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.
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Notes
- 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
Alla, A., Kutz, J.: Randomized model order reduction, submitted (2017). https://arxiv.org/pdf/1611.02316.pdf
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
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)
Antoulas, A.C.: Approximation of Large-Scale Dynamical Systems, Advances in Design and Control. SIAM, Philadelphia (2005)
Atwell, J., King, B.: Proper orthogonal decomposition for reduced basis feedback controllers for parabolic equations. Math. Comput. Model. 33, 1–19 (2001)
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)
Beattie, C., Gugercin, S.: Model reduction by rational interpolation. In: Model Reduction and Approximation: Theory and Algorithms. SIAM, Philadelphia (2017)
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)
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)
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)
Benner, P., Saak, J.: A Galerkin-Newton-ADI method for solving large-scale algebraic Riccati equations. Tech. Rep. 1253, DF Priority program (2010)
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)
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)
Benner, P., Cohen, A., Ohlberger, M., Willcox, K. (eds.): Model Reduction and Approximation: Theory and Algorithms. Computational Science & Engineering. SIAM, Philadelphia (2017)
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)
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)
Druskin, V., Simoncini, V.: Adaptive rational Krylov subspaces for large-scale dynamical systems. Syst. Control Lett. 60, 546–560 (2011)
Falcone, M., Ferretti, R.: Semi-Lagrangian Approximation Schemes for Linear and Hamilton-Jacobi Equations. SIAM, Philadelphia (2014)
Grüne, L., Pannek, J.: Nonlinear Model Predictive Control. Theory and Algorithms. Springer, Berlin (2011)
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)
Gutknecht, M.H.: A completed theory of the unsymmetric Lanczos process and related algorithms. I. SIAM J. Matrix Anal. Appl. 13, 594–639 (1992)
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)
Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S.: Optimization with PDE Constraints. Mathematical Modelling: Theory and Applications. Springer, New York (2009)
Jaimoukha, I.M., Kasenally, E.M.: Krylov subspace methods for solving large Lyapunov equations. SIAM J. Numer. Anal. 31, 227–251 (1994)
Jaimoukha, I.M., Kasenally, E.M.: Oblique projection methods for large scale model reduction. SIAM J. Matrix Anal. Appl. 16, 602–627 (1995)
Jbilou, K.: Block Krylov subspace methods for large algebraic Riccati equations. Numer. Algorithms 34, 339–353 (2003)
Kleinman, D.: On an iterative technique for Riccati equation computations. IEEE Trans. Autom. Control 13, 114–115 (1968)
Kramer, B., Singler, J.: A POD projection method for large-scale algebraic Riccati equations. Numer. Algebra Control Optim. 4, 413–435 (2016)
Kunisch, K., Volkwein, S.: Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics. SIAM J. Numer. Anal. 40, 492–515 (2006)
Kunisch, K., Volkwein, S.: Proper orthogonal decomposition for optimality systems. Math. Model. Numer. Anal. 42, 1–23 (2008)
Lancaster, P., Rodman, L.: Algebraic Riccati Equations. Oxford University Press, Oxford (1995)
Lin, Y., Simoncini, V.: Minimal residual methods for large scale Lyapunov equations. Appl. Numer. Math. 72, 52–71 (2013)
Lin, Y., Simoncini, V.: A new subspace iteration method for the algebraic Riccati equation. Numer. Linear Algebra Appl. 22, 26–47 (2015)
Quarteroni, A., Valli, A.: Numerical Approximation of Partial Differential Equations. Springer Series in Computational Mathematics. Springer, Berlin (1994)
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
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)
Simoncini, V.: Computational methods for linear matrix equations. SIAM Rev. 58, 377–441 (2016)
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)
Sirovich, L.: Turbulence and the dynamics of coherent structures. Parts I-II. Q. Appl. Math. XVL, 561–590 (1987)
Volkwein, S.: Model reduction using proper orthogonal decomposition. Lecture notes, University of Konstanz (2011)
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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
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