Why LMI does not returne a positive matrix?

Hello! I have the following LMI problem: AX-BL+XA'-L'B'>0 And X>0 Where: P = inv(X) And: A, P and X are 4x4 matrices, B is 4x1 and L is 1x4 vectors. I wrote the following script. But P is not positive!? Is it a problem in my script or something else? Thanks in advance!

%%LMIs :
% LMI variables :
setlmis([]);
X       = lmivar(1, [4 1]);
L       = lmivar(2, [1 4]);

% First LMI :
lmiterm([-1 1 1 X],A,1,'s');
lmiterm([ 1 1 1 L],B,1,'s');

% Second LMI :
lmiterm([2 1 1 X],1,1);

% Get LMIs :
lmis    = getlmis;

% P matrix:
[Tmin,Xfeas] = feasp(lmis)
X       = dec2mat(lmis,Xfeas,X);
p       = inv(X)

[Walter Roberson]

It is not obvious to me that P should be non-negative. In order for inv(X) be entirely non-negative, X would have to have only eigenvalues whose real parts are all non-negative. At the moment I do not see why that should necessarily be true in that formula.

[Amine]

Here P must be positive in order to optimize a sliding mode controller surface, where: s = B’Px Would you please give me further explanations? Thanks!

[Michael Hubatka]

The condition X > 0 is defined as

lmiterm([-2 1 1 X], 1, 1);

or

lmiterm([2 1 1 -X], 1, 1);

because lmiterm always uses the '<' condition.

[Amine]

First of all thanks for your interest in my question. Both code lines give the same result a negative P matrix (P > 0) where :

P = inv(X)