A Review of Proper Modeling Techniques

A dynamic system model is proper for a particular application if it achieves the accuracy required by the application with minimal complexity. Because model complexity often—but not always—correlates inversely with simulation speed, a proper model is often alternatively defined as one balancing accuracy and speed. Such balancing is crucial for applications requiring both model accuracy and speed, such as system optimization and hardware-in-the-loop simulation. Furthermore, the simplicity of proper models conduces to control system analysis and design, particularly given the ease with which lower-order controllers can be implemented compared to higher-order ones. The literature presents many algorithms for deducing proper models from simpler ones or reducing complex models until they become proper. This paper presents a broad survey of the proper modeling literature. To simplify the presentation, the algorithms are classified into frequency, projection, optimization, and energy based, based on the metrics they use for obtaining proper models. The basic mechanics, properties, advantages, and limitations of the methods are discussed, along with the relationships between different techniques, with the intention of helping the modeler to identify the most suitable proper modeling method for a given application.

[1]  Leang-San Shieh,et al.  Continued Fraction Inversion by Routh's Algorithm , 1969 .

[2]  Bruce H. Wilson,et al.  A frequency-domain model-order-deduction algorithm for nonlinear systems , 1995, Proceedings of International Conference on Control Applications.

[3]  Anders Lindquist,et al.  The stability and instability of partial realizations , 1982 .

[4]  Walter C. Hurty,et al.  Vibrations of Structural Systems by Component Mode Synthesis , 1960 .

[5]  D. Bernstein,et al.  The optimal projection equations for model reduction and the relationships among the methods of Wilson, Skelton, and Moore , 1985 .

[6]  J. Pal,et al.  Stable padé approximants to multivariable systems using a mixed method , 1980, Proceedings of the IEEE.

[7]  R. C. Rosenberg,et al.  Power-based model insight , 1988 .

[8]  M. Haguenauer,et al.  The real part of the proton-antiproton elastic scattering amplitude at the centre of mass energy of 546 GeV , 1987 .

[9]  Kemin Zhou,et al.  Frequency-weighted model reduction with L ∞ error bounds , 1993 .

[10]  Earl H. Dowell,et al.  Asymptotic modal analysis of dynamical systems: the effect of modal cross-correlation , 2004 .

[11]  A. Megretski.  H-infinity model reduction with guaranteed suboptimality bound , 2006, 2006 American Control Conference.

[12]  J.A. De Abreu-Garcia,et al.  On the computation of reduced order models of nonlinear systems using balancing technique , 1988, Proceedings of the 27th IEEE Conference on Decision and Control.

[13]  C. Lanczos.  An iteration method for the solution of the eigenvalue problem of linear differential and integral operators , 1950 .

[14]  W.S. Gray,et al.  Nonlinear balanced realizations , 2001, Proceedings of the 40th IEEE Conference on Decision and Control (Cat. No.01CH37228).

[15]  Leang-San Shieh,et al.  The generalized matrix continued-fraction descriptions and their application to model simplification† , 1986 .

[16]  R. Skelton.  Dynamic Systems Control: Linear Systems Analysis and Synthesis , 1988 .

[17]  W. Gragg,et al.  On the partial realization problem , 1983 .

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

[19]  V. Sreeram,et al.  A new frequency-weighted balanced truncation method and an error bound , 1997, Proceedings of the 36th IEEE Conference on Decision and Control.

[20]  L. Sirovich.  Turbulence and the dynamics of coherent structures. III. Dynamics and scaling , 1987 .

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

[22]  Zoran Filipi,et al.  Integration and use of diesel engine, driveline and vehicle dynamics models for heavy duty truck simulation , 1999 .

[23]  R. Skelton.  Cost decomposition of linear systems with application to model reduction , 1980 .

[24]  W. Tollmien,et al.  Über Flüssigkeitsbewegung bei sehr kleiner Reibung , 1961 .

[25]  James H. Taylor,et al.  A frequency domain model-order-deduction algorithm for linear systems , 1998 .

[26]  Kyran D. Mish,et al.  Nonlinear Dynamic Response of Frames Using Lanczos Modal Analysis , 1996 .

[27]  S. H. Sung,et al.  Component mode synthesis of a vehicle structural-acoustic system model , 1986 .

