Scilab Function
Last update : April 1999

findBD - initial state and system matrices B and D of a discrete-time system

### Calling Sequence

  `[(x0) (,B (,D)) (,V) (,rcnd)] = findBD(jobx0,comuse (,job),A (,B),C (,D),Y`
  `(,U,tol,printw,ldwork))`

### Parameters

- `jobx0`: integer option to specify whether or not the initial state should be computed:
  - `=` 1 : compute the initial state x0;
  - `=` 2 : do not compute the initial state (possibly, because x0 is known to be zero).
- `comuse`: integer option to specify whether the system matrices B and D should be computed or used:
  - `=` 1 : compute the matrices B and D, as specified by job;
  - `=` 2 : use the matrices B and D, as specified by job;
  - `=` 3 : do not compute/use the matrices B and D.
- `job`: integer option to determine which of the system matrices B and D should be computed or used:
  - `=` 1 : compute/use the matrix B only (D is known to be zero);
  - `=` 2 : compute/use the matrices B and D.

  job must not be specified if jobx0 = 2 and comuse = 2, or if comuse = 3.

- `A`: state matrix of the given system
- `B`: optionnal, input matrix of the given system
- `C`: output matrix of the given system
- `D`: optionnal, direct feedthrough of the given system
- `Y`: the t-by-l output-data sequence matrix. Column j of Y contains the t values of the j-th output component for consecutive time increments.
- `U`: the t-by-m input-data sequence matrix (input when jobx0 = 1 and comuse = 2, or comuse = 1). Column j of U contains the t values of the j-th input component for consecutive time increments.
- `tol`: optionnal, 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; an m-by-n matrix whose estimated condition number is less than 1/tol is considered to be of full rank. Default: m*n*epsilon_machine where epsilon_machine is the relative machine precision.
- `printw`:optionnal, switch for printing the warning messages.
  - `=` 1: print warning messages;
  - `=` 0: do not print warning messages.

  Default: printw = 0.

- `ldwork`: (optional) the workspace size. Default : computed by the formula LDWORK = MAX( minimum workspace size needed, 2*CSIZE/3, CSIZE - ( m + l )*t - 2*n*( n + m + l ) - l*m ) where CSIZE is the cache size in double precision words.
- `x0`: initial state vector
- `Br`: system input matrix
- `Dr`: system direct feedthrough matrix
- `V`: the n-by-n orthogonal matrix which reduces A to a real Schur form (output when jobx0 = 1 or comuse = 1).
- `rcnd`: (optional) the reciprocal condition numbers of the matrices involved in rank decisions.

### Description

findBD function for estimating the initial state and the system matrices B and D of a discrete-time system, using SLICOT routine IB01CD.

         [x0,Br,V,rcnd] = findBD(1,1,1,A,C,Y,U)
       [x0,Br,Dr,V,rcnd] = findBD(1,1,2,A,C,Y,U)
            [Br,V,rcnd] = findBD(2,1,1,A,C,Y,U)
          [B,Dr,V,rcnd] = findBD(2,1,2,A,C,Y,U)
           [x0,V,rcnd] = findBD(1,2,1,A,B,C,Y,U)
           [x0,V,rcnd] = findBD(1,2,2,A,B,C,D,Y,U)
             [x0,rcnd] = findBD(2,2)      // (Set x0 = 0, rcnd = 1)
           [x0,V,rcnd] = findBD(1,3,A,C,Y)

Note: the example lines above may contain at the end the parameters tol, printw, ldwork.

FINDBD estimates the initial state and/or the system matrices Br and Dr of a discrete-time system, given the system matrices A, C, and possibly B, D, and the input and output trajectories of the system.

The model structure is :

             x(k+1) = Ax(k) + Bu(k),   k >= 1,
             y(k)   = Cx(k) + Du(k),

where x(k) is the n-dimensional state vector (at time k),

u(k) is the m-dimensional input vector,

y(k) is the l-dimensional output vector,

and A, B, C, and D are real matrices of appropriate dimensions.

### Comments

  - 1.The n-by-m system input matrix B is an input parameter when jobx0 = 1 and comuse = 2, and it is an output parameter when comuse = 1.
  - 2.The l-by-m system matrix D is an input parameter when jobx0 = 1, comuse = 2 and job = 2, and it is an output parameter when comuse = 1 and job = 2.
  - 3.The n-vector of estimated initial state x(0) is an output parameter when jobx0 = 1, but also when jobx0 = 2 and comuse <= 2, in which case it is set to 0.
  - 4.If ldwork is specified, but it is less than the minimum workspace size needed, that minimum value is used instead.

### 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] = findBD(1,1,2,A,C,Y',U')
    SYS1=syslin(1,A,B,C,D,X0);

    Y1=flts(U,SYS1);
    xbasc();plot2d((1:nsmp)',[Y',Y1'])

### See Also

`inistate`, `findx0BD`, `findABCD`, `findAC`, `findBD`,

### Author

V. Sima, Katholieke Univ. Leuven, Belgium, May 2000. (Revisions: V. Sima, July 2000)