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Control of port-Hamiltonian differential-algebraic systems and applications

Part of: Numerical analysis: Ordinary differential equations

Published online by Cambridge University Press:  11 May 2023

Volker Mehrmann Open the ORCID record for Volker Mehrmann [Opens in a new window]  and
Benjamin Unger
Volker Mehrmann
Affiliation:
Institut für Mathematik, Technische Universität Berlin, Strasse des 17. Juni 136, 10623 Berlin, Germany E-mail: mehrmann@math.tu-berlin.de
Benjamin Unger
Affiliation:
Stuttgart Center for Simulation Science, University of Stuttgart, Universitätsstrasse 32, 70569 Stuttgart, Germany E-mail: benjamin.unger@simtech.uni-stuttgart.de
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Abstract

We discuss the modelling framework of port-Hamiltonian descriptor systems and their use in numerical simulation and control. The structure is ideal for automated network-based modelling since it is invariant under power-conserving interconnection, congruence transformations and Galerkin projection. Moreover, stability and passivity properties are easily shown. Condensed forms under orthogonal transformations present easy analysis tools for existence, uniqueness, regularity and numerical methods to check these properties.

After recalling the concepts for general linear and nonlinear descriptor systems, we demonstrate that many difficulties that arise in general descriptor systems can be easily overcome within the port-Hamiltonian framework. The properties of port-Hamiltonian descriptor systems are analysed, and time discretization and numerical linear algebra techniques are discussed. Structure-preserving regularization procedures for descriptor systems are presented to make them suitable for simulation and control. Model reduction techniques that preserve the structure and stabilization and optimal control techniques are discussed.

The properties of port-Hamiltonian descriptor systems and their use in modelling simulation and control methods are illustrated with several examples from different physical domains. The survey concludes with open problems and research topics that deserve further attention.

MSC classification

Primary: 65L80: Methods for differential-algebraic equations

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Type
Research Article
Information
Acta Numerica , Volume 32 , May 2023 , pp. 395 - 515
Copyright
© The Author(s), 2023. Published by Cambridge University Press

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References

Achleitner, F., Arnold, A. and Carlen, E. A. (2021a), The hypocoercivity index for the short time behavior of linear time-invariant ODE systems . Available at arXiv:2109.10784.Google Scholar
Achleitner, F., Arnold, A. and Mehrmann, V. (2021b), Hypocoercivity and controllability in linear semi-dissipative ODEs and DAEs, ZAMM Z. Angew. Math. Mech. Available at doi:10.1002/zamm.202100171.CrossRefGoogle Scholar
Achleitner, F., Arnold, A. and Mehrmann, V. (2023), Hypocoercivity and hypocontractivity concepts for linear dynamical systems, Electron. J. Linear Algebra 9, 33–61.CrossRefGoogle Scholar
Adrianova, L. Y. (1995), Introduction to Linear Systems of Differential Equations, Vol. 146 of Translations of Mathematical Monographs, American Mathematical Society.CrossRefGoogle Scholar
Afkham, B. M. and Hesthaven, J. S. (2017), Structure preserving model reduction of parametric Hamiltonian systems, SIAM J. Sci. Comput. 39, A2616–A2644.CrossRefGoogle Scholar
Afkham, B. M. and Hesthaven, J. S. (2019), Structure-preserving model-reduction of dissipative Hamiltonian systems, J. Sci. Comput. 81, 3–21.CrossRefGoogle Scholar
Aliyev, N., Mehrmann, V. and Mengi, E. (2020), Computation of stability radii for large-scale dissipative Hamiltonian systems, Adv. Comput. Math. 46, 1–28.CrossRefGoogle Scholar
Altmann, R. and Schulze, P. (2017), A port-Hamiltonian formulation of the Navier–Stokes equations for reactive flows, Systems Control Lett. 100, 51–55.CrossRefGoogle Scholar
Altmann, R., Mehrmann, V. and Unger, B. (2021), Port-Hamiltonian formulations of poroelastic network models, Math. Comput. Model. Dyn. Sys. 27, 429–452.CrossRefGoogle Scholar
Antoulas, A. C. (2005a), Approximation of Large-Scale Dynamical Systems, Advances in Design and Control, Society for Industrial and Applied Mathematics.CrossRefGoogle Scholar
Antoulas, A. C. (2005b), A new result on passivity preserving model reduction, Systems Control Lett. 54, 361–374.CrossRefGoogle Scholar
Antoulas, A. C. (2008), On the construction of passive models from frequency response data, Automatisierungstechnik 56, 447–452.CrossRefGoogle Scholar
Antoulas, A. C., Beattie, C. A. and Gugercin, S. (2020), Interpolatory Methods for Model Reduction, Computational Science and Engineering, Society for Industrial and Applied Mathematics.Google Scholar
Antoulas, A. C., Lefteriu, S. and Ionita, A. C. (2017), Chapter 8: A tutorial introduction to the Loewner framework for model reduction, in Model Reduction and Approximation (Benner, P. et al., eds), Society for Industrial and Applied Mathematics, pp. 335–376.CrossRefGoogle Scholar
Aoues, S., Cardoso-Ribeiro, F. L., Matignon, D. and Alazard, D. (2017), Modeling and control of a rotating flexible spacecraft: A port-Hamiltonian approach, IEEE Trans. Control Sys. Tech. 27, 355–362.CrossRefGoogle Scholar
Aronna, M. S. (2018), Second order necessary and sufficient optimality conditions for singular solutions of partially-affine control problems, Discrete Contin. Dyn. Syst. 11, 1179–1199.Google Scholar
Baaiu, A., Couenne, F., Eberard, D., Jallut, C., Legorrec, Y., Lefèvre, L. and Maschke, B. (2009), Port-based modelling of mass transport phenomena, Math. Comput. Model. Dyn. Sys. 15, 233–254.CrossRefGoogle Scholar
Backes, A. (2006), Optimale Steuerung der linearen DAE im Fall Index 2. PhD thesis, Humboldt-Universität zu Berlin.Google Scholar
Bai, M., Elsworth, D. and Roegiers, J.-C. (1993), Multiporosity/multipermeability approach to the simulation of naturally fractured reservoirs, Water Resour . Res. 29, 1621–1633.Google Scholar
