A Convex Reformulation of the Controller Synthesis Problem for Infinite-Dimensional Systems using Linear Operator Inequalities (LOIs) with Application to MIMO Multi-Delay Systems

In this paper, we propose a formal duality and operator-based framework for controller synthesis of infinite-dimensional systems using convex optimization. Specifically, we propose a class of what we call Linear Operator Inequalities (LOIs) and give conditions under which LOIs are solvable and under which they can be used for controller synthesis. Within this LOI framework, the first technical contribution of the paper is a new dual stability condition under which we can reformulate the controller synthesis problem as an LOI. The second technical contribution is LOIs for both primal and dual stability of systems with multiple delays. The third technical contribution is to show that both these LOIs are solvable. Finally, we use an LMI-based framework to solve these LOIs and demonstrate numerically that the primal and dual stability conditions are functionally equivalent for MIMO systems with multiple delays.