## Copyright (C) 1996 Auburn University.  All Rights Reserved
##
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##
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## under the terms of the GNU General Public License as published by the 
## Free Software Foundation; either version 2, or (at your option) any 
## later version. 
## 
## Octave is distributed in the hope that it will be useful, but WITHOUT 
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## 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 Octave; see the file COPYING.  If not, write to the Free 
## Software Foundation, 59 Temple Place, Suite 330, Boston, MA 02111 USA. 

## -*- texinfo -*-
##@deftypefn {Function File } { } dgkfdemo ( ) 
## Octave Controls toolbox demo: H2/Hinfinity options demos
##@end deftypefn
 
function dgkfdemo ()
## Written by A. S. Hodel June 1995
 
  save_val = page_screen_output;
  page_screen_output = 1;
  while (1)
    clc
    menuopt=0;
    while(menuopt > 10 || menuopt < 1)
      menuopt = menu('Octave H2/Hinfinity options demo', ...
	    'LQ regulator', ...
	    'LG state estimator', ...
	    'LQG optimal control design', ...
	    'H2 gain of a system', ...
	    'H2 optimal controller of a system', ...
	    'Hinf gain of a system', ...
	    'Hinf optimal controller of a SISO system', ...
	    'Hinf optimal controller of a MIMO system', ...
	    'Discrete-time Hinf optimal control by bilinear transform', ...
	    'Return to main demo menu');
    endwhile
    if (menuopt == 1)
      disp('Linear/Quadratic regulator design:')
      disp('Compute optimal state feedback via the lqr command...')
      help lqr
      disp(' ')
      disp('Example:')
      A = [0, 1; -2, -1]
      B = [0; 1]
      Q = [1, 0; 0, 0]
      R = 1
      disp("Q = state penalty matrix; R = input penalty matrix")
      prompt
      disp('Compute state feedback gain k, ARE solution P, and closed-loop')
      disp('poles as follows:');
      cmd = "[k, p, e] = lqr(A,B,Q,R)";
      run_cmd
      prompt
      disp("A similar approach can be used for LTI discrete-time systems")
      disp("by using the dlqr command in place of lqr (see LQG example).")
    elseif (menuopt == 2)
      disp('Linear/Gaussian estimator design:')
      disp('Compute optimal state estimator via the lqe command...')
      help lqe
      disp(' ')
      disp('Example:')
      A = [0, 1; -2, -1]
      disp("disturbance entry matrix G")
      G = eye(2)
      disp("Output measurement matrix C")
      C = [0, 1]
      SigW = [1, 0; 0, 1]
      SigV = 1
      disp("SigW = input disturbance intensity matrix;")
      disp("SigV = measurement noise intensity matrix")
      prompt
      disp('Compute estimator feedback gain k, ARE solution P, and estimator')
      disp('poles via the command: ')
      cmd = "[k, p, e] = lqe(A,G,C,SigW,SigV)";
      run_cmd
      disp("A similar approach can be used for LTI discrete-time systems")
      disp("by using the dlqe command in place of lqe (see LQG example).")
    elseif (menuopt == 3)
      disp('LQG optimal controller of a system:')
      disp('Input accepted as either A,B,C matrices or in system data structure form')
      disp('in both discrete and continuous time.')
      disp("Example 1: continuous time design:")
      prompt
      help lqg
      disp("Example system")
      A = [0, 1; .5, .5];
      B = [0; 2];
      G = eye(2)
      C = [1, 1];
      sys = ss2sys(A, [B, G], C);
      sys = syssetsignals(sys,"in", ...
		       ["control input"; "disturbance 1"; "disturbance 2"]);
      sysout(sys)
      prompt
      disp("Filtering/estimator parameters:")
      SigW = eye(2)
      SigV = 1
      prompt
      disp("State space (LQR) parameters Q and R are:")
      Q = eye(2)
      R = 1
      cmd = "[K,Q1,P1,Ee,Er] = lqg(sys,SigW,SigV,Q,R,1);";
      run_cmd
      disp("Check: closed loop system A-matrix is")
      disp(" [A,      B*Cc]")
      disp(" [Bc*C,   Ac  ]")
      cmd = "[Ac, Bc, Cc] = sys2ss(K);";
      run_cmd
      cmd = "Acl = [A, B*Cc; Bc*C, Ac]";
      run_cmd
      disp("Check: poles of Acl:")
      Acl_poles = sortcom(eig(Acl))
      disp("Predicted poles from design = union(Er,Ee)")
      cmd = "pred_poles = sortcom([Er; Ee])";
      run_cmd
      disp("Example 2: discrete-time example")
      cmd1 = "Dsys = ss2sys(A, [G, B], C, [0, 0, 0], 1);";
      cmd2 = "[K,Q1,P1,Ee,Er] = lqg(Dsys,SigW, SigV,Q,R);";
      disp("Run commands:")
      cmd = cmd1;
      run_cmd
      cmd = cmd2;
      run_cmd
      prompt
      disp("Check: closed loop system A-matrix is")
      disp(" [A,      B*Cc]")
      disp(" [Bc*C,   Ac  ]")
      [Ac,Bc,Cc] = sys2ss(K);
      Acl = [A, B*Cc; Bc*C, Ac]
      prompt
      disp("Check: poles of Acl:")
      Acl_poles = sortcom(eig(Acl))
      disp("Predicted poles from design = union(Er,Ee)")
      pred_poles = sortcom([Er;Ee])
    elseif (menuopt == 4)
      disp('H2 gain of a system: (Energy in impulse response)')
      disp('Example 1: Stable plant:')
      cmd = "A = [0, 1; -2, -1]; B = [0; 1]; C = [1, 0]; sys_poles = eig(A)";
      run_cmd
      disp("Put into Packed system form:")
      cmd = "Asys = ss2sys(A,B,C);";
      run_cmd
      disp("Evaluate system 2-norm (impulse response energy):");
      cmd = "AsysH2 = h2norm(Asys)";
      run_cmd
      disp("Compare with a plot of the system impulse response:")
      tt = 0:0.1:20;
      for ii=1:length(tt)
        ht(ii) = C*expm(A*tt(ii))*B;
      endfor
      plot(tt,ht)
      title("impulse response of example plant")
      prompt
      disp('Example 2: unstable plant')
      cmd = "A = [0, 1; 2, 1]";
      eval(cmd);
      cmd = "B = [0; 1]";
      eval(cmd);
      cmd = "C = [1, 0]";
      eval(cmd);
      cmd = "sys_poles = eig(A)";
      run_cmd
      prompt
      disp('Put into system data structure form:')
      cmd="Bsys = ss2sys(A,B,C);";
      run_cmd
      disp('Evaluate 2-norm:')
      cmd = "BsysH2 = h2norm(Bsys)";
      run_cmd
      disp(' ')
      prompt('NOTICE: program returns a value without an error signal.')
      disp('')

