findx0BD Estimates state and B and D matrices of a discrete-time linear system

## Syntax

                [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

  initial 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
- inistate