Structure Preserving Moment Matching for Port-Hamiltonian Systems: Arnoldi and Lanczos

Structure preserving model reduction of single-input single-output port-Hamiltonian systems is considered by employing the rational Krylov methods. The rational Arnoldi method is shown to preserve (for the reduced order model) not only a specific number of the moments at an arbitrary point in the complex plane but also the port-Hamiltonian structure. Furthermore, it is shown how the rational Lanczos method applied to a subclass of port-Hamiltonian systems, characterized by an algebraic condition, preserves the port-Hamiltonian structure. In fact, for the same subclass of port-Hamiltonian systems the rational Arnoldi method and the rational Lanczos method turn out to be equivalent in the sense of producing reduced order port-Hamiltonian models with the same transfer function.

[1]  Gene H. Golub,et al.  Matrix computations (3rd ed.) , 1996 .

[2]  Boris Lohmann,et al.  Krylov Subspace Methods in Linear Model Order Reduction : Introduction and Invariance Properties , 2002 .

[3]  Jan C. Willems,et al.  Dissipative Dynamical Systems , 2007, Eur. J. Control.

[4]  Serkan Gugercin,et al.  Interpolation-based H 2 Model Reduction for port-Hamiltonian Systems , 2009 .

[5]  Thomas Wolf,et al.  Passivity and Structure Preserving Order Reduction of Linear Port-Hamiltonian Systems Using Krylov Subspaces , 2010, Eur. J. Control.

[6]  Athanasios C. Antoulas,et al.  Approximation of Large-Scale Dynamical Systems , 2005, Advances in Design and Control.

[7]  Paul Van Dooren,et al.  On some recent developments in projection-based model reduction , 1998 .

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

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

[10]  Arjan van der Schaft,et al.  Interpolation-based ℌ2 model reduction for port-Hamiltonian systems , 2009, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.

[11]  Arjan van der Schaft,et al.  Robust Nonlinear Control. Port-controlled Hamiltonian Systems: Towards a Theory for Control and Design of Nonlinear Physical Systems. , 2000 .

[12]  Rostyslav V. Polyuga,et al.  Model reduction of port-Hamiltonian systems , 2010 .

[13]  A. Schaft,et al.  The Hamiltonian formulation of energy conserving physical systems with external ports , 1995 .

[14]  Rostyslav V. Polyuga.  Discussion on: "Passivity and Structure Preserving Order Reduction of Linear Port-Hamiltonian Systems Using Krylov Subspaces" , 2010, Eur. J. Control.

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

[16]  Eric James Grimme,et al.  Krylov Projection Methods for Model Reduction , 1997 .

[17]  Arjan van der Schaft,et al.  Structure-preserving model reduction of complex physical systems , 2009, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.

[18]  Serkan Gugercin,et al.  Interpolation-based H2 model reduction for port-Hamiltonian systems , 2009 .

[19]  GrimmeIntel CorporationSanta Clara.  On Some Recent Developments in Projection-based Model Reduction , 1998 .