Model reduction techniques with a-posteriori error analysis for linear-quadratic optimal control problems

The main focus of this paper is on an a-posteriori analysis for different model-order strategies applied to optimal control problems governed by linear parabolic partial differential equations. Based on a perturbation method it is deduced how far the suboptimal control, computed on the basis of the reduced-order model, is from the (unknown) exact one. For the model-order reduction, $\mathcal H_{2,\alpha}$-norm optimal model reduction (H2), balanced truncation (BT), and proper orthogonal decomposition (POD) are studied. The proposed approach is based on semi-discretization of the underlying dynamics for the state and the adjoint equations as a large scale linear time-invariant (LTI) system. This system is reduced to a lower-dimensional one using Galerkin (POD) or Petrov-Galerkin (H2, BT) projection. The size of the reduced-order system is iteratively increased until the error in the optimal control, computed with the a-posteriori error estimator, satisfies a given accuracy. The method is illustrated with numerical tests.

[1]  K. Hoffmann,et al.  Optimal Control of Partial Differential Equations , 1991 .

[2]  Ekkehard W. Sachs,et al.  A priori error estimates for reduced order models in finance , 2013 .

[3]  S. Volkwein,et al.  MODEL REDUCTION USING PROPER ORTHOGONAL DECOMPOSITION , 2008 .

[4]  D. Kubalinska,et al.  Optimal interpolation-based model reduction , 2008 .

[5]  Stefan Volkwein,et al.  Proper orthogonal decomposition for optimality systems , 2008 .

[6]  Jens Saak,et al.  Efficient Numerical Solution of Large Scale Algebraic Matrix Equations in PDE Control and Model Order Reduction , 2009 .

[7]  B. Moore.  Principal component analysis in linear systems: Controllability, observability, and model reduction , 1981 .

[8]  Peter Benner,et al.  Lyapunov Equations, Energy Functionals, and Model Order Reduction of Bilinear and Stochastic Systems , 2011, SIAM J. Control. Optim..

[9]  J. Hesthaven,et al.  Reduced Basis Approximation and A Posteriori Error Estimation for Parametrized Partial Differential Equations , 2007 .

[10]  S. S. Ravindran,et al.  Reduced-Order Adaptive Controllers for Fluid Flows Using POD , 2000, J. Sci. Comput..

[11]  G. Vossen,et al.  H2,α‐norm optimal model reduction for optimal control problems subject to parabolic and hyperbolic evolution equations , 2014 .

[12]  Edmond A. Jonckheere,et al.  A new set of invariants for linear systems--Application to reduced order compensator design , 1983 .

[13]  Kazufumi Ito,et al.  The primal-dual active set strategy as a semi-smooth Newton method for quadratic problems with affine constraints , 2002 .

[14]  D. Rovas,et al.  Output bounds for reduced-basis approximations of symmetric positive definite eigenvalue problems , 2000 .

[15]  Serkan Gugercin,et al.  H2 Model Reduction for Large-Scale Linear Dynamical Systems , 2008, SIAM J. Matrix Anal. Appl..

[16]  Enrique S. Quintana-Ortí,et al.  Model Reduction Based on Spectral Projection Methods , 2005 .

[17]  Michael Herty,et al.  On an inexact gradient method using Proper Orthogonal Decomposition for parabolic optimal control problems , 2013, Comput. Optim. Appl..

[18]  K. Glover.  All optimal Hankel-norm approximations of linear multivariable systems and their L, ∞ -error bounds† , 1984 .

[19]  Stefan Volkwein,et al.  Error estimates for abstract linear–quadratic optimal control problems using proper orthogonal decomposition , 2008, Comput. Optim. Appl..

[20]  K. Afanasiev,et al.  Adaptive Control Of A Wake Flow Using Proper Orthogonal Decomposition1 , 2001 .

[21]  A. Patera,et al.  Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations , 2007 .

[22]  Mark Kärcher,et al.  Reduced basis a posteriori error bounds for parametrized linear-quadratic elliptic optimal control problems , 2011 .

[23]  Tatjana Stykel,et al.  Gramian-Based Model Reduction for Descriptor Systems , 2004, Math. Control. Signals Syst..

[24]  Stefan Volkwein,et al.  POD a-posteriori error estimates for linear-quadratic optimal control problems , 2009, Comput. Optim. Appl..

[25]  Fredi Tröltzsch,et al.  Error Estimates for the Numerical Approximation of a Semilinear Elliptic Control Problem , 2002, Comput. Optim. Appl..

[26]  Angelika Bunse-Gerstner,et al.  h2-norm optimal model reduction for large scale discrete dynamical MIMO systems , 2010, J. Comput. Appl. Math..

[27]  J. Zowe,et al.  Second-order necessary and sufficient optimality conditions for infinite-dimensional programming problems , 1979 .

[28]  Tatjana Stykel,et al.  A balanced truncation-based strategy for optimal control of evolution problems , 2011, Optim. Methods Softw..

[29]  F. Tröltzsch.  Optimal Control of Partial Differential Equations: Theory, Methods and Applications , 2010 .

[30]  Stefan Volkwein,et al.  A method of a-posteriori error estimation with application to proper orthogonal decomposition , 2012 .

[31]  Athanasios C. Antoulas,et al.  Approximation of Large-Scale Dynamical Systems , 2005, Advances in Design and Control.

[32]  Stefan Volkwein,et al.  Optimality system POD and a-posteriori error analysis for linear-quadratic problems , 2011 .

[33]  Stefan Volkwein,et al.  Galerkin proper orthogonal decomposition methods for parabolic problems , 2001, Numerische Mathematik.

[34]  A. Antoulas,et al.  H 2 Model Reduction for Large-scale Linear Dynamical Systems * , 2022 .

[35]  L. Meier,et al.  Approximation of linear constant systems , 1967, IEEE Transactions on Automatic Control.

[36]  W. Hager,et al.  Optimality, stability, and convergence in nonlinear control , 1995 .

[37]  P. Holmes,et al.  Turbulence, Coherent Structures, Dynamical Systems and Symmetry , 1996 .

[38]  Riaz A. Usmani,et al.  Inversion of a tridiagonal jacobi matrix , 1994 .

[39]  Timo Tonn,et al.  Comparison of the reduced-basis and POD a posteriori error estimators for an elliptic linear-quadratic optimal control problem , 2011 .