Bai, Z. (2002), Krylov subspace techniques for reduced-order modeling of large-scale dynamical systems, Appl. Numer. Math. 43, 9–44.CrossRefGoogle Scholar
Bankmann, D., Mehrmann, V., Nesterov, Y. and Van Dooren, P. (2020), Computation of the analytic center of the solution set of the linear matrix inequality arising in continuous- and discrete-time passivity analysis, Vietnam J. Math. 48, 633–660.CrossRefGoogle ScholarPubMed
Bansal, H., Schulze, P., Abbasi, M. H., Zwart, H., Iapichino, L., Schilders, W. H. A. and Wouw, N. (2021), Port-Hamiltonian formulation of two-phase flow models, Systems Control Lett. 149, 104881.CrossRefGoogle Scholar
Barrault, M., Maday, Y., Nguyen, N. C. and Patera, A. T. (2004), An ‘empirical interpolation’ method: Application to efficient reduced-basis discretization of partial differential equations, C.R . Math. Acad. Sci. Paris 339, 667–672.CrossRefGoogle Scholar
Beattie, C. and Gugercin, S. (2011), Structure-preserving model reduction for nonlinear port-Hamiltonian systems, in 50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC 2011), pp. 6564–6569.Google Scholar
Beattie, C., Gugercin, S. and Mehrmann, V. (2022a), Structure-preserving interpolatory model reduction for port-Hamiltonian differential-algebraic systems, in Realization and Model Reduction of Dynamical Systems (Beattie, C. et al., eds), Springer, pp. 235–254.CrossRefGoogle Scholar
Beattie, C., Mehrmann, V. and Van Dooren, P. (2019), Robust port-Hamiltonian representations of passive systems, Automatica 100, 182–186.CrossRefGoogle Scholar
Beattie, C., Mehrmann, V. and Xu, H. (2022b), Port-Hamiltonian realizations of linear time invariant systems. Available at arXiv:2201.05355.Google Scholar
Beattie, C., Mehrmann, V., Xu, H. and Zwart, H. (2018), Port-Hamiltonian descriptor systems, Math. Control Signals Systems 30, 1–27.CrossRefGoogle Scholar
Beckesch, A. (2018), Pfadverfolgung für Finite-Elemente-Modelle parametrischer mechanischer Systeme. Master’s thesis, Technische Universität Berlin.Google Scholar
Benner, P. and Sokolov, V. I. (2006), Partial realization of descriptor systems, Systems Control Lett. 55, 929–938.CrossRefGoogle Scholar
Benner, P., Cohen, A., Ohlberger, M. and Willcox, K. (2017), Model Reduction and Approximation, Computational Science and Engineering, Society for Industrial and Applied Mathematics.CrossRefGoogle Scholar
Benner, P., Goyal, P. and Van Dooren, P. (2020), Identification of port-Hamiltonian systems from frequency response data, Systems Control Lett. 143, 104741.CrossRefGoogle Scholar
Berger, T. (2012), Bohl exponent for time-varying linear differential-algebraic equations, Internat. J. Control 85, 1433–1451.CrossRefGoogle Scholar
Berger, T. and Reis, T. (2013), Controllability of linear differential-algebraic equations: A survey, in Surveys in Differential-Algebraic Equations I (Ilchmann, A. and Reis, T., eds), Differential-Algebraic Equations Forum, Springer, pp. 1–61.Google Scholar
Bienstock, D. (2015), Electrical Transmission System Cascades and Vulnerability: An Operations Research Viewpoint, MOS-SIAM Series on Optimization, Society for Industrial and Applied Mathematics.CrossRefGoogle Scholar
Biot, M. A. (1941), General theory of three-dimensional consolidation, J. Appl. Phys. 12, 155–164.CrossRefGoogle Scholar
Black, F., Schulze, P. and Unger, B. (2020), Projection-based model reduction with dynamically transformed modes, ESAIM Math. Model. Numer. Anal. 54, 2011–2043.CrossRefGoogle Scholar
Borja, P., Scherpen, J. M. A. and Fujimoto, K. (2023), Extended balancing of continuous LTI systems: A structure-preserving approach, IEEE Trans. Automat. Control 68, 257–271.CrossRefGoogle Scholar
Breedveld, P. C. (2008), Modeling and simulation of dynamic systems using bond graphs, in Control Systems, Robotics and Automation: Modeling and System Identification I (Unbehauen, H., ed.), EOLSS Publishers/UNESCO, pp. 128–173.Google Scholar
Breiten, T. and Schulze, P. (2021), Structure-preserving linear quadratic Gaussian balanced truncation for port-Hamiltonian descriptor systems. Available at arXiv:2111.05065.Google Scholar
Breiten, T. and Unger, B. (2022), Passivity preserving model reduction via spectral factorization, Automatica 142, 110368.CrossRefGoogle Scholar
Breiten, T., Hinsen, D. and Unger, B. (2022a), Towards a modeling class for port-Hamiltonian systems with time-delay. Available at arXiv:2211.10687.Google Scholar
Breiten, T., Morandin, R. and Schulze, P. (2022b), Error bounds for port-Hamiltonian model and controller reduction based on system balancing, Comput. Math. Appl. 116, 100–115.CrossRefGoogle Scholar
Brenan, K. E., Campbell, S. L. and Petzold, L. R. (1996), Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, Society for Industrial and Applied Mathematics.CrossRefGoogle Scholar
Brouwer, J., Gasser, I. and Herty, M. (2011), Gas pipeline models revisited: Model hierarchies, non-isothermal models and simulations of networks, Multiscale Model . Simul. 9, 601–623.Google Scholar
Brown, T., Schlachtberger, D., Kies, A., Schramm, S. and Greiner, M. (2018), Synergies of sector coupling and transmission reinforcement in a cost-optimised, highly renewable European energy system, Energy 160, 720–739.CrossRefGoogle Scholar
Bryson, A. E. and Ho, Y.-C. (2018), Applied Optimal Control: Optimization, Estimation, and Control, Routledge.CrossRefGoogle Scholar
Buchfink, P., Glas, S. and Haasdonk, B. (2021), Symplectic model reduction of Hamiltonian systems on nonlinear manifolds. Available at arXiv:2112.10815.Google Scholar
Bunse-Gerstner, A., Byers, R., Mehrmann, V. and Nichols, N. K. (1991), Numerical computation of an analytic singular value decomposition of a matrix valued function, Numer . Math. 60, 1–39.Google Scholar
Bunse-Gerstner, A., Byers, R., Mehrmann, V. and Nichols, N. K. (1999), Feedback design for regularizing descriptor systems, Linear Algebra Appl. 299, 119–151.CrossRefGoogle Scholar
Byers, R., Geerts, T. and Mehrmann, V. (1997a), Descriptor systems without controllability at infinity, SIAM J. Control Optim. 35, 462–479.CrossRefGoogle Scholar
Byers, R., Kunkel, P. and Mehrmann, V. (1997b), Regularization of linear descriptor systems with variable coefficients, SIAM J. Control Optim. 35, 117–133.CrossRefGoogle Scholar
