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+## Copyright (C) 2026 J. Román <jdaniel.roman2004@gmail.com>
+##
+## This file is part of the control package for GNU Octave.
+##
+## Octave 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 3 of the License, or
+## (at your option) any later version.
+##
+## Octave 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 Octave; see the file COPYING.  If not,
+## see <http://www.gnu.org/licenses/>.
+
+## -*- texinfo -*-
+## @deftypefn {Function File} {[@var{csys}, @var{T}] =} compreal (@var{sys}, @var{realization})
+## Returns the state-space companion controllable and observable form of the input
+## system as well as the transformation matrix. 
+## 
+## Given the state vector
+## @tex
+## $$x(t)$$
+## @end tex
+## @ifnottex
+## @example
+## @group
+##       x(t)
+## @end group
+## @end example
+## @end ifnottex
+##
+## We look for a transformation matrix such that
+##
+## @tex
+## $$ x(t)=Tz(t) $$
+## @end tex
+## @ifnottex
+## @example
+## @group
+##       x(t)=Tz(t)
+## @end group
+## @end example
+## @end ifnottex
+##
+## @tex
+## Where $T$ is our transformation matrix and $z(t)$ is our new state vector.
+## @end tex
+## @ifnottex
+## @group
+## Where T is our transformation matrix and $z(t)$ is our new state vector.
+## @end group
+## @end ifnottex
+## 
+## Our new state-space is given by the following relationships:
+##
+## @tex
+## $$ A_z = T^{-1}AT $$
+## $$ B_z = T^{-1}B $$
+## $$ C_z = CT $$
+## $$ D_z = D $$
+## @end tex
+## @ifnottex
+## @example
+## @group
+## A_z = inv(T)AT
+## B_z = inv(T)B
+## C_z = CT
+## D_z = D
+## @end group
+## @end example
+## @end ifnottex
+##
+## The @var{realization} argument selects the companion form:
+##
+## @table @code
+## @item 'c'
+## Controllable companion form.
+## 
+## @item 'o'
+## Observable companion form.
+## @end table
+##
+## @end deftypefn
+
+## Based on equation (15a) and (15b) of: 
+## T. Kailath, Linear systems. Prentice-Hall Englewood Cliffs, NJ, 1980.
+## ISBM 0-13-536961-4
+
+## Author: J. Román <jdaniel.roman2004@gmail.com>
+## Created: May 2026
+
+function [csys, T] = compreal (sys, realization)
+  
+  ## Validation
+
+  if (nargin < 1 || nargin > 2)
+    print_usage ();
+  endif
+
+  if (issiso (sys) == 0)
+    error ("compreal: system must be SISO");
+  endif
+
+  if (nargin == 1 || strcmp (realization, "c"))
+    realization = "controllable";
+  elseif (strcmp (realization, "o"))
+    realization = "observable";
+  else
+    error ("compreal: realization must be 'c' or 'o'");
+  endif
+
+
+
+  ## Matrix T
+
+  sys = ss (sys);
+
+  switch (realization)
+
+    case "controllable"
+
+      ctrmatrix = ctrb (sys.A, sys.B);
+
+      if (rank (ctrmatrix) < rows (ctrmatrix))
+
+        error ("compreal: system is not controllable");
+
+      endif
+
+      T = ctrmatrix;
+
+    case "observable"
+
+      obsmatrix = obsv (sys.A, sys.C);
+
+      if (rank (obsmatrix) < rows (obsmatrix))
+
+        error ("compreal: system is not observable");
+
+      endif
+
+      T = inv (obsmatrix);
+
+  endswitch
+
+  ## Transformation
+
+  A = T \ sys.A * T;
+
+  B = T \ sys.B;
+
+  C = sys.C * T;
+
+  D = sys.D;
+  
+  csys = ss (A, B, C, D);
+
+endfunction
+
+%!shared sys, A, B, C, D
+%! A = [0 1; -2 -3];
+%! B = [0; 1];
+%! C = [1 0];
+%! D = 0;
+%! sys = ss (A, B, C, D);
+
+%!test
+%! [csys, T] = compreal (sys, 'c');
+%! assert (tfdata(tf(csys)), tfdata(tf(sys)), 1e-6);
+
+%!test 
+%! [csys, T] = compreal (sys, 'c');
+%! assert (T, ctrb (A, B), 1e-10);
+
+%!test
+%! [csys, T] = compreal (sys, 'c');
+%! assert (csys.A, inv (T) * A * T, 1e-10);
+
+%!test
+%! [csys, T] = compreal (sys, 'o');
+%! assert (tfdata (tf (csys)), tfdata (tf (sys)), 1e-6);
+
+%!error <realization must be>
+%! compreal (sys, 'x');
+
+%!error <system must be SISO>
+%! sysmimo = ss (A, B, eye(2), zeros(2,1));
+%! compreal (sysmimo);