Robust Topology Optimization and Optimal Feedback Controller Design for Linear Time-Invariant Systems via Nonlinear Semidefinite Programming

The fundamental goal of this thesis is to show how structural vibrations in
mechanical systems with uncertain time-dependent inputs can be reduced
by using topology optimization and simultaneously designing optimal feedback
controllers via nonlinear semidefinite programming.

Structural vibrations are an important topic to consider in nearly all mechanical
systems of all kinds. Moreover, in real life applications of mechanical structures
there typically appears uncertainty in the form of not exactly known input
data, measurement errors or production tolerances. We use the concept
of robust optimization to control uncertainty in the time-dependent inputs of
the considered mechanical systems. This leads to nonlinear bilevel optimization
problems which are quite difficult to handle. Via the Bounded Real Lemma
we then obtain a semidefinite reformulation of these bilevel problems.

We use a sequential semidefinite programming approach to numerically solve
the resulting semidefinite programming problems, which are nonlinear and
non-convex. We give a full convergence analysis, discuss extension possibilities,
and evaluate the performance of the algorithm. As a practical example,
we apply the developed theory to complex three-dimensional truss structures
under uncertain dynamic loads.