[28]  A. B. Vasil’eva.  ASYMPTOTIC BEHAVIOUR OF SOLUTIONS TO CERTAIN PROBLEMS INVOLVING NON-LINEAR DIFFERENTIAL EQUATIONS CONTAINING A?SMALL PARAMETER MULTIPLYING THE HIGHEST DERIVATIVES , 1963 .

[29]  Demeter G. Fertis,et al.  Mechanical And Structural Vibrations , 1995 .

[30]  J. L. Stein,et al.  System Partitioning and Physical-Domain Model Reduction Through Assessment of Bond Graph Junction Structure , 2005 .

[31]  Michel Loève,et al.  Probability Theory I , 1977 .

[32]  R. Baldini,et al.  Pion pair production in photon-photon collisions at DCI , 1987 .

[33]  Magdi S. Mahmoud,et al.  MODEL SIMPLIFICATION—AN OVERVIEW , 1982 .

[34]  H. Hotelling.  Analysis of a complex of statistical variables into principal components. , 1933 .

[35]  Adhemar Bultheel,et al.  Pade´ techniques for model reduction in linear system theory: a survey , 1986 .

[36]  A. Ran,et al.  Optimal Hankel norm model reductions and Wiener-Hopf factorization II: The non-canonical case , 1987 .

[37]  Ching-An Lin,et al.  Model Reduction via Frequency Weighted Balanced Realization , 1990, 1990 American Control Conference.

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

[39]  A. Galip Ulsoy,et al.  An Input-Output Criterion for Linear Model Deduction , 1996 .

[40]  C. F. Chen.  Model reduction of multivariable control systems by means of matrix continued fractions , 1974 .

[41]  Gregory M. Hulbert,et al.  A Physical-Based Model Reduction Metric with an Application to Vehicle Dynamics , 1998 .

[42]  N. K. Sinha,et al.  A review of some model reduction techniques , 1981, Canadian Electrical Engineering Journal.

[43]  Axel Ruhe.  Rational Krylov algorithms for nonsymmetric eigenvalue problems. II. matrix pairs , 1994 .

[44]  John B. Ferris,et al.  Development of proper models of hybrid systems , 1994 .

[45]  M. Aoki.  Control of large-scale dynamic systems by aggregation , 1968 .

[46]  R. Kálmán.  Irreducible realizations and the degree of a rational matrix. , 1965 .

[47]  K. Han,et al.  Reduction of Transfer Functions by the Stability-Equation Method , 1979 .

[48]  Genki Yagawa,et al.  Large-scale eigenvalue analysis of structures using component mode synthesis , 2001 .

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

[50]  S. Kung,et al.  Optimal Hankel-norm model reductions: Multivariable systems , 1980 .

[51]  Boris Moulin,et al.  Component Mode Synthesis of a Vehicle System Model Using the Fictitious Mass Method , 2003 .

[52]  E. Grimme,et al.  Pade approximation of large-scale dynamic systems with Lanczos methods , 1994, Proceedings of 1994 33rd IEEE Conference on Decision and Control.

[53]  Prasenjit Mohanty,et al.  Operational modal analysis in the presence of harmonic excitation , 2004 .

[54]  D. Cho,et al.  A component mode synthesis applied to mechanisms for an investigation of vibration , 1992 .

[55]  N. Levinson,et al.  Perturbations of discontinuous solutions of non-linear systems of differential equations , 1947, Proceedings of the National Academy of Sciences of the United States of America.

[56]  Jianjun Hu,et al.  Toward a unified and automated design methodology for multi-domain dynamic systems using bond graphs and genetic programming , 2003 .

[57]  David Hickin,et al.  A Unified Theory of Model Reduction for Linear Time Invariant Dynamical Systems , 1978 .

[58]  P. Kokotovic.  Applications of Singular Perturbation Techniques to Control Problems , 1984 .

[59]  R. C. Rosenberg,et al.  Power-based simplification of dynamic system models , 1988 .

[60]  W. Wasow.  Asymptotic expansions for ordinary differential equations , 1965 .

[61]  R. Skelton,et al.  The explicit relation between component cost analysis and balanced coordinates , 1989, Proceedings of the 28th IEEE Conference on Decision and Control,.

[62]  Hassan K. Khalil,et al.  Singular perturbation methods in control : analysis and design , 1986 .

[63]  Augusto Ferrante,et al.  Convergent algorithm for L2 model reduction , 1999, Autom..

