Passive and reciprocal networks: From simple models to simple optimal controllers

Networks constructed out of resistors, inductors, capacitors and transformers form a compelling subclass of simple models. Models constructed out of these basic elements are frequently used to explain phenomena in large-scale applications, from inter-area oscillations in power systems, to the transient behaviour of optimisation algorithms. Furthermore they capture the dynamics of the most commonly applied controllers, including the PID controller. In this paper we show that the inherent structure in these networks can be used to simplify, or even solve analytically, a range of simple optimal control problems. We illustrate these results by designing and synthesising simple, scalable, and globally optimal control laws for solving constrained least squares problems, regulating electrical power systems with stochastic renewable sources, studying the robustness properties of consensus algorithms, and analysing heating networks.