Linear algebra properties of dissipative Hamiltonian descriptor systems. (English) Zbl 1403.15009
Summary: A wide class of matrix pencils connected with dissipative Hamiltonian descriptor systems is investigated. In particular, the following properties are shown: all eigenvalues are in the closed left half plane, the nonzero finite eigenvalues on the imaginary axis are semisimple, the index is at most two, and there are restrictions for the possible left and right minimal indices. For the case that the eigenvalue zero is not semisimple, a structure-preserving method is presented that perturbs the given system into a Lyapunov stable system.
MSC:
| 15A22 | Matrix pencils |
| 15A18 | Eigenvalues, singular values, and eigenvectors |
| 15A21 | Canonical forms, reductions, classification |
Keywords:
descriptor system; Hamiltonian system; matrix pencil; singular pencil; canonical form; Lyapunov stabilityReferences:
| [1] | A. Astolfi, R. Ortega, and A. Venkatraman, A globally exponentially convergent immersion and invariance speed observer for mechanical systems with nonholonomic constraints, Automatica, 46 (2010), pp. 182–189. · Zbl 1213.93090 |
| [2] | C. Beattie, V. Mehrmann, H. Xu, and H. Zwart, Port-Hamiltonian Descriptor Systems, Preprint 06-2017, Institut für Mathematik, TU Berlin, 2017. |
| [3] | P. C. Breedveld, Modeling and Simulation of Dynamic Systems using Bond Graphs, EOLSS Publishers/UNESCO, Oxford, UK, 2008, pp. 128–173. |
| [4] | K. E. Brenan, S. L. Campbell, and L. R. Petzold, Numerical Solution of Initial-Value Problems in Differential Algebraic Equations, 2nd ed., SIAM, Philadelphia, 1996. · Zbl 0844.65058 |
| [5] | L. Dai, Singular Control Systems, Lect. Notes Control Inf. Sci. 118, Springer-Verlag, Berlin, 1989. · Zbl 0669.93034 |
| [6] | F. De Terán, F. Dopico, and D. S. Mackey, Linearizations of singular matrix polynomials and the recovery of minimal indices, Electron. J. Linear Algebra, 18 (2009), pp. 371–402. · Zbl 1190.15015 |
| [7] | P. van Dooren, Reducing Subspaces: Definitions, Properties and Algorithms, Lecture Notes in Mathematics 973, Springer-Verlag, Berlin, 1983. · Zbl 0517.65022 |
| [8] | H. Egger, T. Kugler, B. Liljegren-Sailer, N. Marheineke, and V. Mehrmann, On structure-preserving model reduction for damped wave propagation in transport networks, SIAM J. Sci. Comput., 40 (2018), pp. A331–A365. · Zbl 1383.65110 |
| [9] | E. Emmrich and V. Mehrmann, Operator differential-algebraic equations arising in fluid dynamics, Comput. Methods Appl. Math., 13 (2013), pp. 443–470, . · Zbl 1392.34070 |
| [10] | C. Foias and A. E. Frazho, Positive definite block matrices, in The Commutant Lifting Approach to Interpolation Problems, Springer, New York, 1990, pp. 547–586. |
| [11] | R. W. Freund, The SPRIM algorithm for structure-preserving order reduction of general RCL circuits, in Model Reduction for Circuit Simulation, Springer, New York, 2011, pp. 25–52. |
| [12] | M. Froidevaux, A Structure Preserving Trimmed Linearization for Quadratic Eigenvalue Problems, Master thesis, École Polytechnique Fédérale de Lausanne, Switzerland, 2016. |
| [13] | K. Fujimoto, S. Sakai, and T. Sugie, Passivity based control of a class of Hamiltonian systems with nonholonomic constraints, Automatica, 48 (2012), pp. 3054–3063. · Zbl 1255.93046 |
| [14] | F. R. Gantmacher, Theory of Matrices, vol. 1, Chelsea, New York, 1959. · Zbl 0085.01001 |
| [15] | N. Gillis, V. Mehrmann, and P. Sharma, Computing nearest stable matrix pairs, Numer. Linear Algebra Appl., early view, 2018, e2153. |
| [16] | G. Golo, A. J. van der Schaft, P. C. Breedveld, and B. M. Maschke, Hamiltonian formulation of bond graphs, in Nonlinear and Hybrid Systems in Automotive Control, A. R. R. Johansson, ed., Springer, Heidelberg, 2003, pp. 351–372. |
