Explicit Backstepping Kernel Solutions for Leak Detection in Branched Pipe Flows

Explicit solutions are given for a set of <inline-formula> <tex-math notation="LaTeX">$n+m$ </tex-math></inline-formula> linear hyperbolic observer backstepping kernel equations used for leak detection in branched pipe flows. It is identified that the kernel equations can be separated into <inline-formula> <tex-math notation="LaTeX">$N+1$ </tex-math></inline-formula> distinct Goursat problems for <inline-formula> <tex-math notation="LaTeX">$2{(}j+1{)}$ </tex-math></inline-formula> coupled PDEs each, <inline-formula> <tex-math notation="LaTeX">$j \in \{0,1,\ldots,N\}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$N+1$ </tex-math></inline-formula> being the number of pipes connected via the branching point. Expressing the solutions as infinite matrix power series, the solution to each set of equations is shown to depend on a simplified, scalar Goursat problem, the solution of which is given in terms of derivatives of a modified Bessel function of the first kind. Furthermore, it is shown that the infinite matrix power series expressing the solution writes in terms of modified Bessel functions of the first kind and Marcum Q-functions, as is the case for the previously solved <inline-formula> <tex-math notation="LaTeX">$2\times 2$ </tex-math></inline-formula> constant coefficient case. A numerical example showing adaptive observer gains for leak detection computed via the explicit solutions for multiple operating points of a branched pipe flow is given to illustrate the results.