Boundary control of hyperbolic conservation laws using a frequency domain approach

Litrico, Xavier; Fromion, Vincent

doi:10.1016/j.automatica.2008.09.022

Cited by 53 publications

(27 citation statements)

References 27 publications

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“…The above statement holds true even if B and C are unbounded operators, which is the case considered here. The same has been proved by Litrico and Fromion (2009a) for a very particular hyperbolic system of conservation laws, based on the transfer function decomposition into an unstable finite dimensional part and a stable infinite dimensional part.…”

Section: Definition and Propertiesmentioning

confidence: 75%

“…The graphical representation of these responses can take the form of three-dimensional graphs, taking into account the dependence of the frequency response on both the angular frequency ω and the spatial variable l. Another possibility is the representation in the form of the classical two-dimensional plots, determined for the fixed value of the spatial variable (Jovanović and Bamieh, 2006;Bartecki, 2007;Litrico and Fromion, 2009a). Considering as an example the Bode plot of the frequency response, the expressions for the logarithmic gain and phase take the following form:…”

Section: Frequency Responsesmentioning

confidence: 99%

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A general transfer function representation for a class of hyperbolic distributed parameter systems

Bartecki

2013

International Journal of Applied Mathematics and Computer Science

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Results of transfer function analysis for a class of distributed parameter systems described by dissipative hyperbolic partial differential equations defined on a one-dimensional spatial domain are presented. For the case of two boundary inputs, the closed-form expressions for the individual elements of the 2×2 transfer function matrix are derived both in the exponential and in the hyperbolic form, based on the decoupled canonical representation of the system. Some important properties of the transfer functions considered are pointed out based on the existing results of semigroup theory. The influence of the location of the boundary inputs on the transfer function representation is demonstrated. The pole-zero as well as frequency response analyses are also performed. The discussion is illustrated with a practical example of a shell and tube heat exchanger operating in parallel-and countercurrent-flow modes.

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Section: Definition and Propertiesmentioning

confidence: 75%

Section: Frequency Responsesmentioning

confidence: 99%

A general transfer function representation for a class of hyperbolic distributed parameter systems

Bartecki

2013

International Journal of Applied Mathematics and Computer Science

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“…In [10], it is established that the irrational transfer function relating the Laplace transform û(s)ŵ(s)…”

Section: Criterion For Stability and Performancementioning

confidence: 99%

“…rational) transfer function [4]. It is shown in [10] that the transfer function from the control input u(·)=f (·, 0) to the output component…”

Section: Criterion For Stability and Performancementioning

confidence: 99%

Frequency-domain performance analysis of distributed-parameter systems under periodic sampled-data feedback control

Cantoni

Kao

2015

2015 Proceedings of the Conference on Control and Its Applications

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The stability and performance of distributed-parameter systems operating under periodic sampled-data feedback control is studied via integral-quadratic constraint (IQC) based analysis. Sufficient frequency-domain conditions are derived for verifying a specified bound on the L2-gain of the feedback interconnection of a plant with (irrational) Callier-Desoer class transfer function and a feedback controller obtained via the periodic sample-and-hold discretization of a finitedimensional LTI controller. The analysis is underpinned by a time-varying delay model of the sample-and-hold operation and IQC characterizations of a related system. An illustrative numerial example involving the control of a linear system of hyperbolic conservation laws is presented. IntroductionThe stability and performance characteristics of sampled-data feedback control systems have been studied widely. A tractable characterization of stability for the interconnection of a plant with rational transfer function and a periodic sampled-data feedback controller can be established via finite-dimensional statespace modelling techniques; see e.g. [1]. In [2] and [3], infinite-dimensional state-space models and Lyapunov type analysis are used to study the stability of distributed-parameter plants under periodic sampleddata control.Quadratic measures of performance (e.g. L 2 -gain) can be analysed in a similar fashion. By contrast, in this paper, a purely input-output approach is taken to assess L 2 stability and performance. Attention is restricted to feedback interconnections of a plant with Callier-Desoer class transfer function and the periodic sample-and-hold discretization of a finitedimensional linear time-invariant (LTI) continuous-time controller. Focusing on the case of plant transfer functions that may be irrational, but with at most a fi-

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“…However, this solution is only valid for a specific case, corresponding to rectangular horizontal frictionless channels around a uniform flow regime. This main limitation of the Riemann invariants method has lead to consider an alternative method based on frequency domain approach [5], [4], [6]. Such a method is very close to the one classically used by control engineers: the nonlinear PDE is first linearized around a stationary regime, then the Laplace transform is used to consider the linearized PDE in the frequency domain, and classical frequency domain tools are used to design controllers, in a very similar way as when the system is represented by finite dimensional transfer functions.…”

Section: A Motivationmentioning

confidence: 99%

A link between Riemann invariants and frequency domain approaches for boundary control of open channel flow

Litrico

Fromion²

2008

2008 47th IEEE Conference on Decision and Control

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HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. A link between Riemann invariants and frequency domain approaches for boundary control of open channel flowTo cite this version: A Link Between Riemann Invariants and Frequency Domain Approaches for Boundary Control of Open Channel Flow Xavier Litrico and Vincent FromionAbstract-Open channel flow is traditionally described by hyperbolic conservation laws (the Saint-Venant equations), that can be controlled using boundary conditions. For horizontal frictionless channels, a classical approach consists in using the characteristic form to diagonalize the equations, using socalled Riemann invariants. This elegant approach is much more difficult to apply when friction and slope are not zero, i.e. in the vast majority of cases. On the other hand, a Laplace based method enables to diagonalize the system with nonzero slope and friction, but in the frequency domain. This paper enlightens a link between both methods, showing that the frequency domain method can be considered as an extension of the Riemman invariants form for channels with non zero slope and friction. As an application, we derive explicit expressions for the boundary controls solving the motion planning problem.

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Boundary control of hyperbolic conservation laws using a frequency domain approach

Cited by 53 publications

References 27 publications

A general transfer function representation for a class of hyperbolic distributed parameter systems

A general transfer function representation for a class of hyperbolic distributed parameter systems

Frequency-domain performance analysis of distributed-parameter systems under periodic sampled-data feedback control

A link between Riemann invariants and frequency domain approaches for boundary control of open channel flow

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