rtitr
=====

discrete time response (transfer matrix)



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


::

    [y]=rtitr(Num,Den,u [,up,yp])




Arguments
~~~~~~~~~

:Num,Den polynomial matrices (resp. dimensions : `n`x `m` and `n`x
  `n`)
: :u real matrix (dimension `m`x `(t+1)`
: :up,yp real matrices ( `up` dimension `m`x `(max(degree(Den)))`
  (default values= `0`) , `yp` dimension `n`x `(max(degree(Den)))`)
: :y real matrix
:



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

`y=rtitr(Num,Den,u [,up,yp])` returns the time response of the
discrete time linear system with transfer matrix `Den^-1 Num` for the
input `u`, i.e `y` and `u` are such that `Den y = Num u` at t=0,1,...

If `d1=max(degree(Den))`, and `d2=max(degree(Num))` the polynomial
matrices `Den(z)` and `Num(z)` may be written respectively as:


::

    D(z) = D_0  + D_1  z + ... + D_d1   z^d1
    N(z) = N_0  + N_1  z + ... + N_d2   z^d2


and `Den y = Num u` is interpreted as the recursion:


::

    D(0)y(t)+D(1)y(t+1)+...+ D(d1)y(t+d1)= N(0) u(t) +....+ N(d2) u(t+d2)


It is assumed that `D(d1)` is non singular.

The columns of u are the inputs of the system at t=0,1,...,T:


::

    u=[u(0) , u(1),...,u(T)]


The outputs at `t=0,1,...,T+d1-d2` are the columns of the matrix `y`:


::

    y = [y(0), y(1),  .... y(T+d1-d2)]


`up` and `yp` define the initial conditions for t < 0 i.e


::

    up = [u(-d1), ..., u(-1)  ]
    yp = [y(-d1), ...  y(-1)  ]


Depending on the relative values of `d1` and `d2`, some of the
leftmost components of `up`, `yp` are ignored. The default values of
`up` and `yp` are zero: `up = 0*ones(m,d1), yp=0*ones(n,d1)`



Examples
~~~~~~~~


::

    z=`poly`_(0,'z');
    Num=1+z;Den=1+z;u=[1,2,3,4,5];
    rtitr(Num,Den,u)-u
    //Other examples
    //siso
    //causal
    n1=1;d1=`poly`_([1 1],'z','coeff');       // y(j)=-y(j-1)+u(j-1)
    r1=[0 1 0 1 0 1 0 1 0 1 0];
    r=rtitr(n1,d1,`ones`_(1,10));`norm`_(r1-r,1)
    //hot restart
    r=rtitr(n1,d1,`ones`_(1,9),1,0);`norm`_(r1(2:11)-r)
    //non causal
    n2=`poly`_([1 1 1],'z','coeff');d2=d1;    // y(j)=-y(j-1)+u(j-1)+u(j)+u(j+1)
    r2=[2 1 2 1 2 1 2 1 2];
    r=rtitr(n2,d2,`ones`_(1,10));`norm`_(r-r2,1)
    //hot restart
    r=rtitr(n2,d2,`ones`_(1,9),1,2);`norm`_(r2(2:9)-r,1)
    //
    //MIMO example
    //causal
    d1=d1*`diag`_([1 0.5]);n1=[1 3 1;2 4 1];r1=[5;14]*r1;
    r=rtitr(n1,d1,`ones`_(3,10));`norm`_(r1-r,1)
    //
    r=rtitr(n1,d1,`ones`_(3,9),[1;1;1],[0;0]);
    `norm`_(r1(:,2:11)-r,1)
    //polynomial n1  (same ex.)
    n1(1,1)=`poly`_(1,'z','c');r=rtitr(n1,d1,`ones`_(3,10));`norm`_(r1-r,1)
    //
    r=rtitr(n1,d1,`ones`_(3,9),[1;1;1],[0;0]);
    `norm`_(r1(:,2:11)-r,1)
    //non causal
    d2=d1;n2=n2*n1;r2=[5;14]*r2;
    r=rtitr(n2,d2,`ones`_(3,10));`norm`_(r2-r)
    //
    r=rtitr(n2,d2,`ones`_(3,9),[1;1;1],[10;28]);
    `norm`_(r2(:,2:9)-r,1)
    //
    //  State-space or transfer
    a = [0.21 , 0.63 , 0.56 , 0.23 , 0.31
         0.76 , 0.85 , 0.66 , 0.23 , 0.93
         0 , 0.69 , 0.73 , 0.22 , 0.21
         0.33 , 0.88 , 0.2 , 0.88 , 0.31
         0.67 , 0.07 , 0.54 , 0.65 , 0.36];
    b = [0.29 , 0.5 , 0.92
         0.57 , 0.44 , 0.04
         0.48 , 0.27 , 0.48
         0.33 , 0.63 , 0.26
         0.59 , 0.41 , 0.41];
    c = [0.28 , 0.78 , 0.11 , 0.15 , 0.84
         0.13 , 0.21 , 0.69 , 0.7 , 0.41];
    d = [0.41 , 0.11 , 0.56
         0.88 , 0.2 , 0.59];
    s=`syslin`_('d',a,b,c,d);
    h=`ss2tf`_(s);num=h('num');den=h('den');den=den(1,1)*`eye`_(2,2);
    u=1;u(3,10)=0;r3=`flts`_(u,s);
    r=rtitr(num,den,u);`norm`_(r3-r,1)




See Also
~~~~~~~~


+ `ltitr`_ discrete time response (state space)
+ `exp`_ element-wise exponential
+ `flts`_ time response (discrete time, sampled system)


.. _exp: exp.html
.. _ltitr: ltitr.html
.. _flts: flts.html