Frequency-Limited Balanced Truncation with Low-Rank Approximations

In this article we investigate model order reduction of large-scale systems using frequency-limited balanced truncation, which restricts the well known balanced truncation framework to prescribed frequency regions. The main emphasis is put on the efficient numerical realization of this model reduction approach. We discuss numerical methods to take care of the involved matrix-valued functions. The occurring large-scale Lyapunov equations are solved for low-rank approximations for which we also establish results regarding the eigenvalues of their solutions. These results, and also numerical experiments, will show that the eigenvalues of the Lyapunov solutions in frequency-limited balanced truncation often decay faster than those in standard balanced truncation. Moreover, we show in further numerical examples that frequency-limited balanced truncation generates reduced order models which are significantly more accurate in the considered frequency region.

[1]  Lars Grasedyck,et al.  Existence of a low rank or ℋ︁‐matrix approximant to the solution of a Sylvester equation , 2004, Numer. Linear Algebra Appl..

[2]  Khalide Jbilou,et al.  Block Krylov Subspace Methods for Solving Large Sylvester Equations , 2002, Numerical Algorithms.

[3]  Valeria Simoncini,et al.  A new investigation of the extended Krylov subspace method for matrix function evaluations , 2009, Numer. Linear Algebra Appl..

[4]  Valeria Simoncini,et al.  Krylov subspace methods for projected Lyapunov equations , 2012 .

[5]  Zhongxiao Jia,et al.  A posteriori error estimates of krylov subspace approximations to matrix functions , 2013, Numerical Algorithms.

[6]  B. Moore.  Principal component analysis in linear systems: Controllability, observability, and model reduction , 1981 .

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

[8]  Axel Ruhe.  Rational Krylov sequence methods for eigenvalue computation , 1984 .

[9]  Philip I. Davies,et al.  Computing f (A)b for matrix functions f , 2005 .

[10]  Valeria Simoncini,et al.  Adaptive rational Krylov subspaces for large-scale dynamical systems , 2011, Syst. Control. Lett..

[11]  D. O’Leary.  The block conjugate gradient algorithm and related methods , 1980 .

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

[13]  I. Jaimoukha,et al.  Krylov subspace methods for solving large Lyapunov equations , 1994 .

[14]  Michiel E. Hochstenbach,et al.  Subspace extraction for matrix functions , 2005 .

[15]  Thomas Wolf,et al.  Model Order Reduction by Approximate Balanced Truncation: A Unifying Framework / Modellreduktion durch approximatives Balanciertes Abschneiden: eine vereinigende Formulierung , 2013, Autom..

[16]  I. Moret,et al.  RD-Rational Approximations of the Matrix Exponential , 2004 .

[17]  Jacob K. White,et al.  Low-Rank Solution of Lyapunov Equations , 2004, SIAM Rev..

[18]  P. Benner,et al.  Numerical solution of large and sparse continuous time algebraic matrix Riccati and Lyapunov equations: a state of the art survey , 2013 .

[19]  Muhammad Imran,et al.  Model reduction of descriptor systems using frequency limited Gramians , 2015, J. Frankl. Inst..

[20]  Valeria Simoncini,et al.  Computational Methods for Linear Matrix Equations , 2016, SIAM Rev..

[21]  L. Knizhnerman,et al.  Extended Krylov Subspaces: Approximation of the Matrix Square Root and Related Functions , 1998, SIAM J. Matrix Anal. Appl..

[22]  Valeria Simoncini,et al.  Adaptive Tangential Interpolation in Rational Krylov Subspaces for MIMO Dynamical Systems , 2014, SIAM J. Matrix Anal. Appl..

[23]  Jonathan Baker,et al.  Fast Singular Value Decay for Lyapunov Solutions with Nonnormal Coefficients , 2014, SIAM J. Matrix Anal. Appl..

[24]  Axel Ruhe.  The rational Krylov algorithm for nonsymmetric eigenvalue problems. III: Complex shifts for real matrices , 1994 .

[25]  Thomas Wolf,et al.  The ADI iteration for Lyapunov equations implicitly performs pseudo-optimal model order reduction , 2013, Int. J. Control.

