A Connection between Time Domain Model Order Reduction and Moment Matching

We investigate the time domain model order reduction (MOR) framework using general orthogonal polynomials by Jiang and Chen [1] and extend their idea by exploiting the structure of the corresponding linear system of equations. Identifying an equivalent Sylvester equation, we show a connection to a rational Krylov subspace, and thus to moment matching. This theoretical link between the MOR techniques is illustrated by three numerical examples. For linear time-invariant systems, the link also motivates that the time domain approach can be at best as accurate as moment matching, since the expansion points are fixed by the choice of the polynomial basis, while in moment matching they can be adapted to the system.

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

[2]  B. M. Mohan,et al.  Orthogonal Functions in Systems and Control , 1995 .

[3]  J. Lambert.  Numerical Methods for Ordinary Differential Equations , 1991 .

[4]  Caroline Kulcsar,et al.  Model Reduction of Linear Systems with Random Parameters , 2013, ALCOSP.

[5]  Serkan Gugercin,et al.  Model reduction for systems with inhomogeneous initial conditions , 2016, Syst. Control. Lett..

[6]  David Thomas,et al.  The Art in Computer Programming , 2001 .

[7]  Robert Brawer,et al.  The linear algebra of the pascal matrix , 1992 .

[8]  Thomas Wolf,et al.  H 2 pseudo-optimal model order reduction , 2014 .

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

[10]  James Demmel,et al.  Applied Numerical Linear Algebra , 1997 .

[11]  Rudy Eid,et al.  Time Domain Model Reduction by Moment Matching , 2009 .

[12]  Dag Billger.  The Butterfly Gyro , 2005 .

[13]  Yao-Lin Jiang,et al.  Time Domain Model Order Reduction of General Orthogonal Polynomials for Linear Input-Output Systems , 2012, IEEE Transactions on Automatic Control.

[14]  Antoine Vandendorpe,et al.  Model reduction of linear systems : an interpolation point of view/ , 2004 .

[15]  Athanasios C. Antoulas,et al.  Balanced truncation model reduction for systems with inhomogeneous initial conditions , 2011, Autom..

[16]  Charles R. Johnson,et al.  Topics in Matrix Analysis , 1991 .

[17]  Thilo Penzl.  Algorithms for model reduction of large dynamical systems , 2006 .

[18]  Jens Saak,et al.  Efficient Numerical Solution of Large Scale Algebraic Matrix Equations in PDE Control and Model Order Reduction , 2009 .

[19]  Hyunjoong Kim,et al.  Functional Analysis I , 2017 .

[20]  Taishan Zeng,et al.  OPTIMAL H2 MODEL REDUCTION FOR LARGE SCALE MIMO SYSTEMS VIA TANGENTIAL INTERPOLATION , 2011 .

[21]  Ninoslav Truhar,et al.  An Efficient Method for Estimating the Optimal Dampers' Viscosity for Linear Vibrating Systems Using Lyapunov Equation , 2009, SIAM J. Matrix Anal. Appl..

[22]  Serkan Gugercin,et al.  H2 Model Reduction for Large-Scale Linear Dynamical Systems , 2008, SIAM J. Matrix Anal. Appl..