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

Abstract

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 largescale 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.

Fundamental limitations on the control of lossless systems

“…Since P is invertible, controllability of (A, B) implies that −P AP −1 , P B is also controllable. Therefore B T P, −P −1 A T P is observable, which implies that (C, A) is observable by (4). Observability of (C, A) then implies observability of (C z , A).…”

“…Theorem 1: Assume that A, B, C, and D are as in Problem 1, and that the pair (A, B) is controllable. If there exists a positive definite P such that P A + A T P = 0, P B = C T , and D + D T = 0, (4) then in the H 2 case of Problem 1, γ * H2 = 2 tr (CB). If in addition D = 0, then γ * H∞ = √ 2.…”

“…is optimal for Problem 1, and in the H ∞ case K (s) = − √ 2I is optimal for Problem 1 (for a discussion on how to synthesise these types control laws in a structure exploiting manner, see [4]).…”

“…For example, factorisations involving lossless transfer functions play a crucial role in H ∞ methods [1]. In addition, central control theoretic results, such as the Kalman-Yakubovich-Popov Lemma, simplify significantly in the lossless setting [2], and the state-space and circuit theoretic descriptions of lossless systems have a range of appealing structural properties [3], [4].…”