Reduction of Large Circuit Models Via Low Rank Approximate Gramians

We describe a model reduction algorithm which is well-suited for the reduction of large linear interconnect models. It is an orthogonal projection method which takes as the projection space the sum of the approximate dominant controllable subspace and the approximate dominant observable subspace. These approximate dominant subspaces are obtained using the Cholesky Factor ADI (CF–ADI) algorithm. We describe an improvement upon the existing implementation of CF–ADI which can result in significant savings in computational cost. We show that the new model reduction method matches moments at the negative of the CF–ADI parameters, and that it can be easily adapted to allow for DC matching, as well as for passivity preservation for multi-port RLC circuit models which come from modified nodal analysis.

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

[2]  Jacob K. White,et al.  A coordinate-transformed Arnoldi algorithm for generating guaranteed stable reduced-order models of RLC circuits , 1996, Proceedings of International Conference on Computer Aided Design.

[3]  R. Freund.  Solution of shifted linear systems by quasi-minimal residual iterations , 1993 .

[4]  Eduardo D. Sontag,et al.  Mathematical Control Theory: Deterministic Finite Dimensional Systems , 1990 .

[5]  Roland W. Freund,et al.  Efficient linear circuit analysis by Pade´ approximation via the Lanczos process , 1994, EURO-DAC '94.

[6]  Stable and efficient reduction of substrate model networks using congruence transforms , 1995, ICCAD.

[7]  Jacob K. White,et al.  A coordinate-transformed Arnoldi algorithm for generating guaranteed stable reduced-order models of RLC circuits , 1996, ICCAD 1996.

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

[9]  Paul Van Dooren,et al.  On some modifications of the Lanczos algorithm and the relation with Padé approximations , 1997 .

[10]  L. Silverman,et al.  Model reduction via balanced state space representations , 1982 .

[11]  Eugene L. Wachspress,et al.  Alternating direction implicit iteration for systems with complex spectra , 1991 .

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

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

[14]  M. Safonov,et al.  A Schur method for balanced-truncation model reduction , 1989 .

[15]  Gene H. Golub,et al.  Matrix computations (3rd ed.) , 1996 .

[16]  Eugene L. Wachspress,et al.  Solution of lyapunov equations by alternating direction implicit iteration , 1991 .

[17]  Jacob K. White,et al.  Efficient model reduction of interconnect via approximate system gramians , 1999, 1999 IEEE/ACM International Conference on Computer-Aided Design. Digest of Technical Papers (Cat. No.99CH37051).

[18]  R. W. Freund,et al.  The Look-Ahead Lanczos Process for Large Nonsymmetric Matrices and Related Algorithms , 1993 .

[19]  R. Freund.  Reduced-Order Modeling Techniques Based on Krylov Subspaces and Their Use in Circuit Simulation , 1999 .

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

[21]  P. Chandrasekharan.  Robust Control of Linear Dynamical Systems , 1996 .

[22]  D. Enns.  Model reduction with balanced realizations: An error bound and a frequency weighted generalization , 1984, The 23rd IEEE Conference on Decision and Control.

[23]  I. Postlethwaite,et al.  Truncated balanced realization of a stable non-minimal state-space system , 1987 .

[24]  Lawrence T. Pileggi,et al.  PRIMA: passive reduced-order interconnect macromodeling algorithm , 1998, 1997 Proceedings of IEEE International Conference on Computer Aided Design (ICCAD).

[25]  Mattan Kamon,et al.  A mixed nodal-mesh formulation for efficient extraction and passive reduced-order modeling of 3D interconnects , 1998, Proceedings 1998 Design and Automation Conference. 35th DAC. (Cat. No.98CH36175).