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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')

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