Linear Matrix Inequalities in Control
LMIs recast many control problems as convex feasibility or optimization over positive-definite matrices, solvable reliably by interior-point methods.
What an LMI is
A linear matrix inequality has the form F0 plus the sum of x_i F_i being positive definite, where the F_i are fixed symmetric matrices and x is the decision vector. The feasible set is convex, so any local solution is global and interior-point solvers find it efficiently. Many control conditions that once required Riccati equations become LMIs.
Lyapunov and beyond
Stability of x-dot equals A x holds if a symmetric P is positive definite and A-transpose P plus P A is negative definite. Both are LMIs in P, so stability analysis is a feasibility problem. The bounded-real lemma turns an H-infinity norm bound into an LMI, and the analogous conditions handle H2 cost, passivity, and pole placement in convex regions.
import numpy as np
# Feasibility idea: find P>0 with A^T P + P A < 0 (checked here for a candidate)
A=np.array([[-1.,2.],[0.,-3.]]); P=np.eye(2)
Q=A.T@P+P@A
print('P pd:',np.all(np.linalg.eigvals(P)>0),'AtP+PA nd:',np.all(np.linalg.eigvals(Q)<0))Why controllers benefit
State-feedback synthesis becomes an LMI after a change of variables: substitute the gain times the Lyapunov matrix as a new variable Y, and the bilinear term linearizes. This lets designers add multiple specifications as separate LMIs and solve them jointly, which is the basis of mixed H2/H-infinity and gain-scheduled design.
- Convex feasible set: local optimum is global
- Encodes stability, H2, H-infinity, passivity, pole regions
- Change of variables linearizes state-feedback synthesis
- Multiple specs combine additively in one solver call
The limitation is bilinearity: output-feedback and fixed-order problems produce bilinear matrix inequalities, which are non-convex. Iterative schemes or the elimination lemma recover convexity in special cases.
For multivariable design-stage systems, LMIs let engineers pose robustness and performance requirements as one optimization and read a certificate of success directly from the returned positive-definite matrix. They are the computational backbone of modern robust control.