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Keywords = outer state enclosures

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30 pages, 9016 KB  
Article
Union and Intersection Operators for Thick Ellipsoid State Enclosures: Application to Bounded-Error Discrete-Time State Observer Design
by Andreas Rauh, Auguste Bourgois and Luc Jaulin
Algorithms 2021, 14(3), 88; https://doi.org/10.3390/a14030088 - 14 Mar 2021
Cited by 17 | Viewed by 4947
Abstract
Thick ellipsoids were recently introduced by the authors to represent uncertainty in state variables of dynamic systems, not only in terms of guaranteed outer bounds but also in terms of an inner enclosure that belongs to the true solution set with certainty. Because [...] Read more.
Thick ellipsoids were recently introduced by the authors to represent uncertainty in state variables of dynamic systems, not only in terms of guaranteed outer bounds but also in terms of an inner enclosure that belongs to the true solution set with certainty. Because previous work has focused on the definition and computationally efficient implementation of arithmetic operations and extensions of nonlinear standard functions, where all arguments are replaced by thick ellipsoids, this paper introduces novel operators for specifically evaluating quasi-linear system models with bounded parameters as well as for the union and intersection of thick ellipsoids. These techniques are combined in such a way that a discrete-time state observer can be designed in a predictor-corrector framework. Estimation results are presented for a combined observer-based estimation of state variables as well as disturbance forces and torques in the sense of an unknown input estimator for a hovercraft. Full article
(This article belongs to the Special Issue Algorithms for Reliable Estimation, Identification and Control II)
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21 pages, 1026 KB  
Article
Novel Techniques for a Verified Simulation of Fractional-Order Differential Equations
by Andreas Rauh and Luc Jaulin
Fractal Fract. 2021, 5(1), 17; https://doi.org/10.3390/fractalfract5010017 - 21 Feb 2021
Cited by 13 | Viewed by 3303
Abstract
Verified simulation techniques have been investigated intensively by researchers who are dealing with ordinary and partial differential equations. Tasks that have been considered in this context are the solution to initial value problems and boundary value problems, parameter identification, as well as the [...] Read more.
Verified simulation techniques have been investigated intensively by researchers who are dealing with ordinary and partial differential equations. Tasks that have been considered in this context are the solution to initial value problems and boundary value problems, parameter identification, as well as the solution of optimal control problems in cases in which bounded uncertainty in parameters and initial conditions are present. In contrast to system models with integer-order derivatives, fractional-order models have not yet gained the same attention if verified solution techniques are desired. In general, verified simulation techniques rely on interval methods, zonotopes, or Taylor model arithmetic and allow for computing guaranteed outer enclosures of the sets of solutions. As such, not only the influence of uncertain but bounded parameters can be accounted for in a guaranteed way. In addition, also round-off and (temporal) truncation errors that inevitably occur in numerical software implementations can be considered in a rigorous manner. This paper presents novel iterative and series-based solution approaches for the case of initial value problems to fractional-order system models, which will form the basic building block for implementing state estimation schemes in continuous-discrete settings, where the system dynamics is assumed as being continuous but measurements are only available at specific discrete sampling instants. Full article
(This article belongs to the Special Issue Fractional Order Systems: Deterministic and Stochastic Analysis)
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