Byers, R., Mehrmann, V. and Xu, H. (2007), A structured staircase algorithm for skew-symmetric/symmetric pencils, Electron . Trans. Numer. Anal. 26, 1–13.Google Scholar
Byrnes, C. I., Isidori, A. and Willems, J. C. (1991), Passivity, feedback equivalence, and the global stabilization of minimum phase nonlinear systems, IEEE Trans. Automat. Control 36, 1228–1240.CrossRefGoogle Scholar
Campbell, S. L. (1987), A general form for solvable linear time varying singular systems of differential equations, SIAM J. Math. Anal. 18, 1101–1115.CrossRefGoogle Scholar
Campbell, S. L. and Gear, C. W. (1995), The index of general nonlinear DAEs, Numer . Math. 72, 173–196.Google Scholar
Campbell, S. L., Kunkel, P. and Mehrmann, V. (2012), Regularization of linear and nonlinear descriptor systems, in Control and Optimization with Differential-Algebraic Constraints (Biegler, L. T., Campbell, S. L. and Mehrmann, V., eds), Vol. 23 of Advances in Design and Control, Society for Industrial and Applied Mathematics, pp. 17–36.Google Scholar
Cardoso-Ribeiro, F. L., Matignon, D. and Pommier-Budinger, V. (2017), A port-Hamiltonian model of liquid sloshing in moving containers and application to a fluid-structure system, J. Fluids Structures 69, 402–427.CrossRefGoogle Scholar
Celledoni, E. and Høiseth, E. H. (2017), Energy-preserving and passivity-consistent numerical discretization of port-Hamiltonian systems. Available at arXiv:1706.08621.Google Scholar
Celledoni, E., McLachlan, R. I., McLaren, D. I., Owren, B., Quispel, G. R. W. and Wright, W. M. (2009), Energy-preserving Runge–Kutta methods, ESAIM Math. Model. Numer. Anal. 43, 645–649.CrossRefGoogle Scholar
Chaturantabut, S. and Sorensen, D. (2010), Nonlinear model reduction via discrete empirical interpolation, SIAM J. Sci. Comput. 32, 2737–2764.CrossRefGoogle Scholar
Chaturantabut, S., Beattie, C. A. and Gugercin, S. (2016), Structure-preserving model reduction for nonlinear port-Hamiltonian systems, SIAM J. Sci. Comput. 38, B837–B865.CrossRefGoogle Scholar
Cherifi, K., Gernandt, H. and Hinsen, D. (2022a), The difference between port-Hamiltonian, passive and positive real descriptor systems. Available at arXiv:2204.04990.Google Scholar
Cherifi, K., Mehrmann, V. and Hariche, K. (2019), Numerical methods to compute a minimal realization of a port-Hamiltonian system. Available at arXiv:1903.07042.Google Scholar
Cherifi, K., Mehrmann, V. and Schulze, P. (2022b), Simulations in a digital twin of an electrical machine. Available at arXiv:2207.02171.Google Scholar
Conejo, A. J., Chen, S. and Constante, G. E. (2020), Operations and long-term expansion planning of natural-gas and power systems: A market perspective, Proc . IEEE 108, 1541–1557.CrossRefGoogle Scholar
Dai, L. (1989), Singular Control Systems, Vol. 118 of Lecture Notes in Control and Information Sciences, Springer.CrossRefGoogle Scholar
Desai, U. B. and Pal, D. (1984), A transformation approach to stochastic model reduction, IEEE Trans. Automat. Control 29, 1097–1100.CrossRefGoogle Scholar
Dieci, L. and Eirola, T. (1999), On smooth decompositions of matrices, SIAM J. Matrix Anal. Appl. 20, 800–819.CrossRefGoogle Scholar
Dieci, L. and Van Vleck, E. S. (2002), Lyapunov and other spectra: A survey, in Collected Lectures on the Preservation of Stability under Discretization (Fort Collins, CO, 2001) (Estep, D. and Tavener, S., eds), Society for Industrial and Applied Mathematics, pp. 197–218.Google Scholar
Dieci, L., Russell, R. D. and Van Vleck, E. S. (1997), On the computation of Lyapunov exponents for continuous dynamical systems, SIAM J. Numer. Anal. 34, 402–423.CrossRefGoogle Scholar
Du, N. H., Linh, V. H. and Mehrmann, V. (2013), Robust stability of differential-algebraic equations, in Surveys in Differential-Algebraic Equations I (Ilchmann, A. and Reis, T., eds), Springer, pp. 63–95.CrossRefGoogle Scholar
Duindam, V., Macchelli, A., Stramigioli, S. and Bruyninckx, H. (2009), Modeling and Control of Complex Physical Systems: The Port-Hamiltonian Approach, Springer.CrossRefGoogle Scholar
Egger, H. (2019), Structure preserving approximation of dissipative evolution problems, Numer . Math. 143, 85–106.Google Scholar
Egger, H. and Kugler, T. (2018), Damped wave systems on networks: Exponential stability and uniform approximations, Numer . Math. 138, 839–867.Google Scholar
Egger, H. and Sabouri, M. (2021), On the structure preserving high-order approximation of quasistatic poroelasticity, Math. Comput. Simul. 189, 237–252.CrossRefGoogle Scholar
Egger, H., Kugler, T., Liljegren-Sailer, B., Marheineke, N. and Mehrmann, V. (2018), On structure preserving model reduction for damped wave propagation in transport networks, SIAM J. Sci. Comput. 40, A331–A365.CrossRefGoogle Scholar
Eich-Soellner, E. and Führer, C. (1998), Numerical Methods in Multibody Dynamics, Vieweg and Teubner.CrossRefGoogle Scholar
Emmrich, E. and Mehrmann, V. (2013), Operator differential-algebraic equations arising in fluid dynamics, Comput . Methods Appl. Math 13, 443–470.Google Scholar
Ennsbrunner, H. and Schlacher, K. (2005), On the geometrical representation and interconnection of infinite dimensional port controlled Hamiltonian systems, in 44th IEEE Conference on Decision and Control (CDC 2005), IEEE, pp. 5263–5268.Google Scholar
Faulwasser, T., Maschke, B., Philipp, F., Schaller, M. and Worthmann, K. (2022), Optimal control of port-Hamiltonian descriptor systems with minimal energy supply, SIAM J. Control Optim. 60, 2132–2158.CrossRefGoogle Scholar
Freund, R. W. (2000), Krylov-subspace methods for reduced-order modeling in circuit simulation, J. Comput. Appl. Math. 123, 395–421.CrossRefGoogle Scholar
Freund, R. W. (2005), Padé-type model reduction of second-order and higher-order linear dynamical systems, in Dimension Reduction of Large-Scale Systems (Benner, P., Sorensen, D. C. and Mehrmann, V., eds), Springer, pp. 191–223.CrossRefGoogle Scholar
Freund, R. W. (2011), The SPRIM algorithm for structure-preserving order reduction of general RLC circuits, in Model Reduction for Circuit Simulation (Benner, P. et al., eds), Springer, pp. 25–52.CrossRefGoogle Scholar
