Explicit Backstepping Kernel Solutions for Leak Detection in Pipe Flow Networks Containing Loops

Abstract

A recursive procedure to obtain explicit expressions to a set of observer backstepping kernel equations for an interconnection (cascade) of N+1 systems of 2×2 linear hyperbolic PDEs, N>0 an integer, for use in leak detection in pipe flow networks containing loops is developed. The kernel equations, consisting of two sets each of N+1 pairs of Goursat PDEs defined over a triangular domain, and N(N+1)2 pairs of Goursat PDEs defined over a square domain, interconnected to each other in an overarching triangular structure, is separated into 2(N+1) systems consisting of k+1 pairs of PDEs over a triangular domain interconnected with (N−k2)(k+1) pairs of PDEs over a square domain, k∈{0,1,…,N} . Under the assumption that the mean friction factor of the network may be used in place of individual friction factors for each pipe, it is shown that the solution to each of the simplified kernel equation systems is expressed explicitly in terms of modified Bessel functions of the first kind, and may be constructed recursively. A numerical example is provided to illustrate the results.