armax
=====

armax identification



Calling Sequence
~~~~~~~~~~~~~~~~


::

    [arc,la,lb,sig,resid]=armax(r,s,y,u,[b0f,prf])




Arguments
~~~~~~~~~

:y output process y(ny,n); ( ny: dimension of y , n : sample size)
: :u input process u(nu,n); ( nu: dimension of u , n : sample size)
: :r and s auto-regression orders r >=0 et s >=-1
: :b0f optional parameter. Its default value is 0 and it means that
  the coefficient b0 must be identified. if bof=1 the b0 is supposed to
  be zero and is not identified
: :prf optional parameter for display control. If prf =1, the default
  value, a display of the identified Arma is given.
: :arc a Scilab arma object (see armac)
: :la is the list(a,a+eta,a-eta) ( la = a in dimension 1) ; where eta
  is the estimated standard deviation. , a=[Id,a1,a2,...,ar] where each
  ai is a matrix of size (ny,ny)
: :lb is the list(b,b+etb,b-etb) (lb =b in dimension 1) ; where etb is
  the estimated standard deviation. b=[b0,.....,b_s] where each bi is a
  matrix of size (nu,nu)
: :sig is the estimated standard deviation of the noise and resid=[
  sig*e(t0),....] (
:



Description
~~~~~~~~~~~

armax is used to identify the coefficients of a n-dimensional ARX
process


::

    A(z^-1)y= B(z^-1)u + sig*e(t)


where e(t) is a n-dimensional white noise with variance I. sig an nxn
matrix and A(z) and B(z):


::

    A(z) = 1+a1*z+...+a_r*z^r; ( r=0 => A(z)=1)
    B(z) = b0+b1*z+...+b_s z^s ( s=-1 => B(z)=0)


for the method see Eykhoff in trends and progress in system
identification, page 96. with
`z(t)=[y(t-1),..,y(t-r),u(t),...,u(t-s)]` and `coef=
[-a1,..,-ar,b0,...,b_s]` we can write `y(t)= coef* z(t) + sig*e(t)`
and the algorithm minimises `sum_{t=1}^N ( [y(t)- coef'z(t)]^2)` where
t0=max(max(r,s)+1,1))).



Examples
~~~~~~~~


::

    //-Ex1- Arma model : y(t) = 0.2*u(t-1)+0.01*e(t-1)
    ny=1,nu=1,sig=0.01;
    Arma=`armac`_(1,[0,0.2],[0,1],ny,nu,sig)  //defining the above arma model
    u=`rand`_(1,1000,'normal');     //a random input sequence u
    y=`arsimul`_(Arma,u); //simulation of a y output sequence associated with u.
    Armaest=armax(0,1,y,u);   //Identified model given u and y.
    Acoeff=Armaest('a');    //Coefficients of the polynomial A(x)
    Bcoeff=Armaest('b')     //Coefficients of the polynomial B(x)
    Dcoeff=Armaest('d');    //Coefficients of the polynomial D(x)
    [Ax,Bx,Dx]=`arma2p`_(Armaest)   //Results in polynomial form. 
    
    //-Ex2- Arma1: y_t -0.8*y_{t-1} + 0.2*y_{t-2} =  sig*e(t)
    ny=1,nu=1;sig=0.001;
    // First step: simulation the Arma1 model, for that we define
    // Arma2: y_t -0.8*y_{t-1} + 0.2*y_{t-2} = sig*u(t)
    // with normal deviates for u(t).  
    Arma2=`armac`_([1,-0.8,0.2],sig,0,ny,nu,0);
    //Definition of the Arma2 arma model (a model with B=sig and without noise!)
    u=`rand`_(1,10000,'normal');  // An input sequence for Arma2
    y=`arsimul`_(Arma2,u); // y = output of Arma2 with input u 
    //                     can be seen as output of Arma1.
    // Second step: identification. We look for an Arma model
    // y(t) + a1*y(t-1) + a2 *y(t-2) =  sig*e(t)
    Arma1est=armax(2,-1,y,[]);
    [A,B,D]=`arma2p`_(Arma1est)




See Also
~~~~~~~~


+ `imrep2ss`_ state-space realization of an impulse response
+ `time_id`_ SISO least square identification
+ `arl2`_ SISO model realization by L2 transfer approximation
+ `armax`_ armax identification
+ `frep2tf`_ transfer function realization from frequency response


.. _imrep2ss: imrep2ss.html
.. _arl2: arl2.html
.. _frep2tf: frep2tf.html
.. _time_id: time_id.html
.. _armax: armax.html