Representation of networks and systems with delay: DDEs, DDFs, ODE–PDEs and PIEs

Delay-Differential Equations (DDEs) are the most common representation for systems with delay. However, the DDE representation is limited. In network models with delay, the delayed channels are low-dimensional and accounting for this heterogeneity is not possible in the DDE framework. In addition, DDEs cannot be used to model difference equations. Furthermore, estimation and control of systems in DDE format has proven challenging, despite decades of study. In this paper, we examine alternative representations for systems with delay and provide formulae for conversion between representations. First, we examine the Differential-Difference (DDF) formulation which allows us to represent the low-dimensional nature of delayed information. Next, we examine the coupled ODE–PDE formulation, for which backstepping methods have recently become available. Finally, we consider the algebraic Partial Integral Equation (PIE) representation, which allows the optimal estimation and control problems to be solved efficiently through the use of recent software packages such as PIETOOLS. 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 these representations to 3 archetypical network models.