Backstepping stabilization of a linearized ODE–PDE Rijke tube model

Andrade, Gustavo A. de; Vázquez, Rafael; Pagano, Daniel J.

doi:10.1016/j.automatica.2018.06.034

Cited by 44 publications

(42 citation statements)

References 28 publications

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“…Note that if the delays are not rationally independent, this condition is only sufficient (see [24] for details). Condition (20) requires iterative optimization methods to be tested and is not constructive, in general. For state-feedback synthesis, some numerically tractable sufficient conditions have been proposed using Lyapunov-Krasovskii theory [14,18,33,35,36].…”

Section: Problem Formulationmentioning

confidence: 99%

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Output feedback stabilization of an underactuated cascade network of interconnected linear PDE systems using a backstepping approach

Auriol

2020

Automatica

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We detail in this article the development of a robust stabilizing output feedback control law for an underactuated cascade network of n systems of two heterodirectional linear first-order hyperbolic Partial Differential Equations interconnected through their boundaries. Only one of the subsystems is actuated. The proposed approach combines successive backstepping transformations that present a specific cascade structure. With these transformations it is possible to rewrite the original network system as a simple system for which all the in-domain coupling terms have been removed. One can then design a strictly proper stabilizing control law.The proposed control law is proved to be robust to small delays in the actuation. Finally, a boundary observer is designed, enabling stabilization by output feedback.

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Section: Problem Formulationmentioning

confidence: 99%

“…the design of a virtual flow meter pilot [25] or the case of the Rijke tube [20] (even if, in this case an ODE is sandwiched between two PDEs systems). Finally, freeway traffic can also be modeled by hyperbolic PDEs [10].…”

Section: Introductionmentioning

confidence: 99%

Output feedback stabilization of an underactuated cascade network of interconnected linear PDE systems using a backstepping approach

Auriol

2020

Automatica

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“…In this paper, we consider the problem of exponentially estimating the state of a system of hyperbolic PDEs with linear time-invariant dynamic boundary conditions via boundary measurements. Systems modeled as the interconnection of PDEs and ODEs can be found in numerous applications; see, e.g., [4,Chapter 1], [11,26]. To address this problem, we consider an infinite-dimensional observer that is a copy of the system with a linear boundary injection term to be designed.…”

Section: Contributionmentioning

confidence: 99%

Boundary observer design for cascaded ODE — Hyperbolic PDE systems: A matrix inequalities approach

2020

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Boundary observer design for a system of ODEs in cascade with hyperbolic PDEs is studied. An infinite dimensional observer is used to solve the state estimation problem. The interconnection of the observer and the system is written in estimation error coordinates and analyzed as an abstract dynamical system. The design of the observer is performed to achieve global exponential stability of the estimation error with respect to a suitable norm and with a tunable convergence rate. Sufficient conditions in the form matrix inequalities are given for the design of the observer. The effectiveness of the approach is shown in a numerical example.

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“…Several contributions deal with the stabilization of coupled hyperbolic PDEs with ODEs in either a PDE-ODE structure [14], [5] or an ODE-PDE-ODE structure [21], [13]. The systems considered here arise when there is a lumped element in an otherwise distributed system such as heavy chain systems [19] or the Rijke tube [12]. Stabilizing controllers in particular instances of such systems have been designed in, e.g., [10], [12].…”

Section: Introductionmentioning

confidence: 99%

“…The systems considered here arise when there is a lumped element in an otherwise distributed system such as heavy chain systems [19] or the Rijke tube [12]. Stabilizing controllers in particular instances of such systems have been designed in, e.g., [10], [12]. Most generally, these systems exist in situations where long continuous structures act as conduits for mass, energy, or signal transport constrained under some speed of propagation and acted on by a distributed source, or non-linear damping, term and a single point of actuation or control.…”

Section: Introductionmentioning

confidence: 99%

Robust Control Design of Underactuated 2 × 2 PDE-ODE-PDE Systems

Auriol

Aarsnes

Meglio

et al. 2021

IEEE Control Syst. Lett.

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In this paper, we design a robust stabilizing controller for a system composed of two sets of linear heterodirectional hyperbolic PDEs, with actuation at one boundary of one of the PDEs, and couplings at the middle boundary with ODEs in a PDE-ODE-PDE configuration. The system is underactuated since only one of the PDE systems is actuated. The design approach employs a backstepping transformation to move the undesired system couplings to the proximal boundary (where the actuation is located). We can then express this target system as a time-delay neutral system for which we can design an appropriate control law to obtain an exponentially stable target system.

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Backstepping stabilization of a linearized ODE–PDE Rijke tube model

Cited by 44 publications

References 28 publications

Output feedback stabilization of an underactuated cascade network of interconnected linear PDE systems using a backstepping approach

Output feedback stabilization of an underactuated cascade network of interconnected linear PDE systems using a backstepping approach

Boundary observer design for cascaded ODE — Hyperbolic PDE systems: A matrix inequalities approach

Robust Control Design of Underactuated 2 × 2 PDE-ODE-PDE Systems

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