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, DDE is limited. In network models delay, delayed channels low-dimensional and accounting this heterogeneity not possible in framework. addition, DDEs cannot be used to model difference equations. Furthermore, estimation control of format has proven challenging, despite decades study. paper, we examine alternative representations delay provide formulae conversion between representations. First, Differential-Difference (DDF) formulation which allows us represent nature information. Next, coupled ODE–PDE formulation, backstepping methods have recently become available. Finally, consider algebraic Partial Integral Equation (PIE) representation, optimal problems solved efficiently through use recent software packages such as PIETOOLS. each case, a very general class systems, specifically all four sources — state input output process We then apply these 3 archetypical models.