# 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
- findAC
- findBD
- findBDK
- sorder
- sident