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# 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.

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

### See also

- 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

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