%
% OHKDEMO Optimal Descriptor Hankel Model Reduction Technique & Anticausal
%         Projection demonstration
%

% R. Y. Chiang & M. G. Safonov 3/88
% Copyright (c) 1988 by the MathWorks, Inc.
% All Rights Reserved.
% -------------------------------------------------------------------------
clc
disp('    ')
disp(' ')
disp('            << Demo #2: Optimal Descriptor Hankel Approximation >>')
disp('  ')
disp(' ')
disp('             This demo will show:')
disp(' ')
disp('               1). Optimal Descriptor Hankel Model Reduction')
disp('                   (Optimal Hankel MDA)')
disp('     ')
disp('               2). Optimal Descriptor Anticausal Projection')
disp('  ')
disp(' ')
disp('  ')
disp('  ')
disp(' ')
disp(' ')
disp('                                 (strike a key to continue ...)')
pause
clc
disp(' ')
disp(' ')
disp('                   << Introduction of Hankel Methods >>')
disp('  ')
disp('  ')
disp('    A basis-free descriptor system representation is shown to facilitate')
disp(' ')
disp('    the computation of all minimum-degree and optimal k-th order all-pass')
disp(' ')
disp('    extensions and Hankel-norm approximants. The descriptor representation')
disp(' ')
disp('    has the same simple form for both the optimal and suboptimal cases.')
disp(' ')
disp('    The method makes Hankel model reduction practical for non-minimal and')
disp(' ')
disp('    near non-minimal systems by eliminating the ill-conditioned calculation')
disp(' ')
disp('    of a minimal balanced realization. A simple, numerically sound method')
disp(' ')
disp('    based on singular-value demcompositon enables the results to be expressed')
disp(' ')
disp('    in state-space form.')
disp('  ')
disp('  ')
disp('                                 (strike a key to continue ...)')
pause
clc
format short e
disp(' State-space of the 16-state model to be reduced (stable & nonminimal):')
a1 = [
  -2.0000e+03  -1.1493e+00   1.4858e+03  -6.0345e+02   7.7992e+02   4.5933e+02;
   1.3549e-13  -2.0981e-02  -2.3740e+00   1.2214e+00  -2.4659e+00  -1.1860e+00;
   3.0579e-14  -1.2737e-14  -5.6757e+00   2.2035e+00   2.0629e+01   1.0031e+01;
  -1.5913e-14   5.1825e-15   8.1765e-06  -2.5785e-01   3.6547e+00   3.3445e+00;
   7.7097e-15  -6.4280e-15  -1.0629e-05   7.4497e-05  -3.0000e+01  -1.5516e-04;
   4.0211e-15  -3.9663e-15  -5.2035e-06   3.8799e-05   5.3245e-06  -3.0000e+01;
   1.0512e-26  -1.3191e-26   7.4402e-18  -1.8651e-18  -2.4512e-16  -2.4600e-16;
   1.5017e-26  -6.5896e-27   3.7201e-18  -9.3252e-19  -1.2256e-16  -1.2300e-16;
            0            0            0            0            0            0;
            0            0            0            0            0            0;
            0            0            0            0            0            0;
            0            0            0            0            0            0;
            0            0            0            0            0            0;
            0            0            0            0            0            0;
            0            0            0            0            0            0;
            0            0            0            0            0            0];
a2 = [
   1.1539e-09   5.7695e-10            0            0            0            0;
   7.5319e-13   3.7660e-13            0            0            0            0;
  -4.0627e-12   2.4833e-12            0            0            0            0;
  -2.2146e-12  -2.9492e-12            0            0            0            0;
   1.0331e-11   7.4681e-12            0            0            0            0;
   6.3843e-12   4.5653e-12            0            0            0            0;
  -1.0000e-02   3.9925e-24            0            0            0            0;
   2.9281e-25  -1.0000e-02            0            0            0            0;
            0            0  -1.6184e+01   1.5083e+02   4.8353e+02  -3.7230e+02;
            0            0  -2.1463e+01  -1.1303e+02  -3.9936e+02   4.5213e+02;
            0            0   1.0421e+01   7.2968e+01   2.5251e+02  -2.6766e+02;
