Linear Matrix Inequalities in Control

From truss topology design to LMI’s
First eliminate affine equality constraint A(s)d = f:
minimize f ⊤ (A(s)) −1 f
subject to A(s) ≻ 0, ℓ ⊤ s ≤ v, a ≤ s ≤ b
Push objective to constraints with auxiliary variable γ:
minimize γ
subject to γ > f ⊤ (A(s)) −1 f, A(s) ≻ 0, ℓ ⊤ s ≤ v, a ≤ s ≤ b
Apply Schur lemma to linearize
minimize γ
γ f ⊤
subject to
≻ 0, ℓ
f A(s)
⊤s ≤ v, a ≤ s ≤ b
Note that the latter is an LMI optimization problem as all constraints on s
are formulated as LMI’s!!
Siep Weiland and Carsten Scherer (DISC) Linear Matrix Inequalities in Control Class 1 55 / 59