Theoretical and practical aspects of linear and nonlinear model order reduction techniques

Model order reduction methods have proved to be an important technique for accelerating time-domain simulation in a variety of computer-aided design tools. In this study we present several new techniques for model reduction of the large-scale linear and nonlinear systems. First, we present a method for nonlinear system reduction based on a combination of the trajectory piecewise-linear (TPWL) method with truncated-balanced realizations (TBR). We analyze the stability characteristics of this combined method using perturbation theory. Second, we describe a linear reduction method that approximates TBR model reduction and takes advantage of sparsity of the system matrices or available accelerated solvers. This method is based on AISIAD (approximate implicit subspace iteration with alternate directions) and uses low-rank approximations of a system's gramians. This method is shown to be advantageous over the common approach of independently approximating the controllability and observability gramians, as such independent approximation methods can be inefficient when the gramians do not share a common dominant eigenspace. Third, we present a graph-based method for reduction of parameterized RC circuits. We prove that this method preserves stability and passivity of the models for nominal reduction. We present computational results for large collections of nominal and parameter-dependent circuits. Finally, we present a case study of model reduction applied to electroosmotic flow of a marker concentration pulse in a U-shaped microfluidic channel, where the marker flow in the channel is described by a three-dimensional convection-diffusion equation. First, we demonstrate the effectiveness of the modified AISIAD method in generating a low order models that correctly describe the dispersion of the marker in the linear case; that is, for the case of concentration-independent mobility and diffusion constants. Next, we describe several methods for nonlinear model reduction when the diffusion and mobility constants become concentration-dependent. (Copies available exclusively from MIT Libraries, Rm. 14-0551, Cambridge, MA 02139-4307. Ph. 617-253-5668; Fax 617-253-1690.)

[1]  Yunkai Zhou.  Numerical methods for large scale matrix equations with applications in LTI system model reduction , 2002 .

[2]  E. M. Lifshitz,et al.  Quantum mechanics: Non-relativistic theory, , 1959 .

[3]  H. Aling,et al.  Nonlinear model reduction with application to rapid thermal processing , 1996, Proceedings of 35th IEEE Conference on Decision and Control.

[4]  Munther A. Dahleh,et al.  Lectures on Dynamic Systems and Control , 2004 .

[5]  J.A. De Abreu-Garcia,et al.  A transformation approach for model order reduction of nonlinear systems , 1990, [Proceedings] IECON '90: 16th Annual Conference of IEEE Industrial Electronics Society.

[6]  Ali H. Nayfeh,et al.  A reduced-order model for electrically actuated microbeam-based MEMS , 2003 .

[7]  Steven Paul McCormick,et al.  Modeling and simulation of VLSI interconnections with moments , 1989 .

[8]  Ernest S. Kuh,et al.  Transient simulation of lossy interconnects based on the recursive convolution formulation , 1992 .

[9]  Bernard N. Sheehan,et al.  TICER: Realizable reduction of extracted RC circuits , 1999, 1999 IEEE/ACM International Conference on Computer-Aided Design. Digest of Technical Papers (Cat. No.99CH37051).

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

[11]  Yao-Joe Yang,et al.  Low-order models for fast dynamical simulation of MEMS microstructures , 1997, Proceedings of International Solid State Sensors and Actuators Conference (Transducers '97).

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

[13]  Michal Rewienski,et al.  A trajectory piecewise-linear approach to model order reduction of nonlinear dynamical systems , 2003 .

[14]  Danny C. Sorensen,et al.  Implicit Application of Polynomial Filters in a k-Step Arnoldi Method , 1992, SIAM J. Matrix Anal. Appl..

[15]  Jacob K. White,et al.  Generating nearly optimally compact models from Krylov-subspace based reduced-order models , 2000 .

[16]  Lawrence F. Shampine,et al.  The MATLAB ODE Suite , 1997, SIAM J. Sci. Comput..

[17]  J. Peraire,et al.  AIAA 99 – 1467 LOW ORDER AERODYNAMIC MODELS FOR AEROELASTIC CONTROL OF TURBOMACHINES , 1999 .

