Linear Matrix Inequalities in Control

Truss topology design
Problem
Find s ∈ R N which minimizes elastic energy f ⊤ d subject to the constraints
A(s) ≻ 0, A(s)d = f, a ≤ s ≤ b, ℓ ⊤ s ≤ v
Data: Total volume v > 0, node forces f, bounds a, b, lengths ℓ and
symmetric matrices A1, . . . , AN that define the linear stiffness matrix
A(s) = s1A1 + . . . + sNAN.
Decision variables: Cross sections s and displacements d (both
vectors).
Cost function: stored elastic energy d ↦→ f ⊤ d.
Constraints:
Semi-definite constraint: A(s) ≻ 0
Non-linear equality constraint: A(s)d = f
Linear inequality constraints: a ≤ s ≤ b and ℓ ⊤ s ≤ v.
Siep Weiland and Carsten Scherer (DISC) Linear Matrix Inequalities in Control Class 1 54 / 59