Reduced-order LQG control of a Timoshenko beam model

  • Philipp Braun
  • Erwin Hernández
  • Dante Kalise


We present a computational approach for the construction of reduced-order controllers for the Timoshenko beam model. By means of a space discretization of the Timoshenko equations, we obtain a large-scale, finite-dimensional dynamical system, for which we compute an LQG controller for closed-loop stabilization. The solutions of the algebraic Riccati equations characterizing the LQG controller are then used to construct a balancing transformation which allows the dimensional reduction of the large-scale dynamic compensator. We present numerical tests assessing the stability and performance of the approach.


Timoshenko beam model order reduction LQG control/balancing 

Mathematical subject classification

Primary: 93C20 Secondary: 78M34 


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  1. [1]
    A. Alla, M. Falcone and D. Kalise. An efficient policy iteration algorithm for dynamic programming equations. SIAM J. Sci. Comput., 37(1) (2015), 181–200.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    D.N. Arnold. Discretization by finite element of a model dependent parameter problem. Numer. Math., 37 (1981), 405–421.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    H.T. Banks and K. Kunisch. The linear regulator problem for parabolic systems. SIAM J. Control Optim., 22(5) (1984), 684–698.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    P. Benner. Solving large-scale control problems. IEEE Control Syst. Mag., 14(1) (2004), 44–59.CrossRefGoogle Scholar
  5. [5]
    P. Benner. Balancing-Related Model Reduction for Parabolic Control Systems. Control of Systems Governed by Partial Differential Equations, IFAC, Volume, 1, Part 1, 257–262, (2013).Google Scholar
  6. [6]
    T. Breiten and K. Kunisch. Compensator design for the monodomain equations, to appear in ESAIM: COCV.Google Scholar
  7. [7]
    S. Gugercin and A. Antoulas. A survey of model reduction by balanced truncation and some new results. Int. J. Control, 77(8) (2004), 748–766.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    E. Hernández, D. Kalise and E. Otárola. A locking-free scheme for the LQR control of a Timoshenko beam. J. Comput. Appl. Math., 235(5) (2011), 1383–1393.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    E. Hernández and E. Otárola. A locking-free FEM in active vibration control of a Timoshenko beam. SIAMJ. Numer. Anal., 47 (2009), 2432–2454.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    D. Kalise and A. Kröner. Reduced-order minimum time control of advectionreaction-diffusion systems via dynamic programming. Proceedings of the 21st International Symposium on Mathematical Theory of Networks and Systems, 1196–1202 (2014).Google Scholar
  11. [11]
    R.E. Kalman. Contributions to the theory of optimal control. Bol. Soc.Mat. Mex., 5 (1960), 102–119.MathSciNetzbMATHGoogle Scholar
  12. [12]
    K. Kunisch, S. Volkwein and L. Xie. HJB-POD Based Feedback Design for the Optimal Control of Evolution Problems. SIAMJ. on AppliedDynamical Systems, 4 (2004), 701–722.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    I. Lasiecka and R. Triggiani. Control theory for partial differential equations: continuous and approximations theories. Encyclopedia of mathematics and its applications 74, Cambridge University Press (2000).CrossRefzbMATHGoogle Scholar
  14. [14]
    K.A. Morris. Design of finite-dimensional controllers for infinite-dimensional systems by approximation. J. Math. Systems Estim. Control, 6(2) (1996), 151–180.MathSciNetzbMATHGoogle Scholar
  15. [15]
    D. Mustafa and K. Glover. Controller reduction by H-infinity-balanced truncation. IEEE Trans. Automat. Control, 36(6) (1991), 668–682.MathSciNetCrossRefGoogle Scholar
  16. [16]
    T. Penzl. LYAPACK–Users’ Guide (Version 1.0), available at: Scholar
  17. [17]
    A. Preumont. Vibration Control of Active Structures, 3rd ed., SolidMechanics and its Applications 179, Springer (2011).CrossRefzbMATHGoogle Scholar
  18. [18]
    M. Tadi. Optimal infinite-dimensional estimator and compensator for a Timoshenko Beam. Computers Math. Applic., 27(6) (1994), 19–32.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    M. Tadi. Computational algorithm for controlling a Timoshenko beam. Comput. Methods Appl. Mech. Engrg., 153 (1998), 153–165.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Sociedade Brasileira de Matemática 2016

Authors and Affiliations

  1. 1.Chair of AppliedMathematicsUniversity of BayreuthBayreuthGermany
  2. 2.Department of MathematicsTechnical University Federico Santa MaríaValparaísoChile
  3. 3.Radon Institute for Computational and Applied Mathematics (RICAM)Austrian Academy of SciencesLinzAustria

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