Scilab Function

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)

### Parameters

• 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
```