[64]  C. F. Chen,et al.  A novel approach to linear model simplification , 1968 .

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

[66]  P. Bertrand,et al.  Développements récents sur la réduction de modèles , 1985 .

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

[68]  Guoxiang Gu,et al.  All optimal Hankel-norm approximations and their error bounds in discrete-time , 2005 .

[69]  E. Davison.  A new method for simplifying large linear dynamic systems , 1968 .

[70]  Zu-Qing Qu,et al.  Model Order Reduction Techniques , 2004 .

[71]  K. Karhunen.  Zur Spektraltheorie stochastischer prozesse , 1946 .

[72]  C. T. Leondes,et al.  Multivariable systems reduction via Cauer forms , 1981 .

[73]  Christophe Pierre,et al.  Component Mode Synthesis Using Nonlinear Normal Modes , 2005 .

[74]  M. Chidambara,et al.  Further comments on "A method for simplifying linear dynamic systems" , 1967 .

[75]  Jeffrey L. Stein,et al.  Systematic Model Decoupling Through Assessment of Power-Conserving Constraints: An Engine Dynamics Case Study , 2004 .

[76]  B. Friedland,et al.  Routh approximations for reducing order of linear, time-invariant systems , 1975 .

[77]  Thomas F. Edgar,et al.  An improved method for nonlinear model reduction using balancing of empirical gramians , 2002 .

[78]  E. Davison,et al.  On "A method for simplifying linear dynamic systems" , 1966 .

[79]  B. Anderson,et al.  Frequency weighted balanced reduction technique: a generalization and an error bound , 1995, Proceedings of 1995 34th IEEE Conference on Decision and Control.

[80]  M. Aoki.  Some approximation methods for estimation and control of large scale systems , 1978 .

[81]  E. Jonckheere,et al.  A contraction mapping preserving balanced reduction scheme and its infinity norm error bounds , 1988 .

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

[83]  K. Han,et al.  Model reduction using the stability-equation method and the Padé approximation method , 1980 .

[84]  Thomas F. Edgar,et al.  Balancing Approach to Minimal Realization and Model Reduction of Stable Nonlinear Systems , 2002 .

[85]  Sun-Yuan Kung,et al.  A new identification and model reduction algorithm via singular value decomposition , 1978 .

[86]  F. Alexandro,et al.  Stable partial Pad approximations for reduced-order trnsfer functions , 1984 .

[87]  M. M. Gouda,et al.  Building thermal model reduction using nonlinear constrained optimization , 2002 .

[88]  J. Pal.  Stable reduced-order Padé approximants using the Routh-Hurwitz array , 1979 .

[89]  M. Bampton,et al.  Coupling of substructures for dynamic analyses. , 1968 .

[90]  M. Karplus,et al.  Proteins: A Theoretical Perspective of Dynamics, Structure, and Thermodynamics , 1988 .

[91]  Rein Luus,et al.  Optimization in model reduction , 1980 .

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

[93]  Robert Skelton,et al.  Model reductions using a projection formulation , 1987, 26th IEEE Conference on Decision and Control.

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

[95]  Robert E. Skelton,et al.  Comments on "Balanced gains and their significance for L 2 model reduction" , 1986 .

[96]  G. Verros,et al.  Ride Dynamics of Nonlinear Vehicle Models Using Component Mode Synthesis , 2002 .

[97]  W. Arnoldi.  The principle of minimized iterations in the solution of the matrix eigenvalue problem , 1951 .

[98]  B. Anderson,et al.  Singular perturbation approximation of balanced systems , 1989, Proceedings of the 28th IEEE Conference on Decision and Control,.

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

[100]  G. Wang,et al.  A new frequency-weighted balanced truncation method and an error bound , 1999, IEEE Trans. Autom. Control..

[101]  P. Bertrand,et al.  Representation of linear dynamical systems by aggregated models , 1977 .

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

[103]  Zu-Qing Qu,et al.  Model Order Reduction Techniques with Applications in Finite Element Analysis , 2004 .

[104]  U. Desai,et al.  A transformation approach to stochastic model reduction , 1984 .

[105]  Han-Fu Chen,et al.  Nonconvex Stochastic Optimization for Model Reduction , 2002, J. Glob. Optim..

