Balanced truncation-rational Krylov methods for model reduction in large scale dynamical systems



In this paper, we consider the balanced truncation method for model reductions in large-scale linear and time-independent dynamical systems with multi-inputs and multi-outputs. The method is based on the solutions of two large coupled Lyapunov matrix equations when the system is stable or on the computation of stabilizing positive and semi-definite solutions of some continuous-time algebraic Riccati equations when the dynamical system is not stable. Using the rational block Arnoldi, we show how to compute approximate solutions to these large Lyapunov or algebraic Riccati equations. The obtained approximate solutions are given in a factored form and used to build the reduced order model. We give some theoretical results and present numerical examples with some benchmark models.


Matrix Krylov subspaces Model reduction Dynamical systems 

Mathematics Subject Classification

65F 15A 


  1. Abidi O, Hached M, Jbilou K (2016) Adaptive rational block Arnoldi methods for model reductions in large-scale MIMO dynamical systems. N Tren Math 4(2):227–239CrossRefGoogle Scholar
  2. Abou-Kandil H, Freiling G, Ionescu V, Jank G (2003) Matrix Riccati equations in control and sytems theory. In: Sys. & Contr. Foun. & Appl. Birkhauser, BostonGoogle Scholar
  3. Antoulas AC (2005) Approximation of large-scale dynamical systems. Adv. des. contr. SIAM, PhiladelphiaCrossRefGoogle Scholar
  4. Bartels RH, Stewart GW (1972) Solution of the matrix equation \( AX+XB=C \). Commun ACM 15:820–826CrossRefGoogle Scholar
  5. Benner P, Li J, Penzl T (2008) Numerical solution of large Lyapunov equations, Riccati equations, and linear-quadratic optimal control problems. Numer Lin Alg Appl 15(9):755–777Google Scholar
  6. Bittanti S, Laub A, Willems JC (1991) The Riccati equation. Springer, BerlinCrossRefMATHGoogle Scholar
  7. Datta BN (2003) Numerical methods for linear control systems design and analysis. Elsevier Academic Press, AmsterdamGoogle Scholar
  8. Druskin V, Simoncini V (2011) Adaptive rational Krylov subspaces for large-scale dynamical systems. Syst Contr Lett 60:546–560MathSciNetCrossRefMATHGoogle Scholar
  9. El Guennouni A, Jbilou K, Riquet AJ (2002) Block Krylov subspace methods for solving large Sylvester equations. Numer Alg 29:75–96MathSciNetCrossRefMATHGoogle Scholar
  10. Fortuna L, Nunnari G, Gallo A (1992) Model order reduction techniques with applications in electrical engineering. Springer, LondonCrossRefGoogle Scholar
  11. Glover K (1984) All optimal Hankel-norm approximations of linear multivariable systems and their L-infinity error bounds. Inter J Cont 39:1115–1193MathSciNetCrossRefMATHGoogle Scholar
  12. Gugercin S, Antoulas AC (2004) A survey of model reduction by balanced truncation and some new results. Inter J Cont 77(8):748–766MathSciNetCrossRefMATHGoogle Scholar
  13. Heyouni M, Jbilou K (2009) An extended block Arnoldi algorithm for large-scale solutions of the continuous-time algebraic Riccati equation. Elect Trans Num Anal 33:53–62MathSciNetMATHGoogle Scholar
  14. Jaimoukha IM, Kasenally EM (1994) Krylov subspace methods for solving large Lyapunov equations. SIAM J Matrix Anal Appl 31(1):227–251MathSciNetMATHGoogle Scholar
  15. Jbilou K (2010) ADI preconditioned Krylov methods for large Lyapunov matrix equations. Lin Alg Appl 432(10):2473–2485MathSciNetCrossRefMATHGoogle Scholar
  16. Jbilou K (2006) Low rank approximate solutions to large Sylvester matrix equations Appl. Math Comput 177:365–376MathSciNetMATHGoogle Scholar
  17. Jbilou K (2003) Block Krylov subspace methods for large continuous-time algebraic Riccati equations. Numer Alg 34:339–353MathSciNetCrossRefMATHGoogle Scholar
  18. Jbilou K (2006) An Arnoldi based method for large algebraic Riccati equations. Appl Math Lett 19:437–444MathSciNetCrossRefMATHGoogle Scholar
  19. Kleinman BN (1968) On an iterative technique for Riccati equation computations. IEEC Trans Autom Contr 13:114–115Google Scholar
  20. Laub AJ (1979) A Schur method for solving algebraic Riccati equations. IEEE Trans Automat control 24:913–921Google Scholar
  21. Mehrmann V, Penzl T (1998) Benchmark collections in SLICOT. Technical report SLWN1998- 5, SLICOT working note, ESAT, KU Leuven, K. Mercierlaan 94, Leuven-Heverlee 3100, Belgium, 1998.
  22. Moore BC (1981) Principal component analysis in linear systems: controllability, observability and model reduction. IEEE Trans. Autom. Contr. 26:17–32Google Scholar
  23. Mullis CT, Roberts RA, Trans IEEE Acoust Speec Signal Process 24 (1976)Google Scholar
  24. Penzl T (2012) LYAPACK A MATLAB toolbox for large Lyapunov and Riccati equations. In: Model reduction problems, and linear-quadratic optimal control problems.
  25. Simoncini V (2007) A new iterative method for solving large-scale Lyapunov matrix equations. SIAM J Sci Comput 29(3):1268–1288MathSciNetCrossRefMATHGoogle Scholar

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© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2016

Authors and Affiliations

  1. 1.Université du Littoral, Côte d’OpaleCalais CedexFrance

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