observer

Scilab Function
Last update : 12/01/2004

observer - observer design

### Calling Sequence

  `Obs=observer(Sys,J)`
  `[Obs,U,m]=observer(Sys [,flag,alfa])`

### Parameters

- `Sys` : `syslin` list (linear system)
- `J` : nx x ny constant matrix (output injection matrix)
- `flag` : character strings (`'pp'` or `'st'` (default))
- `alfa` : location of closed-loop poles (optional parameter, default=-1)
- `Obs` : linear system (`syslin` list), the observer
- `U` : orthogonal matrix (see `dt_ility` )
- `m` : integer (dimension of unstable unobservable (`st` ) or unobservable (`pp` ) subspace)

### Description

`Obs=observer(Sys,J)` returns the observer `Obs=syslin(td,A+J*C,[B+J*D,-J],eye(A))` obtained from `Sys` by a `J` output injection. (td is the time domain of `Sys` ). More generally, `observer` returns in `Obs` an observer for the observable part of linear system `Sys` : `dotx=A x + Bu, y=Cx + Du` represented by a `syslin` list. `Sys` has `nx` state variables, `nu` inputs and `ny` outputs. `Obs` is a linear system with matrices `[Ao,Bo,Identity]` , where `Ao` is `no x no` , `Bo` is `no x (nu+ny)` , `Co` is `no x no` and `no=nx-m` .

Input to `Obs` is `[u,y]` and output of `Obs` is:

xhat=estimate of x modulo unobservable subsp. (case `flag='pp'` ) or

xhat=estimate of x modulo unstable unobservable subsp. (case `flag='st'` )

case `flag='st'` : z=H*x can be estimated with stable observer iff `H*U(:,1:m)=0` and assignable poles of the observer are set to `alfa(1),alfa(2),...`

case `flag='pp'` : z=H*x can be estimated with given error spectrum iff `H*U(:,1:m)=0` all poles of the observer are assigned and set to `alfa(1),alfa(2),...`

If H satifies the constraint: `H*U(:,1:m)=0` (ker(H) contains unobs-subsp. of Sys) one has H*U=[0,H2] and the observer for z=H*x is H2*Obs with H2=H*U(:,m+1:nx) i.e. Co, the C-matrix of the observer for H*x, is Co=H2.

In the particular case where the pair `(A,C)` of `Sys` is observable, one has `m=0` and the linear system `U*Obs` (resp. `H*U*Obs` ) is an observer for `x` (resp. `Hx` ). The error spectrum is `alpha(1),alpha(2),...,alpha(nx)` .

### Examples

    nx=5;nu=1;ny=1;un=3;us=2;Sys=ssrand(ny,nu,nx,list('dt',us,us,un));
    //nx=5 states, nu=1 input, ny=1 output,
    //un=3 unobservable states, us=2 of them unstable.
    [Obs,U,m]=observer(Sys);  //Stable observer (default)
    W=U';H=W(m+1:nx,:);[A,B,C,D]=abcd(Sys);  //H*U=[0,eye(no,no)];
    Sys2=ss2tf(syslin('c',A,B,H))  //Transfer u-->z
    Idu=eye(nu,nu);Sys3=ss2tf(H*U(:,m+1:$)*Obs*[Idu;Sys])
    //Transfer u-->[u;y=Sys*u]-->Obs-->xhat-->HUxhat=zhat  i.e. u-->output of Obs
    //this transfer must equal Sys2, the u-->z transfer  (H2=eye).

### See Also

`dt_ility` , `unobs` , `stabil` ,

### Author

F.D.