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


.. _sorder: sorder.html
.. _findBD: findBD.html