Sign convention for enforcing positive definite matrix P ( 𝑃 ⪰ 𝜀 𝐼 ) using lmiterm in MATLAB LMI Toolbox (LMI)

%% Common Lyapunov LMI using MATLAB LMI Toolbox

 Solver for LMI feasibility problems L(x) < R(x)
    This solver minimizes  t  subject to  L(x) < R(x) + t*I
    The best value of t should be negative for feasibility

 Iteration   :    Best value of t so far 
 
     1                       -0.226331

 Result:  best value of t:    -0.226331
          f-radius saturation:  0.000% of R =  1.00e+09
 

Feasible P:

    0.4133    0.0704
    0.0704    0.2539

Eigenvalues of P:

    0.2273
    0.4400

% Example system matrices (can be replaced by A_rule{k})
A_rule{1} = [-1  2; 
              0 -3];
A_rule{2} = [-2  1;
             -1 -4];

N = numel(A_rule);
n = size(A_rule{1},1);

setlmis([])

% Decision variable: symmetric P
p = lmivar(1,[n 1]);

% Stability LMIs: A_k' P + P A_k <= 0
for k = 1:N
    L = newlmi;
    lmiterm([L 1 1 p],1,A_rule{k},'s');
end

% Positivity constraint: P >= eps*I ?
eps = 1e-6;
S = newlmi;

% ---- QUESTION IS ABOUT THESE TWO LINES ----
lmiterm([-S 1 1 p],1,1);         % term involving P
lmiterm([-S 1 1 0],eps*eye(n));  % constant term
% ------------------------------------------

lmis = getlmis;
[tmin,xfeas] = feasp(lmis);

if tmin < 1e-6
    P = dec2mat(lmis,xfeas,p);
    disp('Feasible P:');
    disp(P)
    disp('Eigenvalues of P:');
    disp(eig(P))
else
    disp('No feasible solution.');
end

In the code above, I want to enforce the Lyapunov positivity constraint P⪰εI.

Since all LMIs in the MATLAB LMI Toolbox are interpreted as

LHS≤0,

this corresponds to

−P+εI≤0.

Do the two lines

lmiterm([-S 1 1 p],1,1);
lmiterm([-S 1 1 0],eps*eye(n));

correctly enforce P⪰εI according to the official LMI Toolbox convention?

If not, what is the recommended and unambiguous way to impose this constraint (e.g., using explicit negative coefficients on P)?

Answer 1

[Sam Chak]

Hi @Manish Kumar

I believe I made an error in my previous comment from another thread. LMIs appear deceptively simple but can be logically unforgiving, particularly regarding signs and certain command lines, without a clear understanding of the underlying mathematics.

Although your code does not affect the result, the following commands will effectively enforce the Lyapunov positivity constraint::

lmiterm([-S 1 1 p], 1, 1);          % -S places   "P" on the RHS of the inequality:   0 < P
% lmiterm([-S 1 1 0], eps*eye(n));  % -S places "ε*I" → 0 < P + ε*I, which implies -ε*I < P (weak condition)
lmiterm([ S 1 1 0], eps*eye(n));    % +S places "ε*I" on the LHS of the inequality: ε*I < P

[Manish Kumar]

Thank you! This really helped clarify things.