Stability Radii for Linear Hamiltonian Systems with Dissipation Under Structure-Preserving Perturbations

Dissipative Hamiltonian (DH) systems are an important concept in energy based modeling of dynamical systems. One of the major advantages of the DH formulation is that system properties are encoded in an algebraic way. For instance, the algebraic structure of DH systems guarantees that the system is automatically stable. In this paper the question is discussed when a linear constant coefficient DH system is on the boundary of the region of asymptotic stability, i.e., when it has purely imaginary eigenvalues, or how much it has to be perturbed to be on this boundary. For unstructured systems this distance to instability (stability radius) is well understood. In this paper, explicit formulas for this distance under structure-preserving perturbations are determined. It is also shown (via numerical examples) that under structure-preserving perturbations the asymptotical stability of a DH system is much more robust than under general perturbations, since the distance to instability can be much larger when struc...

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

[2]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

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

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

[5]  Caren Tischendorf,et al.  Structural analysis of electric circuits and consequences for MNA , 2000, Int. J. Circuit Theory Appl..

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

[7]  Peter Lancaster,et al.  The theory of matrices , 1969 .

[8]  F. R. Gantmakher.  The Theory of Matrices , 1984 .

[9]  Arjan van der Schaft,et al.  Port-Hamiltonian Systems on Graphs , 2011, SIAM J. Control. Optim..

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

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

[12]  Punit Sharma,et al.  Structured Eigenvalue Backward Errors of Matrix Pencils and Polynomials with Hermitian and Related Structures , 2014, SIAM J. Matrix Anal. Appl..

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

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

[15]  Götz Trenkler.  Matrices Which Take a Given Vector into a Given Vector—Revisited , 2004, Am. Math. Mon..

[16]  D. Hinrichsen,et al.  Real and Complex Stability Radii: A Survey , 1990 .

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

[18]  K. Veselic,et al.  Damped Oscillations of Linear Systems: A Mathematical Introduction , 2011 .

[19]  N. Martins.  Efficient Eigenvalue and Frequency Response Methods Applied to Power System Small-Signal Stability Studies , 1986, IEEE Transactions on Power Systems.

[20]  Volker Mehrmann,et al.  Numerical methods for parametric model reduction in the simulation of disk brake squeal , 2016 .

[21]  Werner Schiehlen,et al.  Multibody Systems Handbook , 2012 .

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

[23]  Françoise Tisseur,et al.  Structured Mapping Problems for Matrices Associated with Scalar Products. Part I: Lie and Jordan Algebras , 2007, SIAM J. Matrix Anal. Appl..

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

[25]  Joost Rommes,et al.  Exploiting structure in large-scale electrical circuit and power system problems , 2009 .

[26]  Arjan van der Schaft,et al.  Interconnection and damping assignment passivity-based control of port-controlled Hamiltonian systems , 2002, Autom..

[27]  Alastair Spence,et al.  A Newton-based method for the calculation of the distance to instability , 2011 .

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

[29]  C. Loan.  How Near is a Stable Matrix to an Unstable Matrix , 1984 .

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

[31]  P. Pellanda,et al.  Computation of Transfer Function Dominant Zeros With Applications to Oscillation Damping Control of Large Power Systems , 2007, IEEE Transactions on Power Systems.

[32]  R. Byers.  A Bisection Method for Measuring the Distance of a Stable Matrix to the Unstable Matrices , 1988 .

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

[34]  N. Martins,et al.  Determination of suitable locations for power system stabilizers and static VAr compensators for damping electromechanical oscillations in large scale power systems , 1989, Conference Papers Power Industry Computer Application Conference.

[35]  D. Hinrichsen,et al.  Stability radii of linear systems , 1986 .

[36]  Michael Karow,et al.  μ-Values and Spectral Value Sets for Linear Perturbation Classes Defined by a Scalar Product , 2011, SIAM J. Matrix Anal. Appl..

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

[38]  A. V. der,et al.  An Intrinsic Hamiltonian Formulation of Network Dynamics : Non-standard Poisson Structures and Gyrators , 2001 .

[39]  G. Alistair Watson,et al.  An Algorithm for Computing the Distance to Instability , 1998, SIAM J. Matrix Anal. Appl..

[40]  A. Schaft,et al.  On Representations and Integrability of Mathematical Structures in Energy-Conserving Physical Systems , 1999 .

[41]  Ji-guang Sun.  Backward perturbation analysis of certain characteristic subspaces , 1993 .

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