findBD — initial state and system matrices B and D of a discrete-time system
[(x0) (,B (,D)) (,V) (,rcnd)] = findBD(jobx0,comuse (,job),A (,B),C (,D),Y (,U,tol,printw,ldwork))
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).
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.
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.
state matrix of the given system
optionnal, input matrix of the given system
output matrix of the given system
optionnal, direct feedthrough of the given system
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.
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.
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.
optionnal, switch for printing the warning messages.
1: print warning messages;
0: do not print warning messages.
Default: printw = 0.
(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.
initial state vector
system input matrix
system direct feedthrough matrix
the n-by-n orthogonal matrix which reduces A to a real Schur form (output when jobx0 = 1 or comuse = 1).
(optional) the reciprocal condition numbers of the matrices involved in rank decisions.
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.
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.
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.
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.
If ldwork is specified, but it is less than the minimum workspace size needed, that minimum value is used instead.
//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'])