Contents

function DEMO_RI_T_HYBRID(istest)
% Computes the solution of the Hinf Riccati equation for a random generated
% generalized system. The computations are done using the Newton method for
% the LQG step and afterwards the RADI. Afterwards, the real residual
% norm is shown and compared to the set tolerance.
%
% Input:
% istest  decides whether the function runs as an interactive demo or a
%         continuous integration test. (optional; defaults to 0, i.e.
%         interactive demo)
%

%
% This file is part of the M-M.E.S.S. project
% (http://www.mpi-magdeburg.mpg.de/projects/mess).
% Copyright © 2009-2022 Jens Saak, Martin Koehler, Peter Benner and others.
% All rights reserved.
% License: BSD 2-Clause License (see COPYING)
%
if nargin<1, istest=0; end

Construction of system data.

if exist('OCTAVE_VERSION', 'builtin')
    rand('seed', 1.0); %#ok<RAND>
    eqn.A_ = rand(500) - 250 * eye(500);

    rand('seed', 2.0); %#ok<RAND>
    eqn.B2 = rand(500, 2);
    rand('seed', 3.0); %#ok<RAND>
    B1 = rand(500, 2);

    rand('seed', 4.0); %#ok<RAND>
    eqn.C2 = rand(3, 500);
    rand('seed', 5.0); %#ok<RAND>
    C1 = rand(3, 500);
else
    rng(1.0);
    eqn.A_ = rand(500) - 250 * eye(500);

    rng(2.0);
    eqn.B2 = rand(500, 2);
    rng(3.0);
    B1 = rand(500, 2);

    rng(4.0);
    eqn.C2 = rand(3, 500);
    rng(5.0);
    C1 = rand(3, 500);
end

gam = 5; % Scaling term for disturbances.

Set operator.

oper = operatormanager('default');

Construction of options struct.

ADI settings.

opts.adi.maxiter          = 200;
opts.adi.res_tol           = 1.0e-12;
opts.adi.rel_diff_tol            = 0;
opts.adi.info             = 1;
opts.adi.compute_sol_fac  = 1;
opts.adi.accumulateK      = 0;
opts.adi.accumulateDeltaK = 0;

% Shift options.
opts.shifts.num_desired     = 5;
opts.shifts.method = 'projection';

% % NM settings.
opts.nm.maxiter       = 50;
opts.nm.res_tol        = 1.0e-10;
opts.nm.rel_diff_tol         = 1.0e-12;
opts.nm.info          = 1;
opts.nm.linesearch    = 0;
opts.nm.accumulateRes = 1;

% NM projection settings.
opts.nm.projection      = [];
opts.nm.projection.freq = 0;
opts.nm.res.maxiter     = 10;
opts.nm.res.tol         = 1.0e-06;
opts.nm.res.orth        = 1;

% RADI settings.
opts.radi.maxiter = opts.adi.maxiter;
opts.radi.res_tol  = opts.nm.res_tol;
opts.radi.rel_diff_tol   = 1.0e-16;
opts.radi.info    = 1;

% RI settings.
opts.ri.riccati_solver = 'radi';
opts.ri.lqg_solver     = 'newton';
opts.ri.maxiter        = 10;
opts.ri.res_tol         = 1.0e-10;
opts.ri.rel_diff_tol          = 1.0e-16;
opts.ri.compres_tol    = 1.0e-16;
opts.ri.info           = 1;

% global options
opts.norm           = 2;

% %% Call Riccati iteration with Newton solver.
t_RI_call = tic;
eqn.type = 'T';
eqn.B1   = 1/gam * B1;
eqn.C1   = C1;
[out, eqn, opts, ~] = mess_lrri(eqn, opts, oper);
t_elapsed = toc(t_RI_call);
fprintf(1,'mess_lrri took %6.2f seconds \n' , t_elapsed);
ADI step:    1 normalized residual: 9.007645e-01 
		 normalized outer residual: 4.569837e+00
ADI step:    2 normalized residual: 8.986356e-01 
		 normalized outer residual: 2.041060e+01
ADI step:    3 normalized residual: 8.773219e-01 
		 normalized outer residual: 7.625133e+02
ADI step:    4 normalized residual: 8.752302e-01 
		 normalized outer residual: 8.924871e+02
ADI step:    5 normalized residual: 1.213059e-12 
		 normalized outer residual: 9.515593e+05
ADI step:    7 normalized residual: 8.735569e-13 
		 normalized outer residual: 9.515593e+05

 NM step:    1  normalized residual: 9.515593e+05
               relative change in K: 1.000000e+00
               number of ADI steps: 7 

ADI step:    1 normalized residual: 2.227100e+03 
		 normalized outer residual: 2.367773e+05
ADI step:    2 normalized residual: 2.219935e+03 
		 normalized outer residual: 2.367809e+05
ADI step:    3 normalized residual: 2.212820e+03 
		 normalized outer residual: 2.367845e+05
ADI step:    5 normalized residual: 2.198504e+03 
		 normalized outer residual: 2.367916e+05
ADI step:    6 normalized residual: 1.505244e-16 
		 normalized outer residual: 2.378896e+05