Friedrichs, K. O. and Lax, P. D. (1971), Systems of conservation equations with a convex extension, Proc . Nat. Acad. Sci. USA 68, 1686–1688.CrossRefGoogle Scholar
Fujimoto, K. and Kajiura, H. (2007), Balanced realization and model reduction of port-Hamiltonian systems, in 2007 American Control Conference, pp. 930–934.Google Scholar
Gantmacher, F. R. (1959), The Theory of Matrices, Vol. 2, Chelsea.Google Scholar
Gay-Balmaz, F. and Yoshimura, H. (2018), Dirac structures in nonequilibrium thermodynamics, J. Math. Phys. 59, 012701.CrossRefGoogle Scholar
Gay-Balmaz, F. and Yoshimura, H. (2019), From Lagranian mechanics to nonequilibrium thermodynamics: A variational perspective, Entropy 21, 8.CrossRefGoogle Scholar
Gernandt, H. and Haller, F. E. (2021), On the stability of port-Hamiltonian descriptor systems, IFAC-PapersOnLine 54(19), 137–142.CrossRefGoogle Scholar
Gernandt, H., Haller, F. E. and Reis, E. (2021), A linear relation approach to port-Hamiltonian differential-algebraic equations, SIAM J. Matrix Anal. Appl. 42, 1011–1044.CrossRefGoogle Scholar
Gillis, N. and Sharma, P. (2017), On computing the distance to stability for matrices using linear dissipative Hamiltonian systems, Automatica 85, 113–121.CrossRefGoogle Scholar
Gillis, N., Mehrmann, V. and Sharma, P. (2018), Computing the nearest stable matrix pairs, Numer . Linear Algebra Appl. 25, e2153.CrossRefGoogle Scholar
Gräbner, N., Mehrmann, V., Quraishi, S., Schröder, C. and von Wagner, U. (2016), Numerical methods for parametric model reduction in the simulation of disc brake squeal, ZAMM Z. Angew. Math. Mech. 96, 1388–1405.CrossRefGoogle Scholar
Grimme, E. J. (1997), Krylov projection methods for model reduction. PhD thesis, University of Illinois, Urbana-Champaign.Google Scholar
Grmela, M. and Öttinger, H. C. (1997), Dynamics and thermodynamics of complex fluids, I: Development of a general formalism, Phys. Rev. E 56, 6620–6632.CrossRefGoogle Scholar
Güdücü, C., Liesen, J., Mehrmann, V. and Szyld, D. (2022), On non-Hermitian positive (semi)definite linear algebraic systems arising from dissipative Hamiltonian DAEs, SIAM J. Sci. Comput. 44, A2871–A2894.CrossRefGoogle Scholar
Gugercin, S. and Antoulas, A. C. (2004), A survey of model reduction by balanced truncation and some new results, Internat. J. Control 77, 748–766.CrossRefGoogle Scholar
Gugercin, S., Antoulas, A. C. and Beattie, C. A. (2008), ${\mathrm{\mathscr{H}}}_2$ model reduction for large-scale linear dynamical systems, SIAM J. Matrix Anal. Appl. 30, 609–638.CrossRefGoogle Scholar
Gugercin, S., Polyuga, R. V., Beattie, C. and van der Schaft, A. (2009), Interpolation-based ${\mathrm{\mathscr{H}}}_2$ model reduction for port-Hamiltonian systems, in 48th IEEE Conference on Decision and Control (CDC 2009), pp. 5362–5369.Google Scholar
Gugercin, S., Polyuga, R. V., Beattie, C. and van der Schaft, A. (2012), Structure-preserving tangential interpolation for model reduction of port-Hamiltonian systems, Automatica 48, 1963–1974.CrossRefGoogle Scholar
Gugercin, S., Stykel, T. and Wyatt, S. (2013), Model reduction of descriptor systems by interpolatory projection methods, SIAM J. Sci. Comput. 35, B1010–B1033.CrossRefGoogle Scholar
Guglielmi, N. and Mehrmann, V. (2022), Computation of the nearest structured matrix triplet with common null space, Electron . Trans. Numer. Anal. 55, 508–531.Google Scholar
Guiver, C. and Opmeer, M. R. (2013), Error bounds in the gap metric for dissipative balanced approximations, Linear Algebra Appl. 439, 3659–3698.CrossRefGoogle Scholar
Günther, M., Bartel, A., Jacob, B. and Reis, T. (2021), Dynamic iteration schemes and port-Hamiltonian formulation in coupled differential-algebraic equation circuit simulation, Int J. Circ. Theor. Appl. 49, 430–452.CrossRefGoogle Scholar
Haasdonk, B. (2017), Chapter 2: Reduced basis methods for parametrized PDEs: A tutorial introduction for stationary and instationary problems, in Model Reduction and Approximation (Benner, P. et al., eds), Society for Industrial and Applied Mathematics, pp. 65–136.CrossRefGoogle Scholar
Hairer, E. and Wanner, G. (1996), Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, second edition, Springer.CrossRefGoogle Scholar
Hairer, E., Lubich, C. and Roche, M. (1989), The Numerical Solution of Differential-Algebraic Systems by Runge–Kutta Methods, Springer.CrossRefGoogle Scholar
Hairer, E., Lubich, C. and Wanner, G. (2002), Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, Springer.CrossRefGoogle Scholar
Hamann, P. and Mehrmann, V. (2008), Numerical solution of hybrid systems of differential-algebraic equations, Comput . Meth. Appl. Mech. Eng. 197, 693–705.CrossRefGoogle Scholar
Hauschild, S.-A., Marheineke, N. and Mehrmann, V. (2019), Model reduction techniques for linear constant coefficient port-Hamiltonian differential-algebraic systems, Control Cybernet. 48, 125–152.Google Scholar
Heinkenschloss, M., Sorensen, D. C. and Sun, K. (2008), Balanced truncation model reduction for a class of descriptor systems with application to the Oseen equations, SIAM J. Sci. Comput. 30, 1038–1063.CrossRefGoogle Scholar
Hesthaven, J. S., Rozza, G. and Stamm, B. (2016), Certified Reduced Basis Methods for Parametrized Partial Differential Equations, Springer Briefs in Mathematics, Springer.CrossRefGoogle Scholar
Hinrichsen, D. and Pritchard, A. J. (2005), Mathematical Systems Theory I: Modelling, State Space Analysis, Stability and Robustness, Springer.Google Scholar
Hou, M. (1994), A three-link planar manipulator model. Technical report, Sicherheitstechnische Regelungs- und Meßtechnik, Bergische Universität–GH Wuppertal, Germany.Google Scholar
Hou, M. and Müller, P. C. (1994), $LQ$ and tracking control of descriptor systems with application to constrained manipulator. Technical report, Sicherheitstechnische Regelungs- und Meßtechnik, Universität Wuppertal.Google Scholar
Hughes, T. J. R. (2012), The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, Courier.Google Scholar
Ionescu, T. C. and Astolfi, A. (2013), Families of moment matching based, structure preserving approximations for linear port Hamiltonian systems, Automatica 49, 2424–2434.CrossRefGoogle Scholar
