Port-Hamiltonian descriptor systems

The modeling framework of port-Hamiltonian systems is systematically extended to constrained dynamical systems (descriptor systems, differential-algebraic equations). A new algebraically and geometrically defined system structure is derived. It is shown that this structure is invariant under equivalence transformations, and that it is adequate also for the modeling of high-index descriptor systems. The regularization procedure for descriptor systems to make them suitable for simulation and control is modified to deal with the port-Hamiltonian structure. The relevance of the new structure is demonstrated with several examples.

[1]  A. J. van der Schaft,et al.  Port-Hamiltonian Differential-Algebraic Systems , 2013 .

[2]  Saifon Chaturantabut,et al.  Structure-preserving model reduction for nonlinear port-Hamiltonian systems , 2011, IEEE Conference on Decision and Control and European Control Conference.

[3]  Arjan van der Schaft,et al.  Structure preserving model reduction of port-Hamiltonian systems by moment matching at infinity , 2010, Autom..

[4]  Andreas Kugi,et al.  Automatic Control of Mechatronic Systems , 2001 .

[5]  Herbert Egger,et al.  Damped wave systems on networks: exponential stability and uniform approximations , 2016, Numerische Mathematik.

[6]  Johann Bals,et al.  Virtual Iron Bird - A Multidisciplinary Modelling and Simulation Platform for New Aircraft System Architectures , 2005 .

[7]  Romeo Ortega,et al.  Putting energy back in control , 2001 .

[8]  S. Campbell.  Linearization of DAEs along trajectories , 1995 .

[9]  Hans Zwart,et al.  Linear Port-Hamiltonian Systems on Infinite-dimensional Spaces , 2012 .

[10]  Arjan van der Schaft,et al.  Structure-preserving tangential interpolation for model reduction of port-Hamiltonian systems , 2011, Autom..

[11]  René Lamour,et al.  Differential-Algebraic Equations: A Projector Based Analysis , 2013 .

[12]  Volker Mehrmann,et al.  Analysis of Over- and Underdetermined Nonlinear Differential-Algebraic Systems with Application to Nonlinear Control Problems , 2001, Math. Control. Signals Syst..

[13]  Volker Mehrmann,et al.  Analysis and Numerical Solution of Control Problems in Descriptor Form , 2001, Math. Control. Signals Syst..

[14]  Volker Mehrmann,et al.  Self-adjoint differential-algebraic equations , 2014, Math. Control. Signals Syst..

[15]  Stephen L. Campbell,et al.  A general form for solvable linear time varying singular systems of differential equations , 1987 .

[16]  P. Rentrop,et al.  Differential-Algebraic Equations , 2006 .

[17]  Bernhard Maschke,et al.  Bond graph for dynamic modelling in chemical engineering , 2008 .

[18]  Jan C. Willems,et al.  Introduction to mathematical systems theory: a behavioral approach, Texts in Applied Mathematics 26 , 1999 .

[19]  Roland W. Freund,et al.  The SPRIM Algorithm for Structure-Preserving Order Reduction of General RCL Circuits , 2011 .

[20]  P. C. Breedveld,et al.  Modeling and Simulation of Dynamic Systems using Bond Graphs , 2008, ICRA 2008.

[21]  A. Schaft.  Port-Hamiltonian Systems: Network Modeling and Control of Nonlinear Physical Systems , 2004 .

[22]  Linda R. Petzold,et al.  Numerical solution of initial-value problems in differential-algebraic equations , 1996, Classics in applied mathematics.

[23]  Volker Mehrmann,et al.  Nonlinear eigenvalue and frequency response problems in industrial practice , 2011 .

[24]  Volker Mehrmann,et al.  Regularization of linear and nonlinear descriptor systems , 2011 .

[25]  Werner Schiehlen,et al.  Advanced Multibody System Dynamics , 1899 .

[26]  Volker Mehrmann,et al.  A Matlab Toolbox for the Regularization of Descriptor Systems Arising from Generalized Realization Procedures , 2015 .

[27]  Kathleen Saint-Onge.  Passivity , 2019, Discovering FranÇoise Dolto.

[28]  Claus Führer,et al.  Numerical Methods in Multibody Dynamics , 2013 .

[29]  Nicole Marheineke,et al.  On Structure-Preserving Model Reduction for Damped Wave Propagation in Transport Networks , 2017, SIAM J. Sci. Comput..

[30]  Volker Mehrmann,et al.  Differential-Algebraic Equations: Analysis and Numerical Solution , 2006 .

[31]  A. Schaft.  Port-Hamiltonian systems: an introductory survey , 2006 .

[32]  Arjan van der Schaft,et al.  Hamiltonian formulation of bond graphs , 2003 .

[33]  E. Fuehrer C. Eich,et al.  Numerical Methods in Multibody Dynamies , 1992 .

[34]  A. Bunse-Gerstner,et al.  Feedback design for regularizing descriptor systems , 1999 .

[35]  L. Dai,et al.  Singular Control Systems , 1989, Lecture Notes in Control and Information Sciences.

[36]  D. Hinrichsen,et al.  Mathematical Systems Theory I: Modelling, State Space Analysis, Stability and Robustness , 2010 .

[37]  Bernhard Maschke,et al.  An intrinsic Hamiltonian formulation of network dynamics: Non-standard Poisson structures and gyrators , 1992 .

[38]  Diederich Hinrichsen,et al.  Mathematical Systems Theory I , 2006, IEEE Transactions on Automatic Control.

[39]  Volker Mehrmann,et al.  Regularization of Linear Descriptor Systems with Variable Coefficients , 1997 .

[40]  F. Callier.  Introduction to Mathematical Systems Theory: A Behavioural Approach, by Jan Willem Polderman and Jan C. Willems, Springer, New York, NY, 1998, Texts in Applied Mathematics Vol. 26, 424pp. , 2002 .

[41]  A. J. V. D. Schafta,et al.  Hamiltonian formulation of distributed-parameter systems with boundary energy flow , 2002 .

[42]  A. Schaft,et al.  Hamiltonian formulation of distributed-parameter systems with boundary energy flow , 2002 .

[43]  A. Isidori,et al.  Passivity, feedback equivalence, and the global stabilization of minimum phase nonlinear systems , 1991 .

[44]  Manfred Hiller,et al.  Multibody system dynamics and mechatronics , 2006 .