    elseif (menuopt == 5)
      disp('H2 optimal controller of a system: command = h2syn:')
      prompt
      help h2syn
      prompt
      disp("Example system: double integrator with output noise and")
      disp("input disturbance:")
      disp(" ");
      disp("       -------------------->y2");
      disp("       |   _________");
      disp("u(t)-->o-->| 1/s^2 |-->o-> y1");
      disp("       ^   ---------   ^");
      disp("       |               |");
      disp("      w1(t)           w2(t)");
      disp(" ")
      disp("w enters the system through B1, u through B2")
      disp("z = [y1; y2] is obtained through C1, y=y1 through C2");
      disp(" ")
      cmd = "A = [0, 1; 0, 0];  B1 = [0, 0; 1, 0]; B2 = [0; 1];";
      disp(cmd)
      eval(cmd);
      cmd = "C1 = [1, 0; 0, 0]; C2 = [1, 0];    D11 = zeros(2);";
      disp(cmd)
      eval(cmd);
      cmd = "D12 = [0; 1];  D21 = [0, 1];  D22 = 0; D = [D11, D12; D21, D22];";
      disp(cmd)
      eval(cmd);
      disp("Design objective: compute U(s)=K(s)Y1(s) to minimize the closed")
      disp("loop impulse response from w(t) =[w1; w2] to z(t) = [y1; y2]");
      prompt
      disp("First: pack system:")
      cmd="Asys = ss2sys(A, [B1, B2], [C1; C2], D);";
      run_cmd
      disp("Open loop multivariable Bode plot: (will take a moment)")
      cmd="bode(Asys);";
      run_cmd
      prompt("Press a key to close plot and continue");
      closeplot
      disp("Controller design command: (only need 1st two output arguments)")
      cmd="[K,gain, Kc, Kf, Pc,  Pf] = h2syn(Asys,1,1);";
      run_cmd
      disp("Controller is:")
      cmd = "sysout(K)";
      run_cmd
      disp(["returned gain value is: ",num2str(gain)]);
      disp("Check: close the loop and then compute h2norm:")
      prompt
      cmd="K_loop = sysgroup(Asys,K);";
      run_cmd
      cmd = "Kcl = sysconnect(K_loop,[3,4],[4,3]);";
      run_cmd
      cmd = "Kcl = sysprune(Kcl,[1,2],[1,2]);";
      run_cmd
      cmd="gain_Kcl = h2norm(Kcl)";
      run_cmd
      cmd="gain_err = gain_Kcl - gain";
      run_cmd
      disp("Check: multivarible bode plot:")
      cmd="bode(Kcl);";
      run_cmd
      prompt
      disp("Related functions: is_dgkf, is_controllable, is_stabilizable,")
      disp("                is_observable, is_detectable")
    elseif (menuopt == 6)
      disp('Hinfinity gain of a system: (max gain over all j-omega)')
      disp('Example 1: Stable plant:')
      cmd = "A = [0, 1; -2, -1]; B = [0; 1]; C = [1, 0]; sys_poles = eig(A)";
      run_cmd
      disp('Pack into system format:')
      cmd = "Asys = ss2sys(A,B,C);";
      run_cmd
      disp('The infinity norm must be computed iteratively by')
      disp('binary search.  For this example, we select tolerance tol = 0.01, ')
      disp('min gain gmin = 1e-2, max gain gmax=1e4.')
      disp('Search quits when upper bound <= (1+tol)*lower bound.')
      cmd = "tol = 0.01; gmin = 1e-2; gmax = 1e+4;";
      run_cmd
      cmd = "[AsysHinf,gmin,gmax] = hinfnorm(Asys,tol,gmin,gmax)"
      run_cmd
      disp("Check: look at max value of magntude Bode plot of Asys:");
      [M,P,w] = bode(Asys);
      xlabel('Omega')
      ylabel('|Asys(j omega)| ')
      grid();
      semilogx(w,M);
      disp(["Max magnitude is ",num2str(max(M)), ...
	", compared with gmin=",num2str(gmin)," and gmax=", ...
        num2str(gmax),"."])
      prompt
      disp('Example 2: unstable plant')
      cmd = "A = [0, 1; 2, 1]; B = [0; 1]; C = [1, 0]; sys_poles = eig(A)";
      run_cmd
      disp("Pack into system format:")
      cmd = "Bsys = ss2sys(A,B,C);";
      run_cmd
      disp('Evaluate with BsysH2 = hinfnorm(Bsys,tol,gmin,gmax)')
      BsysH2 = hinfnorm(Bsys,tol,gmin,gmax)
      disp(' ')
      disp('NOTICE: program returns a value without an error signal.')
      disp('')