Ionescu, T. C. and Scherpen, J. M. A. (2007), Positive real balancing for nonlinear systems, in Scientific Computing in Electrical Engineering (Ciuprina, G. and Ioan, D., eds), Vol. 11 of Mathematics in Industry, Springer, pp. 153–159.CrossRefGoogle Scholar
Ionutiu, R., Rommes, J. and Antoulas, A. C. (2008), Passivity-preserving model reduction using dominant spectral-zero interpolation, IEEE Trans. Computer-Aided Design of Integr. Circ. Syst. 27, 2250–2263.CrossRefGoogle Scholar
Jacob, B. and Zwart, H. (2012), Linear Port-Hamiltonian Systems on Infinite-Dimensional Spaces, Vol. 223 of Operator Theory: Advances and Applications, Birkhäuser.Google Scholar
Kailath, T. (1980), Linear Systems, Vol. 156 of Information and System Sciences Series, Prentice-Hall.Google Scholar
Kannan, R., Hendry, S., Higham, N. J. and Tisseur, F. (2014), Detecting the causes of ill-conditioning in structural finite element models, Computers & Structures 133, 79–89.CrossRefGoogle Scholar
Karsai, A. (2022), Structure-preserving control of port-Hamiltonian systems. Master’s thesis, Technische Universität Berlin.Google Scholar
Kawano, Y. and Scherpen, J. M. A. (2018), Structure preserving truncation of nonlinear port Hamiltonian systems, IEEE Trans. Automat. Control 63, 4286–4293.CrossRefGoogle Scholar
Kotyczka, P. (2019), Numerical methods for distributed parameter port-Hamiltonian systems. structure-preserving approaches for simulation and control. Habilitation thesis, TU Munich.Google Scholar
Kotyczka, P. and Lefèvre, L. (2018), Discrete-time port-Hamiltonian systems based on Gauss–Legendre collocation, IFAC-PapersOnLine 51(3), 125–130.CrossRefGoogle Scholar
Kotyczka, P., Maschke, B. and Lefèvre, L. (2018), Weak form of Stokes–Dirac structures and geometric discretization of port-Hamiltonian systems, J. Comput. Phys. 361, 442–476.CrossRefGoogle Scholar
Kunkel, P. and Mehrmann, V. (1991), Smooth factorizations of matrix valued functions and their derivatives, Numer . Math. 60, 115–131.Google Scholar
Kunkel, P. and Mehrmann, V. (1996), A new class of discretization methods for the solution of linear differential-algebraic equations with variable coefficients, SIAM J. Numer. Anal. 33, 1941–1961.CrossRefGoogle Scholar
Kunkel, P. and Mehrmann, V. (1998), Regular solutions of nonlinear differential-algebraic equations and their numerical determination, Numer . Math. 79, 581–600.Google Scholar
Kunkel, P. and Mehrmann, V. (2001), Analysis of over- and underdetermined nonlinear differential-algebraic systems with application to nonlinear control problems, Math. Control Signals Systems 14, 233–256.CrossRefGoogle Scholar
Kunkel, P. and Mehrmann, V. (2006), Differential-Algebraic Equations: Analysis and Numerical Solution, European Mathematical Society.CrossRefGoogle Scholar
Kunkel, P. and Mehrmann, V. (2007), Stability properties of differential-algebraic equations and spin-stabilized discretizations, Electron . Trans. Numer. Anal. 26, 385–420.Google Scholar
Kunkel, P. and Mehrmann, V. (2008), Optimal control for unstructured nonlinear differential-algebraic equations of arbitrary index, Math. Control Signals Systems 20, 227–269.CrossRefGoogle Scholar
Kunkel, P. and Mehrmann, V. (2011), Formal adjoints of linear DAE operators and their role in optimal control, Electron. J. Linear Algebra 22, 672–693.CrossRefGoogle Scholar
Kunkel, P. and Mehrmann, V. (2018), Regular solutions of DAE hybrid systems and regularization techniques, BIT Numer. Math. 58, 1049–1077.CrossRefGoogle Scholar
Kunkel, P. and Mehrmann, V. (2023), Local and global canonical forms for differential-algebraic equations with symmetries, Vietnam J. Math. 51, 177–198.CrossRefGoogle Scholar
Kunkel, P., Mehrmann, V. and Rath, W. (2001), Analysis and numerical solution of control problems in descriptor form, Math. Control Signals Systems 14, 29–61.CrossRefGoogle Scholar
Kunkel, P., Mehrmann, V. and Scholz, L. (2014), Self-adjoint differential-algebraic equations, Math. Control Signals Systems 26, 47–76.CrossRefGoogle Scholar
Kurina, G. A. and März, R. (2004), On linear-quadratic optimal control problems for time-varying descriptor systems, SIAM J. Control Optim. 42, 2062–2077.CrossRefGoogle Scholar
Kurula, M. (2020), Well-posedness of time-varying linear systems, IEEE Trans. Automat. Control 65, 4075–4089.CrossRefGoogle Scholar
Kurula, M., Zwart, H., van der Schaft, A. and Behrndt, J. (2010), Dirac structures and their composition on Hilbert spaces, J. Math. Anal. Appl. 372, 402–422.CrossRefGoogle Scholar
La Salle, J. and Lefschetz, S. (1961), Stability by Liapunov’s Direct Method, Mathematics in Science and Engineering, Academic Press.Google Scholar
La Salle, J. P. (1976), The Stability of Dynamical Systems, CBMS-NSF Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics.Google Scholar
Lamour, R., März, R. and Tischendorf, C. (2013), Differential-Algebraic Equations: A Projector Based Analysis, Differential-Algebraic Equations Forum, Springer.CrossRefGoogle Scholar
Lang, S. (2012), Differential and Riemannian Manifolds, Vol. 160 of Graduate Texts in Mathematics, Springer.Google Scholar
Layton, W. (2008), Introduction to the Numerical Analysis of Incompressible Viscous Flows, Computational Science and Engineering, Society for Industrial and Applied Mathematics.CrossRefGoogle Scholar
Le Gorrec, Y., Zwart, H. and Maschke, B. (2005), Dirac structures and boundary control systems associated with skew-symmetric differential operators, SIAM J. Control Optim. 44, 1864–1892.CrossRefGoogle Scholar
Liljegren-Sailer, B. (2020), On port-Hamiltonian modeling and structure-preserving model reduction. PhD thesis, Universität Trier.Google Scholar
Linh, V. H. and Mehrmann, V. (2009), Lyapunov, Bohl and Sacker–Sell spectral intervals for differential-algebraic equations, J. Dynam. Differ. Equations 21, 153–194.CrossRefGoogle Scholar
Linh, V. H. and Mehrmann, V. (2011), Spectral analysis for linear differential-algebraic equations, Discrete Contin . Dyn. Syst. 2011, 991–1000.Google Scholar
