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# 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.

### 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'])

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