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See the recommended documentation of this function

# findBD

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

### Syntax

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

### Arguments

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

optional, input matrix of the given system

C

output matrix of the given system

D

optional, 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

optional, 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

optional, 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.

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);
clf();plot2d((1:nsmp)',[Y',Y1'])```

• inistate — Estimates the initial state of a discrete-time system
• findx0BD — Estimates state and B and D matrices of a discrete-time linear system
• findABCD — discrete-time system subspace identification
• findAC — discrete-time system subspace identification
• findBD — initial state and system matrices B and D of a discrete-time system