| [17] | G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed, Johns Hopkins University Press, Baltimore, 1996. · Zbl 0865.65009 |
| [18] | N. Gräbner, V. Mehrmann, S. Quraishi, C. Schröder, and U. von Wagner, Numerical methods for parametric model reduction in the simulation of disc brake squeal, Z. Angew. Math. Mech., 96 (2016), pp. 1388–1405. · Zbl 1538.65081 |
| [19] | J. Y. Ishihara and M. H. Terra, Generalized Lyapunov theorems for rectangular descriptor systems, in Proceedings of the 40th IEEE Conference on Decision and Control, Orlando, FL, 2001, pp. 2858–2859. |
| [20] | J. Y. Ishihara and M. H. Terra, On the Lyapunov theorem for singular systems, IEEE Trans. Automat. Control, 47 (2002), pp. 1926–1930. · Zbl 1364.93570 |
| [21] | B. Jacob and H. Zwart, Linear Port-Hamiltonian Systems on Infinite-Dimensional Spaces, Oper. Theory Adv. Appl. 223, Birkhäuser/Springer, Cham, Switzerland, 2012. · Zbl 1254.93002 |
| [22] | P. Lancaster and M. Tismenetsky, The Theory of Matrices, 2nd ed., Academic Press, Orlando, FL, 1985. · Zbl 0558.15001 |
| [23] | J. Maddocks and M. Overton, Stability theory for dissipatively perturbed Hamiltonian systems, Commun. Pure Appl. Math., 48 (1995), pp. 583–610. · Zbl 0828.70010 |
| [24] | T. Maehara and K. Murota, Simultaneous singular value decomposition, Linear Algebra Appl., 435 (2011), pp. 106–116. · Zbl 1233.65034 |
| [25] | C. Mehl, V. Mehrmann, and P. Sharma, Stability radii for linear Hamiltonian systems with dissipation under structure-preserving perturbations, SIAM Matrix Anal. Appl., 37 (2016), pp. 1625–1654. · Zbl 1349.93332 |
| [26] | V. Mehrmann and F. Poloni, An inverse-free ADI algorithm for computing Lagrangian invariant subspaces, Numer. Lin. Alg. Appl., 23 (2016), pp. 147–168. · Zbl 1438.65079 |
| [27] | R. Ortega, A. J. van der Schaft, Y. Mareels, and B. M. Maschke, Putting energy back in control, Control Syst. Mag., 21 (2001), pp. 18–ñ33. |
| [28] | C. Paige and C. F. Van Loan, A Schur decomposition for Hamiltonian matrices, Linear Algebra Appl., 14 (1981), pp. 11–32. · Zbl 1115.15316 |
| [29] | L. Scholz, Condensed Forms for Linear Port-Hamiltonian Descriptor Systems, Preprint 09–2017, Institut für Mathematik, TU Berlin, 2017. |
| [30] | A. J. van der Schaft, Port-Hamiltonian systems: Network modeling and control of nonlinear physical systems, in Advanced Dynamics and Control of Structures and Machines, CISM Courses and Lect. 444, Springer Cham, Switzerland, 2004. |
| [31] | A. J. van der Schaft, Port-Hamiltonian systems: An introductory survey, in Proc. of the International Congress of Mathematicians, vol. III, Invited Lectures, J. L. V. M. Sanz-Sole and J. Verdura, eds., Madrid, Spain, 2006, pp. 1339–1365. · Zbl 1108.93012 |
| [32] | A. J. van der Schaft, Port-Hamiltonian Differential-Algebraic Systems, in Surveys in Differential-Algebraic Equations I, Springer-Verlag, New York, 2013, pp. 173–226. · Zbl 1275.34002 |
| [33] | T. Stykel, Analysis and Numerical Solution of Generalized Lyapunov Equations, thesis, Technische Universität Berlin, Berlin, Germany, 2002. · Zbl 1097.65074 |
| [34] | K. Takaba, N. Morihira, and T. Katayama, A generalized Lyapunov theorem for descriptor system, Syst. Control Lett., 24 (1995), pp. 49–51. · Zbl 0883.93035 |
| [35] | L. Taslaman, Strongly damped quadratic matrix polynomials, SIAM Matrix Anal. Appl., 36 (2015), pp. 461–475. · Zbl 1320.15007 |
| [36] | R. Thompson, The characteristic polynomial of a principal subpencil of a Hermitian matrix pencil, Linear Algebra Appl., 14 (1976), pp. 135–177. · Zbl 0386.15011 |
| [37] | F. Tisseur and K. Meerbergen, The quadratic eigenvalue problem, SIAM Rev., 43 (2001), pp. 235–286. · Zbl 0985.65028 |
| [38] | K. Veselić, Damped Oscillations of Linear Systems. A Mathematical Introduction, Springer-Verlag, Berlin, 2011. · Zbl 1232.37004 |
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.
