Fluid networks are characterized by complex interconnected flows, involving high order nonlinear dynamics and transport phenomena. Classical lumped models typically capture the interconnections and nonlinear effects but ignore the transport phenomena, which may strongly affect the transient response. To control such flows with regulators of reduced complexity, we improve a classical lumped model (obtained by combining Kirchhoff's laws and graph theory) by introducing the effect of advection as a time delay. The model is based on the isothermal Euler equations to describe the dynamics of the fluid through the pipe. The resulting hyperbolic system of partial differential equations (PDEs) is diagonalized using Riemann invariants to find a solution in terms of delayed equations, obtained analytically using the method of the characteristics. Conservation principles are applied at the nodes of the network to describe the dynamics as a set of (possibly non linear) delay differential equations. Both linearized and nonlinear Euler equations are considered.
Rodriguez, D. F. N., Witrant, E., & Sename, O. (2015). Control-oriented modeling of fluid networks: A time-delay approach. In Recent Results on Nonlinear Delay Control Systems: In honor of Miroslav Krstic (Vol. 4, pp. 275–289). Springer International Publishing. https://doi.org/10.1007/978-3-319-18072-4_14