[18]  Sarma B. K. Vrudhula,et al.  Stochastic analysis of interconnect performance in the presence of process variations , 2004, IEEE/ACM International Conference on Computer Aided Design, 2004. ICCAD-2004..

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

[20]  Luís Miguel Silveira,et al.  Guaranteed passive balancing transformations for model order reduction , 2002, DAC '02.

[21]  Luís Miguel Silveira,et al.  Poor man's TBR: a simple model reduction scheme , 2004, Proceedings Design, Automation and Test in Europe Conference and Exhibition.

[22]  J. Marsden,et al.  A subspace approach to balanced truncation for model reduction of nonlinear control systems , 2002 .

[23]  Frank Wang,et al.  An efficient Lyapunov equation-based approach for generating reduced-order models of interconnect , 1999, DAC '99.

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

[25]  Karen Willcox,et al.  Reduced-order aerodynamic models for aeroelastic control of turbomachines , 1999 .

[26]  Rob A. Rutenbar,et al.  Scalable trajectory methods for on-demand analog macromodel extraction , 2005, Proceedings. 42nd Design Automation Conference, 2005..

[27]  Luís Miguel Silveira,et al.  A convex programming approach for generating guaranteed passive approximations to tabulated frequency-data , 2004, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems.

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

[29]  Jing-Rebecca Li.  Model reduction of large linear systems via low rank system gramians , 2000 .

[30]  N. P. van der Meijs,et al.  Reduced RC models for IC interconnections with coupling capacitances , 1992 .

[31]  P. Holmes,et al.  The Proper Orthogonal Decomposition in the Analysis of Turbulent Flows , 1993 .

[32]  Danny C. Sorensen,et al.  Projection Methods for Balanced Model Reduction , 2001 .

[33]  A. West,et al.  Electrokinetic flow control for composition modulation in a microchannel , 2002 .

[34]  Jacob K. White,et al.  A multiparameter moment-matching model-reduction approach for generating geometrically parameterized interconnect performance models , 2004, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems.

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

[36]  T. Stykel.  Analysis and Numerical Solution of Generalized Lyapunov Equations , 2002 .

[37]  Jian-Ming Jin,et al.  Karhunen-Loeve based model order reduction of nonlinear systems , 2002, IEEE Antennas and Propagation Society International Symposium (IEEE Cat. No.02CH37313).

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

[39]  Lawrence T. Pileggi,et al.  Parameterized interconnect order reduction with explicit-and-implicit multi-parameter moment matching for inter/intra-die variations , 2005, ICCAD-2005. IEEE/ACM International Conference on Computer-Aided Design, 2005..

[40]  Jacob K. White,et al.  A trajectory piecewise-linear approach to model order reduction and fast simulation of nonlinear circuits and micromachined devices , 2001, IEEE/ACM International Conference on Computer Aided Design. ICCAD 2001. IEEE/ACM Digest of Technical Papers (Cat. No.01CH37281).

[41]  Yehea I. Ismail,et al.  Realizable RLCK circuit crunching , 2003, Proceedings 2003. Design Automation Conference (IEEE Cat. No.03CH37451).

[42]  RewieÅ ski,et al.  A trajectory piecewise-linear approach to model order reduction of nonlinear dynamical systems , 2003 .

[43]  N. P. van der Meijs,et al.  Including higher-order moments of RC interconnections in layout-to-circuit extraction , 1996, Proceedings ED&TC European Design and Test Conference.

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

[45]  Sung-Mo Kang,et al.  An algorithm for automatic model-order reduction of nonlinear MEMS devices , 2000, 2000 IEEE International Symposium on Circuits and Systems. Emerging Technologies for the 21st Century. Proceedings (IEEE Cat No.00CH36353).

[46]  Yong P. Chen,et al.  A Quadratic Method for Nonlinear Model Order Reduction , 2000 .

[47]  A. Semlyen,et al.  Rational approximation of frequency domain responses by vector fitting , 1999 .

