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)

Parameters

• 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);
xbasc();plot2d((1:nsmp)',[Y',Y1'])
 

See Also

findBD  inistate