Condensed Forms for Linear Port-Hamiltonian Descriptor Systems

Motivated by the structure which arises in the port-Hamiltonian formulation of constraint dynamical systems, structure preserving condensed forms for skew-adjoint differential-algebraic equations (DAEs) are derived. Moreover, structure preserving condensed forms under constant rank assumptions for linear port-Hamiltonian differential-algebraic equations are developed. These condensed forms allow for the further analysis of the properties of port-Hamiltonian DAEs and to study, e.g., existence and uniqueness of solutions or to determine the index. It can be shown that under certain conditions for regular port-Hamiltonian DAEs the strangeness index is bounded by $\mu\leq1$.

[1]  V. Mehrmann,et al.  Canonical forms for linear differential-algebraic equations with variable coefficients , 1994 .

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

[3]  Hans Zwart,et al.  Port-Hamiltonian descriptor systems , 2017, 1705.09081.

[4]  Lena Scholz,et al.  The Signature Method for DAEs arising in the modeling of electrical circuits , 2018, J. Comput. Appl. Math..

[5]  M. A. Akanbi,et al.  Numerical solution of initial value problems in differential - algebraic equations , 2005 .

[6]  R. C. Thompson,et al.  Pencils of complex and real symmetric and skew matrices , 1991 .

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

[8]  Volker Mehrmann,et al.  A new look at pencils of matrix valued functions , 1994 .

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

[10]  Seamus D. Garvey,et al.  Dynamics of Rotating Machines , 2010 .

[11]  Werner C. Rheinboldt,et al.  Nonholonomic motion of rigid mechanical systems from a DAE viewpoint , 1987 .

[12]  Timo Eirola,et al.  On Smooth Decompositions of Matrices , 1999, SIAM J. Matrix Anal. Appl..

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

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

[15]  E. Griepentrog,et al.  Differential-algebraic equations and their numerical treatment , 1986 .

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

[17]  L. Biegler,et al.  Control and Optimization with Differential-Algebraic Constraints , 2012 .

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

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

[20]  Volker Mehrmann,et al.  A STRUCTURED STAIRCASE ALGORITHM FOR SKEW-SYMMETRIC / SYMMETRIC , 2005 .

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

[22]  E. Hairer,et al.  Geometric Numerical Integration: Structure Preserving Algorithms for Ordinary Differential Equations , 2004 .

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

[24]  Ricardo Riaza,et al.  Differential-Algebraic Systems: Analytical Aspects and Circuit Applications , 2008 .

[25]  P. Lancaster,et al.  Overdamped and Gyroscopic Vibrating Systems , 1992 .

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