inistate Estimates the initial state of a discrete-time system

## Syntax

                X0 = inistate(A, C, Y)
                X0 = inistate(A, B, C, Y, U)
                X0 = inistate(SYS, Y, U, TOL, PRINTW)

                [x0, V, rcnd] = inistate(SYS, Y, U, TOL, PRINTW)

## Arguments

SYS

  given system, syslin(dt,A,B,C,D)

Y

  the output of the system

U

  the input of the system

TOL

  TOL is 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

  PRINTW is a switch for printing the warning messages.

  =

    1: print warning messages;

  =

    0: do not print warning messages.

  Default: PRINTW = 0.

X0

  the estimated initial state vector

V

  orthogonal matrix which reduces the system state matrix A to a real Schur form

rcnd

  estimate of the reciprocal condition number of the coefficient matrix of the least squares problem solved.

## Description

inistate Estimates the initial state of a discrete-time system, given the (estimated) system matrices, and a set of input/output data.

X0 = inistate(SYS,Y,U,TOL,PRINTW) estimates the initial state X0 of the discrete-time system SYS = (A,B,C,D), using the 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.

Instead of the first input parameter SYS (an syslin object), equivalent information may be specified using matrix parameters, for instance, X0 = inistate(A,B,C,Y,U); or X0 = inistate(A,C,Y);

[x0,V,rcnd] = inistate(SYS,Y,U,TOL,PRINTW) returns, besides x0, the orthogonal matrix V which reduces the system state matrix A to a real Schur form, as well as an estimate of the reciprocal condition number of the coefficient matrix of the least squares problem solved.

## 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;
    [R,N1,SVAL] = findR(S,Y',U');
    N=3;
    SYS1 = findABCD(S,N,1,R) ;
    SYS1.dt=0.1;

    inistate(SYS1,Y',U')

## See also

- findBD
- findx0BD