[26]  Peter Eberhard,et al.  Greedy‐based approximation of frequency‐weighted Gramian matrices for model reduction in multibody dynamics , 2013 .

[27]  Hermann Mena,et al.  On the benefits of the LDLT factorization for large-scale differential matrix equation solvers , 2015 .

[28]  Thilo Penzl.  Eigenvalue decay bounds for solutions of Lyapunov equations: the symmetric case , 2000 .

[29]  Peter Benner,et al.  Self-Generating and Efficient Shift Parameters in ADI Methods for Large Lyapunov and Sylvester Equations , 2014 .

[30]  Valeria Simoncini,et al.  Minimal residual methods for large scale Lyapunov equations , 2013 .

[31]  Y. Zhou,et al.  On the decay rate of Hankel singular values and related issues , 2002, Syst. Control. Lett..

[32]  Y. Saad.  Analysis of some Krylov subspace approximations to the matrix exponential operator , 1992 .

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

[34]  L. Knizhnerman.  Calculation of functions of unsymmetric matrices using Arnoldi's method , 1991 .

[35]  Nicholas J. Higham,et al.  Functions of matrices - theory and computation , 2008 .

[36]  S. Nash,et al.  Numerical methods and software , 1990 .

[37]  Peter Benner,et al.  On the Parameter Selection Problem in the Newton-ADI Iteration for Large Scale Riccati Equations , 2007 .

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

[39]  Thilo Penzl,et al.  A Cyclic Low-Rank Smith Method for Large Sparse Lyapunov Equations , 1998, SIAM J. Sci. Comput..

[40]  Serkan Gugercin,et al.  Smith-Type Methods for Balanced Truncation of Large Sparse Systems , 2005 .

[41]  Boris Lohmann,et al.  Model Order Reduction by Approximate Balanced Truncation : A Unifying Framework , 2013 .

[42]  A. Antoulas,et al.  A Survey of Model Reduction by Balanced Truncation and Some New Results , 2004 .

[43]  Peter Benner,et al.  Computing real low-rank solutions of Sylvester equations by the factored ADI method , 2014, Comput. Math. Appl..

[44]  Hans Richter,et al.  Zum Logarithmus einer Matrix , 1949 .

[45]  Peter Benner,et al.  A Reformulated Low‐Rank ADI Iteration with Explicit Residual Factors , 2013 .

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

[47]  Peter Benner,et al.  On the ADI method for Sylvester equations , 2009, J. Comput. Appl. Math..

[48]  Marlis Hochbruck,et al.  Preconditioning Lanczos Approximations to the Matrix Exponential , 2005, SIAM J. Sci. Comput..

[49]  Jer-Nan Juang,et al.  Model reduction in limited time and frequency intervals , 1990 .

[50]  I. Koltracht,et al.  Triangular factors of Cauchy and Vandermonde matrices , 1996 .

[51]  T. Wolf.  Modell Order Reduction by Approximate Balanced Truncation: A Unifying Framework , 2013 .

[52]  Ninoslav Truhar,et al.  Bounds on the trace of a solution to the Lyapunov equation with a general stable matrix , 2007, Syst. Control. Lett..

[53]  Daniel Petersson,et al.  A Nonlinear Optimization Approach to H2-Optimal Modeling and Control , 2013 .

[54]  M. Fiedler.  Hankel and loewner matrices , 1984 .

[55]  Für Dynamik Komplexer,et al.  Balanced Truncation of Linear Time-Invariant Systems at a Single Frequency , 2013 .

[56]  S. Güttel.  Rational Krylov approximation of matrix functions: Numerical methods and optimal pole selection , 2013 .

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

[58]  Valeria Simoncini,et al.  A New Iterative Method for Solving Large-Scale Lyapunov Matrix Equations , 2007, SIAM J. Sci. Comput..

[59]  Hermann Mena,et al.  An LDLT factorization based ADI algorithm for solving large‐scale differential matrix equations , 2014 .

[60]  VALERIA SIMONCINI,et al.  MATRIX FUNCTIONS , 2006 .