A structured model reduction method for large scale networks

Mathematical models of networked systems often take the form of a set of complex large-scale differential equations. Model reduction is a commonly used technique of producing a simplified, yet accurate, description of these systems. Most available model reduction techniques require state transformations, which can cause the structural information of the system to be lost. In this paper, a systematic methodology is proposed for reducing linear network system models without employing state transformations. The proposed method is based on minimising the Hankel error norm between the original system and the reduced order model while ensuring that the state vector in the reduced model is a subset of the original state vector, which preserves the model structure. An error bound between the original and reduced models is ensured and the steady-state behaviour of the system is also preserved. The methodology can be automated so that it be applied to large scale networks. The proposed method can be extended to uncertain systems described by linear parameter varying models. The effectiveness of the proposed methods is demonstrated through simulation examples.

[1]  Robert E. Skelton,et al.  Stability tests for constrained linear systems , 2001 .

[2]  Li Li,et al.  Structured coprime factor model reduction based on LMIs , 2005, Autom..

[3]  Sheldon X.-D. Tan,et al.  Advanced Model Order Reduction Techniques in VLSI Design , 2007 .

[4]  Maamar Bettayeb,et al.  Characterization of the solution to the optimal H∞ model reduction problem , 1993 .

[5]  A. Tzafriri,et al.  The total quasi-steady-state approximation is valid for reversible enzyme kinetics. , 2004, Journal of theoretical biology.

[6]  Tobias Breiten,et al.  Krylov subspace methods for model order reduction of bilinear control systems , 2010, Syst. Control. Lett..

[7]  Karen Willcox,et al.  Krylov projection framework for Fourier model reduction , 2008, Autom..

[8]  Petar V. Kokotovic,et al.  Singular perturbations and time-scale methods in control theory: Survey 1976-1983 , 1982, Autom..

[9]  Khalide Jbilou,et al.  Matrix Krylov subspace methods for large scale model reduction problems , 2006, Appl. Math. Comput..

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

[11]  John J. Tyson,et al.  Modeling Networks of Coupled Enzymatic Reactions Using the Total Quasi-Steady State Approximation , 2007, PLoS Comput. Biol..

[12]  Petar V. Kokotovic,et al.  Singular perturbations and order reduction in control theory - An overview , 1975, at - Automatisierungstechnik.

[13]  Ali Zilouchian,et al.  Balanced structures and model reduction of unstable systems , 1991, IEEE Proceedings of the SOUTHEASTCON '91.

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

[15]  J. Liao,et al.  Lumping analysis of biochemical reaction systems with time scale separation , 1988, Biotechnology and bioengineering.

[16]  Fen Wu,et al.  Computationally Efficient Algorithm For Frequency‐Weighted Optimal H∞ Model Reduction , 2003 .

[17]  Luigi Fortuna,et al.  Model Order Reduction Techniques with Applications in Electrical Engineering , 1992 .

[18]  K. Grigoriadis.  Optimal H ∞ model reduction via linear matrix inequalities: continuous- and discrete-time cases , 1995 .

[19]  Paul Van Dooren,et al.  Model Reduction of Interconnected Systems , 2008 .

[20]  Alison S. Tomlin,et al.  A systematic lumping approach for the reduction of comprehensive kinetic models , 2005 .

[21]  Joe H. Chow,et al.  Singular perturbation and iterative separation of time scales , 1979, Autom..

[22]  Antonis Papachristodoulou,et al.  Structured model reduction for dynamical networked systems , 2010, 49th IEEE Conference on Decision and Control (CDC).

[23]  John T. Spanos,et al.  A new algorithm for L2 optimal model reduction , 1992, Autom..

[24]  Bing Chu,et al.  A model for using control theory to design sustainable policies for greenhouse gas emissions , 2011, 2011 IEEE International Conference on Control Applications (CCA).

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

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

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

[28]  Antonis Papachristodoulou,et al.  Model decomposition and reduction tools for large-scale networks in systems biology , 2011, Autom..

[30]  R. M. Murray,et al.  Model reduction of interconnected linear systems , 2009 .

[31]  Joel A. Tropp,et al.  Greed is good: algorithmic results for sparse approximation , 2004, IEEE Transactions on Information Theory.

[32]  M. Safonov,et al.  Optimal Hankel model reduction for nonminimal systems , 1990 .

[33]  Emrah Biyik,et al.  Area Aggregation and Time Scale Modeling for Sparse Nonlinear Networks , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

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

[35]  Nikolaos Kazantzis,et al.  Model reduction and coarse-graining approaches for multiscale phenomena , 2006 .