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# inistate

Estimates the initial state of a discrete-time system

### Syntax

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

[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')```