[106]  Vlad Ionescu,et al.  The class of suboptimal and optimal solutions to the discrete Nehari problem : characterization and computation , 1996 .

[107]  Loucas S. Louca,et al.  Modelling and reduction techniques for studies of integrated hybrid vehicle systems , 2006 .

[108]  D. Wilson.  Optimum solution of model-reduction problem , 1970 .

[109]  L. Sirovich.  Turbulence and the dynamics of coherent structures. I. Coherent structures , 1987 .

[110]  W. Hurty.  Dynamic Analysis of Structural Systems Using Component Modes , 1965 .

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

[112]  Rein Luus,et al.  Model reduction by minimization of integral square error performance indices , 1990 .

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

[114]  P. Kabamba.  Balanced gains and their significance for L^{2} model reduction , 1985 .

[115]  Vlad Ionescu,et al.  The four-block Adamjan–Arov–Krein problem for discrete-time systems , 2001 .

[116]  Margolis,et al.  Reduction of models of large scale lumped structures using normal modes and bond graphs , 1977 .

[117]  M. Kreĭn,et al.  ANALYTIC PROPERTIES OF SCHMIDT PAIRS FOR A HANKEL OPERATOR AND THE GENERALIZED SCHUR-TAKAGI PROBLEM , 1971 .

[118]  Hiroshi Mori,et al.  Balancing the Speed and Fidelity of Automotive Powertrain Models Through Surrogation , 2004 .

[119]  Naresh K. Sinha,et al.  Aggregation matrices for a class of low-order models for large-scale systems , 1975 .

[120]  K. Khatwani,et al.  On lossless bandpass ladder networks , 1978 .

[121]  Michael Green,et al.  Balanced stochastic realizations , 1988 .

[122]  Keith Glover,et al.  A state-space algorithm for the superoptimal Hankel-Norm approximation problem , 1993 .

[123]  J. Hickin,et al.  Model reduction for linear multivariable systems , 1980 .

[124]  K. Han,et al.  Model reduction using the stability-equation method and the continued-fraction method , 1980 .

[125]  A. Antoulas,et al.  A comparative study of 7 algorithms for model reduction , 2000, Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187).

[126]  M. Chidambara,et al.  Further remarks on simplifying linear dynamic systems , 1967, IEEE Transactions on Automatic Control.

[127]  J. M. A. Scherpen,et al.  Balancing for nonlinear systems , 1993 .

[128]  L. Sirovich.  Turbulence and the dynamics of coherent structures. II. Symmetries and transformations , 1987 .

[129]  R. Skelton,et al.  Component cost analysis of large scale systems , 1983 .

[130]  Serkan Gugercin,et al.  A new approach to model reduction which preserves stability and passivity: an overview , 2002, Proceedings of the 41st IEEE Conference on Decision and Control, 2002..

[131]  L. Chau,et al.  Further comments on K0(K0) → 2γ , 1987 .

[132]  Chunguo Wu,et al.  A note on equivalence of proper orthogonal decomposition methods , 2003 .

[133]  E. Assuncao,et al.  A global optimization approach for the H/sub 2/-norm model reduction problem , 1999, Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No.99CH36304).

[134]  Jianjun Hu,et al.  A novel evolutionary engineering design approach for mixed-domain systems , 2004 .

[135]  E. E. Ungar.  Statistical energy analysis , 1997 .

[136]  Frank J. Alexandro,et al.  Stable Partial Pade Approximations for Reduced Order Transfer Functions , 1983, 1983 American Control Conference.

[137]  J. L. Stein,et al.  Fundamental concordances between balanced truncation and activity-based model reduction , 2005 .

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

[139]  P. Kokotovic,et al.  Singular perturbation method for reducing the model order in optimal control design , 1968 .

[140]  Ian Postlethwaite,et al.  Multivariable Feedback Control: Analysis and Design , 1996 .

[141]  Y. Shamash.  Stable reduced-order models using Padé-type approximations , 1974 .

[142]  Y. Shamash.  Model reduction using the Routh stability criterion and the Padé approximation technique , 1975 .

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

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

[145]  Edmond A. Jonckheere,et al.  A new set of invariants for linear systems--Application to reduced order compensator design , 1983 .

[146]  M. Green,et al.  A relative error bound for balanced stochastic truncation , 1988 .

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

[148]  Evagoras G. Xydas,et al.  Model Reduction of Modal Representations , 2006 .