Skip to main content
SpringerLink
Log in

Order Reduction Approaches for the Algebraic Riccati Equation and the LQR Problem

  • Chapter
  • First Online:
Numerical Methods for Optimal Control Problems

Part of the book series: Springer INdAM Series ((SINDAMS,volume 29))

Abstract

We explore order reduction techniques to solve the algebraic Riccati equation (ARE), and investigate the numerical solution of the linear-quadratic regulator problem (LQR). A classical approach is to build a low dimensional surrogate model of the dynamical system, for instance by means of balanced truncation, and then solve the corresponding ARE. Alternatively, iterative methods can be used to directly solve the ARE and use its approximate solution to estimate quantities associated with the LQR. We propose a class of Petrov-Galerkin strategies based on Krylov subspaces that simultaneously reduce the dynamical system while approximately solving the ARE by projection. This methodology significantly generalizes a recently developed Galerkin method, based on Krylov subspaces, by using a pair of projection spaces, as it is often done in model order reduction (MOR) of dynamical systems. Numerical experiments illustrate the advantages of the new class of methods over classical approaches when dealing with large matrices.

Part of this work was supported by the Indam-GNCS 2017 Project “Metodi numerici avanzati per equazioni e funzioni di matrici con struttura”. The author “Valeria Simoncini” is a member of the Italian INdAM Research group GNCS.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 89.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 119.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    We are unaware of any available implementation of rational Krylov subspace based approaches for large scale BT either with single or coupled bases, that simultaneously performs the balanced truncation while approximating the Gramians.

References

  1. Alla, A., Kutz, J.: Randomized model order reduction, submitted (2017). https://arxiv.org/pdf/1611.02316.pdf

  2. Alla, A., Graessle, C., Hinze, M.: A posteriori snapshot location for POD in optimal control of linear parabolic equations. ESAIM Math Model. Numer Anal. (2018). https://doi.org/10.1051/m2an/2018009

  3. Alla, A., Schmidt, A., Haasdonk, B.: Model order reduction approaches for infinite horizon optimal control problems via the HJB equation. In: Benner, P., Ohlberger, M., Patera, A., Rozza, G., Urban, K. (eds.) Model Reduction of Parametrized Systems, pp. 333–347. Springer International Publishing, Cham (2017)

    Chapter  Google Scholar 

  4. Antoulas, A.C.: Approximation of Large-Scale Dynamical Systems, Advances in Design and Control. SIAM, Philadelphia (2005)

    Google Scholar 

  5. Atwell, J., King, B.: Proper orthogonal decomposition for reduced basis feedback controllers for parabolic equations. Math. Comput. Model. 33, 1–19 (2001)

    Article  MathSciNet  Google Scholar 

  6. Barkouki, H., Bentbib, A.H., Jbilou, K.: An adaptive rational method Lanczos-type algorithm for model reduction of large scale dynamical systems. J. Sci. Comput. 67, 221–236 (2015)

    Article  Google Scholar 

  7. Beattie, C., Gugercin, S.: Model reduction by rational interpolation. In: Model Reduction and Approximation: Theory and Algorithms. SIAM, Philadelphia (2017)

    Google Scholar 

  8. Benner, P., Bujanović, Z.: On the solution of large-scale algebraic Riccati equations by using low-dimensional invariant subspaces. Linear Algebra Appl. 488, 430–459 (2016)

    Article  MathSciNet  Google Scholar 

  9. Benner, P., Heiland, J.: LQG-balanced truncation low-order controller for stabilization of laminar flows. In: King, R. (ed.) Active Flow and Combustion Control 2014, vol. 127 of Notes on Numerical Fluid Mechanics and Multidisciplinary Design, pp. 365–379. Springer International Publishing, Cham (2015)

    MATH  Google Scholar 

  10. Benner, P., Hein, S.: MPC/LQG for infinite-dimensional systems using time-invariant linearizations. In: Tröltzsch, F., Hömberg, D. (eds.) System Modeling and Optimization. IFIP AICT, vol. 291, pp. 217–224. Springer, Berlin (2013)