Linh, V. H. and Mehrmann, V. (2012), Spectra and leading directions for linear DAEs, in Control and Optimization with Differential-Algebraic Constraints (Biegler, L. T., Campbell, S. L. and Mehrmann, V., eds), Vol. 23 of Advances in Design and Control, Society for Industrial and Applied Mathematics, pp. 59–78.CrossRefGoogle Scholar
Linh, V. H. and Mehrmann, V. (2014), Efficient integration of strangeness-free non-stiff differential-algebraic equations by half-explicit methods, J. Comput. Appl. Math. 262, 346–360.CrossRefGoogle Scholar
Linh, V. H., Mehrmann, V. and Van Vleck, E. S. (2011), $QR$ methods and error analysis for computing Lyapunov and Sacker–Sell spectral intervals for linear differential-algebraic equations, Adv. Comput. Math. 35, 281–322.CrossRefGoogle Scholar
Macchelli, A. and Maschke, B. (2009), Chapter 4: Infinite-dimensional port-Hamiltonian systems, in Modeling and Control of Complex Physical Systems: The Port-Hamiltonian Approach (Duindam, V. et al., eds), Springer, pp. 211–272.Google Scholar
Macchelli, A., Melchiorri, C. and Bassi, L. (2005), Port-based modelling and control of the Mindlin plate, in 44th IEEE Conference on Decision and Control (CDC 2005), pp. 5989–5994.Google Scholar
Macchelli, A., van der Schaft, A. and Melchiorri, C. (2004a), Port Hamiltonian formulation of infinite dimensional systems, I: Modeling, in 43rd IEEE Conference on Decision and Control (CDC 2004), Vol. 4, pp. 3762–3767.CrossRefGoogle Scholar
Macchelli, A., van der Schaft, A. and Melchiorri, C. (2004b), Port Hamiltonian formulation of infinite dimensional systems, II: Boundary control by interconnection, in 43rd IEEE Conference on Decision and Control (CDC 2004), Vol. 4, pp. 3768–3773.CrossRefGoogle Scholar
Machowski, J., Lubosny, Z., Bialek, J. W. and Bumby, J. R. (2020), Power System Dynamics: Stability and Control, Wiley.Google Scholar
Manuoglu, M. and Mehrmann, V. (2019), A robust iterative scheme for symmetric indefinite systems, SIAM J. Sci. Comput. 41, A1733–A1752.CrossRefGoogle Scholar
Matignon, D. and Hélie, T. (2013), A class of damping models preserving eigenspaces for linear conservative port-Hamiltonian systems, Eur. J. Control 19, 486–494.CrossRefGoogle Scholar
Mayo, A. J. and Antoulas, A. C. (2007), A framework for the solution of the generalized realization problem, Linear Algebra Appl. 425, 634–662.CrossRefGoogle Scholar
Mehl, C., Mehrmann, V. and Sharma, P. (2016), Stability radii for linear Hamiltonian systems with dissipation under structure-preserving perturbations, SIAM J. Matrix Anal. Appl. 37, 1625–1654.CrossRefGoogle Scholar
Mehl, C., Mehrmann, V. and Wojtylak, M. (2018), Linear algebra properties of dissipative Hamiltonian descriptor systems, SIAM J. Matrix Anal. Appl. 39, 1489–1519.CrossRefGoogle Scholar
Mehl, C., Mehrmann, V. and Wojtylak, M. (2021), Distance problems for dissipative Hamiltonian systems and related matrix polynomials, Linear Algebra Appl. pp. 335–366.Google Scholar
Mehrmann, V. (1991), The Autonomous Linear Quadratic Control Problem, Theory and Numerical Solution, Vol. 163 of Lecture Notes in Control and Information Sciences, Springer.CrossRefGoogle Scholar
Mehrmann, V. (2015), Index concepts for differential-algebraic equations, in Encyclopedia of Applied and Computational Mathematics, Springer, pp. 676–681.CrossRefGoogle Scholar
Mehrmann, V. and Dooren, P. V. (2020), Optimal robustness of port-Hamiltonian systems, SIAM J. Matrix Anal. Appl. 41, 134–151.CrossRefGoogle Scholar
Mehrmann, V. and Morandin, R. (2019), Structure-preserving discretization for port-Hamiltonian descriptor systems, in 58th IEEE Conference on Decision and Control (CDC 2019), pp. 6863–6868.Google Scholar
Mehrmann, V. and Stykel, T. (2005), Balanced truncation model reduction for large-scale systems in descriptor form, in Dimension Reduction of Large-Scale Systems (Benner, P., Sorensen, D. C. and Mehrmann, V., eds), Springer, pp. 83–116.CrossRefGoogle Scholar
Mehrmann, V. and van der Schaft, A. (2023), Differential–algebraic systems with dissipative Hamiltonian structure, Math. Control Signals Systems. Available at doi:10.1007/s00498-023-00349-2.CrossRefGoogle Scholar
Mehrmann, V. and Xu, H. (2000), Numerical methods in control, J. Comput. Appl. Math. 123, 371–394.CrossRefGoogle Scholar
Montbrun-Di Filippo, J., Delgado, M., Brie, C. and Paynter, H. M. (1991), A survey of bond graphs: Theory, applications and programs, J. Franklin Institute 328, 565–606.CrossRefGoogle Scholar
Morandin, R., Nicodemus, J. and Unger, B. (2022), Port-Hamiltonian dynamic mode decomposition. Available at arXiv:2204.13474.Google Scholar
Moser, T. and Lohmann, B. (2020), A new Riemannian framework for efficient ${\mathrm{\mathscr{H}}}_2$ -optimal model reduction of port-Hamiltonian systems, in 59th IEEE Conference on Decision and Control (CDC 2020), pp. 5043–5049.Google Scholar
Moser, T., Schwerdtner, P., Mehrmann, V. and Voigt, M. (2022), Structure-preserving model order reduction for index two port-Hamiltonian descriptor systems. Available at arXiv:2206.03942.Google Scholar
Badlyan, A. Moses and Zimmer, C. (2018), Operator-GENERIC formulation of thermodynamics of irreversible processes. Available at arXiv:1807.09822.Google Scholar
Badlyan, A. Moses, Maschke, B., Beattie, C. and Mehrmann, V. (2018), Open physical systems: From GENERIC to port-Hamiltonian systems, in 23rd International Symposium on Mathematical Theory of Systems and Networks, pp. 204–211.Google Scholar
Nedialkov, N., Pryce, J. D. and Scholz, L. (2022), An energy-based, always index $\le 1$ and structurally amenable electrical circuit model, SIAM J. Sci. Comput. 44, B1122–B1147.CrossRefGoogle Scholar
Öttinger, H. C. (2006), Nonequilibrium thermodynamics for open systems, Phys. Rev. E 73, 036126.CrossRefGoogle ScholarPubMed
Öttinger, H. C. and Grmela, M. (1997), Dynamics and thermodynamics of complex fluids, II: Illustrations of a general formalism, Phys. Rev. E 56, 6633–6655.CrossRefGoogle Scholar
Pantelides, C. C. (1988), The consistent initialization of differential-algebraic systems, SIAM J. Sci. Statist. Comput. 9, 213–231.CrossRefGoogle Scholar
Paynter, H. M. (1961), Analysis and Design of Engineering Systems, MIT Press.Google Scholar
