Interval observers for PDEs: approximation approach

Kharkovskaya, Tatiana; Efimov, Denis; Polyakov, Andrey; Richard, Jean‐Pierre

doi:10.1016/j.ifacol.2016.10.283

Cited by 9 publications

(8 citation statements)

References 15 publications

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“…If simplified sparse solutions for H according to (11) shall be optimized by a minimization of µ ∞ ≥ 0 according to the constraint (21) with the help of the H ∞ design described above, the LMIs M (Σ) ≺ 0 in (19) with (20)…”

Section: Between the Open-loop Statementioning

confidence: 99%

“…At least for the last two decades, interval observers have been designed for several different types of dynamic system models. Such system models can be characterized into continuous-and discrete-time state-space representations of systems with finite-dimensional dynamics as well as into special types of partial differential equations (PDEs) [11,14,18,19,34,35,43]. Especially for the case of finite-dimensional systems, linear time-invariant, linear parameter-varying, linear time-varying and (special types) of nonlinear dynamics have been accounted for [6,24,40,41].…”

Section: Introductionmentioning

confidence: 99%

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From Verified Parameter Identification to the Design of Interval Observers and Cooperativity-Preserving Controllers

Rauh

Kersten

2020

Acta Cybern

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One of the most important advantages of interval observers and the associated trajectory computation is their capability to provide estimates for a given dynamic system model in terms of guaranteed state bounds which are compatible with measured data subject to bounded uncertainty. However, the inevitable requirement for being able to produce such verified bounds is the knowledge about a dynamic system model in which possible uncertainties and inaccuracies are themselves represented by guaranteed bounds. For that reason, classical point-valued parameter identification schemes are often not sufficient or should, at least, be handled with sufficient care if safety critical applications are of interest. This paper provides an application-oriented description of the major steps leading from a control-oriented system model with an associated interval-valued parameter and disturbance identification to a verified design of interval observers which provide the basis for the development and implementation of cooperativity-preserving feedback controllers. Such combined control and observer structures allow for forecasting guaranteed lower and upper state bounds that can be determined by solving initial value problems for crisp-parameter models. As such, they replace the significantly more demanding task of computing tubes of reachable states by means of general-purpose interval methods. The corresponding computational steps for the cooperativity-preserving control and observer synthesis are described and visualized for the temperature control of a laboratory-scale test rig available at the Chair of Mechatronics at the University of Rostock.

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Section: Between the Open-loop Statementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

From Verified Parameter Identification to the Design of Interval Observers and Cooperativity-Preserving Controllers

Rauh

Kersten

2020

Acta Cybern

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“…Remark 7. Note that since for calculation of solutions the finite-element discretization/approximation methods are use, then their error of approximation has to be taken into account in the final estimates in order to ensure the desired interval inclusion property for all x ∈ I and t ∈ T , see Kharkovskaya et al (2016) where the result from Wheeler (1973) was applied for an evaluation of this error.…”

Section: Examplementioning

confidence: 99%

“…The idea of interval observer design has been proposed rather recently in Gouzé et al (2000), but it has already received numerous extensions for various classes of dynamical models. Interval observers for systems described by PDEs have been proposed in Perez and Moura (2015); Kharkovskaya et al (2016). The finite-dimensional approximation approach was applied in Kharkovskaya et al (2016) using the discretization error estimates from Wheeler (1973).…”

Section: Introductionmentioning

confidence: 99%

“…Interval observers for systems described by PDEs have been proposed in Perez and Moura (2015); Kharkovskaya et al (2016). The finite-dimensional approximation approach was applied in Kharkovskaya et al (2016) using the discretization error estimates from Wheeler (1973). In Perez and Moura (2015) the sensitivity function interval estimates are used for an interval observer design.…”

Section: Introductionmentioning

confidence: 99%

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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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Design of interval observers and controls for PDEs using finite-element approximations

et al. 2018

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Synthesis of interval state estimators is investigated for the systems described by a class of parabolic Partial Differential Equations (PDEs). First, a finite-element approximation of a PDE is constructed and the design of an interval observer for the derived ordinary differential equation is given. Second, the interval inclusion of the state function of the PDE is calculated using the error estimates of the finite-element approximation. Finally, the obtained interval estimates are used to design a dynamic output stabilizing control. The results are illustrated by numerical experiments with an academic example and the Black-Scholes model of financial market.

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Interval observers for PDEs: approximation approach

Cited by 9 publications

References 15 publications

From Verified Parameter Identification to the Design of Interval Observers and Cooperativity-Preserving Controllers

From Verified Parameter Identification to the Design of Interval Observers and Cooperativity-Preserving Controllers

On design of interval observers for parabolic PDEs

Design of interval observers and controls for PDEs using finite-element approximations

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