[48]  Alexandre Megretski,et al.  A Quasi-Convex Optimization Approach to Parameterized Model Order Reduction , 2005, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems.

[49]  P. S. Bauer.  Dissipative Dynamical Systems: I. , 1931, Proceedings of the National Academy of Sciences of the United States of America.

[50]  Luca Daniel,et al.  Parameterized model order reduction of nonlinear dynamical systems , 2005, ICCAD-2005. IEEE/ACM International Conference on Computer-Aided Design, 2005..

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

[52]  Ying Liu,et al.  Model order-reduction of RC(L) interconnect including variational analysis , 1999, DAC '99.

[53]  J. Phillips,et al.  Analog Macromodeling using Kernel Methods , 2003, ICCAD 2003.

[54]  Hugo De Man,et al.  An analytic Volterra-series-based model for a MEMS variable capacitor , 2003, IEEE Trans. Comput. Aided Des. Integr. Circuits Syst..

[55]  Joel R. Phillips,et al.  Variational interconnect analysis via PMTBR , 2004, IEEE/ACM International Conference on Computer Aided Design, 2004. ICCAD-2004..

[56]  Luís Miguel Silveira,et al.  Robust rational function approximation algorithm for model generation , 1999, Proceedings 1999 Design Automation Conference (Cat. No. 99CH36361).

[57]  J. Peraire,et al.  Balanced Model Reduction via the Proper Orthogonal Decomposition , 2002 .

[58]  B. Gustavsen,et al.  Enforcing Passivity for Admittance Matrices Approximated by Rational Functions , 2001, IEEE Power Engineering Review.

[59]  Danny C. Sorensen,et al.  A Modified Low-Rank Smith Method for Large-Scale Lyapunov Equations , 2004, Numerical Algorithms.

[60]  Yong Chen.  Model order reduction for nonlinear systems , 1999 .

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

[62]  D. Sorensen,et al.  A Survey of Model Reduction Methods for Large-Scale Systems , 2000 .

[63]  H. Nicholson,et al.  On the cross-Gramian for symmetric MIMO systems , 1985 .

[64]  Alan V. Oppenheim,et al.  Signals & systems (2nd ed.) , 1996 .

[65]  Jacob K. White,et al.  An Arnoldi approach for generation of reduced-order models for turbomachinery , 2000 .

[66]  Rob A. Rutenbar,et al.  Interval-Valued Reduced-Order Statistical Interconnect Modeling , 2004, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems.

[67]  A. Messiah.  Quantum Mechanics , 1961 .

[68]  K. Poolla,et al.  NUMERICAL SOLUTION OF THE LYAPUNOV EQUATION BY APPROXIMATE POWER ITERATION , 1996 .

[69]  Jacob K. White,et al.  Perturbation analysis of TBR model reduction in application to trajectory-piecewise linear algorithm for MEMS structures. , 2004 .

[70]  Lawrence F. Shampine,et al.  A friendly Fortran DDE solver , 2006 .

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

[72]  P. V. Doorenb,et al.  Sylvester equations and projection-based model reduction , 2003 .

[73]  Z. Bai.  Krylov subspace techniques for reduced-order modeling of large-scale dynamical systems , 2002 .

[74]  Michel S. Nakhla,et al.  Model order reduction of nonuniform transmission lines using integrated congruence transform , 2003, Proceedings 2003. Design Automation Conference (IEEE Cat. No.03CH37451).

[75]  Tatjana Stykel,et al.  Gramian-Based Model Reduction for Descriptor Systems , 2004, Math. Control. Signals Syst..

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

[77]  Ning Dong,et al.  Piecewise polynomial nonlinear model reduction , 2003, Proceedings 2003. Design Automation Conference (IEEE Cat. No.03CH37451).

[78]  Brian D. O. Anderson,et al.  Singular perturbation approximation of balanced systems , 1989 .

[79]  Jacob K. White,et al.  A TBR-based trajectory piecewise-linear algorithm for generating accurate low-order models for nonlinear analog circuits and MEMS , 2003, Proceedings 2003. Design Automation Conference (IEEE Cat. No.03CH37451).