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.