# findR

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

### Syntax

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

### 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

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