    Chapter  Google Scholar 

  11. Benner, P., Saak, J.: A Galerkin-Newton-ADI method for solving large-scale algebraic Riccati equations. Tech. Rep. 1253, DF Priority program (2010)

    Google Scholar 

  12. Benner, P., Saak, J.: Numerical solution of large and sparse continuous time algebraic matrix Riccati and Lyapunov equations: a state of the art survey. GAMM-Mitteilungen 36, 32–52 (2013)

    Article  MathSciNet  Google Scholar 

  13. Benner, P., Li, J.-R., Penzl, T.: Numerical solution of large-scale Lyapunov equations, Riccati equations, and linear-quadratic optimal control problems. Numer. Linear Algebra Appl. 15, 1–23 (2008)

    Article  MathSciNet  Google Scholar 

  14. Benner, P., Cohen, A., Ohlberger, M., Willcox, K. (eds.): Model Reduction and Approximation: Theory and Algorithms. Computational Science & Engineering. SIAM, Philadelphia (2017)

    MATH  Google Scholar 

  15. Borggaard, J., Gugercin, S.: Model reduction for DAEs with an application to flow control. In: King, R. (ed.) Active Flow and Combustion Control 2014. Notes on Numerical Fluid Mechanics and Multidisciplinary Design, vol. 127, pp. 381–396. Springer, Cham (2015)

    MATH  Google Scholar 

  16. Braun, P., Hernández, E., Kalise, D.: Reduced-order LQG control of a Timoshenko beam model. Bull. Braz. Math. Soc. N. Ser. 47, 143–155 (2016)

    Article  MathSciNet  Google Scholar 

  17. Druskin, V., Simoncini, V.: Adaptive rational Krylov subspaces for large-scale dynamical systems. Syst. Control Lett. 60, 546–560 (2011)

    Article  MathSciNet  Google Scholar 

  18. Falcone, M., Ferretti, R.: Semi-Lagrangian Approximation Schemes for Linear and Hamilton-Jacobi Equations. SIAM, Philadelphia (2014)

    MATH  Google Scholar 

  19. Grüne, L., Pannek, J.: Nonlinear Model Predictive Control. Theory and Algorithms. Springer, Berlin (2011)

    Google Scholar 

  20. Gugercin, S., Antoulas, A.C., Beattie, C.: \(\mathcal {H}_2\) model reduction for large-scale linear dynamical systems. SIAM J. Matrix Anal. Appl. 30, 609–638 (2008)

    Google Scholar 

  21. Gutknecht, M.H.: A completed theory of the unsymmetric Lanczos process and related algorithms. I. SIAM J. Matrix Anal. Appl. 13, 594–639 (1992)

    Article  MathSciNet  Google Scholar 

  22. Heyouni, M., Jbilou, K.: An extended Block Krylov method for large-scale continuous-time algebraic Riccati equations. Electron. Trans. Numer. Anal. 33, 53–62 (2008–2009)

    MATH  Google Scholar 

  23. Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S.: Optimization with PDE Constraints. Mathematical Modelling: Theory and Applications. Springer, New York (2009)

    Google Scholar 

  24. Jaimoukha, I.M., Kasenally, E.M.: Krylov subspace methods for solving large Lyapunov equations. SIAM J. Numer. Anal. 31, 227–251 (1994)

    Article  MathSciNet  Google Scholar 

  25. Jaimoukha, I.M., Kasenally, E.M.: Oblique projection methods for large scale model reduction. SIAM J. Matrix Anal. Appl. 16, 602–627 (1995)

    Article  MathSciNet  Google Scholar 

  26. Jbilou, K.: Block Krylov subspace methods for large algebraic Riccati equations. Numer. Algorithms 34, 339–353 (2003)

    Article  MathSciNet  Google Scholar 

  27. Kleinman, D.: On an iterative technique for Riccati equation computations. IEEE Trans. Autom. Control 13, 114–115 (1968)

