Stability Analysis for a Class of Linear $2\times 2$ Hyperbolic PDEs Using a Backstepping Transform

Saba, David Bou; Bribiesca-Argomedo, Federico; Auriol, Jean; Loreto, Michaël Di; Meglio, Florent Di

doi:10.1109/tac.2019.2934384

Cited by 13 publications

(2 citation statements)

References 49 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In [12] the kernel equations for 2 × 2 linear hyperbolic PDEs with constant coefficients were solved in closed form, which is exploited in [9] for application to leak detection. In [13] explicit solutions are given for a 4 × 4 system of kernel equations.…”

Section: A Backgroundmentioning

confidence: 99%

Explicit Backstepping Kernel Solutions for Leak Detection in Branched Pipe Flows

Wilhelmsen

Aamo

2023

IEEE Control Syst. Lett.

View full text Add to dashboard Cite

Explicit solutions are given for a set of n+m 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 N + 1 distinct Goursat problems for 2( j + 1) coupled PDEs each, j ∈ {0, 1, . . . , N} and N + 1 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 2 × 2 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.

show abstract

Section: A Backgroundmentioning

confidence: 99%

Explicit Backstepping Kernel Solutions for Leak Detection in Branched Pipe Flows

Wilhelmsen

Aamo

2023

IEEE Control Syst. Lett.

View full text Add to dashboard Cite

show abstract

“…In parallel, the backstepping methodology [16] has proven to be a powerful design method for boundary feedback control of interconnected PDE systems [17,18,19]. Based on 30 invertible state transformations (usually Volterra integral transforms), it consists of mapping the original system into a simpler form (called target system) amenable to analysis, control, and observer design [20]. One of the main difficulties with the backstepping method lies in finding a suitable 35 target system.…”

Section: Introductionmentioning

confidence: 99%