            0            0   4.2729e+00   5.0112e+01   1.6975e+02  -1.6710e+02;
            0            0   1.3789e+01   1.0265e+02   3.5676e+02  -3.7530e+02;
            0            0  -5.9738e+01  -3.3114e+02  -1.1675e+03   1.2905e+03;
            0            0   2.0960e+03   1.1873e+04   4.1764e+04  -4.5988e+04;
            0            0   1.6822e+03  -1.6399e+04  -5.2599e+04   4.0665e+04];
a3 = [
            0            0            0            0;
            0            0            0            0;
            0            0            0            0;
            0            0            0            0;
            0            0            0            0;
            0            0            0            0;
            0            0            0            0;
            0            0            0            0;
   2.3930e-01  -3.3054e-01  -6.5690e-03   1.2279e+00;
  -6.8475e+00   6.3611e+00   1.5087e+00   4.1866e-02;
   2.6085e+00  -2.4529e+00  -5.6231e-01   3.1315e-01;
   5.9270e-01  -5.8463e-01  -1.1822e-01   3.5004e-01;
   2.9069e+00  -5.8875e+00  -9.0978e-01   8.2380e-01;
  -1.1292e+01   1.7963e+01   2.2323e+00   5.8339e-01;
   3.6532e+02  -5.8425e+02  -4.9193e+01   1.9730e-01;
  -9.0859e+00   9.8929e+00   1.0133e+00  -3.2254e+01];
a = [a1 a2 a3];
b =[
  -9.3933e+02   1.1326e+03;
   6.3293e+01  -1.5312e+01;
  -2.7062e-04   4.0846e-03;
   2.6371e-03   1.5393e-03;
  -5.5859e-04   8.9453e-04;
  -3.6171e-04   5.5788e-04;
   7.5916e-16  -1.3986e-15;
   3.7954e-16  -6.9930e-16;
   1.5533e-02  -1.5516e-01;
   1.7881e-01   5.2796e-03;
  -8.6917e-02  -1.8393e-02;
  -3.9949e-02  -2.3137e-02;
  -1.2324e-01  -2.8050e-02;
   4.7783e-01   3.8904e-02;
  -1.6720e+01  -1.5469e+00;
  -1.5476e+00   1.6728e+01];
c1 = [
  -9.1517e-08  -5.2591e-11   3.1223e-03  -1.2700e-03  -1.9162e-05  -4.5190e-04;
   9.8922e-07   5.6846e-10  -1.7184e-03   6.9980e-04   9.5218e-04   5.0433e-04];
c2 = [
   3.6519e-16  -2.0788e-17  -2.0891e+01  -1.1933e+02  -4.1720e+02   4.6024e+02;
   9.2135e-16   1.6474e-15  -1.6124e+01   1.6319e+02   5.2691e+02  -4.0617e+02];
c3 = [
  -4.8842e+00   5.2742e+00   1.4817e+00   7.4520e-03;
  -2.8106e+00  -1.8255e+00  -4.2506e-02   1.3483e+00];
c = [c1 c2 c3];
d = [
   1.6729e-01   1.5477e-02
   1.5477e-02  -1.6729e-01];
disp('     First 6 column of the A matrix:')
a1
disp('                                 (strike a key to continue ...)')
pause
clc
disp('     Column 7 to 12 of the A matrix:')
a2
disp('                                 (strike a key to continue ...)')
pause
clc
disp('     Column 13 to 16 of the A matrix:')
a3
disp('                                 (strike a key to continue ...)')
pause
clc
b
disp('                                 (strike a key to continue ...)')
pause
clc
c
disp('                                 (strike a key to continue ...)')
pause
clc
d
disp('                                 (strike a key to continue ...)')
pause
clc
disp('                 << Example 1. Optimal Descriptor Hankel MDA >>')
disp('       -----------------------------------------------------------------')
disp('        First let us compute the Hankel singular values of the model...')
disp('  ')
disp('        [hsv] = hksv(a,b,c);    % Computing Hankel singular values')
disp('       -----------------------------------------------------------------')
[hsv] = hksv(a,b,c);
hsv = hsv';
format compact
hsv
format loose
disp(' ')
disp('      ----------------------------------------------------------------')
disp('       Obviously, the "balanced" version of Hankel MDA fails immedi-')
disp('       ately due to the last few Hankel singular vlaues being "0" ...')
disp('      ----------------------------------------------------------------')
disp(' ')
disp('        (strike a key to see a bar chart of the Hankel Singular Values ...)')
pause
bar(hsv)
title('Hankel Singular Values')
pause
clc
disp(' ')
disp(' ')
disp('      ---------------------------------------------------------------')
disp('       If we truncate the model to 6-state, the infinity-norm error ') 
disp('       bound can be computed beforehand ...')
