lmisolver
=========

Solve linear matrix inequations.



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


::

    [XLISTF[,OPT]] = lmisolver(XLIST0,evalfunc [,options])




Arguments
~~~~~~~~~

:XLIST0 a list of containing initial guess (e.g.
  `XLIST0=list(X1,X2,..,Xn)`)
: :evalfunc a Scilab function ("external" function with specific
  syntax) The syntax the function `evalfunc` must be as follows:
  `[LME,LMI,OBJ]=evalfunct(X)` where `X` is a list of matrices, `LME,
  LMI` are lists and `OBJ` a real scalar.
: :XLISTF a list of matrices (e.g. `XLIST0=list(X1,X2,..,Xn)`)
: :options optional parameter. If given, `options` is a real row
  vector with 5 components `[Mbound,abstol,nu,maxiters,reltol]`
:



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

`lmisolver` solves the following problem:

minimize `f(X1,X2,...,Xn)` a linear function of Xi's

under the linear constraints: `Gi(X1,X2,...,Xn)=0` for i=1,...,p and
LMI (linear matrix inequalities) constraints:

`Hj(X1,X2,...,Xn) > 0` for j=1,...,q

The functions f, G, H are coded in the Scilab function `evalfunc` and
the set of matrices Xi's in the list X (i.e. `X=list(X1,...,Xn)`).

The function `evalfun` must return in the list `LME` the matrices
`G1(X),...,Gp(X)` (i.e. `LME(i)=Gi(X1,...,Xn),` i=1,...,p). `evalfun`
must return in the list `LMI` the matrices `H1(X0),...,Hq(X)` (i.e.
`LMI(j)=Hj(X1,...,Xn)`, j=1,...,q). `evalfun` must return in `OBJ` the
value of `f(X)` (i.e. `OBJ=f(X1,...,Xn)`).

`lmisolver` returns in `XLISTF`, a list of real matrices, i. e.
`XLIST=list(X1,X2,..,Xn)` where the Xi's solve the LMI problem:

Defining `Y,Z` and `cost` by:

`[Y,Z,cost]=evalfunc(XLIST)`, `Y` is a list of zero matrices,
`Y=list(Y1,...,Yp)`, `Y1=0, Y2=0, ..., Yp=0`.

`Z` is a list of square symmetric matrices, `Z=list(Z1,...,Zq)`, which
are semi positive definite `Z1>0, Z2>0, ..., Zq>0` (i.e. `spec(Z(j))`
> 0),

`cost` is minimized.

`lmisolver` can also solve LMI problems in which the `Xi's` are not
matrices but lists of matrices. More details are given in the
documentation of LMITOOL.



Examples
~~~~~~~~


::

    //Find diagonal matrix X (i.e. X=diag(diag(X), p=1) such that
    //A1'*X+X*A1+Q1 < 0, A2'*X+X*A2+Q2 < 0 (q=2) and trace(X) is maximized 
    n  = 2;
    A1 = `rand`_(n,n);
    A2 = `rand`_(n,n);
    Xs = `diag`_(1:n);
    Q1 = -(A1'*Xs+Xs*A1+0.1*`eye`_());
    Q2 = -(A2'*Xs+Xs*A2+0.2*`eye`_());
    
    function [LME, LMI, OBJ]=evalf(Xlist)
      X   = Xlist(1)
      LME = X-`diag`_(`diag`_(X))
      LMI = `list`_(-(A1'*X+X*A1+Q1),-(A2'*X+X*A2+Q2))
      OBJ = -`sum`_(`diag`_(X))
    endfunction
    
    X=lmisolver(`list`_(`zeros`_(A1)),evalf);
    
    X=X(1)
    [Y,Z,c]=evalf(X)




See Also
~~~~~~~~


+ `lmitool`_ Graphical tool for solving linear matrix inequations.


.. _lmitool: lmitool.html