sident

discrete-time state-space realization and Kalman gain

Calling Sequence

[[A,C][,B[,D]][,K,Q,Ry,S][,rcnd]] = sident(meth,job,s,n,l,R[,tol,t,Ai,Ci,printw])

Arguments

meth integer option to determine the method to use:  := 1 : MOESP method with past inputs and outputs; : := 2 : N4SID method; : := 3 : combined method: A and C via MOESP, B and D via N4SID. :

: :job integer option to determine the calculation to be performed:

:= 1 : compute all system matrices, A, B, C, D; : := 2 : compute the matrices A and C only; : := 3 : compute the matrix B only; : := 4 : compute the matrices B and D only. :

: :s the number of block rows in the processed input and output block

Hankel matrices. s > 0.

: :n integer, the order of the system : :l integer, the number of the system outputs : :R the 2*(m+l)*s-by-2*(m+l)*s part of R contains the processed upper

triangular factor R from the QR factorization of the concatenated block-Hankel matrices, and further details needed for computing system matrices.

: :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.

: :t (optional) the total number of samples used for calculating the

covariance matrices. Either t = 0, or t >= 2*(m+l)*s. This parameter is not needed if the covariance matrices and/or the Kalman predictor gain matrix are not desired. If t = 0, then K, Q, Ry, and S are not computed. Default: t = 0.

: :Ai real matrix : :Ci real matrix : :printw (optional) switch for printing the warning messages.

= 1:print warning messages;

: := 0: do not print warning messages. :

Default: printw = 0. : :A real matrix : :C real matrix : :B real matrix : :D real matrix : :K real matrix, kalman gain : :Q (optional) the n-by-n positive semidefinite state covariance

matrix used as state weighting matrix when computing the Kalman gain.

: :RY (optional) the l-by-l positive (semi)definite output covariance

matrix used as output weighting matrix when computing the Kalman gain.

: :S (optional) the n-by-l state-output cross-covariance matrix used

as cross-weighting matrix when computing the Kalman gain.

: :rcnd (optional) vector of length lr, containing estimates of the

reciprocal condition numbers of the matrices involved in rank decisions, least squares, or Riccati equation solutions, where lr = 4, if Kalman gain matrix K is not required, and lr = 12, if Kalman gain matrix K is required.

:

Description

SIDENT function for computing a discrete-time state-space realization (A,B,C,D) and Kalman gain K using SLICOT routine IB01BD.

[A,C,B,D] = sident(meth,1,s,n,l,R)
[A,C,B,D,K,Q,Ry,S,rcnd] = sident(meth,1,s,n,l,R,tol,t)
    [A,C] = sident(meth,2,s,n,l,R)
        B = sident(meth,3,s,n,l,R,tol,0,Ai,Ci)
[B,K,Q,Ry,S,rcnd] = sident(meth,3,s,n,l,R,tol,t,Ai,Ci)
    [B,D] = sident(meth,4,s,n,l,R,tol,0,Ai,Ci)
[B,D,K,Q,Ry,S,rcnd] = sident(meth,4,s,n,l,R,tol,t,Ai,Ci)

SIDENT computes a state-space realization (A,B,C,D) and the Kalman predictor gain K of a discrete-time system, given the system order and the relevant part of the R factor of the concatenated block-Hankel matrices, using subspace identification techniques (MOESP, N4SID, or their combination).

The model structure is :

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

e(k) is the l-dimensional disturbance vector,

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

Comments

1. The n-by-n system state matrix A, and the p-by-n system output matrix C are computed for job <= 2.

2. The n-by-m system input matrix B is computed for job <> 2.
3. The l-by-m system matrix D is computed for job = 1 or 4.

4. The n-by-l Kalman predictor gain matrix K and the covariance matrices Q, Ry, and S are computed for t > 0.

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'));

S = 15;
N = 3;
METH=1;
[R,N1] = `findR`_(S,Y',U',METH);
[A,C,B,D,K] = sident(METH,1,S,N,1,R);
SYS1=`syslin`_(1,A,B,C,D);
SYS1.X0 = `inistate`_(SYS1,Y',U');

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

METH = 2;
[R,N1,SVAL] = `findR`_(S,Y',U',METH);
tol = 0;
t = `size`_(U',1)-2*S+1;

[A,C,B,D,K] = sident(METH,1,S,N,1,R,tol,t)
SYS1=`syslin`_(1,A,B,C,D)
SYS1.X0 = `inistate`_(SYS1,Y',U');

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

See Also

• findBD initial state and system matrices B and D of a discrete- time system
• sorder computing the order of a discrete-time system