Philipp, F., Schaller, M., Faulwasser, T., Maschke, B. and Worthmann, K. (2021), Minimizing the energy supply of infinite-dimensional linear port-Hamiltonian systems, IFAC-PapersOnLine 54(19), 155–160.CrossRefGoogle Scholar
Polderman, J. W. and Willems, J. C. (1998), Introduction to Mathematical Systems Theory: A Behavioural Approach, Texts in Applied Mathematics, Springer.CrossRefGoogle Scholar
Polyuga, R. V. and van der Schaft, A. (2010), Structure preserving model reduction of port-Hamiltonian systems by moment matching at infinity, Automatica 46, 665–672.CrossRefGoogle Scholar
Polyuga, R. V. and van der Schaft, A. (2011), Structure-preserving moment matching for port-Hamiltonian systems: Arnoldi and Lanczos, IEEE Trans. Automat. Control 56, 1458–1462.CrossRefGoogle Scholar
Polyuga, R. V. and van der Schaft, A. (2012), Effort- and flow-constraint reduction methods for structure preserving model reduction of port-Hamiltonian systems, Systems Control Lett. 61, 412–421.CrossRefGoogle Scholar
Pryce, J. D. (2001), A simple structural analysis method for DAEs, BIT Numer. Math. 41, 364–394.CrossRefGoogle Scholar
Quarteroni, A., Manzoni, A. and Negri, F. (2015), Reduced Basis Methods for Partial Differential Equations: An Introduction, UNITEXT, Springer International Publishing.CrossRefGoogle Scholar
Quispel, G. R. W. and McLaren, D. I. (2008), A new class of energy-preserving numerical integration methods, J. Phys. A 41, 045206.CrossRefGoogle Scholar
Rabier, P. J. and Rheinboldt, W. C. (1996a), Classical and generalized solutions of time-dependent linear differential-algebraic equations, Linear Algebra Appl. 245, 259–293.CrossRefGoogle Scholar
Rabier, P. J. and Rheinboldt, W. C. (1996b), Time-dependent linear DAEs with discontinuous inputs, Linear Algebra Appl. 247, 1–29.CrossRefGoogle Scholar
Rabier, P. J. and Rheinboldt, W. C. (2000), Nonholonomic Motion of Rigid Mechanical Systems from a DAE Viewpoint, Society for Industrial and Applied Mathematics.CrossRefGoogle Scholar
Estay, M. H. Ramirez (2019), Modeling and control of irreversible thermodynamic processes and systems described by partial differential equations: A port-Hamiltonian approach. Habilitation thesis, Université Bourgogne Franche-Comté.Google Scholar
Ramsebner, J., Haas, R., Ajanovic, A. and Wietschel, M. (2021), The sector coupling concept: A critical review, WIREs Energy and Environment 10, e396.CrossRefGoogle Scholar
Rannacher, R. (2000), Finite element methods for the incompressible Navier–Stokes equations, in Fundamental Directions in Mathematical Fluid Mechanics (Galdi, P., Heywood, J. and Rannacher, R., eds), Birkhäuser, pp. 191–293.CrossRefGoogle Scholar
Rapoport, D. (1978), A nonlinear Lanczos algorithm and the stationary Navier–Stokes equation. PhD thesis, Department of Mathematics, Courant Institute, New York University.Google Scholar
Rashad, R., Califano, F., van der Schaft, A. and Stramigioli, S. (2020), Twenty years of distributed port-Hamiltonian systems: A literature review, IMA J. Math. Control I. pp. 1–23.Google Scholar
Reich, S. (1990), On a geometrical interpretation of differential-algebraic equations, Circuits Systems Signal Process. 9, 367–382.CrossRefGoogle Scholar
Reis, T. and Stykel, T. (2010a), PABTEC: Passivity-preserving balanced truncation for electrical circuits, IEEE Trans. Circuits and Systems 29, 1354–1367.Google Scholar
Reis, T. and Stykel, T. (2010b), Passivity-preserving balanced truncation model reduction of circuit equations, in Scientific Computing in Electrical Engineering SCEE 2008 (Roos, J. and Costa, L., eds), Vol. 14 of Mathematics in Industry, Springer, pp. 483–490.CrossRefGoogle Scholar
Reis, T. and Voigt, M. (2015), The Kalman–Yakubovich–Popov inequality for differential-algebraic systems: Existence of nonpositive solutions, Systems Control Lett. 86, 1–8.CrossRefGoogle Scholar
Reis, T., Rendel, O. and Voigt, M. (2015), The Kalman–Yakubovich–Popov inequality for differential-algebraic systems, Linear Algebra Appl. 485, 153–193.CrossRefGoogle Scholar
Rheinboldt, W. C. (1984), Differential-algebraic systems as differential equations on manifolds, Math. Comp. 43, 473–482.CrossRefGoogle Scholar
Romer, A., Berberich, J., Köhler, H. and Allgöwer, F. (2019), One-shot verification of dissipativity properties from input–output data, IEEE Control Systems Lett. 3, 709–714.CrossRefGoogle Scholar
Romer, A., Montenbruck, J. M. and Allgöwer, F. (2017), Determining dissipation inequalities from input-output samples, IFAC-PapersOnLine 50, 7789–7794.CrossRefGoogle Scholar
Saad, Y. (2003), Iterative Methods for Sparse Linear Systems, second edition, Society for Industrial and Applied Mathematics.CrossRefGoogle Scholar
Saak, J., Köhler, M. and Benner, P. (2021), M-M.E.S.S.-2.1: The matrix equations sparse solvers library. See also https://www.mpi-magdeburg.mpg.de/projects/mess.Google Scholar
Sato, K. and Sato, H. (2018), Structure-preserving ${H}^2$ optimal model reduction based on the Riemannian trust-region method, IEEE Trans. Automat. Control 63, 505–512.CrossRefGoogle Scholar
Schaller, M., Philipp, F., Faulwasser, T., Worthmann, K. and Maschke, B. (2021), Control of port-Hamiltonian systems with minimal energy supply, Eur. J. Control 62, 33–40.CrossRefGoogle Scholar
Scheuermann, T. M., Kotyczka, P. and Lohmann, B. (2019), On parametric structure preserving model order reduction of linear port-Hamiltonian systems, Automatisierungstechnik 67, 521–525.CrossRefGoogle Scholar
Schiffer, J., Fridman, E., Ortega, R. and Raisch, J. (2016), Stability of a class of delayed port-Hamiltonian systems with application to microgrids with distributed rotational and electronic generation, Automatica 74, 71–79.CrossRefGoogle Scholar
Schöberl, M. and Schlacher, K. (2017), Variational principles for different representations of Lagranian and Hamiltonian systems, in Dynamics and Control of Advanced Structures and Machines (Irschik, H., Belyaev, A. and Krommer, M., eds), Springer, pp. 65–73.CrossRefGoogle Scholar
Scholz, L. (2019), Condensed forms for linear port-Hamiltonian descriptor systems, Electron. J. Linear Algebra 35, 65–89.CrossRefGoogle Scholar
Schulze, P. (2023), Energy-based model reduction of transport-dominated phenomena. PhD thesis, Institut für Mathematik, Technische Universität, Berlin.Google Scholar
