soellner E, Führer C (1998) Numerical methods in multibody dynamics. B. G. Teubner, Stuttgart · Zbl 0899.70001 [21] Freitas, F; Rommes, J; Martins, N, Gramian-based reduction method applied to large sparse power system descriptor models, IEEE Trans Power Syst, 23, 1258-1270, (2008) [22] Guglielmi, N; Gürbüzbalaban, M; Overton, ML, Fast approximation of the $$H_∞$$ norm via optimization of spectral value sets, SIAM J Matrix Anal Appl, 34, 709-737, (2013) · Zbl 1271.93057 [23] Guglielmi N, Kressner D, Lubich C (2013) Low-rank differential equations for Hamiltonian matrix nearness problems, Oberwolfach Preprint OWP 2013-01, Mathematisches Forschungsinstitut Oberwolfach, Jan. 2013. Available from http://www.mfo.de/scientific-programme/publications/owp/2013/OWP2013_01.pdf · Zbl 0699.93021 [24] Guglielmi N, Lubich C (2013) Low-rank dynamics for computing extremal points of real pseudospectra. SIAM J Matrix Anal Appl (in press) · Zbl 1272.65032 [25] Guglielmi N, Overton ML Local convergence analysis of [22]- private communication with M. L. Overton [26] Guglielmi, N; Overton, ML, Fast algorithms for the approximation of the pseudospectral abscissa and pseudospectral radius of a matrix, SIAM J Matrix Anal Appl, 32, 1166-1192, (2011) · Zbl 1248.65034 [27] Heinkenschloss, M; Sorensen, DC; Sun, K, Balanced truncation model reduction for a class of descriptor systems with application to the Oseen equations, SIAM J Sci Comput, 30, 1038-1063, (2008) · Zbl 1216.76015 [28] Hinrichsen D, Pritchard AJ (1986) Stability radii of linear systems. Syst Control Lett 7:1-10 · Zbl 0631.93064 [29] Hinrichsen, D; Pritchard, AJ, Stability radius for structured perturbations and the algebraic Riccati equation, Syst Control Lett, 8, 105-113, (1986) · Zbl 0626.93054 [30] Hinrichsen D, Pritchard AJ (1990) Real and complex stability radii: a survey. In: Progress in systems and Control Theory, vol 6. Birkhäuser, Boston, pp 119-162 · Zbl 0729.93056 [31] Kunkel P, Mehrmann V (2006) Differential-Algebraic Equations—Analysis and Numerical Solution. In: Textbooks in Mathematics. European Mathematical Society, Zürich · Zbl 1095.34004 [32] Leibfritz F (2004) Compl$$_{e}$$ib: Constraint matrix-optimization problem library—a collection of test examples for nonlinear semidefinite programs, control system design and related problems. Technical report, 2004. Available from http://www.friedemann-leibfritz.de/COMPlib_Data/COMPlib_Main_Paper.pdf · Zbl 0704.93014 [33] Leibfritz F, Lipinski W (2004) Compl$$_{e}$$ib 1.0 - user manual and quick reference. Technical report, 2004. Available from http://www.friedemann-leibfritz.de/COMPlib_Data/COMPlib_User_Guide.pdf · Zbl 1042.65060 [34] Martins, N; Pellanda, PC; Rommes, J, Computation of transfer function dominant zeros with applications to oscillation damping control of large power systems, IEEE Trans Power Syst, 22, 1657-1664, (2007) [35] Mehrmann, V; Schröder, C; Simoncini, V, An implicitly-restarted Krylov subspace method for real symmetric/skew-symmetric eigenproblems, Linear Algebra Appl, 436, 4070-4087, (2012) · Zbl 1247.65050 [36] Mehrmann V, Stykel T (2005) Balanced truncation model reduction for large-scale systems in descriptor form. In: Benner P, Mehrmann V, Sorensen D (eds) Dimension Reduction of Large-Scale Systems, vol 45 of Lecture Notes Comput. Sci. Eng. Springer, Berlin, Heidelberg, New York, ch. 3, pp 89-116 · Zbl 1107.93013 [37] Overton ML, Van Dooren P (2005) On computing the complex passivity radius. In: Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference 2005, pp 7960-7964 [38] Reis, T, Circuit synthesis of passive descriptor systems—a modified nodal approach, Int J Circ Theor Appl, 38, 44-68, (2010) · Zbl 1191.94159 [39] Reis T, Stykel T (2010) PABTEC: Passivity-preserving balanced truncation for electrical circuits. IEEE Trans Computer-Aided Design Integr Circuits Syst 29:1354-1367 [40] Reis, T; Stykel, T, Lyapunov balancing for passivity-preserving model reduction of RC circuits, SIAM J Appl Dyn Syst, 10, 1-34, (2011) · Zbl 1213.78038 [41] Riedel, KS, Generalized epsilon-pseudospectra, SIAM J Numer Anal, 31, 1219-1225, (1994) · Zbl 0805.15005 [42] Rommes, J, Arnoldi and Jacobi-Davidson methods for generalized eigenvalue problems $$Ax=λ Bx$$ with singular $$B$$, Math Comp, 77, 995-1015, (2008) · Zbl 1133.65020 [43] Rommes, J; Martins, N, Efficient computation of multivariate transfer function dominant poles using subspace acceleration, IEEE Trans Power Syst, 21, 1471-1483, (2006) [44] Rommes, J; Martins, N, Efficient computation of transfer function dominant poles using subspace acceleration, IEEE Trans Power Syst, 21, 1218-1226, (2006) [45] Rommes, J; Sleijpen, GLG, Convergence of the dominant pole algorithm and Rayleigh quotient iteration, SIAM J Matrix Anal Appl, 30, 346-363, (2008) · Zbl 1165.65016 [46] Stewart GW, Sun J-G (1990) Matrix Perturbation Theory. In: Computer Science and Scientific Computing. Academic Press, London [47] Weickert J (1997) Applications of the Theory of Differential-Algebraic Equations to Partial Differential Equations of Fluid Dynamics. PhD thesis, Chemnitz University of Technology, Department of Mathematics, Germany · Zbl 0917.65083 [48] Wright TG (2002) Eigtool, 2002. Available from http://www.comlab.ox.ac.uk/pseudospectra/eigtool/ [49] Zhou K, Doyle JD (1998) Essentials of Robust Control, 1st edn. Prentice Hall, Upper Saddle River
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