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Delay-Differential Equations (DDEs) are the most common representation forsystems with delay. However, the DDE representation is limited. In networkmodels with delay, the delayed channels are low-dimensional and accounting forthis heterogeneity is not possible in the DDE framework. In addition, DDEscannot be used to model difference equations. Furthermore, estimation andcontrol of systems in DDE format has proven challenging, despite decades ofstudy. In this paper, we examine alternative representations for systems withdelay and provide formulae for conversion between representations. First, weexamine the Differential-Difference (DDF) formulation which allows us torepresent the low-dimensional nature of delayed information. Next, we examinethe coupled ODE-PDE formulation, for which backstepping methods have recentlybecome available. Finally, we consider the algebraic Partial-Integral Equation(PIE) representation, which allows the optimal estimation and control problemsto be solved efficiently through the use of recent software packages such asPIETOOLS. In each case, we consider a very general class of delay systems,specifically accounting for all four possible sources of delay - state delay,input delay, output delay, and process delay. We then apply theserepresentations to 3 archetypical network models.

Representation of Networks and Systems with Delay: DDEs, DDFs, ODE-PDEs and PIEs