sident discrete-time state-space realization and Kalman gain

## Syntax

    [[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
- sorder