Feedback Networks

Abstract

This paper presents a survey of some recent results which are primarily concerned with the action of relatively simple controllers working in complex networks. Feedback system design is all about getting the right amount of gain into the right places. In the design of a controller for a given linear multivariable system (with maybe an uncertain frequency response, but with a known topology of inputs and outputs), this means getting the right amount of gain into the right frequency ranges and in the right directions. $\mathcal{H}_\infty$ theory, to which Keith Glover has of course contributed a great deal, tells us a lot about managing these tradeoffs (the $\mathcal{H}_\infty$ loop shaping design method in particular [1]). Here though, we are not concerned with complex multivariable systems, but rather simple scalar systems which are perfectly well controlled by simple controllers – or rather they would be if only they weren’t also connected to other systems who are in turn connected with more systems in a complex web of feedback interactions. A further complication in the applications we have in mind is that we rarely have information about the whole network, we only know the local interactions. A natural question again is “how do we get the right amount of gain into the right places”.