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.