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


.. _inistate: inistate.html
.. _findBD: findBD.html