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