    Article  Google Scholar 

  28. Kramer, B., Singler, J.: A POD projection method for large-scale algebraic Riccati equations. Numer. Algebra Control Optim. 4, 413–435 (2016)

    MathSciNet  MATH  Google Scholar 

  29. Kunisch, K., Volkwein, S.: Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics. SIAM J. Numer. Anal. 40, 492–515 (2006)

    Article  MathSciNet  Google Scholar 

  30. Kunisch, K., Volkwein, S.: Proper orthogonal decomposition for optimality systems. Math. Model. Numer. Anal. 42, 1–23 (2008)

    Article  MathSciNet  Google Scholar 

  31. Lancaster, P., Rodman, L.: Algebraic Riccati Equations. Oxford University Press, Oxford (1995)

    MATH  Google Scholar 

  32. Lin, Y., Simoncini, V.: Minimal residual methods for large scale Lyapunov equations. Appl. Numer. Math. 72, 52–71 (2013)

    Article  MathSciNet  Google Scholar 

  33. Lin, Y., Simoncini, V.: A new subspace iteration method for the algebraic Riccati equation. Numer. Linear Algebra Appl. 22, 26–47 (2015)

    Article  MathSciNet  Google Scholar 

  34. Quarteroni, A., Valli, A.: Numerical Approximation of Partial Differential Equations. Springer Series in Computational Mathematics. Springer, Berlin (1994)

    Google Scholar 

  35. Schmidt, A., Haasdonk, B.: Reduced basis approximation of large scale parametric algebraic Riccati equations. ESAIM Control Optim. Calc. Var. (2017). https://doi.org/10.1051/cocv/2017011

  36. Simoncini, V.: Analysis of the rational Krylov subspace projection method for large-scale algebraic Riccati equations. SIAM J. Matrix Anal. Appl. 37, 1655–1674 (2016)

    Article  MathSciNet  Google Scholar 

  37. Simoncini, V.: Computational methods for linear matrix equations. SIAM Rev. 58, 377–441 (2016)

    Article  MathSciNet  Google Scholar 

  38. Simoncini, V., Szyld, D.B., Monslave, M.: On two numerical methods for the solution of large-scale algebraic Riccati equations. IMA J. Numer. Anal. 34, 904–920 (2014)

    Article  MathSciNet  Google Scholar 

  39. Sirovich, L.: Turbulence and the dynamics of coherent structures. Parts I-II. Q. Appl. Math. XVL, 561–590 (1987)

    Google Scholar 

  40. Volkwein, S.: Model reduction using proper orthogonal decomposition. Lecture notes, University of Konstanz (2011)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, PUC-Rio, Rio De Janeiro, Brazil

    Alessandro Alla

  2. Alma Mater Studiorum - Universita’ di Bologna, Bologna, Italy

    Valeria Simoncini

  3. IMATI-CNR, Pavia, Italy

    Valeria Simoncini

Authors
  1. Alessandro Alla
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Valeria Simoncini
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alessandro Alla .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Sapienza University of Rome, Roma, Italy

    Maurizio Falcone

  2. Department of Mathematics & Physics, Roma Tre University, Rome, Italy

    Roberto Ferretti

  3. Mathematical Institut, Universität Bayreuth, Bayreuth, Bayern, Germany

    Lars Grüne

  4. Department of Mechanical and Aerospace Engineering, University of California, San Diego, La Jolla, CA, USA

    William M. McEneaney

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Alla, A., Simoncini, V. (2018). Order Reduction Approaches for the Algebraic Riccati Equation and the LQR Problem. In: Falcone, M., Ferretti, R., Grüne, L., McEneaney, W. (eds) Numerical Methods for Optimal Control Problems. Springer INdAM Series, vol 29. Springer, Cham. https://doi.org/10.1007/978-3-030-01959-4_5

Download citation

Publish with us

Policies and ethics

Search

Navigation