Adaptive Control of Parabolic PDEs

Smyshlyaev, Andrey; Krstić, Miroslav

doi:10.1515/9781400835362

Cited by 298 publications

(221 citation statements)

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“…Once the existence of a solutionṽ that belongs to an appropriate solution space is clarified, we can show the exponential stability of theṽ-system (15)- (17) with respect to the V norm · V in a similar manner to Smyshlyaev and Krstic (2010);Liu (2003). Indeed, the temporal derivative of (1/2) w(·, t) 2 V is given by (A ww (·, t),w(·, t)) V for all t > 0.…”

Section: Convergence Of Errormentioning

confidence: 99%

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Backstepping observer design for parabolic PDEs with measurement of weighted spatial averages

Tsubakino

Hara

2015

Automatica

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This paper is concerned with the observer design for one-dimensional linear parabolic partial differential equations whose output is a weighted spatial average of the state over the entire spatial domain. We focus on the backstepping approach, which provides a systematic procedure to design an observer gain for systems with boundary measurement. If the output is not a boundary value of the state, the backstepping approach is not directly applicable to obtaining an observer gain that stabilizes the error dynamics. Therefore, we attempt to convert the error system into another system to which backstepping is applicable. The conversion is successfully achieved for a class of weighting functions, and the resultant observer realizes exponential convergence of the estimation error with an arbitrary decay rate in terms of the L 2 norm. In addition, an explicit expression of the observer gain is available in a special case. The effectiveness of the proposed observer is also confirmed by numerical simulations.

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Section: Convergence Of Errormentioning

confidence: 99%

“…With the aid of the method explained in Smyshlyaev and Krstic (2010), we can obtain the solution to the kernel PDE (31)-(33) for λ 0 and h given by (40) as…”

Section: Explicit Observer Gainsmentioning

confidence: 99%

Backstepping observer design for parabolic PDEs with measurement of weighted spatial averages

Tsubakino

Hara

2015

Automatica

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“…This approach is natural when modeling errors are assumed to be small: the observer can then effectively filter out measurement noise. State observers for partial differential equations are extensively treated in [29] which may be used as a starting point for further work in this direction.…”

Section: Closed Loop Observermentioning

confidence: 99%

Real-time physics-model-based simulation of the current density profile in tokamak plasmas

et al. 2011

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Abstract.A new paradigm is presented to reconstruct the plasma current density profile in a tokamak in real-time. The traditional method of basing the reconstruction on real-time diagnostics combined with a real-time GradShafranov solver suffers from the difficulty of obtaining reliable internal current profile measurements with sufficient spatial and temporal accuracy to have a complete picture of the profile evolution at all times. A new methodology is proposed in which the plasma current density profile is simulated in real-time by solving the first-principle physics-based equations determining its evolution. Effectively, an interpretative transport simulation similar to those run today in post-plasma shot analysis is performed in real-time. This provides realtime reconstructions of the current density profile with spatial and temporal resolution constrained only by the capabilities of the computational platform used and not by the available diagnostics or the choice of basis functions. The diagnostic measurements available in real-time are used to constrain and improve the accuracy of the simulated profiles. Estimates of other plasma quantities, related to the current density profile, become available in real-time as well. The implementation of the proposed paradigm in the TCV tokamak is discussed, and its successful use in plasma experiments is demonstrated. This framework opens up the possibility of unifying q profile reconstructions across different tokamaks using a common physics model and will support a wealth of applications in which improved real-time knowledge of the plasma state is used for feedback control, disruption avoidance, scenario monitoring, and external disturbance estimation.

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“…sound, heat, electrostatics, electrodynamics, fluid flow, elasticity, or quantum mechanics), whose distributed nature introduces additional level of complexity in design. That is why control and estimation of PDEs is a very popular direction of research nowadays Bredies et al (2013); Smyshlyaev and Krstic (2010). Frequently, for design of a state estimator or control, the finitedimensional approximation approach is used Alvarez and This work is partially supported by the Government of Russian Federation (Grant 074-U01) and the Ministry of Education and Science of Russian Federation (Project 14.Z50.31.0031).…”

Section: Introductionmentioning

confidence: 99%

On design of interval observers for parabolic PDEs

et al. 2017

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The problem of state estimation for heat non-homogeneous equations under distributed in space measurements is considered. An interval observer is designed, described by Partial Differential Equations (PDEs), for uncertain distributed parameter systems without application of finite-element approximations. Conditions of boundedness of solutions of interval observer with non-zero boundary conditions and measurement noise are proposed. The results are illustrated by numerical experiments with an academic example.

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Adaptive Control of Parabolic PDEs

Cited by 298 publications

References 0 publications

Backstepping observer design for parabolic PDEs with measurement of weighted spatial averages

Backstepping observer design for parabolic PDEs with measurement of weighted spatial averages

Real-time physics-model-based simulation of the current density profile in tokamak plasmas

On design of interval observers for parabolic PDEs

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