    elseif (menuopt == 7)
      disp('Hinfinity optimal controller of a system: command = hinfsyn:')
      prompt
      help hinfsyn
      prompt
      disp("Example system: double integrator with output noise and")
      disp("input disturbance:")
      A = [0, 1; 0, 0]
      B1 = [0, 0; 1, 0]
      B2 = [0; 1]
      C1 = [1, 0; 0, 0]
      C2 = [1, 0]
      D11 = zeros(2);
      D12 = [0; 1];
      D21 = [0, 1];
      D22 = 0;
      D = [D11, D12; D21, D22]
      prompt
      disp("First: pack system:")
      cmd="Asys = ss2sys(A, [B1, B2], [C1; C2], D);";
      run_cmd
      prompt
      disp("Open loop multivariable Bode plot: (will take a moment)")
      cmd="bode(Asys);";
      run_cmd
      prompt
      disp("Controller design command: (only need 1st two output arguments)")
      gmax = 1000
      gmin = 0.1
      gtol = 0.01
      cmd="[K,gain] = hinfsyn(Asys,1,1,gmin,gmax,gtol);";
      run_cmd
      disp("Check: close the loop and then compute h2norm:")
      prompt
      cmd="K_loop = sysgroup(Asys,K);";
      run_cmd
      cmd = "Kcl = sysconnect(K_loop,[3,4],[4,3]);";
      run_cmd
      cmd = "Kcl = sysprune(Kcl,[1,2],[1,2]);";
      run_cmd
      cmd="gain_Kcl = hinfnorm(Kcl)";
      run_cmd
      cmd="gain_err = gain_Kcl - gain";
      run_cmd
      disp("Check: multivarible bode plot:")
      cmd="bode(Kcl);";
      run_cmd
      prompt
      disp("Related functions: is_dgkf, is_controllable, is_stabilizable,")
      disp("                   is_observable, is_detectable, buildssic")
    elseif (menuopt == 8)
      disp('Hinfinity optimal controller of MIMO system: command = hinfsyn:')
      prompt
      help hinfsyn
      prompt
      disp("Example system: Boeing 707-321 airspeed/pitch angle control")
      disp(" ")
      hinfdemo
    elseif (menuopt == 9)
      disp("Discrete time H-infinity control via bilinear transform");
      prompt
      dhinfdemo
    elseif (menuopt == 10)
      return
    endif
    prompt
  endwhile  
  page_screen_output = save_val;
endfunction

