Control of Large-Scale Delayed Networks: DDEs, DDFs and PIEs

Delay-Differential Equations (DDEs) are the most common representation for systems with delay. However, the DDE representation has limitations. In network models with delay, the delayed channels are typically low-dimensional and accounting for this heterogeneity is challenging in the DDE framework. In addition, DDEs cannot be used to model difference equations. In this paper, we examine alternative representations for networked 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 consider the coupled ODE-PDE framework and extend this to the recently developed Partial-Integral Equation (PIE) representation. The PIE framework has the advantage that it allows the H_∞-optimal estimation and control problems to be solved efficiently using the recently developed software package PIETOOLS. In each case, we consider a very general class of networks, specifically accounting for four sources of delay - state delay, input delay, output delay, and process delay. Finally, we use a scalable network model of temperature control to show that the use of the DDF/PIE formulation allows for optimal control of a network with 40 users, 80 states, 40 delays, 40 inputs, and 40 disturbances.