 NM step:    2  normalized residual: 2.378896e+05
               relative change in K: 9.999991e-01
               number of ADI steps: 6 

ADI step:    1 normalized residual: 1.020530e-01 
		 normalized outer residual: 5.947217e+04
ADI step:    2 normalized residual: 1.859404e-05 
		 normalized outer residual: 5.947217e+04
ADI step:    4 normalized residual: 5.128794e-13 
		 normalized outer residual: 5.947217e+04

 NM step:    3  normalized residual: 5.947217e+04
               relative change in K: 9.999943e-01
               number of ADI steps: 4 

ADI step:    1 normalized residual: 1.013988e-01 
		 normalized outer residual: 1.486782e+04
ADI step:    2 normalized residual: 1.088172e-04 
		 normalized outer residual: 1.486782e+04
ADI step:    3 normalized residual: 2.126873e-08 
		 normalized outer residual: 1.486782e+04
ADI step:    4 normalized residual: 3.594234e-12 
		 normalized outer residual: 1.486782e+04
ADI step:    5 normalized residual: 5.970466e-16 
		 normalized outer residual: 1.486782e+04
ADI step:    7 normalized residual: 1.200218e-20 
		 normalized outer residual: 1.486782e+04

 NM step:    4  normalized residual: 1.486782e+04
               relative change in K: 9.999735e-01
               number of ADI steps: 7 

ADI step:    1 normalized residual: 1.001011e-01 
		 normalized outer residual: 3.716729e+03
ADI step:    2 normalized residual: 1.822499e-05 
		 normalized outer residual: 3.716729e+03
ADI step:    3 normalized residual: 3.050870e-09 
		 normalized outer residual: 3.716729e+03
ADI step:    5 normalized residual: 8.768742e-17 
		 normalized outer residual: 3.716729e+03

 NM step:    5  normalized residual: 3.716729e+03
               relative change in K: 9.998864e-01
               number of ADI steps: 5 

ADI step:    1 normalized residual: 9.754853e-02 
		 normalized outer residual: 9.289569e+02
ADI step:    2 normalized residual: 1.775524e-05 
		 normalized outer residual: 9.289570e+02
ADI step:    3 normalized residual: 2.876211e-09 
		 normalized outer residual: 9.289570e+02
ADI step:    5 normalized residual: 7.622568e-17 
		 normalized outer residual: 9.289570e+02

 NM step:    6  normalized residual: 9.289570e+02
               relative change in K: 9.995304e-01
               number of ADI steps: 5 

ADI step:    1 normalized residual: 9.261102e-02 
		 normalized outer residual: 2.320141e+02
ADI step:    2 normalized residual: 1.685715e-05 
		 normalized outer residual: 2.320142e+02
ADI step:    3 normalized residual: 2.730957e-09 
		 normalized outer residual: 2.320142e+02
ADI step:    5 normalized residual: 7.263355e-17 
		 normalized outer residual: 2.320142e+02

 NM step:    7  normalized residual: 2.320142e+02
               relative change in K: 9.980932e-01
               number of ADI steps: 5 

ADI step:    1 normalized residual: 8.338454e-02 
		 normalized outer residual: 5.777910e+01
ADI step:    2 normalized residual: 1.517248e-05 
		 normalized outer residual: 5.777920e+01
ADI step:    3 normalized residual: 2.456734e-09 
		 normalized outer residual: 5.777920e+01
ADI step:    5 normalized residual: 1.924532e-16 
		 normalized outer residual: 5.777920e+01

 NM step:    8  normalized residual: 5.777920e+01
               relative change in K: 9.923460e-01
               number of ADI steps: 5 

ADI step:    1 normalized residual: 6.737154e-02 
		 normalized outer residual: 1.422321e+01
ADI step:    2 normalized residual: 1.222722e-05 
		 normalized outer residual: 1.422331e+01
ADI step:    3 normalized residual: 1.978799e-09 
		 normalized outer residual: 1.422331e+01
ADI step:    5 normalized residual: 1.612239e-15 
		 normalized outer residual: 1.422331e+01

 NM step:    9  normalized residual: 1.422331e+01
               relative change in K: 9.698033e-01
               number of ADI steps: 5 

ADI step:    1 normalized residual: 4.397716e-02 
		 normalized outer residual: 3.344867e+00
ADI step:    2 normalized residual: 7.806692e-06 
		 normalized outer residual: 3.344989e+00
ADI step:    3 normalized residual: 1.261477e-09 
		 normalized outer residual: 3.344989e+00
ADI step:    5 normalized residual: 8.233107e-15 
		 normalized outer residual: 3.344989e+00

 NM step:   10  normalized residual: 3.344989e+00
               relative change in K: 8.871467e-01
               number of ADI steps: 5 

