On the embedding of state space realizations

In this paper, the generality of the particular model reduction method, known as the projection of state space realization, is investigated. Given two transfer functions, one wants to find the necessary and sufficient conditions for the embedding of a state-space realization of the transfer function of smaller McMillan degree into a state-space realization of the transfer function of larger McMillan degree. Two approaches are considered, both in the MIMO case. First, when the difference of the McMillan degree between the transfer functions is equal to one and there is no common pole, necessary and sufficient conditions are provided. Then, the generic case is considered using a pencil approach. Finally, it is shown that the condition of embedding is related to the eigen structure of a pencil that appears in the framework of tangential interpolation.

[1]  H. Rosenbrock,et al.  State-space and multivariable theory, , 1970 .

[2]  R. C. Thompson.  Interlacing inequalities for invariant factors , 1979 .

[3]  E. M. Sá,et al.  Imbedding conditions for λ-matrices , 1979 .

[4]  Thomas Kailath,et al.  Linear Systems , 1980 .

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

[6]  採編典藏組.  Society for Industrial and Applied Mathematics(SIAM) , 2008 .

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

[8]  P. Dooren,et al.  A reduced order observer for descriptor systems , 1986 .

[9]  Sabine Mondié,et al.  Assigning the Kronecker invariants of a matrix pencil by row or column completions , 1998 .

[10]  Andras Varga,et al.  Enhanced Modal Approach for Model Reduction , 1995 .

[11]  Paul Van Dooren,et al.  Model reduction of state space systems via an implicitly restarted Lanczos method , 1996, Numerical Algorithms.

[12]  Yoram Halevi,et al.  ON MODEL ORDER REDUCTION VIA PROJECTION , 2002 .

[13]  Paul Van Dooren,et al.  Model Reduction of MIMO Systems via Tangential Interpolation , 2005, SIAM J. Matrix Anal. Appl..

[14]  Danny C. Sorensen,et al.  The Sylvester equation and approximate balanced reduction , 2002 .

[15]  B. Anderson,et al.  Model reduction for control system design , 1984 .

[16]  P. Dooren.  The generalized eigenstructure problem in linear system theory , 1981 .

[17]  J. Doyle,et al.  Robust and optimal control , 1995, Proceedings of 35th IEEE Conference on Decision and Control.

[18]  Brian D. O. Anderson,et al.  On the problem of stable rational interpolation , 1989 .

[19]  Y. Halevi.  Can any reduced order model be obtained via projection? , 2004, Proceedings of the 2004 American Control Conference.

[20]  K. Glover.  All optimal Hankel-norm approximations of linear multivariable systems and their L, ∞ -error bounds† , 1984 .

[21]  Kyle A. Gallivan,et al.  Model reduction via truncation: an interpolation point of view , 2003 .

[22]  P. Dooren,et al.  Asymptotic Waveform Evaluation via a Lanczos Method , 1994 .

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

[24]  Paul Van Dooren,et al.  Projection of state space realizations , 2004 .

[25]  R. Freund.  Krylov-subspace methods for reduced-order modeling in circuit simulation , 2000 .

[26]  I. Jaimoukha,et al.  Implicitly Restarted Krylov Subspace Methods for Stable Partial Realizations , 1997, SIAM J. Matrix Anal. Appl..

[27]  P. Van Dooren,et al.  Model reduction via projection of generalized state space systems , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

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

[29]  Paul Van Dooren,et al.  A generalized state-space approach for the additive decomposition of a transfer matrix , 1992 .