function LQR_rail_Rosenbrock(k)
% Computes the optimal feedback via the low-rank Rosenbrock[1,2] methods for % the selective cooling of Steel profiles application described in [3,4,5]. % % Usage: LQR_Rail(k,shifts,inexact,Galerkin,istest) % % Inputs: % % k k-stage Rosenbrock method % possible values: 1, 2 % (optional, defaults to 2) % % References: % % [1] N. Lang, H. Mena, J. Saak, On the benefits of the LDLT factorization % for large-scale differential matrix equation solvers, Linear Algebra % Appl. 480 (2015) 44–71. https://doi.org/10.1016/j.laa.2015.04.006 % % [2] N. Lang, Numerical methods for large-scale linear time-varying % control systems and related differential matrix equations, % Dissertation, Technische Universität Chemnitz, Chemnitz, Germany, % logos-Verlag, Berlin, ISBN 978-3-8325-4700-4 (Jun. 2017). % URL https://www.logos-verlag.de/cgi-bin/buch/isbn/4700 % % [3] J. Saak, Effiziente numerische Lösung eines % Optimalsteuerungsproblems für die Abkühlung von Stahlprofilen, % Diplomarbeit, Fachbereich 3/Mathematik und Informatik, Universität % Bremen, D-28334 Bremen (Sep. 2003). % https://doi.org/10.5281/zenodo.1187040 % % [4] P. Benner, J. Saak, A semi-discretized heat transfer model for % optimal cooling of steel profiles, in: P. Benner, V. Mehrmann, D. % Sorensen (Eds.), Dimension Reduction of Large-Scale Systems, Vol. 45 % of Lecture Notes in Computational Science and Engineering, Springer-Verlag, Berlin/Heidelberg, % Germany, 2005, pp. 353–356. https://doi.org/10.1007/3-540-27909-1_19 % % [5] J. Saak, Efficient numerical solution of large scale algebraic matrix % equations in PDE control and model order reduction, Dissertation, % Technische Universität Chemnitz, Chemnitz, Germany (Jul. 2009). % URL http://nbn-resolving.de/urn:nbn:de:bsz:ch1-200901642 % % % This file is part of the M-M.E.S.S. project % (http://www.mpi-magdeburg.mpg.de/projects/mess). % Copyright (c) 2009-2025 Jens Saak, Martin Koehler, Peter Benner and others. % All rights reserved. % License: BSD 2-Clause License (see COPYING) % narginchk(0, 1); if nargin < 1 k = 2; end
set operation
opts = struct(); [oper, opts] = operatormanager(opts, 'default'); % Problem data eqn = mess_get_linear_rail(0);
opts.norm = 'fro'; % ADI tolerances and maximum iteration number opts.adi.maxiter = 100; opts.adi.res_tol = 1e-14; opts.adi.rel_diff_tol = 1e-16; opts.adi.info = 0; opts.adi.compute_sol_fac = true; opts.adi.accumulateK = true; eqn.type = 'T';
Heuristic shift parameters via projection
opts.shifts.num_desired = 7;
opts.shifts.method = 'projection';
Rosenbrock parameters
opts.rosenbrock.time_steps = 0:50:4500; opts.rosenbrock.stage = k; opts.rosenbrock.info = 1; opts.rosenbrock.gamma = 1 + 1 / sqrt(2); opts.rosenbrock.save_solution = false;
t_mess_rosenbrock_dre = tic; [out_ros] = mess_rosenbrock_dre(eqn, opts, oper); t_elapsed = toc(t_mess_rosenbrock_dre); mess_fprintf(opts, 'mess_rosenbrock_dre took %6.2f seconds \n', t_elapsed); y = zeros(1, length(out_ros.Ks)); for i = 1:length(out_ros.Ks) y(i) = out_ros.Ks{i}(1, 77); end x = opts.rosenbrock.time_steps; figure(1); plot(x, y, 'LineWidth', 3); title('evolution of component (1,77) of the optimal feedback'); xlabel('time'); ylabel('magnitude');
Warning: Initial condition factor L0 is not defined or corrupted. Setting it to the zero vector. ↳ In <a href="matlab:opentoline('/builds/mess/mmess/_release/package/package.m',13)">package (line 13)</a> ↳ In <a href="matlab:opentoline('/builds/mess/mmess/_release/publish_demos.m',18)">publish_demos (line 18)</a> ↳ In <a href="matlab:opentoline('/matlab/R2020b/toolbox/matlab/codetools/publish.p',0)">publish (line 0)</a> ↳ In <a href="matlab:opentoline('',21)">evalmxdom (line 21)</a> ↳ In <a href="matlab:opentoline('',109)">instrumentAndRun (line 109)</a> ↳ In <a href="matlab:opentoline('/builds/mess/mmess/DEMOS/Rail/LQR_rail_Rosenbrock.m',87)">LQR_rail_Rosenbrock (line 87)</a> ↳ In <a href="matlab:opentoline('/builds/mess/mmess/mat-eqn-solvers/mess_rosenbrock_dre.m',233)">mess_rosenbrock_dre (line 233)</a> Warning: Initial condition factor D0 is not defined or corrupted. Setting it to the identity matrix. ↳ In <a href="matlab:opentoline('/builds/mess/mmess/_release/package/package.m',13)">package (line 13)</a> ↳ In <a href="matlab:opentoline('/builds/mess/mmess/_release/publish_demos.m',18)">publish_demos (line 18)</a> ↳ In <a href="matlab:opentoline('/matlab/R2020b/toolbox/matlab/codetools/publish.p',0)">publish (line 0)</a> ↳ In <a href="matlab:opentoline('',21)">evalmxdom (line 21)</a> ↳ In <a href="matlab:opentoline('',109)">instrumentAndRun (line 109)</a> ↳ In <a href="matlab:opentoline('/builds/mess/mmess/DEMOS/Rail/LQR_rail_Rosenbrock.m',87)">LQR_rail_Rosenbrock (line 87)</a> ↳ In <a href="matlab:opentoline('/builds/mess/mmess/mat-eqn-solvers/mess_rosenbrock_dre.m',241)">mess_rosenbrock_dre (line 241)</a>
Reference to non-existent field 'R'. Error in mess_accumulateK (line 112) K_update = K_update / eqn.R; Error in mess_lradi (line 708) mess_accumulateK(eqn, opts, oper, out, pc, V); Error in mess_rosenbrock_dre (line 449) adiout1 = mess_lradi(eqn, opts, oper); Error in LQR_rail_Rosenbrock (line 87) [out_ros] = mess_rosenbrock_dre(eqn, opts, oper);