function LQR_rail_BDF(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-step BDF method % possible values: 1, ..., 6 % (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) 4471. 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 fr 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. 353356. 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(1);
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.cc_info = 0; eqn.type = 'T';
Heuristic shift parameters via basic Arnoldi
n = oper.size(eqn, opts);
opts.shifts.num_desired = 7;
opts.shifts.num_Ritz = 50;
opts.shifts.num_hRitz = 25;
opts.shifts.method = 'heur';
opts.shifts.b0 = ones(n, 1);
Newton tolerances and maximum iteration number
opts.nm.maxiter = 8;
opts.nm.res_tol = 1e-10;
opts.nm.rel_diff_tol = 1e-16;
opts.nm.info = 0;
opts.norm = 'fro';
opts.nm.accumulateRes = true;
opts.nm.linesearch = true;
BDF parameters
opts.bdf.time_steps = 0:50:4500; opts.bdf.step = k; opts.bdf.info = 1; opts.bdf.save_solution = 0; opts.bdf.startup_iter = 7;
t_mess_bdf_dre = tic;
[out_bdf] = mess_bdf_dre(eqn, opts, oper);
t_elapsed = toc(t_mess_bdf_dre);
mess_fprintf(opts, 'mess_bdf_dre took %6.2f seconds \n', t_elapsed);
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_BDF.m',104)">LQR_rail_BDF (line 104)</a> ↳ In <a href="matlab:opentoline('/builds/mess/mmess/mat-eqn-solvers/mess_bdf_dre.m',310)">mess_bdf_dre (line 310)</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_BDF.m',104)">LQR_rail_BDF (line 104)</a> ↳ In <a href="matlab:opentoline('/builds/mess/mmess/mat-eqn-solvers/mess_bdf_dre.m',318)">mess_bdf_dre (line 318)</a> BDF step 4450 s Newton steps: 2 Rank 1 BDF step 4400 s Newton steps: 2 Rank 76 BDF step 4350 s Newton steps: 2 Rank 78 BDF step 4300 s Newton steps: 2 Rank 79 BDF step 4250 s Newton steps: 2 Rank 81 BDF step 4200 s Newton steps: 2 Rank 82 BDF step 4150 s Newton steps: 2 Rank 84 BDF step 4100 s Newton steps: 2 Rank 85 BDF step 4050 s Newton steps: 2 Rank 87 BDF step 4000 s Newton steps: 2 Rank 88 BDF step 3950 s Newton steps: 2 Rank 88 BDF step 3900 s Newton steps: 2 Rank 89 BDF step 3850 s Newton steps: 2 Rank 91 BDF step 3800 s Newton steps: 2 Rank 91 BDF step 3750 s Newton steps: 2 Rank 92 BDF step 3700 s Newton steps: 2 Rank 93 BDF step 3650 s Newton steps: 2 Rank 93 BDF step 3600 s Newton steps: 2 Rank 94 BDF step 3550 s Newton steps: 2 Rank 94 BDF step 3500 s Newton steps: 2 Rank 95 BDF step 3450 s Newton steps: 2 Rank 96 BDF step 3400 s Newton steps: 2 Rank 97 BDF step 3350 s Newton steps: 2 Rank 97 BDF step 3300 s Newton steps: 2 Rank 97 BDF step 3250 s Newton steps: 2 Rank 98 BDF step 3200 s Newton steps: 2 Rank 98 BDF step 3150 s Newton steps: 2 Rank 98 BDF step 3100 s Newton steps: 2 Rank 99 BDF step 3050 s Newton steps: 2 Rank 99 BDF step 3000 s Newton steps: 2 Rank 100 BDF step 2950 s Newton steps: 2 Rank 100 BDF step 2900 s Newton steps: 2 Rank 100 BDF step 2850 s Newton steps: 2 Rank 101 BDF step 2800 s Newton steps: 2 Rank 101 BDF step 2750 s Newton steps: 2 Rank 101 BDF step 2700 s Newton steps: 2 Rank 101 BDF step 2650 s Newton steps: 2 Rank 102 BDF step 2600 s Newton steps: 2 Rank 102 BDF step 2550 s Newton steps: 2 Rank 102 BDF step 2500 s Newton steps: 2 Rank 103 BDF step 2450 s Newton steps: 2 Rank 103 BDF step 2400 s Newton steps: 2 Rank 103 BDF step 2350 s Newton steps: 2 Rank 103 BDF step 2300 s Newton steps: 2 Rank 104 BDF step 2250 s Newton steps: 2 Rank 104 BDF step 2200 s Newton steps: 2 Rank 104 BDF step 2150 s Newton steps: 2 Rank 104 BDF step 2100 s Newton steps: 2 Rank 104 BDF step 2050 s Newton steps: 2 Rank 104 BDF step 2000 s Newton steps: 2 Rank 104 BDF step 1950 s Newton steps: 2 Rank 104 BDF step 1900 s Newton steps: 2 Rank 104 BDF step 1850 s Newton steps: 2 Rank 104 BDF step 1800 s Newton steps: 2 Rank 104 BDF step 1750 s Newton steps: 2 Rank 106 BDF step 1700 s Newton steps: 2 Rank 106 BDF step 1650 s Newton steps: 2 Rank 106 BDF step 1600 s Newton steps: 2 Rank 106 BDF step 1550 s Newton steps: 2 Rank 106 BDF step 1500 s Newton steps: 2 Rank 106 BDF step 1450 s Newton steps: 2 Rank 106 BDF step 1400 s Newton steps: 2 Rank 106 BDF step 1350 s Newton steps: 2 Rank 106 BDF step 1300 s Newton steps: 2 Rank 106 BDF step 1250 s Newton steps: 2 Rank 106 BDF step 1200 s Newton steps: 2 Rank 106 BDF step 1150 s Newton steps: 2 Rank 106 BDF step 1100 s Newton steps: 2 Rank 106 BDF step 1050 s Newton steps: 2 Rank 106 BDF step 1000 s Newton steps: 2 Rank 106 BDF step 950 s Newton steps: 2 Rank 106 BDF step 900 s Newton steps: 2 Rank 106 BDF step 850 s Newton steps: 2 Rank 106 BDF step 800 s Newton steps: 2 Rank 106 BDF step 750 s Newton steps: 2 Rank 106 BDF step 700 s Newton steps: 2 Rank 106 BDF step 650 s Newton steps: 2 Rank 107 BDF step 600 s Newton steps: 2 Rank 107 BDF step 550 s Newton steps: 2 Rank 107 BDF step 500 s Newton steps: 2 Rank 107 BDF step 450 s Newton steps: 2 Rank 107 BDF step 400 s Newton steps: 2 Rank 107 BDF step 350 s Newton steps: 2 Rank 107 BDF step 300 s Newton steps: 2 Rank 107 BDF step 250 s Newton steps: 2 Rank 107 BDF step 200 s Newton steps: 2 Rank 107 BDF step 150 s Newton steps: 2 Rank 107 BDF step 100 s Newton steps: 2 Rank 107 BDF step 50 s Newton steps: 2 Rank 108 BDF step 0 s Newton steps: 2 Rank 108 mess_bdf_dre took 66.56 seconds
y = zeros(1, length(out_bdf.Ks)); for i = 1:length(out_bdf.Ks) y(i) = out_bdf.Ks{i}(1, 77); end x = opts.bdf.time_steps; figure(1); plot(x, y, 'LineWidth', 3); title('evolution of component (1,77) of the optimal feedback');
