Abstract
We present a method for model reduction based on ideas from the behavioral theory of dissipative systems, in which the reduced order model is required to reproduce a subset of the set of trajectories of minimal dissipation of the original system. The passivity-preserving model reduction method of Antoulas (Syst Control Lett 54:361–374, 2005) and Sorensen (Syst Control Lett 54:347–360, 2005) is shown to be a particular case of this more general class of model reduction procedures.
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References
Antoulas AC (2004) Approximation of large-scale dynamical systems. SIAM Press, Philadelphia
Antoulas AC (2005) A new result on passivity-preserving model reduction. Syst Control Lett 54: 361–374
Belur MN (2003) Control in a behavioral context. Dissertation, University of Groningen
Bultheel A, van Baren M (1986) Padé techniques for model reduction in linear system theory: a survey. J Comp Appl Math 14: 401–438
Chen X, Wen JT (1995) Positive realness preserving model reduction with H ∞ norm error bounds. IEEE Trans Circuits Syst 42(1): 23–29
Kim S-Y, Gopal N, Pillage LT (1994) Time-domain macromodels for VLSI interconnected analysis. IEEE Trans Comput Aided Des Integr Circuit Syst 13: 1257–1270
Knockaert L, de Zutter D (1999) Passive reduced-order multiport modelling: the Padé-Laguerre, Krylov–Arnoldi–SVD connection. Int J Electron Commun 53: 254–260
Glover K (1984) All optimal Hankel-norm approximations of linear multivariable systems and their L ∞-error bounds. Int J Control 39: 1115–1193
Minh HB (2009) Model reduction in a behavioral framework. Ph.D. Thesis, University of Groningen, The Netherlands
Krajewski W, Lepschy A, Viaro U (1994) Approximation of continuous-time linear systems using Markov parameters and energy indices. IEEE Trans Autom Control 39: 2126–2129
Fanizza G, Karlsson J, Lindquist A, Nagamune R (2007) Passivity preserving model reduction by analytic interpolation. Linear Algebra Appl 425: 608–633
Meinsma G (1993) Frequency domain methods in H ∞ control. Ph.D. Thesis, University of Twente
Moore BC (1981) Principal components analysis in linear systems: controllability, observability, and model reduction. IEEE Trans Autom Control AC-26: 17–32
Ober R (1991) Balanced parametrization of classes of linear systems. SIAM J Control Optim 29: 1251–1287
Opdenacker PC, Jonckheere EA (1988) A contraction mapping preserving balanced reduction scheme and its infinity norm error bounds. IEEE Trans Circuits Syst 35(2): 184–189
Polderman JW, Willems JC (1997) Introduction to mathematical system theory: a behavioral approach. Springer, Berlin
Rapisarda P, Willems JC (1997) State maps for linear systems. SIAM J Control Optim 35(3): 1053–1091
Schumacher JM (1988) Linear systems under external equivalence. Linear Algebra Appl 102: 1–33
Sorensen DC (2005) Passivity preserving model reduction via interpolation of spectral zeros. Syst Control Lett 54: 347–360
Trentelman HL, Willems JC (1991) The dissipation inequality and the algebraic Riccati equation. In: Bittanti S, Laub AJ, Willems JC(eds) The Riccati equation. Springer, Berlin
Trentelman HL, Willems JC (1997) Every storage function is a state function. Syst Control Lett 32: 249–259
Trentelman HL, Stoorvogel AA, Hautus MLJ (2001) Control theory for linear systems. Springer, London
Weiland S (1991) Theory of approximation and disturbance attenuation for linear system. Dissertation, University of Groningen
Yousuff A, Wagie DA, Skelton RE (1985) Linear system approximation via covariance equivalent realizations. J Math Anal Appl 106: 91–115
Wang JM, Chu C-C, Yu Q, Kuh ES (2002) On projection-based algorithms for model-order reduction of interconnects. IEEE Trans Circuits Syst I Fundam Theory Appl CAS-49: 1564–1585
Willems JC (1983) Input–output and state-space representations of finite-dimensional linear time-invariant systems. Linear Algebra Appl 50: 581–608
Willems JC, Trentelman HL (1991) The dissipation inequality and the algebraic Riccati equation. The Riccati equation. Springer, Berlin
Willems JC (1972) Dissipative dynamical systems—Part I: General theory. Arch Ration Mech Anal 45: 321–351
Willems JC (1972) Dissipative dynamical systems—Part II: Linear systems with quadratic supply rates. Arch Ration Mech Anal 45: 352–393
Willems JC, Trentelman HL (1998) On quadratic differential forms. SIAM J Control Optim 36(5): 1703–1749
Willems JC, Trentelman HL (2002) Synthesis of dissipative systems using quadratic differential forms—Part I. IEEE Trans Autom Control 47(1): 53–69
Zhou K (1996) Robust and optimal control. Prentice-Hall, New Jersey
Acknowledgments
The authors would like to thank Prof. A. C. Antoulas for some extremely useful discussions on the topic of this paper.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Trentelman, H.L., Minh, H.B. & Rapisarda, P. Dissipativity preserving model reduction by retention of trajectories of minimal dissipation. Math. Control Signals Syst. 21, 171–201 (2009). https://doi.org/10.1007/s00498-009-0043-6
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DOI: https://doi.org/10.1007/s00498-009-0043-6