On design of interval observers for parabolic PDEs

Kharkovskaia, Tatiana; Efimov, Denis; Fridman, Emilia; Polyakov, Andrey; Richard, Jean‐Pierre

doi:10.1016/j.ifacol.2017.08.723

Cited by 16 publications

(7 citation statements)

References 27 publications

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“…The interval width is linked to the size of the model uncertainty. There are several approaches to design interval/set-membership estimators for PDE systems within which we can cite [83][84][85].…”

Section: Robust Synthesismentioning

confidence: 99%

A Concise Review of State Estimation Techniques for Partial Differential Equation Systems

2021

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While state estimation techniques are routinely applied to systems represented by ordinary differential equation (ODE) models, it remains a challenging task to design an observer for a distributed parameter system described by partial differential equations (PDEs). Indeed, PDE systems present a number of unique challenges related to the space-time dependence of the states, and well-established methods for ODE systems do not translate directly. However, the steady progresses in computational power allows executing increasingly sophisticated algorithms, and the field of state estimation for PDE systems has received revived interest in the last decades, also from a theoretical point of view. This paper provides a concise overview of some of the available methods for the design of state observers, or software sensors, for linear and semilinear PDE systems based on both early and late lumping approaches.

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Section: Robust Synthesismentioning

confidence: 99%

A Concise Review of State Estimation Techniques for Partial Differential Equation Systems

2021

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“…It is worth to highlight that here such a restriction on shape functions is not related with any early lumping procedure. Some preliminary results on an interval PDE observer have been proposed in [33].…”

Section: Introductionmentioning

confidence: 99%

Interval observer design and control of uncertain non-homogeneous heat equations

et al. 2020

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The problems of state estimation and observer-based control for heat non-homogeneous equations under distributed in space point measurements are considered. First, an interval observer is designed in the form of Partial Differential Equations (PDEs), without Galerkin projection. Second, conditions of boundedness of the interval observer solutions with non-zero boundary conditions and measurement noise are proposed. Third, the obtained interval estimates are used to design a dynamic outputfeedback stabilizing controller. The advantages of the PDE-based interval observer over the approximation-based one are clearly shown in the numerical example.

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“…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%

“…Although the summary above was mainly focused on finite-dimensional system models, also sets of PDEs such as the parabolic differential equation of heat conduction were investigated. There, two fundamentally different approaches could be distinguished, namely, techniques relying on a replacement of the infinite-dimensional dynamics by a finite-dimensional system model before the observer design (early lumping) and the design of observers and its stability proof on the basis of the PDE model (late lumping) [18,20].…”

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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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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On design of interval observers for parabolic PDEs

Cited by 16 publications

References 27 publications

A Concise Review of State Estimation Techniques for Partial Differential Equation Systems

A Concise Review of State Estimation Techniques for Partial Differential Equation Systems

Interval observer design and control of uncertain non-homogeneous heat equations

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

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