disp(' ')
disp('       bndsch = 2*sum(hsv(1,7:16));  % Infinity-norm error bound')
disp('      ---------------------------------------------------------------')
bndsch = 2*sum(hsv(1,7:16))
disp('                                 (strike a key to continue ...)')
pause
clc
disp(' ')
disp('  ')
disp('     ------------------------------------------------------------------')
disp('      Let us start Optimal Descriptor Hankel Model Reduction to get a')
disp('      6-state reduced order model ...')
disp(' ')
disp('      [ah,bh,ch,dh,bndhkl,hsv] = ohklmr(a,b,c,d,1,6); % Run OHKLMR.M')
disp('     ------------------------------------------------------------------')
[ah,bh,ch,dh,bndhkl,hsv] = ohklmr(a,b,c,d,1,6);
disp('                                                 ^')
disp('        - - - Computing SV Bode plot of G(s) and G(s) - - - ')
w = logspace(-4,4,100);
svg = sigma(a,b,c,d,1,w); svg = 20*log10(svg);
svhkl = sigma(ah,bh,ch,dh,1,w); svhkl = 20*log10(svhkl);
[aher,bher,cher,dher] = addss(a,b,c,d,ah,bh,-ch,-dh);
disp('  ')
disp('        - - - Computing SV Bode plot of the error system - - -')
sverhkl = sigma(aher,bher,cher,dher,1,w); sverhkl = 20*log10(sverhkl);
bndhkl = 20*log10(bndhkl)*ones(w);
disp(' ')
disp('                                                               ^')
disp('                       (strike a key to see the plot of G(s) & G(s)...)')
pause
clg
semilogx(w,svg,w,svhkl)
title('Optimal Descriptor Hankel Model Reduction (16-state --> 6-state)')
xlabel('Frequency - Rad/Sec')
ylabel('SV - db')
text(0.5,-120,'Original 16-State Model')
text(0.001,-40,'6-State Approximation')
grid
pause
disp(' ')
disp('                                                 ^')
disp('           (strike a key to see the plot of G(s)-G(s) and its bound...)')
pause
clg
semilogx(w,bndhkl,w,sverhkl)
title('Optimal Descriptor Hankel MDA (Infinity Error Bound)')
xlabel('Frequency - Rad/Sec')
ylabel('SV - db')
text(1,-45,'Error Bound')
text(1000,-100,'Error')
grid
pause
clc
disp(' ')
disp('  ')
disp('       << Example 2. Optimal Descriptor Hankel Anticausal Projection >>')
disp('     -------------------------------------------------------------------')
disp('       Let us try the same algorithm but do an anticausal projection')
disp('       this time. This is also referred to the 0-th order Hankel MDA ..')
disp(' ')
disp('       [ahx,bhx,chx,dhx,ahy,bhy,chy,dhy,aug] = ohkapp(a,b,c,d,1,0); ')
disp('                                                     % Run OHKAPP.M')
disp('       State-space of the anticausal approximant: (ahy,bhy,chy,dhy)')
disp('     -------------------------------------------------------------------')
[ahx,bhx,chx,dhx,ahy,bhy,chy,dhy,aug] = ohkapp(a,b,c,d,1,0); 
disp(' ')
disp(' ')
disp('                                 (strike a key to continue ...)')
pause
clc
disp(' ')
disp('    -----------------------------------------------------------------')
disp('     Let us show that the maximum singular value of the total error ')
disp('          ^')
disp('     G(s)-G(s)  is an all-pass function ...')
disp('             +')
disp('           ^')
disp('     where G(s)  is the anticausal Hankel approximant of G(s).')
disp('              +')
disp('    -----------------------------------------------------------------')
disp('      ')
disp(' ')
disp('                                                         ^')
disp('        - - - Computing singular value Bode plot of G(s)-G(s) - - -')
disp('                                                            +')
[aher,bher,cher,dher] = addss(a,b,c,d,ahy,bhy,-chy,-dhy);
svher = sigma(aher,bher,cher,dher,1,w); svher = 20*log10(svher);
disp(' ')
disp(' ')
disp('                (strike a key to see the "all-pass" error function ...)')
pause
subplot(211)
semilogx(w,svher(1,:))
title('Optimal Descriptor Hankel Anticausal Projection (Error System)')
xlabel('Frequency - Rad/Sec')
ylabel('max(SV) - db')
subplot(212)
semilogx(w,svher(2,:))
xlabel('Frequency - Rad/Sec')
ylabel('min(SV) - db')
grid
pause
subplot(111)
%
% ----- End of OHKDEMO.M --- RYC/MGS 
