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 % (optinal, defaults to 2) % % References: % % [1] N. Lang, H. Mena, J. Saak, On the benefits of the LDLT factorization % for largescale 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 Universitt 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 Lsung eines % Optimalsteuerungsproblems fr die Abkhlung von Stahlprofilen, % Diplomarbeit, Fachbereich 3/Mathematik und Informatik, Universitt % Bremen, D-28334 Bremen (Sep. 2003). % % [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 Lect. Notes Comput. Sci. Eng., 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 Universitt Chemnitz, Chemnitz, Germany (Jul. 2009). % URL http://nbn-resolving.de/urn:nbn:de:bsz:ch1-200901642 % % % This program is free software; you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation; either version 2 of the License, or % (at your option) any later version. % % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program; if not, see <http://www.gnu.org/licenses/>. % % Copyright (C) Jens Saak, Martin Koehler, Peter Benner and others % 2009-2020 %
narginchk(0,1); if nargin<1, k =2; end
set operation
oper = operatormanager('default'); % Problem data eqn=getrail(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 = 1; 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 = 1;
opts.nm.linesearch = 1;
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;
tic; [out_bdf]=mess_bdf_dre(eqn,opts,oper); toc;
Warning: Initial condition factor L0 is not defined or corrupted. Setting it to the zero vector. Warning: Initial condition factor D0 is not defined or corrupted. Setting it to the identity matrix. BDF step 4450 s Newton steps: 2 Rank 1 BDF step 4400 s Newton steps: 2 Rank 79 BDF step 4350 s Newton steps: 2 Rank 79 BDF step 4300 s Newton steps: 2 Rank 82 BDF step 4250 s Newton steps: 2 Rank 84 BDF step 4200 s Newton steps: 2 Rank 85 BDF step 4150 s Newton steps: 2 Rank 86 BDF step 4100 s Newton steps: 2 Rank 88 BDF step 4050 s Newton steps: 2 Rank 89 BDF step 4000 s Newton steps: 2 Rank 90 BDF step 3950 s Newton steps: 2 Rank 92 BDF step 3900 s Newton steps: 2 Rank 92 BDF step 3850 s Newton steps: 2 Rank 92 BDF step 3800 s Newton steps: 2 Rank 94 BDF step 3750 s Newton steps: 2 Rank 94 BDF step 3700 s Newton steps: 2 Rank 95 BDF step 3650 s Newton steps: 2 Rank 97 BDF step 3600 s Newton steps: 2 Rank 97 BDF step 3550 s Newton steps: 2 Rank 97 BDF step 3500 s Newton steps: 2 Rank 98 BDF step 3450 s Newton steps: 2 Rank 98 BDF step 3400 s Newton steps: 2 Rank 98 BDF step 3350 s Newton steps: 2 Rank 99 BDF step 3300 s Newton steps: 2 Rank 100 BDF step 3250 s Newton steps: 2 Rank 100 BDF step 3200 s Newton steps: 2 Rank 101 BDF step 3150 s Newton steps: 2 Rank 101 BDF step 3100 s Newton steps: 2 Rank 102 BDF step 3050 s Newton steps: 2 Rank 102 BDF step 3000 s Newton steps: 2 Rank 102 BDF step 2950 s Newton steps: 2 Rank 102 BDF step 2900 s Newton steps: 2 Rank 102 BDF step 2850 s Newton steps: 2 Rank 103 BDF step 2800 s Newton steps: 2 Rank 103 BDF step 2750 s Newton steps: 2 Rank 103 BDF step 2700 s Newton steps: 2 Rank 104 BDF step 2650 s Newton steps: 2 Rank 104 BDF step 2600 s Newton steps: 2 Rank 104 BDF step 2550 s Newton steps: 2 Rank 105 BDF step 2500 s Newton steps: 2 Rank 105 BDF step 2450 s Newton steps: 2 Rank 105 BDF step 2400 s Newton steps: 2 Rank 105 BDF step 2350 s Newton steps: 2 Rank 105 BDF step 2300 s Newton steps: 2 Rank 105 BDF step 2250 s Newton steps: 2 Rank 106 BDF step 2200 s Newton steps: 2 Rank 106 BDF step 2150 s Newton steps: 2 Rank 106 BDF step 2100 s Newton steps: 2 Rank 106 BDF step 2050 s Newton steps: 2 Rank 106 BDF step 2000 s Newton steps: 2 Rank 106 BDF step 1950 s Newton steps: 2 Rank 107 BDF step 1900 s Newton steps: 2 Rank 107 BDF step 1850 s Newton steps: 2 Rank 107 BDF step 1800 s Newton steps: 2 Rank 107 BDF step 1750 s Newton steps: 2 Rank 107 BDF step 1700 s Newton steps: 2 Rank 107 BDF step 1650 s Newton steps: 2 Rank 108 BDF step 1600 s Newton steps: 2 Rank 108 BDF step 1550 s Newton steps: 2 Rank 108 BDF step 1500 s Newton steps: 2 Rank 108 BDF step 1450 s Newton steps: 2 Rank 108 BDF step 1400 s Newton steps: 2 Rank 108 BDF step 1350 s Newton steps: 2 Rank 108 BDF step 1300 s Newton steps: 2 Rank 108 BDF step 1250 s Newton steps: 2 Rank 108 BDF step 1200 s Newton steps: 2 Rank 108 BDF step 1150 s Newton steps: 2 Rank 108 BDF step 1100 s Newton steps: 2 Rank 108 BDF step 1050 s Newton steps: 2 Rank 108 BDF step 1000 s Newton steps: 2 Rank 109 BDF step 950 s Newton steps: 2 Rank 109 BDF step 900 s Newton steps: 2 Rank 109 BDF step 850 s Newton steps: 2 Rank 109 BDF step 800 s Newton steps: 2 Rank 109 BDF step 750 s Newton steps: 2 Rank 109 BDF step 700 s Newton steps: 2 Rank 109 BDF step 650 s Newton steps: 2 Rank 109 BDF step 600 s Newton steps: 2 Rank 109 BDF step 550 s Newton steps: 2 Rank 109 BDF step 500 s Newton steps: 2 Rank 109 BDF step 450 s Newton steps: 2 Rank 109 BDF step 400 s Newton steps: 2 Rank 109 BDF step 350 s Newton steps: 2 Rank 109 BDF step 300 s Newton steps: 2 Rank 110 BDF step 250 s Newton steps: 2 Rank 110 BDF step 200 s Newton steps: 2 Rank 110 BDF step 150 s Newton steps: 2 Rank 110 BDF step 100 s Newton steps: 2 Rank 110 BDF step 50 s Newton steps: 2 Rank 110 BDF step 0 s Newton steps: 2 Rank 110 Elapsed time is 36.836962 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); title('evolution of component (1,77) of the optimal feedback');
