Numerical methods to compute a minimal realization of a port-Hamiltonian system

Port-Hamiltonian (pH) systems are a very important modeling tool in almost all areas of systems and control, in particular in network based model of multi-physics multi-scale systems. They lead to remarkably robust models that can be easily interconnected. This paper discusses the derivation of pH models from time-domain input-output data. While a direct construction of pH models is still an open problem, we present three different indirect numerical methods for the realization of pH systems. The algorithms are implemented in MATLAB and their performance is illustrated via several numerical examples.

[1]  Timo Reis,et al.  Positive real and bounded real balancing for model reduction of descriptor systems , 2010, Int. J. Control.

[2]  Peter Benner,et al.  Model Reduction of Descriptor Systems with the MORLAB Toolbox , 2018 .

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

[4]  Leiba Rodman,et al.  Algebraic Riccati equations , 1995 .

[5]  J. Willems.  Dissipative dynamical systems part I: General theory , 1972 .

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

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

[8]  Kamel Hariche,et al.  Review of MIMO Minimal Realization Techniques and Case Study on SCARA Robot Manipulator , 2017 .

[9]  J. Willems.  Least squares stationary optimal control and the algebraic Riccati equation , 1971 .

[10]  H. Rosenbrock,et al.  State-space and multivariable theory, , 1970 .

[11]  Paul Van Dooren,et al.  Robust port-Hamiltonian representations of passive systems , 2018, Autom..

[12]  Juan Luis Varona,et al.  Port-Hamiltonian systems: an introductory survey , 2006 .

[13]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[14]  Rik W. De Doncker,et al.  Port Controlled Hamiltonian Modeling and IDA-PBC Control of Dual Active Bridge Converters for DC Microgrids , 2019, IEEE Transactions on Industrial Electronics.

[15]  Volker Mehrmann,et al.  A Behavioral Approach to Time-Varying Linear Systems. Part 2: Descriptor Systems , 2005, SIAM J. Control. Optim..

[16]  Punit Sharma,et al.  Computing the nearest stable matrix pairs , 2017, Numer. Linear Algebra Appl..

[17]  Paul Van Dooren,et al.  Computing the regularization of a linear differential-algebraic system , 2015, Syst. Control. Lett..

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

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

[20]  Volker Mehrmann,et al.  Linear Algebra Properties of Dissipative Hamiltonian Descriptor Systems , 2018, SIAM J. Matrix Anal. Appl..

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

[22]  Gene H. Golub,et al.  Matrix computations , 1983 .

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

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

[25]  Pia Domschke,et al.  Modellierung von Gasnetzwerken: Eine Übersicht , 2017 .

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

[27]  Yun Zeng,et al.  Direct modeling method of Generalized Hamiltonian system and simulation simplified , 2012 .

[28]  Chi-Tsong Chen,et al.  Linear System Theory and Design , 1995 .

[29]  Hans Zwart,et al.  Linear port-Hamiltonian descriptor systems , 2018, Math. Control. Signals Syst..

[30]  Punit Sharma,et al.  Finding the Nearest Positive-Real System , 2017, SIAM J. Numer. Anal..

[31]  V. Mehrmann,et al.  Open physical systems: from GENERIC to port-Hamiltonian systems , 2018, 1804.04064.

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

[33]  Saeed Ghadimi,et al.  Accelerated gradient methods for nonconvex nonlinear and stochastic programming , 2013, Mathematical Programming.

[34]  P. Dooren.  The Computation of Kronecker's Canonical Form of a Singular Pencil , 1979 .

[35]  Roberto Kawakami Harrop Galvão,et al.  Conversion of descriptor representations to state-space form: an extension of the shuffle algorithm , 2018, Int. J. Control.

[36]  Arjan van der Schaft,et al.  Port-Hamiltonian Systems Theory: An Introductory Overview , 2014, Found. Trends Syst. Control..

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

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

[39]  Brian D. O. Anderson,et al.  Spectral factorization with imaginary-axis zeros☆ , 1997 .

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

[41]  A. Antoulas,et al.  A framework for the solution of the generalized realization problem , 2007 .

[42]  T. Berger,et al.  Hamburger Beiträge zur Angewandten Mathematik Controllability of linear differential-algebraic systems-A survey , 2012 .

[43]  B. Schutter,et al.  Minimal state-space realization in linear system theory: an overview , 2000 .

[44]  P. Lancaster,et al.  The Algebraic Riccati Equation , 1995 .

[45]  Thomas Kailath,et al.  Linear Systems , 1980 .

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

[47]  Françoise Couenne,et al.  The port Hamiltonian approach to modeling and control of Continuous Stirred Tank Reactors , 2011 .

[48]  Robert Altmann,et al.  On the Port-Hamiltonian Structure of the Navier-Stokes Equations for Reactive Flows , 2016 .

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

[50]  Volker Mehrmann,et al.  Port-Hamiltonian Realizations of Linear Time Invariant Systems , 2022, ArXiv.

[51]  Sanda Lefteriu,et al.  Chapter 8: A Tutorial Introduction to the Loewner Framework for Model Reduction , 2017 .

[52]  Benjamin Peherstorfer,et al.  Data-Driven Reduced Model Construction with Time-Domain Loewner Models , 2017, SIAM J. Sci. Comput..

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

[54]  Heinrich H. Bülthoff,et al.  Port Hamiltonian Modeling of a Cable Driven Robot , 2018 .

[55]  Nicole Marheineke,et al.  Model reduction techniques for linear constant coefficient port-Hamiltonian differential-algebraic systems , 2019, 1901.10242.

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

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

[58]  Sabine Van Huffel,et al.  SLICOT—A Subroutine Library in Systems and Control Theory , 1999 .

[59]  Arjan van der Schaft,et al.  Generalized port-Hamiltonian DAE systems , 2018, Syst. Control. Lett..

[60]  Punit Sharma,et al.  Stability Radii for Linear Hamiltonian Systems with Dissipation Under Structure-Preserving Perturbations , 2016, SIAM J. Matrix Anal. Appl..