ADI step:    1 normalized residual: 1.984205e-02 
		 normalized outer residual: 6.603327e-01
ADI step:    2 normalized residual: 3.298348e-06 
		 normalized outer residual: 6.604783e-01
ADI step:    3 normalized residual: 5.162608e-10 
		 normalized outer residual: 6.604783e-01
ADI step:    5 normalized residual: 1.556940e-14 
		 normalized outer residual: 6.604783e-01

 NM step:   11  normalized residual: 6.604783e-01
               relative change in K: 6.474732e-01
               number of ADI steps: 5 

ADI step:    1 normalized residual: 5.298135e-03 
		 normalized outer residual: 7.052487e-02
ADI step:    2 normalized residual: 1.058165e-06 
		 normalized outer residual: 7.065377e-02
ADI step:    3 normalized residual: 1.622901e-10 
		 normalized outer residual: 7.065379e-02
ADI step:    5 normalized residual: 6.499298e-15 
		 normalized outer residual: 7.065379e-02

 NM step:   12  normalized residual: 7.065379e-02
               relative change in K: 2.661734e-01
               number of ADI steps: 5 

ADI step:    1 normalized residual: 7.679310e-03 
		 normalized outer residual: 7.551888e-03
ADI step:    2 normalized residual: 2.009181e-06 
		 normalized outer residual: 1.310633e-03
ADI step:    3 normalized residual: 2.505526e-10 
		 normalized outer residual: 1.310721e-03
ADI step:    5 normalized residual: 1.710551e-15 
		 normalized outer residual: 1.310721e-03

 NM step:   13  normalized residual: 1.310721e-03
               relative change in K: 3.750155e-02
               number of ADI steps: 5 

ADI step:    1 normalized residual: 9.727785e-03 
		 normalized outer residual: 9.727063e-03
ADI step:    2 normalized residual: 2.849486e-06 
		 normalized outer residual: 2.816420e-06
ADI step:    3 normalized residual: 4.139584e-10 
		 normalized outer residual: 4.797276e-07
ADI step:    5 normalized residual: 1.104932e-15 
		 normalized outer residual: 4.797468e-07

 NM step:   14  normalized residual: 4.797468e-07
               relative change in K: 7.179093e-04
               number of ADI steps: 5 

ADI step:    1 normalized residual: 9.781351e-03 
		 normalized outer residual: 9.781046e-03
ADI step:    2 normalized residual: 2.919786e-06 
		 normalized outer residual: 2.919786e-06
ADI step:    3 normalized residual: 5.111351e-10 
		 normalized outer residual: 5.111314e-10
ADI step:    5 normalized residual: 5.893132e-16 
		 normalized outer residual: 6.180658e-14

 NM step:   15  normalized residual: 6.180658e-14
               relative change in K: 2.578733e-07
               number of ADI steps: 5 

RI step:    1 normalized residual: 4.001782e-02 relative change in Z: 1.000000e+00
               number of Newton steps:   15

RADI step:    1 pc: -2.988561e+02 + 0.000000e+00i normalized residual: 6.618973e-03 relative change in Z: 1.000000e+00
RADI step:    2 pc: -2.503193e+02 + 0.000000e+00i normalized residual: 1.614140e-06 relative change in Z: 8.854109e-02
RADI step:    3 pc: -3.026763e+02 + 0.000000e+00i normalized residual: 1.814056e-08 relative change in Z: 1.362125e-03
RADI step:    4 pc: -2.520618e+02 + 0.000000e+00i normalized residual: 5.525809e-12 relative change in Z: 1.478580e-04
RI step:    2 normalized residual: 1.927978e-05 relative change in Z: 1.894557e-02
               number of RADI steps:    4

RADI step:    1 pc: -3.022722e+02 + 0.000000e+00i normalized residual: 7.811222e-03 relative change in Z: 1.000000e+00
RADI step:    2 pc: -2.452339e+02 + 0.000000e+00i normalized residual: 1.375602e-06 relative change in Z: 9.716098e-02
RADI step:    4 pc: -2.469850e+02 + -1.821623e+00i normalized residual: 3.865302e-13 relative change in Z: 1.277212e-03
RI step:    3 normalized residual: 4.541195e-12 relative change in Z: 9.182042e-06
               number of RADI steps:    4

mess_lrri took   0.77 seconds 

Compute real residuals.

abserr = norm(eqn.A_' * (out.Z * out.Z') + (out.Z * out.Z') * eqn.A_ ...
    + (out.Z * out.Z') * (1/gam^2 * (B1 * B1') ...
    - eqn.B2 * eqn.B2') * (out.Z * out.Z') + C1' * C1, 2);
relerr = abserr / norm(C1 * C1', 2);
fprintf(1, '\nset tolerance vs. real residual: %e | %e\n', ...
    opts.ri.res_tol, relerr);

if istest
    assert(relerr < opts.ri.res_tol, ...
        'MESS:TEST:accuracy','unexpectedly inaccurate result');
end
set tolerance vs. real residual: 1.000000e-10 | 4.492133e-12