findR
=====

Preprocessor for estimating the matrices of a linear time-invariant
dynamical system



Calling Sequence
~~~~~~~~~~~~~~~~


::

    [R,N [,SVAL,RCND]] = findR(S,Y,U,METH,ALG,JOBD,TOL,PRINTW)
    [R,N] = findR(S,Y)




Arguments
~~~~~~~~~

:S the number of block rows in the block-Hankel matrices.
: :Y
: :U
: :METH an option for the method to use:
    :1 MOESP method with past inputs and outputs;
    : :2 N4SI15 0 1 1 1000D method.
    :
Default: METH = 1.
: :ALG an option for the algorithm to compute the triangular factor of
the concatenated block-Hankel matrices built from the input-output
data:
    :1 Cholesky algorithm on the correlation matrix;
    : :2 fast QR algorithm;
    : :3 standard QR algorithm.
    :
Default: ALG = 1.
: :JOBD an option to specify if the matrices B and D should later be
computed using the MOESP approach:
    := 1 : the matrices B and D should later be computed using the MOESP
      approach;
    : := 2 : the matrices B and D should not be computed using the MOESP
      approach.
    :
Default: JOBD = 2. This parameter is not relevant for METH = 2.
: :TOL a vector of length 2 containing tolerances:
    :TOL (1) is the tolerance for estimating the rank of matrices. If
      TOL(1) > 0, the given value of TOL(1) is used as a lower bound for the
      reciprocal condition number. Default: TOL(1) =
      prod(size(matrix))*epsilon_machine where epsilon_machine is the
      relative machine precision.
    : :TOL (2) is the tolerance for estimating the system order. If TOL(2)
      >= 0, the estimate is indicated by the index of the last singular
      value greater than or equal to TOL(2). (Singular values less than
      TOL(2) are considered as zero.) When TOL(2) = 0, then
      S*epsilon_machine*sval(1) is used instead TOL(2), where sval(1) is the
      maximal singular value. When TOL(2) < 0, the estimate is indicated by
      the index of the singular value that has the largest logarithmic gap
      to its successor. Default: TOL(2) = -1.
    :

: :PRINTW a switch for printing the warning messages.
    := 1: print warning messages;
    : := 0: do not print warning messages.
    :
Default: PRINTW = 0.
: :R
: :N the order of the discrete-time realization
: :SVAL singular values SVAL, used for estimating the order.
: :RCND vector of length 2 containing the reciprocal condition numbers
  of the matrices involved in rank decisions or least squares solutions.
:



Description
~~~~~~~~~~~

findR Preprocesses the input-output data for estimating the matrices
of a linear time-invariant dynamical system, using Cholesky or (fast)
QR factorization and subspace identification techniques (MOESP or
N4SID), and estimates the system order.

[R,N] = findR(S,Y,U,METH,ALG,JOBD,TOL,PRINTW) returns the processed
upper triangular factor R of the concatenated block-Hankel matrices
built from the input-output data, and the order N of a discrete-time
realization. The model structure is:


::

    x(k+1) = Ax(k) + Bu(k) + w(k),   k >= 1,
    y(k)   = Cx(k) + Du(k) + e(k).


The vectors y(k) and u(k) are transposes of the k-th rows of Y and U,
respectively.

[R,N,SVAL,RCND] = findR(S,Y,U,METH,ALG,JOBD,TOL,PRINTW) also returns
the singular values SVAL, used for estimating the order, as well as,
if meth = 2, the vector RCND of length 2 containing the reciprocal
condition numbers of the matrices involved in rank decisions or least
squares solutions.

[R,N] = findR(S,Y) assumes U = [] and default values for the remaining
input arguments.



Examples
~~~~~~~~


::

    //generate data from a given linear system
    A = [ 0.5, 0.1,-0.1, 0.2;
          0.1, 0,  -0.1,-0.1;      
         -0.4,-0.6,-0.7,-0.1;  
          0.8, 0,  -0.6,-0.6];      
    B = [0.8;0.1;1;-1];
    C = [1 2 -1 0];
    SYS=`syslin`_(0.1,A,B,C);
    U=(`ones`_(1,1000)+`rand`_(1,1000,'normal')); 
    Y=(`flts`_(U,SYS)+0.5*`rand`_(1,1000,'normal'));
    // Compute R
    
    [R,N,SVAL] = findR(15,Y',U');
    SVAL
    N




See Also
~~~~~~~~


+ `findABCD`_ discrete-time system subspace identification
+ `findAC`_ discrete-time system subspace identification
+ `findBD`_ initial state and system matrices B and D of a discrete-
  time system
+ `findBDK`_ Kalman gain and B D system matrices of a discrete-time
  system
+ `sorder`_ computing the order of a discrete-time system
+ `sident`_ discrete-time state-space realization and Kalman gain


.. _findBDK: findBDK.html
.. _sident: sident.html
.. _findAC: findAC.html
.. _sorder: sorder.html
.. _findABCD: findABCD.html
.. _findBD: findBD.html