Schulze, P. and Unger, B. (2018), Model reduction for linear systems with low-rank switching, SIAM J. Control Optim. 56, 4365–4384.CrossRefGoogle Scholar
Schulze, P., Unger, B., Beattie, C. and Gugercin, S. (2018), Data-driven structured realization, Linear Algebra Appl. 537, 250–286.CrossRefGoogle Scholar
Schwerdtner, P. (2021), Port-Hamiltonian system identification from noisy frequency response data. Available at arXiv:2106.11355.Google Scholar
Schwerdtner, P. and Voigt, M. (2020), SOBMOR: Structured optimization-based model order reduction . Available at arXiv:2011.07567.Google Scholar
Schwerdtner, P. and Voigt, M. (2021), Adaptive sampling for structure-preserving model order reduction of port-Hamiltonian systems, IFAC-PapersOnLine 54(19), 143–148.CrossRefGoogle Scholar
Schwerdtner, P., Moser, T., Mehrmann, V. and Voigt, M. (2022), Structure-preserving model order reduction for index one port-Hamiltonian descriptor systems. Available at arXiv:2206.01608.Google Scholar
Serhani, A., Matignon, D. and Haine, G. (2019), A partitioned finite element method for the structure-preserving discretization of damped infinite-dimensional port-Hamiltonian systems with boundary control, in Geometric Science of Information (Nielsen, F. and Barbaresco, F., eds), Springer, pp. 549–558.CrossRefGoogle Scholar
Sharma, H., Wang, Z. and Kramer, B. (2022), Hamiltonian operator inference: Physics-preserving learning of reduced-order models for canonical Hamiltonian systems, Phys. D 431, 133122.Google Scholar
Simeon, B. (2013), Computational Flexible Multibody Dynamics: A Differential-Algebraic Approach, Differential-Algebraic Equations Forum, Springer.CrossRefGoogle Scholar
Sobey, I., Eisenträger, A., Wirth, B. and Czosnyka, M. (2012), Simulation of cerebral infusion tests using a poroelastic model, Int. J. Numer. Anal. Model. Ser. B 3, 52–64.Google Scholar
Sorensen, D. C. (2005), Passivity preserving model reduction via interpolation of spectral zeros, Systems Control Lett. 54, 347–360.CrossRefGoogle Scholar
Stykel, T. (2002), Stability and inertia theorems for generalized Lyapunov equations, Linear Algebra Appl. 355, 297–314.CrossRefGoogle Scholar
Stykel, T. (2006), On some norms for descriptor systems, IEEE Trans. Automat. Control 51, 842–847.CrossRefGoogle Scholar
Temam, R. (1977), Navier–Stokes Equations: Theory and Numerical Analysis, North-Holland.Google Scholar
Trenn, S. (2013), Solution concepts for linear DAEs: A survey, in Surveys in Differential-Algebraic Equations I (Ilchmann, A. and Reis, T., eds), Differential-Algebraic Equations Forum, Springer, pp. 137–172.CrossRefGoogle Scholar
Trenn, S. and Unger, B. (2019), Delay regularity of differential-algebraic equations, in 58th IEEE Conference on Decision and Control (CDC 2019), pp. 989–994.Google Scholar
Tully, B. and Ventikos, Y. (2011), Cerebral water transport using multiple-network poroelastic theory: Application to normal pressure hydrocephalus, J. Fluid Mech. 667, 188–215.CrossRefGoogle Scholar
Unger, B. (2020), Well-posedness and realization theory for delay differential-algebraic equations. PhD thesis, Technische Universität Berlin.Google Scholar
Unger, B. and Gugercin, S. (2019), Kolmogorov n-widths for linear dynamical systems, Adv. Comput. Math. 45, 2273–2286.CrossRefGoogle Scholar
van der Schaft, A. (2000), L2-gain and Passivity Techniques in Nonlinear Control, Springer.CrossRefGoogle Scholar
van der Schaft, A. (2013), Port-Hamiltonian differential-algebraic systems, in Surveys in Differential-Algebraic Equations I (Ilchmann, A. and Reis, T., eds), Differential-Algebraic Equations Forum, Springer, pp. 173–226.CrossRefGoogle Scholar
van der Schaft, A. and Jeltsema, D. (2014), Port-Hamiltonian systems theory: An introductory overview, Found . Trends Systems Control 1, 173–378.CrossRefGoogle Scholar
van der Schaft, A. and Maschke, B. (2002), Hamiltonian formulation of distributed parameter systems with boundary energy flow, J. Geom. Phys. 42, 166–174.CrossRefGoogle Scholar
van der Schaft, A. and Maschke, B. (2018), Generalized port-Hamiltonian DAE systems, Systems Control Lett. 121, 31–37.CrossRefGoogle Scholar
van der Schaft, A. and Maschke, B. (2020), Dirac and Lagrange algebraic constraints in nonlinear port-Hamiltonian systems, Vietnam J. Math. 48, 929–939.CrossRefGoogle Scholar
van Waarde, H. J., Camlibel, M. K., Rapisarda, P. and Trentelman, H. L. (2022), Data-driven dissipativity analysis: Application of the matrix S-lemma, IEEE Control Syst. Mag. 42, 140–149.CrossRefGoogle Scholar
Villegas, J. A. (2007), A port-Hamiltonian approach to distributed parameter systems. PhD thesis, University of Twente.Google Scholar
Widlund, O. (1978), A Lanczos method for a class of nonsymmetric systems of linear equations, SIAM J. Numer. Anal. 15, 801–812.CrossRefGoogle Scholar
Willems, J. C. (1971), Least squares stationary optimal control and the algebraic Riccati equation, IEEE Trans. Automat. Control 16, 621–634.CrossRefGoogle Scholar
Wolf, T., Lohmann, B., Eid, R. and Kotyczka, P. (2010), Passivity and structure preserving order reduction of linear port-Hamiltonian systems using Krylov subspaces, Eur. J. Control 16, 401–406.CrossRefGoogle Scholar
Wu, Y., Hamroun, B., Le Gorrec, Y. and Maschke, B. (2014), Structure preserving reduction of port Hamiltonian system using a modified LQG method, in Proceedings of the 33rd Chinese Control Conference, pp. 3528–3533.Google Scholar
Wu, Y., Hamroun, B., Le Gorrec, Y. and Maschke, B. (2018), Reduced order LQG control design for port Hamiltonian systems, Automatica 95, 86–92.CrossRefGoogle Scholar
Yoshimura, H. and Marsden, J. E. (2006a), Dirac structures in Lagrangian mechanics, I: Implicit Lagranian systems, J. Geom. Phys. 57, 133–156.Google Scholar
Yoshimura, H. and Marsden, J. E. (2006b), Dirac structures in Lagrangian mechanics, II: Variational structures, J. Geom. Phys. 57, 209–250.CrossRefGoogle Scholar
Zienkiewicz, O. C. and Taylor, R. L. (2005), The Finite Element Method for Solid and Structural Mechanics, Elsevier.Google Scholar
Zoback, M. D. (2010), Reservoir Geomechanics, Cambridge University Press.Google Scholar