Minimum time control of heterodirectional linear coupled hyperbolic PDEs

Abstract

International audienceWe solve the problem of stabilizing a general class of linear first-order hyperbolic systems. Considered systems feature an arbitrary number of coupled transport PDEs convecting in either direction. Using the backstepping approach, we derive a full-state feedback law and a boundary observer enabling stabilization by output feedback. Unlike previous results, finite-time convergence to zero is achieved in the theoretical lower bound for control time

Disturbance rejection in general heterodirectional 1-D linear hyperbolic systems using collocated sensing and control

“…The requirement (11) that all λ and µ must be distinct is a slightly stronger assumption than in previous literature (e.g. Auriol & Di Meglio, 2016, Hu et al, in press), and is due to the construction of a collocated observer in the present paper, as opposed to the anti-collocated design in Auriol and Di Meglio (2016) and Hu et al (in press).…”

“…The results have been demonstrated in a simulation. Two possible extensions are to port the results to the minimum time controller and observer recently developed in Auriol and Di Meglio (2016), shortening the time needed for disturbance rejection, and to extend the current results to also cover spatially varying coefficients. The latter results in more complicated kernel equations in the sense that the coefficients are spatially varying.…”

Delay-Robust Control Design for Two Heterodirectional Linear Coupled Hyperbolic PDEs

“…More precisely, we show that imposing finite-time convergence by completely canceling the proximal reflection (i.e the reflection at the actuated boundary) yields, in some cases, zero robustness margins to arbitrarily small delays in the actuation path. In particular, the control laws in recent contributions (see for instance [9], [17], [18], [26], [31]) can have very poor to no robustness to delays due to the cancellation of the proximal reflection. To overcome this problem, we propose some changes in the design of target system to preserver a small amount of this reflection and ensure delay-robustness.…”

“…The backstepping approach [18], [31] has enabled the design of stabilizing full-state feedback laws for these systems. The generalization of these stabilization results for a large number of systems has been a focus point in the recent literature (details in [9], [17], [18], [31]). The main objective of these controllers is to ensure convergence in the minimum achievable time (as defined in [36]), thereby neglecting the robustness aspects that are essential for practical applications.…”

Boundary Observer for Congested Freeway Traffic State Estimation via Aw-Rascle-Zhang model

“…Boundary control of the ARZ model has been studied through many recent efforts including Belletti, et al (2015) Zhang & Prieur (2017) Karafyllis, et al (2017) Yu & Krstic (2018a) Yu & Krstic (2018b). Boundary control and observer design using PDE backstepping method have been developed for 2×2 coupled hyperbolic systems Coron, et al (2013) Yu & Krstic (2017) and the theoretical result for more general hetero-directional hyperbolic systems developed in Long, et al (2016) Florent, et al (2013) Auriol & Di Meglio. (2016.…”