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  
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BDF step 4300 s  
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BDF step 4250 s  
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BDF step 4200 s  
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BDF step 4150 s  
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BDF step 4100 s  
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BDF step 4050 s  
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BDF step 4000 s  
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BDF step 3950 s  
	 Newton steps:  2   
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BDF step 3900 s  
	 Newton steps:  2   
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BDF step 3850 s  
	 Newton steps:  2   
	 Rank 91  
BDF step 3800 s  
	 Newton steps:  2   
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BDF step 3750 s  
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BDF step 3700 s  
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BDF step 3650 s  
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BDF step 3600 s  
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BDF step 3550 s  
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BDF step 3500 s  
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BDF step 3450 s  
	 Newton steps:  2   
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BDF step 3400 s  
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BDF step 3350 s  
	 Newton steps:  2   
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BDF step 3300 s  
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BDF step 3250 s  
	 Newton steps:  2   
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BDF step 3200 s  
	 Newton steps:  2   
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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   
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BDF step 3000 s  
	 Newton steps:  2   
	 Rank 100  
BDF step 2950 s  
	 Newton steps:  2   
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BDF step 2900 s  
	 Newton steps:  2   
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BDF step 2850 s  
	 Newton steps:  2   
	 Rank 101  
BDF step 2800 s  
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BDF step 2750 s  
	 Newton steps:  2   
	 Rank 101  
BDF step 2700 s  
	 Newton steps:  2   
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BDF step 2650 s  
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BDF step 2600 s  
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BDF step 2550 s  
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BDF step 2500 s  
	 Newton steps:  2   
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BDF step 2450 s  
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BDF step 2400 s  
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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   
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BDF step 2050 s  
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BDF step 2000 s  
	 Newton steps:  2   
	 Rank 104  
BDF step 1950 s  
	 Newton steps:  2   
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BDF step 1900 s  
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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   
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BDF step 1500 s  
	 Newton steps:  2   
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BDF step 1450 s  
	 Newton steps:  2   
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	 Newton steps:  2   
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	 Newton steps:  2   
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BDF step  850 s  
	 Newton steps:  2   
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BDF step  800 s  
	 Newton steps:  2   
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BDF step  750 s  
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	 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   
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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   
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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');

Published with MATLAB® R2020b