findx0BD

Estimates state and B and D matrices of a discrete-time linear system

Calling Sequence

[X0,B,D] = findx0BD(A,C,Y,U,WITHX0,WITHD,TOL,PRINTW)
[x0,B,D,V,rcnd] = findx0BD(A,C,Y,U)

Arguments

:A state matrix of the system : :C C matrix of the system : :Y system output : :U system input : :WITHX0 a switch for estimating the initial state x0.

= 1:estimate x0;

: := 0: do not estimate x0. :

Default: WITHX0 = 1. : :WITHD a switch for estimating the matrix D.

= 1:estimate the matrix D;

: := 0: do not estimate the matrix D. :

Default: WITHD = 1. : :TOL the tolerance used for estimating the rank of matrices. If TOL

> 0, then the given value of TOL is used as a lower bound for the reciprocal condition number. Default: prod(size(matrix))*epsilon_machine where epsilon_machine is the relative machine precision.

: :PRINTW a switch for printing the warning messages.

= 1:print warning messages;

: := 0: do not print warning messages. :

Default: PRINTW = 0. : :X0 intial state of the estimated linear system. : :B B matrix of the estimated linear system. : :D D matrix of the estimated linear system. : :V orthogonal matrix which reduces the system state matrix A to a

real Schur form

: :rcnd estimates of the reciprocal condition numbers of the matrices

involved in rank decisions.

:

Description

findx0BD Estimates the initial state and/or the matrices B and D of a discrete-time linear system, given the (estimated) system matrices A, C, and a set of input/output data.

[X0,B,D] = findx0BD(A,C,Y,U,WITHX0,WITHD,TOL,PRINTW) estimates the initial state X0 and the matrices B and D of a discrete-time system using the system matrices A, C, output data Y and the input data U. The model structure is :

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

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

[x0,B,D,V,rcnd] = findx0BD(A,C,Y,U) also returns the orthogonal matrix V which reduces the system state matrix A to a real Schur form, as well as some estimates of the reciprocal condition numbers of the matrices involved in rank decisions.

B = findx0BD(A,C,Y,U,0,0)  returns B only, `and`_
[B,D] = findx0BD(A,C,Y,U,0)    returns B `and`_ D only.

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);
nsmp=100;
U=`prbs_a`_(nsmp,nsmp/5);
Y=(`flts`_(U,SYS)+0.3*`rand`_(1,nsmp,'normal'));

// Compute R
S=15;L=1;
[R,N,SVAL] = `findR`_(S,Y',U');

N=3;
METH=3;TOL=-1;
[A,C] = `findAC`_(S,N,L,R,METH,TOL);

[X0,B,D,V,rcnd] = findx0BD(A,C,Y',U');
SYS1=`syslin`_(1,A,B,C,D,X0);

Y1=`flts`_(U,SYS1);
`clf`_();`plot2d`_((1:nsmp)',[Y',Y1'])

See Also

• findBD initial state and system matrices B and D of a discrete- time system
• inistate Estimates the initial state of a discrete-time system